Download EFFECTIVE METHODOLOGIES FOR SUPPLIER SELECTION AND

The Pennsylvania State University
The Graduate School
EFFECTIVE METHODOLOGIES FOR SUPPLIER SELECTION
AND ORDER QUANTITY ALLOCATION
A Thesis in
Industrial Engineering and Operations Research
by
Abraham Mendoza
c 2007 Abraham Mendoza
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
December 2007
The thesis of Abraham Mendoza was reviewed and approved∗ by the following:
José A. Ventura
Professor of Industrial Engineering
Thesis Advisor, Chair of Committee
A. Ravindran
Professor of Industrial Engineering
Tao Yao
Assistant Professor of Industrial Engineering
Terry P. Harrison
Professor of Supply Chain and Information Systems
Richard J. Koubek
Professor of Industrial Engineering
Head of the Harold and Inge Marcus Department of Industrial
and Manufacturing Engineering
∗
Signatures are on file in the Graduate School.
Abstract
Supplier selection is an essential task within the purchasing function. A well-selected set
of suppliers makes a strategic difference to an organization’s ability to reduce costs and
improve the quality of its end products. This realization drives the search for new and
better ways to evaluate and select suppliers.
First, this research presents a three-phase methodology that integrates the various
steps of the supplier selection process. This helps decision makers reduce a base of
potential suppliers to a manageable number and make the final selection and order
quantity allocation by means of multi-criteria techniques, such as the ideal solution
approach, analytical hierarchy process (AHP), and goal programming. The first two,
respectively, are used to reduce a large number of potential suppliers. The last one
is used to decide the final order allocation. For illustrative purposes this three-phase
methodology was applied to a manufacturing facility located in Tijuana, Mexico.
Second, this research considers the importance of inventory management in determining the optimal order quantity from selected suppliers. Two mixed integer nonlinear
programming models are proposed to obtain optimal inventory policies that simultaneously determine how much, how often, and from which suppliers to order. They minimize
the setup, holding, and purchasing costs per time unit under suppliers’ capacity and quality constraints. The first model allows independent order quantities for each supplier and
multiple orders from selected suppliers within an order cycle. This model outperforms
an existing model in the literature. The second model restricts all order quantities to be
of equal size, as required in a multi-stage [supply chain] inventory model. A closed-form
solution is derived for the second model to determine the optimal inventory policy for the
case when two potential suppliers are considered. Both proposed models allow the user
to control the length of the order cycle time to streamline the inventory management
process.
Next, the two optimization models discussed in the previous paragraph are extended
to consider transportation cost. This consideration is important because it has been
repeatedly overlooked in supplier selection literature. Since they are neither continuous
nor convex, LTL transportation freight rates are approximated using either a linear or
a power function to obtain near-optimal inventory policies. To obtain optimal policies
for small to medium-size problems, actual LTL transportation costs are modeled with a
piecewise linear function using binary variables. In the numerical example illustrated,
the total cost per time unit obtained using the power function to estimate actual freight
rates was only 1.4% greater than the optimal total cost per time unit.
iii
Finally, given the prevalence of both supplier selection and inventory control problems
in supply chain management, this research addresses these problems simultaneously by
developing a mathematical model for an N -stage serial system. The model determines
an optimal inventory policy that coordinates the different stages of the system while
allocating orders to selected suppliers in Stage 1. A lower bound on the optimal total cost
per time unit is obtained and a 98% effective power-of-two inventory policy is derived.
iv
Table of Contents
List of Figures
viii
List of Tables
x
Acknowledgments
xii
Chapter 1 Introduction and Overview
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Supply Chain Management . . . . . . . . . . . . . . . .
1.1.2 Role of Purchasing within the Supply Chain . . . . . .
1.1.3 Supplier Selection Process . . . . . . . . . . . . . . . .
1.1.4 Inventory Management and Transportation in Supplier
lection Decisions . . . . . . . . . . . . . . . . . . . . .
1.1.5 Uncertainty in Supplier Selection . . . . . . . . . . . .
1.1.6 Recent Trends in Supplier Selection . . . . . . . . . . .
1.2 Research Objectives and Contributions . . . . . . . . . . . . .
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 2 Literature Review
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Previous Reviews of Supplier Selection . . . . . . . . .
2.3 Decision Support Models . . . . . . . . . . . . . . . . .
2.3.1 Problem Definition and Formulation of Criteria
2.3.2 Pre-qualification of Potential Suppliers . . . . .
2.3.3 Final Selection . . . . . . . . . . . . . . . . . .
2.3.4 Combined Approaches . . . . . . . . . . . . . .
2.4 Inventory Models with Transportation Costs . . . . . .
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2.5
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Multi-Stage Inventory Models . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 3 A Three-Phase Multi-Criteria Methodology for Supplier Selection
37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 The Three-Phase Multi-criteria Methodology for Supplier Selection 38
3.2.1 Phase 1: Screening Process with an Lp Metric . . . . . . . . 38
3.2.2 Phase 2: Criteria Weights and Ranking of Suppliers with
AHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2.1 AHP Algorithm . . . . . . . . . . . . . . . . . . . . 43
3.2.3 Phase 3: Order Quantity Allocation with a Preemptive GP
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.3.1 Goal Constraints . . . . . . . . . . . . . . . . . . . 49
3.2.3.2 Real Constraints . . . . . . . . . . . . . . . . . . . 52
3.3 Application and Analysis . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.1 Computational Results . . . . . . . . . . . . . . . . . . . . . 55
3.3.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 4 Analytical Models for Supplier Selection and Order
Quantity Allocation
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Problem Description and Assumptions . . . . . . . . . . . . . . .
4.3 Different-Size Order Quantities and Dependent Holding Costs . .
4.3.1 Illustrative Example . . . . . . . . . . . . . . . . . . . . .
4.4 Equal-Size Order Quantities and Dependent Holding Costs . . . .
4.4.1 Illustrative Example . . . . . . . . . . . . . . . . . . . . .
4.5 Equal-Size Order Quantities and Constant Holding Costs . . . . .
4.5.1 Closed-Form Solution Analysis for Two Suppliers . . . . .
4.5.1.1 Development of the Closed-Form Solution . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5 Incorporating Transportation Costs into Supplier Selection and Order Quantity Allocation
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Actual Transportation Freight Rates . . . . . . . . . . . . . . . .
5.3 Problem Description and Assumptions . . . . . . . . . . . . . . .
5.4 Freight Rate Continuous Functions . . . . . . . . . . . . . . . . .
5.5 Transportation-Inclusive Models with Equal-Size Order Quantities
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5.5.1
5.5.2
5.5.3
5.5.4
5.5.5
5.5.6
5.6
5.7
Estimating Transportation Costs . . . . . . . . . . . . . . . 101
In-Transit Inventory . . . . . . . . . . . . . . . . . . . . . . 102
Model Considering Continuous Functions . . . . . . . . . . . 103
Linearizing Actual LTL Freight Rates . . . . . . . . . . . . . 104
Model Considering Actual Freight Rates . . . . . . . . . . . 107
Illustrative Example . . . . . . . . . . . . . . . . . . . . . . 108
5.5.6.1 Data and Parameters . . . . . . . . . . . . . . . . . 109
5.5.6.2 Analysis of Results . . . . . . . . . . . . . . . . . . 110
5.5.7 Use of Multiple Trucks . . . . . . . . . . . . . . . . . . . . . 116
5.5.7.1 Illustrative Example . . . . . . . . . . . . . . . . . 119
Transportation-Inclusive Models with Different–Size Order Quantities 120
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Chapter 6 A Serial Inventory System with Supplier Selection and
Order Quantity Allocation
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Problem Description and Assumptions . . . . . . . . . . . . . . .
6.3 Multi-Stage Serial Inventory Model . . . . . . . . . . . . . . . . .
6.3.1 Power-of-Two Inventory Policy . . . . . . . . . . . . . . .
6.4 Illustrative Example and Analysis . . . . . . . . . . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 7 Conclusions and Future Research
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7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Appendix A Transportation-Inclusive Models with
Order Quantities
A.1 Model Considering Continuous Functions . . . . .
A.2 Model Considering Actual Freight Rates . . . . .
A.3 Illustrative Example . . . . . . . . . . . . . . . .
A.3.1 Analysis of Results . . . . . . . . . . . . .
Different-Size
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Appendix B A Serial Inventory System with Supplier Selection
at All Stages
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B.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Bibliography
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vii
List of Figures
1.1
1.2
1.3
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1.6
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A Typical Serial Supply Chain . . . . . . . . . . . . . . . . . . . .
Example of Supply Chain Flows . . . . . . . . . . . . . . . . . . .
Purchasing Process Activities . . . . . . . . . . . . . . . . . . . .
Purchased Materials and Services as a Percentage of Cost of Goods
Sold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supplier Evaluation and Selection Process . . . . . . . . . . . . .
Logistics Cost as a Percentage of Gross Domestic Product . . . .
Breakdown of Logistics Cost . . . . . . . . . . . . . . . . . . . . .
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2.2
Decision Steps in Supplier Selection . . . . . . . . . . . . . . . . . .
Decision Models Used in Supplier Selection . . . . . . . . . . . . . .
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Phase 1 – Screening the Initial List of Suppliers . . . . . . . . . .
AHP Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Growth in the Number of Questions . . . . . . . . . . . . . . . . .
Phase 2 – Defining the Weights with AHP and Supplier Screening
Supplier Selection Criteria Weights . . . . . . . . . . . . . . . . .
Consistency Test Results for the Pairwise Comparison Matrix . .
Phase 3 – Goal Programming . . . . . . . . . . . . . . . . . . . .
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System Under Consideration . . . . . . . . . . . . . . . . . .
Order Cycle for Three Selected Suppliers . . . . . . . . . . .
Total Monthly Cost for Different M Values . . . . . . . . . .
Total Monthly Cost Versus M Values for Multiple Equal-Size
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Orders
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Freight Rate Vs. Weight Shipped . . . . . . . . . . . . . .
Total Transportation Cost Structure as Typically Stated .
Total Transportation Cost Function as Typically Charged .
Langley’s and Power Function Estimates . . . . . . . . . .
LTL Rate Structure . . . . . . . . . . . . . . . . . . . . . .
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Serial System with Three Stages and Multiple Potential Suppliers . 124
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Synchronized Inventory Levels at Three Stages . . . . . . . . . . . . 127
Inventory Levels for All Stages (Separate Inventory Policies) . . . . 141
B.1 Supplier Selection Decisions at All Stages . . . . . . . . . . . . . . . 158
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List of Tables
2.1
Supplier Selection Criteria . . . . . . . . . . . . . . . . . . . . . . .
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3.1
3.2
3.3
3.4
3.5
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Ideal Values for Each Criterion . . . . . . . . . .
Initial Suppliers’ Data . . . . . . . . . . . . . . .
Normalized Suppliers’ Data . . . . . . . . . . . .
Ranking Ordering of Suppliers Based on L2 Value
Rating Scale for Pairwise Comparison . . . . . . .
Pairwise Comparison Matrix . . . . . . . . . . . .
Normalized Matrix . . . . . . . . . . . . . . . . .
Random Index (RI) Values . . . . . . . . . . . .
Problem Notation . . . . . . . . . . . . . . . . . .
GP Model Priorities . . . . . . . . . . . . . . . .
Input Data for the GP Model . . . . . . . . . . .
Orders Allocated to Each Supplier . . . . . . . . .
Goal Achievement . . . . . . . . . . . . . . . . . .
Analysis of Scenarios . . . . . . . . . . . . . . . .
Allocation for the Different Scenarios . . . . . . .
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4.1
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Supplier’s Data for the Illustrative Example .
Detailed Solutions for the Illustrative Example
Supplier’s Data for the Illustrative Example .
Cases Considered in the Closed-Form Solution
Closed-Form Solution of Feasible Cases . . . .
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Example of Nominal Freight Rates .
Actual Freight Rate Schedule . . . .
Nominal and Actual Freight Rates for
Supplier’s Data . . . . . . . . . . . .
Nominal and Actual Freight Rates for
Nominal and Actual Freight Rates for
Nominal and Actual Freight Rates for
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Supplier
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Supplier
Supplier
Supplier
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Summary of Freight Rate Continuous Estimates
Solutions to Illustrative Example (Same Q’s) . .
Analysis of Transportation Costs . . . . . . . .
Analysis of Variable Costs ($/month) . . . . . .
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Data for Stages . . . . . . . . . . . . . . . . . . . .
Suppliers’ Data . . . . . . . . . . . . . . . . . . . .
Optimal Solution to Problem (P6.1) . . . . . . . . .
Coordinated Inventory Policy for Different Values of
Adjusted Data for Stages . . . . . . . . . . . . . . .
Details of the Proposed Algorithm . . . . . . . . . .
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A.1 Summary of Freight Rate Continuous Estimates . . . . . . . . . . . 153
A.2 Solutions to Illustrative Example (Different Qi ’s) . . . . . . . . . . 155
A.3 Analysis of Transportation Costs . . . . . . . . . . . . . . . . . . . 157
xi
Acknowledgments
I would like to express enormous gratitude to my advisor, Dr. José A. Ventura.
This dissertation would not have been possible without your patience, support, and
guidance. I also want to thank my committee members, Dr. A. Ravindran, Dr.
Terry P. Harrison, and Dr. Tao Yao for your helpful comments and suggestions.
I would like to thank Dr. A. Ravindran and Eduardo Santiago for providing
substantial input as co-authors of a research paper that is the base of Chapter 3
in my dissertation. I would like to express my sincere gratitude to Dr. Robert C.
Voigt who always believed in my work and supported me from a professional and
personal perspective. I am also very grateful to my friend Eugenio Longoria Sáenz
for his support and help in proofreading the final drafts of my dissertation.
I would like to thank the Consejo Nacional de Ciencia y Tecnologı́a for the
financial support that made all of this possible. I am also grateful to Universidad
Panamericana for providing additional financial assistance throughout my graduate
studies at Penn State.
I will always be very grateful to Vicente Saucedo, Humberto Ramı́rez, and
Francisco Villanueva for believing in me and encouraging me to pursue graduate
studies.
xii
Dedication
A Dios por todas sus bendiciones. Quia tu es, Deus, fortitudo mea.
A mi esposa Marisol, gracias por creer en mı́ y por todo el tiempo que nos has
regalado durante estos años. Sin duda tu amor, tu paciencia y tu dedicación son
la pieza fundamental para que todo esto haya sido posible.
A mis hijos Abraham, Santiago, Josemarı́a y Emilio, gracias por la
alegrı́a y las bendiciones que han traı́do a nuestro hogar. Ustedes son y serán
siempre mi inspiración y motivación en todo lo que hago.
A mi padre, gracias por darme tu ejemplo de trabajo y por darme lo mejor que
pudiste siempre. Ahora que has partido, tus enseñanzas permanecerán por siempre
en mi corazón.
A mi madre, gracias por darme ejemplo de fortaleza, de fe y de amor. Gracias
por creer en mı́ y por apoyarme en cada paso de mi vida.
A Sol, gracias por tu gran amor y apoyo incondicional para nuestra familia durante
todos estos años.
xiii
Chapter
1
Introduction and Overview
1.1
Introduction
Many factors in today’s global market have influenced companies to search for
a competitive advantage by focusing attention on their entire supply chain. Of
the various activities involved in supply chain management, purchasing is one of
the most strategic because it provides companies with opportunities to reduce
costs and, consequently, increase profits. An essential task within the purchasing
function is supplier selection. In most industries, the cost of raw materials and
component parts represents the largest percentage of the total product cost. For
instance, in high technology firms, purchased materials and services account for up
to 80% of the total product cost (Weber et al. [1]). Therefore, selecting the right
suppliers is key to the procurement process and represents a major opportunity
for companies to reduce costs across its entire supply chain.
For many years, the traditional approach to supplier selection has been to select
suppliers solely on the basis of price (Degraeve and Roodhooft [2]). However, as
companies have learned that the sole emphasis on price as a single criterion for
supplier selection is not efficient, they have turned into to a more comprehensive
multi-criteria approach. Recently, these criteria have become increasingly complex
as environmental, social, political, and customer satisfaction concerns have been
added to the traditional factors of quality, delivery, cost, and service.
The realization that a well-selected set of suppliers can make a strategic difference to an organization’s ability to provide continued improvement in customer sat-
2
isfaction drives the search for new and better ways to evaluate and select suppliers.
This research reviews the supplier selection literature concerning existing models
and methodologies supporting the supplier selection process, identifies some important opportunities, and presents new and efficient decision-making tools aimed at
helping companies select the most effective suppliers. This research also supports
multiple-sourcing over single-sourcing strategies. The use of multiple suppliers provides greater flexibility due to the diversification of the firm’s total requirements
and fosters competitiveness among alternative suppliers (Jayaraman [3]).
In this chapter, Sections 1.1.1–1.1.6 present an overview of supply chain management, the role of supplier selection within the supply chain, the risks involved
in the supplier selection process, and the new trends in supplier selection. Section 1.2 describes the major contributions of this research and Section 1.3 provides
an overview of this dissertation.
1.1.1
Supply Chain Management
Over the last years there has been an increasing interest in supply chain related
issues. Chopra and Meindl [4] formally define a supply chain as all the stages that
are directly or indirectly involved in satisfying customer demand. These stages
include customers, retailers, wholesalers/distributors, manufacturers, and suppliers. From an organizational standpoint, the supply chain comprises all functions
involved in fulfilling customer’s requirements and needs. These functions include
purchasing, product development, marketing, operations, finance, and customer
service.
In a typical supply chain, raw materials are first procured in order for the manufacturer to be able to produce. Finished products are then shipped to warehouses
for storage and finally shipped to distributors, retailers, and customers. Figure 1.1
shows a typical example of a serial supply chain structure.
Notice that Figure 1.1 does not exhibit a network representation of the supply
chain. Instead, it is included to highlight the fact that a supply chain is not a
set of isolated and individual entities, but the sum relationship of all the entities
involved in the production and distribution of a product. In most industries,
however, supply chains are comprised of several suppliers, manufacturing sites,
3
Supplier
Manufacturer
Distributor
Retailer
Figure 1.1. A Typical Serial Supply Chain
†
distribution centers, retailers, etc.
The three major flows that occur in a supply chain are physical, information,
and money (Lee [5]). Figure 1.2 shows a pictorial representation of these flows.
Physical Flow
Information Flow
Raw
Materials
Research
&
Development
Factory
Distribution
Center
Money Flow
Marketing
Retailer
Customer
Figure 1.2. Example of Supply Chain Flows
To satisfy customer demand in a typical supply chain, raw materials are procured from diverse companies. These raw materials flow through a series of production and distribution stages until the final customer is reached with a finished
product. This is what typically represents the physical flow. Next, in order to
efficiently coordinate the physical flows in a supply chain, information flow plays
an important role. Information flow involves transmitting orders and updating
the status of delivery. For instance, information about customer demand must be
available at each stage involved in the production and distribution process. Last,
money flows from the customer upstream to each one of the stages involved in
the supply chain. For example, customers transfer money to retailers and retailers transfer money to the distributors. Similarly, different transactions involving
†
Adapted from http://www.directalliance.com/public/solutions/dsp scm logistics.cfm
4
money take place across all the stages of the supply chain.
The goal of any supply chain is to maximize supply chain profitability. According to Chopra and Meindl [4] profitability is defined as “the difference between revenue generated from the customer and the overall cost across the supply
chain” (p.5). Hence, the profitability in a supply chain is represented by the total
profit shared across all members of the supply chain. This implies that the success of any supply chain should be measured as a whole and not as the success of
each separate member involved. Consequently, in order to increase profitability,
reduce costs, and improve customer satisfaction, effective supply chain strategies
must take into account the individual stages of the supply chain as well as the
interaction among them.
The four major drivers in a supply chain are: inventory, transportation, facilities, and information, as discussed in Chopra and Meindl [4]. High inventory
levels increase the responsiveness of the supply chain but decrease its cost efficiency
because of holding-inventory costs. Hence, an important problem in supply chain
management is to determine the appropriate levels of inventory (e.g. inventory
policy) at the various stages. Another important problem involves the mode of
transportation used in moving goods within the supply chain. Facility location
also has a significant impact on the supply chain. The specific location of facilities influences the mode of transportation and, consequently, costs and delivery
leadtimes. A reliable information system is necessary for optimal management and
coordination of a supply chain.
Control in supply chain management is characterized in two ways: centralized
versus decentralized. In a centralized supply chain, decisions are made by a single
decision maker at a central location for the entire supply chain system. The typical
objective in a centralized supply chain is to minimize the total cost of the system.
In a decentralized supply chain, each entity decides its own effective strategy without considering the impact on the other entities of the supply chain system. In
this way, centralized decisions lead to global optimization, whereas decentralized
decisions lead to local optimization.
5
1.1.2
Role of Purchasing within the Supply Chain
Purchasing within an organization usually encompasses all activities related to the
buying process. According to Van Weele [6], these activities are: determining the
need, selecting the supplier, arriving at a proper price, specifying terms and conditions, issuing the contract or order, and ensuring proper delivery. The increasing
importance of supply chain management is motivating companies to fit purchasing
and sourcing strategies into their supply chain objectives. Figure 1.3 illustrates
the main activities within the purchasing function.
Purchasing Function
Internal
Customer
Determining
Specification
Selecting
Supplier
Contracting
Ordering
Sourcing
Expediting
and
Evaluation
Follow-up
and
Evaluation
Supplier
Supply
Buying
Procurement
Figure 1.3. Purchasing Process Activities (Van Weele [6])
One of the purchasing functions is selecting suppliers capable of procuring the
demanded items that meet the required specifications. Monczka et al. [7] defined
supplier selection as an essential task of purchasing. Moreover, Ellram and Carr [8]
concluded that purchasing plays a key role in corporate strategic success through
the appropriate selection of suppliers supporting the company’s long term strategy
and competitive positioning.
It is not difficult to see the impact that suppliers have on a company’s total
cost. As mentioned in the introduction of this chapter, the cost of raw materials
and component parts represents the largest percentage of the total product’s cost
in most industries. For example, Van Weele [6] presents an analysis on the average
percentage of purchased materials and services as a percentage of cost of goods
sold. Figure 1.4 shows the results of the analysis for different industries.
It can be concluded that important savings can be realized through effective
purchasing strategies. On the one hand, selecting the right suppliers significantly
6
60-85
60-80
50-70
60-80
50
25 50
25-50
Percentage
100
10 40
10-40
0
Retailers
Computers
Consumer
Electronics
Automotive
Pharma
Service
Industry
Figure 1.4. Purchased Materials and Services as a Percentage of Cost of Goods Sold
(Van Weele [6])
affects the total cost of a product and helps companies improve corporate competitiveness (Willis et al. [9]). On the other hand, selecting the wrong suppliers can
cause operational and financial problems (Degraeve and Roodhoft [2]). As companies become more dependant on suppliers, the direct and indirect consequences of
poor decision making become more severe (De Boer et al. [10]). Apart from mere
cost reduction, companies continuously work with suppliers to remain competitive
by reducing product development time, improving product quality, and reducing
leadtimes. For instance, a qualified base of suppliers helps a company achieve
greater innovation through improved product design and increased flexibility.
1.1.3
Supplier Selection Process
This section presents the steps involved in the supplier selection process, as addressed by Monczka et al. [7]. The quality of the final set of suppliers largely
depends on the quality of all the steps involved in the selection process. The first
part of this research proposes a methodology for supplier selection that integrates
the various steps of the selection process. Figure 1.5 depicts the supplier selection
and evaluation process.
7
STEP 1:
Recognize the
Need for
Supplier Selection
STEP 2:
Identify Key Sourcing Requirements
and Criteria
STEP 5:
Limit Suppliers
in Selection
Pool
STEP 3:
Determine Sourcing
Strategy
STEP 4:
Identify Potential
Supply Sources
STEP 6:
Determine Method
for Final
Selection
STEP 7:
Select Suppliers
and Reach
Agreement
Figure 1.5. Supplier Evaluation and Selection Process
Step 1: Recognize the Need for Supplier Selection
The first step in supplier selection usually implies the identification of the need
for a specific product or service. Different situations may trigger the need for
supplier selection. For example, new product development, modifications to a set
of existing suppliers due to a bad performance, the end of a contract, expansion to
different markets, current suppliers’ capacity is not sufficient to satisfy increases
in demand. These situations are particular to every company.
Step 2: Identify Key Sourcing Requirements and Criteria
As mentioned in the introduction of this chapter, supplier selection is complicated
because of the multiple criteria involved in the decision process. Additionally,
many times these criteria may conflict each other. Therefore, defining the proper
criteria becomes critical.
Some of the most widely used criteria in supplier selection are supplier’s capacity, quality, and purchasing price. However, the set of criteria (e.g., Stamm
and Golhar [11], Ellram [12], Weber et al. [1], Kingsman et al. [13], Easton and
Moodie [14], and Mummalaneni et al. [15]) to be chosen largely depends on the
company’s objectives and the type of industry in which the company competes.
These criteria are discussed further in Chapter 2.
Step 3: Determine Sourcing Strategy
Sourcing requires that companies clearly define the strategy approach to be taken
during the supplier selection process. Examples of sourcing strategies are: single
versus multiple suppliers, domestic versus international, and short term versus long
8
term supplier contracts.
This research assumes that single sourcing may not be an appropriate strategy in most purchasing situations. Single sourcing tends to minimize total costs
by determining the best supplier for each purchased part or product. However,
dependency on a single supplier exposes the buying company to a greater risk of
supply interruption. An example of realized supply risk resulting from a single
sourcing strategy is the case of Toyota’s 1977 brake valve crisis. Toyota’s assembly
plants in Japan were forced to shut down for several days after a fire at its only
supplier’s (Aisin Seiki) main plant. This facility was the only source for valves that
were used in all Toyota vehicles (Nishiguchi and Beaudet [16]). The estimated cost
of this single event was $195 million and 70,000 units of production. Thereafter,
Toyota sought at least two suppliers for each part (Treece [17]).
Multiple sourcing strategies provide a greater flexibility due to the diversification of the firm’s total requirements. In addition to ensuring product availability,
working with multiple suppliers is important because suppliers are motivated to
be competitive in factors such as price and quality (Jayaraman [3]).
Step 4: Identify Potential Supply Sources
The importance of the item under consideration influences the resources spent
on identifying potential suppliers. For example, major resources are spent when
potential suppliers are needed for an item of high strategic importance. Guidelines
are provided in Monczka et al. [7].
Step 5: Limit Suppliers in Selection Pool
Given the limited resources of a company, a purchaser needs to pre-screen the
potential suppliers to reduce their number before proceeding with a more detailed
analysis and evaluation. The supplier selection criteria determined in Step 2 plays
a key role in this reduction process. Howard [18] defined this reduction process
as the process by which suppliers satisfy certain ‘entry qualifiers’ before further
analysis.
9
Step 6: Determine Method for Final Selection
There exists many different ways to evaluate and select suppliers. Since this research is devoted to developing effective decision-making methodologies and models
capturing important aspects of the supplier selection problem, Chapter 2 presents
an extensive literature review on decision methods and models for final supplier
selection.
Step 7: Select Suppliers and Reach Agreement
The final step of the supplier evaluation and selection process is to clearly select
those suppliers that best meet the company’s sourcing strategy. This decision
is often accompanied with determining the order quantity allocation to selected
suppliers.
1.1.4
Inventory Management and Transportation in Supplier Selection Decisions
Tactical decisions about inventory levels of a supply chain are an important part
of inventory management. It is important to determine how much inventory to
order (e.g. order quantity) and when best to place an order (inventory policy).
Otherwise, if inventory levels of the materials procured from suppliers are high,
capital investment and storage costs will be high. Inversely, if inventory levels are
too low, shortages may occur resulting in consumer dissatisfaction and possible
future loss of sales.
Order quantity allocation in the supplier selection problem is considered in this
research. To derive optimal inventory policies that simultaneously determine how
much, how often, and from which suppliers to order, typical inventory costs are
considered. These include holding, ordering, and purchasing. Additionally, criteria
relevant to supplier selection (quality and capacity) are incorporated.
The models used to determine the order quantity allocation are further extended to consider transportation costs. Considering inventory and transportation
costs simultaneously requires that a trade-off be made. Companies often need to
determine if it is more cost effective to order smaller shipments more frequently at a
10
higher cost per-unit shipping cost, or to order larger, but less expensive, shipments
less frequently. Traditionally, companies have ignored transportation costs in inventory management decisions and, in the process, have failed to take advantage
of economies of scale in shipping (Natarajan [19]).
Logistics costs (comprised of inventory, transportation, and logistics administration costs) currently represents 9.5% of the gross domestic product (GDP)
(Cook [20]). This shows how significant transportation and inventory costs are.
Figure 1.6 shows logistics costs as a percentage of GDP.
10.5
%o
of GDP
10.0
9.5
90
9.0
8.5
8.0
94 95 96 97 98 99 00 01 02 03 04 05
Year
Figure 1.6. Logistics Cost as a Percentage of Gross Domestic Product
The reduction shown from mid nineties to 2003 is mainly explained by the
fact that Just-In-Time initiatives were undertaken by many U.S. companies during that period of time. However, after 2003 the total logistics costs have started
to rise again mainly due to an increase in inventory holding and transportation
costs. The increase in holding cost is caused by the fact that companies are storing
more goods in response to longer, often unpredictable transit times (Cook [20]).
Also, contributing to the accumulation of inventory is a shift from central, megawarehouses to smaller distribution centers across the country. Although this strategy has allowed companies to improve delivery times and reliability, it has forced
shippers to hold product in more locations.
Regarding the increase in transportation costs, transportation expenses rose
from $509 billion in 2004 to $583 billion in 2005. An additional factor explaining
the increase in transportation costs is gas and diesel rising fuel costs.
11
In 2005 the logistics cost as a percent of the GDP was 9.5% and was valued at
$1.183 trillion. The break down for each component of the logistics cost is shown
in Figure 1.7.
% of f GDP
Administration Costs
Transportation Costs
Inventory Carrying
Costs
0
200
400
600
800
Cost ($ Billions)
Figure 1.7. Breakdown of Logistics Cost
The estimated transportation cost was $744 billion, whereas inventory carrying
cost and logistics administration costs were estimated to be about $393 billion and
$46 billion, respectively. The transportation cost accounted for nearly 6% of the
GDP. Out of the $744 billion estimated in transportation cost, $583 billion were
spent on transportation using trucks. This clearly supports the fact that road
mode is still the preferred transportation mode in the U.S.
From the preceding paragraphs, it is apparent the importance of incorporating
transportation costs into the inventory replenishment decisions. Inventory decisions made without incorporating transportation costs would not take advantage
of the discounts offered by transportation companies as the weight shipped increases. Additionally, failing to do so may result in suboptimal solutions, i.e.
those with much higher than minimal total logistics costs (Warsing [21]).
1.1.5
Uncertainty in Supplier Selection
Supplier selection in supply chain systems is made even more difficult because
supply chains are operated in uncertain environments where disruptions can affect
the short and long-term performance of a company.
12
A well known case of a supply disruption is that of Ericsson and Nokia (see
Sheffi [22]). In 2000, these two companies experienced a disruption in the supply of
chips used in their phones after a fire destroyed the plant of one of their suppliers
(Philips Electronics). Nokia was able to overcome the problem by making use
of existing partners’ inventory, whereas Ericsson’s inability to obtain the needed
parts from alternate suppliers led to its delayed response. From this, it is clear
that proper risk management strategies must be in place so that companies are
able to react accordingly.
Supply chain risk management (SCRM) is the area concerned with the study
of supply chain risks. SCRM is defined as “the management of supply chain risks
through coordination or collaboration among the supply chain partners so as to
ensure profitability and continuity” (Tang [23]). Risks affect supply chain management in two ways: (1) operational risks (Tang [23]) arising from coordinating
supply and demand (e.g., uncertain customer demand), and (2) disruption risks
(Kleindorfer and Saad [24]) arising from disruptions to normal activities (e.g., natural disasters). For the purpose of this research, the focus is on those operational
risks found within supply management.
One way to account for risks in supply management, particularly in the supplier selection process, is to model these when determining the order allocation in
the final choice step. The following operational risks have been modeled in the
literature:
• Uncertain demands occur due to markets’ changing conditions. To model
this, Moinzadeh and Nahmias [25] present a modified continuous review policy, (s1 , s2 , Q1 , Q2 ). This policy maintains a regular and an emergency supply.
When the inventory reaches s1 , Q1 units are ordered (regular order). If the
inventory reaches s2 within the leadtime of the regular order, Q2 units are
ordered as an emergency.
• Uncertain supply yields occur when order quantities from selected suppliers
are received incomplete due to disruptions in a supplier’s production or manufacturing system. Agrawal and Nahmias [26] present a model for evaluating
the costs associated with yield loss. Their model determines the optimal
number of suppliers with different yields when the demand is known.
13
• Uncertain supply leadtimes arise due to disruptions that occur in the gap
between the time the order is placed and is received. Ramasesh et al. [27]
proposed an (s, Q) ordering policy that splits the order quantity Q evenly
between two suppliers. Sedarage et al. [28] extends this work further by considering more than two suppliers and a non-even split order quantity among
them. Generally, the exact analysis of multiple suppliers with stochastic supply leadtimes is intractable (Tang [23]). Therefore, deterministic demand is
assumed to model this case.
• Uncertain supply costs occur when a cost is imposed by an upstream supply
chain partner or when uncertain currency exchange rates take place. Gurnani
and Tang [29] determine the optimal ordering policy for a retailer who has
two instants to order a seasonal product from a manufacturer prior to a single
selling season. Although the demand is uncertain, the retailer can improve
the forecast by utilizing the market signals observed between the first and
second instants. However, the unit cost at the second instant is uncertain
and could be higher or lower than the unit cost at the first instant. The
system modeled is a 2-period dynamic programming model that optimally
allocates the corresponding order quantity for the first and second instants.
Even though risks can be modeled quantitatively, it is difficult to completely
avoid disruptions in practice. Successful companies are those that are able to positively react to disruptions in a quick manner. Sheffi [22] defines this as resilience,
“the ability of a supply chain to bounce back and continue normal operations
after high-impact, unanticipated disruption”. Sheffi explains a seven-step plan
for companies to avoid the impact of disruptive events, and therefore, reduce the
vulnerability of their supply chains.
Although the importance of considering uncertainty in the supplier selection
problem is briefly addressed in the preceding paragraphs, modeling risk is beyond
the scope of this research. Nonetheless, Chapter 7 introduces some ideas on how
to expand the models in this research to include risk.
14
1.1.6
Recent Trends in Supplier Selection
Most recently, the internet and related information technology systems began impacting purchasing operations. Internet-based procurement, commonly referred
to as e-procurement, is being used by both suppliers and buyers to manage their
procurement relationships. E-procurement involves the use of the internet for
activities such as procuring materials, transportation, and warehousing (Kameshwaran et al. [30]). In addition, e-procurement is concerned with selecting suppliers
among different alternatives and determining the nature of contracts with them.
A typical e-procurement system consists of the following major steps: (1)
request-for-quote (RFQ) generation and distribution by the buyer company to
all potential suppliers; (2) the submission of bids by interested suppliers, and (3)
the evaluation of bids to determine the winning bids.
According to Kameshwaran et al. [30], the business logic used in current eprocurement systems is broadly categorized as:
• Reverse auctions are “auctions in which the auctioneer, on behalf of a buyer,
solicits bids from a group of potential suppliers” (Chen-Ritzo et al. [31]).
The primary objective is to drive purchase prices down allowing the lowest
bidder to win. Typically, reverse auctions have focused on price as a single
attribute. Hohner et al. [32] present an implementation of reverse auctions
at Mars, producer of confectionary, pet food, and rice brands. Efficiencies
resulting from the implementation of reverse auctions come from matching
suppliers’ capabilities to Mar’s needs and specifications.
• Multi-attribute auctions combine multi-criteria decision analysis and auction
mechanisms. Chen-Ritzo et al. [31] propose a multi-attribute auction mechanism where bidders can specify price and levels of quality and leadtime. The
performance of this mechanism is compared to a price-only auction mechanism.
• Optimization techniques take into account various business rules and constraints, e.g. exclusion constraints, aggregation constraints, exposure constraints, business objectives constraints. Companies like Emptoris‡ uses op‡
http://www.emptoris.com
15
timization techniques in their commercial bid software. In 2000, Motorola’s
global procurement function selected one of Emptori’s negotiation platforms
to determine its procurement strategy. Motorola reported over $200 million
in savings after implementation (Metty et al. [33]).
• Configurable bids “enable suppliers to specify multiple values and price markups for each attribute” (Bichler and Kalagnanam [34]). This is basically
an extension of multi-attribute auctions allowing for configurability in bids.
Procter & Gamble implemented a solution from CombineNet§ , a software
company, using configurable bids (Sandholm et al. [35]). Savings, since implementation, have amounted to $294.8 million.
1.2
Research Objectives and Contributions
In this research, the primary objective is to develop efficient inventory policies for
order quantity allocation in the supplier selection problem while simultaneously
considering inventory and transportation costs. The specific contributions of this
research are:
1. A three-phase methodology that integrates all the steps involved in the supplier selection process while also considering the multi-criteria nature of the
problem.
2. An order quantity allocation model that considers typical inventory costs
for a single-stage system with multiple suppliers. Additional criteria such as
quality and capacity are considered as constraints. This model outperforms
an existing model in the literature (Ghodsypour and O’Brien [36]).
3. A closed-form solution is derived to determine the optimal order quantity
allocation for the single-stage system considering two suppliers.
4. Under the assumption that shipments from suppliers are LTL, near-optimal
inventory policies are provided for the single-stage system with multiple suppliers. LTL transportation freight rates are approximated using a linear and
a power functions.
§
http://www.combinenet.com/
16
5. Under the assumptions that shipments from suppliers are LTL, an optimal
inventory policy is provided for the single-stage system with multiple suppliers. Actual transportation costs are modeled as continuous piecewise linear
functions using binary variables.
6. An optimal order allocation policy is provided for the case involving multiple
TL or a combination of TL and LTL.
7. A mathematical model to determine the optimal inventory policy that coordinates the different stages of a serial system while allocating orders to
selected suppliers in Stage 1.
8. A power-of-two policy that is within 2% of an analytical lower bound is
developed for the serial inventory system.
These results provide an effective way of selecting suppliers and properly allocating the corresponding order quantities to them while optimizing inventory levels
of the parts being procured.
1.3
Overview
The remainder of this dissertation is organized as follows. Chapter 2 discusses
the relevant contributions from the literature. In particular, three main research
areas are reviewed: (1) decision support models for supplier selection; (2) inventory models with transportation costs; and (3) inventory models for multi-stage
inventory systems.
Chapter 3 presents a method to solve the supplier selection problem. Multicriteria techniques (ideal solution approach, analytical hierarchy process and goal
programming) are used to reduce the base of potential suppliers to a manageable
number and to optimize the allocation of a predetermined order quantity (singleperiod problem) among selected suppliers.
Chapter 4 presents a order quantity allocation model for a single-stage system
with multiple suppliers. The objective is to select the best suppliers that minimizes
the holding, setup, and purchasing costs. The problem is constrained by suppliers
17
quality level and capacity. A closed-form solution is derived to determine the
optimal inventory policy for the case of two suppliers.
Chapter 5 extends the single-stage system in Chapter 4 to consider transportation costs in addition to the inventory costs. This chapter focuses on the usage of
trucks as a means of transporting goods and incorporates the transportation cost
as a function of the shipment quantity. Assuming that all shipments from suppliers are shipped using LTL, approximate and optimal policies can be obtained. To
obtain approximate policies, LTL transportation freight rates are modeled using
continuous functions. To obtain optimal policies, transportation costs are modeled
using continuous piecewise linear functions using binary variables. The case where
more than one TL might be needed to transport items from suppliers is considered.
A procedure is provided to determine the number of TL’s or the combination of
TL and LTL needed to ship the orders from suppliers.
Chapter 6 addresses the supplier selection and inventory control problems simultaneously by developing a mathematical model for an N -stage serial system.
This model determines an optimal inventory policy that coordinates the different
stages of the system while properly allocating orders to selected suppliers in Stage
1. A lower bound on the optimal total cost per time unit is obtained and a 98%
effective power-of-two inventory policy is derived for the system under consideration.
Chapter 7 provides concluding remarks for the results obtained in this research
and several directions for future research.
Chapter
2
Literature Review
2.1
Introduction
In this chapter, contributions related to (1) decision support models for supplier
selection, (2) inventory models with transportation costs, and (3) inventory models for multi-stage inventory systems, are reviewed.
The supplier selection process requires efficient and systematic methods to help
purchasers make sound decisions. The focus of this review is on operations research
(OR) techniques that have been used for supplier selection. OR offers methods and
techniques that support the decision maker in dealing with complexities involved
in the supplier selection process (De Boer et al. [10]).
This chapter is organized as follows. Section 2.2 presents previous reviews on
supplier selection literature. Section 2.3 discusses decision support models for supplier selection. A framework that classifies the literature according to the different
steps involved in the supplier selection process is used. Section 2.4 reviews the literature pertaining to inventory models with transportation costs. In Section 2.5, a
review of multi-stage inventory models is discussed. The conclusions drawn from
the literature review are presented in Section 2.6.
19
2.2
Previous Reviews of Supplier Selection
There has been a comprehensive effort to develop decision methods and techniques
for supplier selection. Some previous reviews of these decision methods have been
presented by Weber et al. [1], Holt [37], Degraeve et al. [38], and De Boer et al. [10].
Weber et al. [1] reviewed and classified 74 articles that appeared in the literature since 1966 in terms of the particular criteria mentioned in each article,
the purchasing environment, and the decision techniques used to select the best
suppliers.
Holt [37] presented a review of contractor evaluation and selection modeling
methodologies. Some of these methodologies included: multi-attribute analysis,
multi-attribute utility theory, and cluster analysis. Appropriate applications of
each one of these techniques were also discussed.
Degraeve et al. [38] used the Total Cost of Ownership (TCO) concept as a
framework for comparing supplier selection models. The TCO approach basically
considers all relevant costs involved in the purchasing process of a good or service
from a particular supplier. Some advantages and limitations of TCO are provided
in Bhutta and Huq [39].
De Boer et al. [10] studied the supplier selection literature in a more comprehensive manner. They extended previous reviews by classifying the existing literature in a framework. This framework recognized several decision-making steps in
the supplier selection process prior to the ultimate choice step. These steps are:
problem definition, formulation of selection criteria, pre-qualification (preliminary
screening), and final selection. Figure 2.1 shows how these steps proposed by De
Boer et al. relate to those presented by Monczka et al. [7].
In addition, De Boer et al.’s framework classified the supplier selection literature according to different purchasing situations such as first-time buys, modified
rebuys, and straight rebuys.
2.3
Decision Support Models
The literature on decision support models for supplier selection presented next is
discussed in relationship to the four steps of the supplier selection process concep-
20
STEP 1:
Recognize
g
the
Need for
Supplier Selection
STEP 2:
Identify Key
Sourcing
Requirements
STEP 1:
Problem
Definition
STEP 2:
Formulation
of Criteria
STEP 3:
Determine Sourcing
Strategy
STEP 4:
Identify Potential
Supply Sources
STEP 3:
Pre-qualification
STEP 5:
Limit Suppliers
In Selection
Pool
STEP 6:
Determine Method
of Supplier
Selection
STEP 4:
Final Selection
STEP 7:
S
Select Suppliers
and Reach
Agreement
Figure 2.1. Decision Steps in Supplier Selection
tualized by De Boer et al. [10], see Figure 2.1. Previously, the last literature review
in this area was published in 2001. The goal of this section is to update the most
relevant supplier selection literature.
2.3.1
Problem Definition and Formulation of Criteria
Problem definition, although an integral step in the supplier selection process, is
not a complex issue. It simply assumes the need for suppliers based on a company’s
demand (De Boer et al. [10]). Generally, this demand poses a supplier selection
problem for the company and consequently, requires that the company formulate
the particular criteria to be used throughout the remaining steps of the supplier
selection process.
Relying on a single criterion makes the supplier selection process risky. Therefore, a multi-criteria approach is recommended. A pioneering work in supplier
selection criteria was that of Dickson [40] in 1966. He identified and ranked 23 cri-
21
teria collected from responses to a questionnaire completed by purchasing agents.
In 1991, Weber et al. [1] reprioritized the 23 criteria identified by Dickson based
on 74 articles that appeared in the literature since 1966 (see Table 2.1).
Table 2.1. Supplier Selection Criteria
Rank
Dickson [40]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Weber et al. [1]
3
2
10
23
4
1
6
9
16
18
8
21
7
14
11
12
20
13
17
5
22
15
19
Criteria
Quality
Delivery
Performance History
Warranties and Claim Policies
Production Facilities and Capabilities
Net Price
Technical Capability
Financial Position
Bidding Procedural Compliance
Communication System
Reputation and Position in Industry
Desire for Business
Management and Organization
Operational Controls
Repair Service
Attitude
Impression
Packaging Ability
Labor Relations Records
Geographical Location
Amount of Past Business
Training Aids
Reciprocal Arrangements
Notice that in both rankings price, delivery, and quality continued to be considered important criteria. The most significant difference in the rankings is ‘Geographical Location’. With economic globalization, companies can choose suppliers
from anywhere in the world. For instance, developing countries are becoming
more competitive given their low labor and operating costs. Not only have criteria
changed over time in terms of importance, but in definition and meaning. Net
price was once considered the price offered by each vendor including discounts and
freight charges (Dickson [40]). Today, net price per se is not longer sufficient and
total cost has become a more accurate term. The total cost may include the fixed
cost (Current and Weber [41]), the inventory holding cost (Tempelmeier [42]), and
the technology cost (Bhutta and Huq [39]).
Current emphasis on reducing a supply chain’s cost has led to global sourcing
becoming a common practice among companies. Consequently, political stability,
22
foreign exchange rates, and tariffs (Motwani et al. [43]) have become important criteria. Even environmental issues have forced companies to increase their awareness
by integrating related criteria into their supplier selection decisions. Examples of
these criteria are air emission, water waste disposal, recycling, and chemical waste
(Humphreysa et al. [44]).
Mandal and Deshmukh [45] proposed interpretative structural modeling (ISM)
as a technique to help decision makers formulate and identify criteria in a systematic fashion. ISM separates dependent from independent criteria and identifies
the step in the supplier selection process where the criteria will be considered.
Recently, Chan [46] presented an interactive selection model using the analytic
hierarchy process to systemize Steps 1 and 2 of the supplier selection process.
2.3.2
Pre-qualification of Potential Suppliers
Pre-qualification is the process of reducing an initial set of potential suppliers
(pre-screening). Narrowing down the options facilitates an effective analysis and a
further more comprehensive investigation of the remaining suppliers. This reduces
the possibility of rejecting good suppliers early in the supplier selection process.
The methods employed for pre-qualification are: categorical methods, data envelopment analysis, and cluster analysis.
Categorical methods are qualitative models that help decision makers evaluate
their suppliers’ performance on a set of criteria using historical data and buyers’
experience. First, the suppliers’ performance on each criterion is categorized as
‘positive’, ‘neutral’, or ‘negative’. Second, and after the set of criteria has been
evaluated, the suppliers receive an overall rating, again labeled as either ‘positive’,
‘neutral’, or ‘negative’. Timmerman [47] discussed this method thoroughly.
Data Envelopment Analysis (DEA) is a classification system that splits suppliers between two categories, ‘efficient’ or ‘inefficient’. Suppliers are judged on
two sets of criteria, benefits (output) and costs (input). A supplier’s efficiency is
described as the ratio of the weighted sum of its outputs to the weighted sum of
its inputs (De Boer et al. [10]). For an example of how to use DEA as a tool for
negotiating with inefficient suppliers, see Weber et al. [48]. Liu et al. [49], Weber
and Ellram [50], and Weber et al. [48] presented some other applications of DEA
to the supplier selection problem.
23
Cluster Analysis (CA) is a method for statistical data analysis. Its purpose is
to separate a set of potential suppliers into smaller clusters where those grouped
together are most similar to each other and unlike those from other clusters. This
classification is used to reduce a larger set of suppliers into smaller more manageable subsets. Holt [37] suggested Euclidean distances as a way of establishing
degree of difference between suppliers.
2.3.3
Final Selection
Most decision models are final selection models. They are primarily concerned with
the allocation of final order quantities to selected suppliers. Figure 2.2 outlines the
different steps of the supplier selection process and the decision models associated
with each step. Of particular importance at this point, are the decision models
associated with Step 4.
Step 1: Problem Definition
Step 2: Formulation of Criteria
- Interpretative Structural Modeling (ISM)
- Interactive Selection Model
- Categorical Methods
Literature Review
Framework
S
Step
3:
3 Pre-qualification
f
of Suppliers
- Data
D t Envelopment
E
l
t Analysis
A l i (DEA)
- Cluster Analysis (CA)
- Linear Weighting Models
- Total Cost of Ownership (TCO)
Step 4: Final Selection
- Statistical Models
- Mathematical Programming Models
- Single Objective
- Multiple Objectives
- Artificial-Intelligence Based Models
Figure 2.2. Decision Models Used in Supplier Selection
Linear weighting models place a numerical weight on each selection criterion
(generally subjectively determined) and provide a total score for each supplier by
summing up the supplier’s performance on the criteria multiplied by these weights.
Although these approaches are very simple, they heavily depend on human judgment and proper scaling of criteria values.
24
The analytic hierarchy process (AHP) is considered one of the most widely
used linear weighting techniques. AHP provides a framework to cope with multiple
criteria (Saaty [51], and Saaty [52]). A hierarchical structure captures the criteria,
subcriteria, and alternative suppliers. The final AHP outcome is a score for each
supplier. The main advantage of AHP is that it handles both quantitative and
qualitative criteria. Examples of the use of AHP in supplier selection are provided
by Narasimhan [53], and Ghodsypour and O’Brien [54].
The multi-attribute utility theory (MAUT) proposed by Min [55] is also considered a linear weighting technique. MAUT handles multiple conflicting criteria
and allows decision makers to evaluate “what-if” scenarios.
Total cost of ownership (TCO) models look beyond price to include other major
costs affecting purchases. A taxonomy of TCO models is presented by Ellram [56].
Degraeve et al. [38] proposed the use of TCO as a basis for comparing supplier
selection models. Timmerman [47] proposed a method called cost-ratio which
collects all costs related to quality, service, and delivery, and express them as a
percentage of the total unit price.
Statistical models capture the uncertainty related to the supplier selection problem, for example, uncertain demand and stochastic lead times. As an approach to
capture uncertainty, Ding et al. [57] proposed a simulation optimization methodology for supplier selection. The methodology consists of three modules: (1)
a genetic algorithm (GA) optimizer that continuously searches for new supplier
portfolios; (2) using the output from the GA optimizer, a discrete-event simulation model is run to evaluate suppliers on pre-selected key performance indicators
(KPI’s); (3) after simulation runs, a fitness value is calculated based on the KPI’s.
The fitness is returned to the GA optimizer to search for the next supplier portfolio.
More recently, Liao and Rittscher [58] presented a multi-objective supplier selection
model under stochastic demand conditions with capacity and demand satisfaction
constraints. They developed a GA algorithm to find alternative solutions.
Mathematical programming models allow decision makers to consider different
constraints in selecting the best set of suppliers. Some of these constraints are
the minimum or maximum number of suppliers to be selected, limits on quantities allocated to suppliers, and quality levels. Most importantly, mathematical
programming models are ideal for solving the supplier selection problem because
25
they can optimize results using either single objective models or multiple objective
models.
Single objective models focus mainly on minimizing costs or maximizing profits.
This research separates them into the following categories:
• Linear programming: Moore and Fearon [59] stated that price, quality, and
delivery are important criteria for supplier selection. They discussed the
use of linear programming in the decision making process. Anthony and
Buffa [60] developed a single objective linear programming model to support
strategic purchasing scheduling (SPS). This model minimizes the total cost
(purchasing and storage) while considering purchasing budget and suppliers’ capacities as constraints. Turner [61] employed a single objective linear
programming model for evaluating alternative suppliers and allocating order
quantities to them. This model minimized the total discounted price by considering, as constraints, suppliers’ capacities, maximum and minimum order
quantities, demand, and regional allocated bounds. Pan [62] proposed multiple sourcing to improve the reliability of supply for critical materials. He
formulated a single objective linear programming model to select suppliers
based on: price, quality, and service.
• Nonlinear programming: Pirkul and Aras [63] analyzed the problem of determining order quantities for multiple items by considering all-units quantity
discounts on the purchasing price. They proposed a nonlinear mathematical
model with the objective of minimizing purchasing, holding, and ordering
costs. Additionally, they developed a Lagrangian relaxation procedure to
solve the model. Benton [64] introduced a nonlinear program and a heuristic
procedure using Lagrangian relaxation for supplier selection with multiple
items, multiple suppliers, resource limitations, and quantity discounts.
• Mixed integer programming: Bender et al. [65] applied single objective programming to develop a commercial computerized model for supplier selection at IBM. They used mixed integer programming to minimize purchasing,
transportation, and inventory costs. Narasimhan and Stoynoff [66] applied
a single objective mixed integer programming model to a large manufacturing firm in the Midwest to optimize the allocation procurement for a group
26
of suppliers. Chaudry et al.[67] used mixed integer linear programming to
minimize the total cost of supplier selection by considering price breaks. The
constraints included were capacity, delivery performance, and quality. This
model can be solved using commercial optimization software. Rosenthal et
al. [68] developed a mixed integer linear programming model to minimize
the total purchasing cost over a single period. The constraints considered in
this model were capacity of suppliers, demand satisfaction, delivery requirements, and quality. Later, Sarkis and Semple [69] reformulated Rosenthal
et al.’s model to reduce the computational effort and eliminate some limitations of the original model. Kasilingam and Lee [70] proposed a mixed
integer programming model to select suppliers and determine their order
quantities. This model considered the stochastic nature of the demand, the
quality of the supplied parts, the purchasing cost, the fixed cost of establishing new suppliers, and the cost of poor quality. Jayaraman et al. [3] proposed
a mixed integer linear programming model for supplier selection and order
quantity allocation. They considered quality, production capacity, storage
capacity, and demand satisfaction as constraints. Rosenblatt et al. [71] presented a supplier selection model with capacity as a constraint to find the
best acquisition policy. The costs in the objective function of their model
are purchasing, setup, holding, and supplier management cost. This model
uses the structure of the well known single-sink, fixed-charge transportation
(SSFCT) problem, which only studies the impact of suppliers’ capacity in
order quantity allocation. Recently, Chang [72] proposed an extension of
Rosenblatt’s work by adding a good relationship with suppliers and warehouse space capacity as constraints. His approach finds the global optimal
solution of the model. Tempelmeier [42] formulated a mixed integer linear
optimization model for supplier selection and order quantity determination
for a single item under dynamic conditions. Ghodsypour and O’Brien [36]
applied a mixed integer nonlinear programming model to select and properly
allocate order quantities to suppliers while minimizing the total annual ordering, holding, and purchasing costs. Quality and capacity were considered
as constraints to the problem. Their model was restricted to allocate only
one order to each selected supplier per order cycle.
27
• Dynamic programming: Alidaee and Kochenberger [73] developed a dynamic
programming algorithm that efficiently solved Rosenblatt et al.’s [71] formulation. Basnet and Leung [74] presented a multi-period inventory lot-sizing
scenario for multiple products and multiple suppliers. They provided an
enumerative search algorithm and a solution heuristic based on the WagnerWithin algorithm to obtain effective solutions to the problem.
Multi-objective models deal with optimization problems involving two or more
conflicting criteria. This research separates them into the following categories:
• Goal programming: Buffa and Jackson [75] presented a multi-criteria linear
goal programming model that considers two set of factors. In the first set,
supplier attributes included quality, price, service experience, and deliveries.
In the second set, the buyers’ specifications included material requirements
and safety stock. Chaudry et al. [76] suggested the use of goal programming
to select suppliers and allocate specific order quantities to these suppliers.
The criteria considered in their model are leadtime, service, and quality performance. Karpak et al. [77] used visual interacting goal programming as a
tool to approach the supplier selection problem. They applied it to the hydraulic pump division of a manufacturing company interested in identifying
the best suppliers and allocating orders to them while minimizing product
acquisition costs and maximizing quality and delivery reliability.
• Multi-objective programming: Weber and Current [78] used multi-objective
mixed integer programming to minimize the total purchasing price, late deliveries, and rejected units. An actual case was used to illustrate the model.
Weber and Ellram [50] also employed multi-objective programming in a justin-time environment considering the simultaneous trade-offs of price, delivery,
and quality. Narasimhan et al. [79] proposed a multi-objective programming
model to select suppliers and allocate order quantities to them. They assumed that the relative importance of supplier selection criteria varies according to the product’s life cycle. The resulting order quantities are determined after a bidding process across different periods.
Finally, it is worth noting that Wadhwa and Ravindran [80] presented and
compared several multi-objective optimization methods to solve the supplier se-
28
lection problem and allocation of order quantities. They used weighted objective,
goal programming, and compromise programming. The objective is to determine
the order quantity allocation while minimizing three conflicting criteria: price,
leadtime, and quality.
Artificial-Intelligence (AI) models are computer-based systems trained by the
decision maker using historical data and experience. In this way, the system is
able to replicate certain human decisions. These types of systems usually cope
very well with the complexity and uncertainty involved in the supplier selection
process. Albino and Garavelli [81] presented a decision support system for rating
subcontractors in a construction environment. Khoo et al. [82] discussed the concept of internet-based technology, intelligent software agents (ISAs), to automate
procurement decisions. Vokurka et al. [83] developed a system able to incorporate
the strategic partnership considerations of supplier selection.
2.3.4
Combined Approaches
Some authors have combined decision models from different steps in the supplier
selection process, for example, Weber et al. [48] combined DEA from Step 3 (prequalification of suppliers) and mathematical programming models from Step 4
(final selection). This combination provided decision makers with a tool for negotiating with suppliers. Degraeve and Roodhoft [2] developed a model combining mathematical programming model and TCO. They derived the inventory
management policy using activity-based costing information. Ghodsypour and
O’Brien [54] used AHP and mathematical programming to determine the best
order quantity allocation while considering qualitative criteria into the analysis.
Finally, Xia and Wu [84] presented an integrated approach of AHP improved by
rough sets theory and multi-objective mixed integer programming.
2.4
Inventory Models with Transportation Costs
Chapter 1 highlights the relevance of incorporating transportation costs into the
order quantity allocation decisions. Several researchers have also emphasized this
fact, e.g. Langley [85], Hall [86], Carter and Ferrin [87], Buffa [88].
Existing literature in supplier selection and order quantity allocation has typi-
29
cally assumed that: (1) transportation costs are managed by suppliers and, therefore, considered to be a part of the unit price; or (2) transportation costs are managed by the buyer and, therefore, considered to be a part of the setup/ordering
cost. These assumptions are clearly unrealistic because models do not consider
the effect of the shipment quantity on the per shipment cost of transportation, for
example, those times when goods are moved in smaller-sized, less-than-truckload
shipments (Warsing [89]).
To my knowledge, no publication currently presents a model that effectively
links the issue of order quantity allocation in the supplier selection problem with
multiple suppliers while considering inventory and actual transportation costs. One
difficulty in trying to incorporate transportation costs into the analysis is the nature of the actual freight rate structure. Trucking companies offer discounts on
the freight rate to encourage shippers to buy in larger quantities. Two problems
have been recognized when trying to incorporate actual freight rates into inventory models (Natarajan [19]): (1) determining the exact rates between origin and
destination is time consuming and expensive; and (2) the freight rate function is
not differentiable. A more detailed discussion of this issue is presented in Chapter 5. The remaining review examines different ways in which researchers have
incorporated the transportation cost into inventory management decisions.
Baumol and Vinod [90] proposed the inventory theoretic model integrating
transportation and inventory costs. Their approach incorporated three elements
of transportation: cost of shipping (constant shipping cost/unit), speed (mean lead
time), and reliability (variance of lead time). Demand and leadtime were treated as
random variables. It was assumed that demand follows a Poisson distribution and
that leadtime was normally distributed. Safety stock was calculated using a normal
approximation to the Poisson distribution. This continued to assume a constant
unit shipping cost and did not deal with freight rate discounts. Other researchers
have used theoretic models as foundations for further development. Das [91] extended Baumol and Vinod’s model to allow for independent order quantity and
safety stock decisions. Buffa and Reynolds [92] extended the inventory theoretic
model to include the rates for LTL, TL, and Carload (CL) shipments. Although
the transportation cost was still considered to be constant per unit shipped, they
used indifference curves to perform a sensitivity analysis by changing the param-
30
eters of the transportation factors. They concluded that the order quantity was
sensitive to tariff rate, moderately sensitive to variability in lead time, and insensitive to mean lead time. Another extension of the theoretic model is presented
by Constable and Whybark [93]. They developed a model incorporating backorder costs and transportation costs linearly related to volume. They provided
an enumeration method to get exact solutions and a heuristic to avoid the total
enumeration of all possible order quantities. Rieksts [94] investigated models with
TL and LTL transportation costs. He derived optimal policies for both infinite
and finite planning horizons that allowed a combination of the two transportation
modes as an alternative to using a unique option exclusively. The LTL rates were
assumed to be constant per unit shipped.
Langley [85] used an explicit enumeration procedure to determine the optimal
order quantity for a transportation step function (equivalent to freight rate discounts). From this analysis, Larson [95], Tyworth [96], and Carter and Ferrin [87]
continued to use enumeration techniques to determine the optimal order quantity
while explicitly considering the actual freight rate structure.
Other authors have alternatively proposed complex algorithms to deal with the
same problem. Burwell et al. [97] incorporated quantity and freight discounts in
inventory decision making when demand was dependent on price. An algorithm
was developed and implemented in a computer program to determine the optimal
order quantity and selling price for a class of demand functions, including constant
price-elasticity and linear demand. Lee [98] extended the basic EOQ model to
incorporate the freight cost as part of the setup cost. In addition, he considered
discounts in the freight rates in order to exploit economies of scale. He studied the
(1) all units discount structure, (2) the incremental discount structure, and (3) the
case of a stepwise freight cost (proportional to the number of trucks used). Tersine
and Barman [99] incorporated freight discounts and quantity discounts into the
order quantity decision in a deterministic economic order quantity environment.
Tersine et al. [100] used the inventory theoretic formulation as a base to develop two
optimal inventory-transportation decision support algorithms for freight discount.
The freight rates were modeled using weight discounts for both the all-weight and
incremental freight rate discount structures. Their models calculated the optimal
order quantity while minimizing the long-term costs.
31
Due to the complexity of the actual transportation freight rates, the use of
freight rate continuous functions to estimate actual freight rates has been repeatedly presented in the literature. Advantages of using continuous functions as stated
in Warsing [89] are: (1) they do not require the explicit specification of rate break
points for varying shipment sizes nor do they require any embedded analysis to determine if it is economical to increase, over-declare, the shipping weight on a given
route; and (2) continuous functions may be used in a wide variety of optimization
models.
The first author presenting an analysis of continuous functions was Langley [85].
He used a linear approximation to the transportation freight rates. Later on,
Ballou [101] proposed a linear approximation of trucking rates. Swenseth and
Godfrey [102] studied five alternative freight rate functions: constant, proportional,
exponential, inverse, and adjusted inverse. They evaluated these functions on how
well they emulate the actual freight rates. This is measured by the minimum
mean squared difference between rates obtained by the proposed functions and the
actual freight rates. Later, Swenseth and Godfrey [103] recommended the use of
the inverse and adjusted inverse freight rate functions to approximate actual freight
rates while determining the optimal order quantity. They proposed a heuristic to
predict the shipping weight at which the shipment should be over-declared as a TL.
In this case, the function that best emulates the TL cost is the inverse function.
Conversely, LTL is best emulated by means of the adjusted inverse function. In
overcoming some of the lack of fit from the functions proposed by Swenseth and
Godfrey, especially in the case of LTL, Tyworth and Zeng [104] and Tyworth and
Ruiz-Torres [105] proposed the use of a power function to model the freight rates
within a truck load (LTL).
Some interesting applications of continuous functions have been presented by
DiFilippo [106], Mysore [107], and Natarajan [19]. DiFilippo [106] considered a
multi-criteria optimization approach to model the supply chain as a single warehouse, multiple retailer network. Transportation was considered as a criterion
along with capital invested in inventory and annual number of orders. He used
the proportional function introduced by Swenseth and Godfrey [103] to develop
his theoretical results. Since the actual transportation freight rates were known,
he showed that Swenseth and Godfrey’s function can be reduced to Langley’s [85]
32
function. Mysore [107] studied a three-stage supply chain system where multiple
modes of transportation were considered. Natarajan [19] modeled the supply chain
as a single warehouse, multi-retailer problem. He assumed that LTL was used to
procure units from the warehouse to the different retailers. In order to estimate the
actual freight rates, he used the adjusted inverse function proposed by Swenseth
and Godfrey [103]. Whenever an order was placed it was assumed that TL was
used as the mode of transportation, therefore, the inverse function proposed by
Swenseth and Godfrey [103] is used to emulate the actual freight rates. It is clear
that the assumptions concerning the type of transportation, LTL for the retailers
and TL for the warehouse, determine the type of function to be used. However,
if after solving the problem, the shipping quantity for the warehouse can be overdeclared as a TL, then the specific function used might be overestimating the total
annual transportation cost. The same applies to the case of the warehouse. In
order to overcome this problem, a formulation with an additional binary variable
indicates when to use the proportional function and when to use the inverse function. Although Natarajan assumes the proportional function for LTL to develop
his theoretical results, a power function is used in the numerical examples and case
studies. The reason is that a power function is a better fit for the data used. This
function is similar to the one proposed by Tyworth and Zeng [104], and Tyworth
and Ruiz-Torres [105].
2.5
Multi-Stage Inventory Models
The first inventory policies for multi-stage systems were presented by Clark and
Scarf [108] and Hadley and Within [109]. Determination of optimal inventory policies for multi-stage inventory systems is made difficult by the complex interaction
between different levels, even in the cases where demand is deterministic. Given
this, several researchers have developed different approaches to find effective solutions to these problems. Schwarz [110] concentrated on a class of policies called
the basic policy and showed that the optimal policy can be found in a set of basic policies. He proposed a heuristic solution to solve the general one-warehouse
multi-retailer problem. Rangarajan and Ravindran [111] introduced a base period
policy for a decentralized supply chain. This policy states that every retailer orders
33
in integer multiples of some base period, which is arbitrarily set by the warehouse.
Most recently, Natarajan [19] proposed a modified base period policy for the onewarehouse, multi-retailer system. He formulated the system as a multi-criteria
problem and considered transportation costs between the echelons.
Roundy [112] introduced the power-of-two policies. He presented a 98% effective power-of-two policy for a one-warehouse, multi-retailer inventory system
with constant demand rate. In this class of policies, the time between consecutive
orders at each facility is a power-of-two of some base period. Several researchers
have used the power-of-two policies for multi-stage inventory systems that do not
incorporate supplier selection. These policies have proven to be useful in supply
chain management since they are computationally efficient and easy to implement.
Maxwell and Muckstadt [113] developed a power-of-two policy for a productiondistribution system. Roundy [114] extended his original 98% effective policy to
a general multi-product, multi-stage production/inventory system where a serial
system is a special case. Federgruen and Zheng [115] introduced algorithms for finding optimal power-of-two policies for production/distribution systems with general
joint setup cost. For the stochastic cases, Chen and Zheng [116] presented lower
bounds for the serial, assembly, and one-warehouse multi-retailer systems.
For the serial inventory system, Schwarz and Schrage [117] and Love [118]
proved that an optimal policy must be nested and follow the zero-ordering inventory policy. A policy is nested provided that if a stage orders at any given
time, every downstream stage must order at this time as well. The zero-ordering
inventory policy refers to the case when orders only occur at an inventory level
of zero. Muckstadt and Roundy [119] developed a power-of-two policy for a serial assembly system and proved that such a policy cannot exceed the cost of any
other policy by more than 2% for a variable base period. They introduced an algorithm to solve the problem along with the corresponding analysis of the worst case
behavior. Sun and Atkins [120] presented a power-of-two policy for a serial system that includes backlogging. They reduced the problem with backlogging to an
equivalent one without backlogging and used Muckstadt and Roundy’s algorithm
to solve this transformed problem. For serial systems with stochastic demand,
an echelon-stock (R, nQ) policy for compound Poisson demand was introduced by
Chen and Zheng [121].
34
Most recently, Rieksts et al. [122] developed power-of-two policies for a serial
inventory system with a constant demand rate and incremental quantity discounts
at the most upstream stage. They provided a 94% effective policy for a fixed base
planning period and a 98% effective policy for a variable base planning period.
Chapter 6 proposes a serial supply chain system with supplier selection and
order quantity considerations at the first stage. A power-of-two policy is proposed
that provides near-optimal solutions for the system.
Some authors have considered multi-criteria approaches to multi-stage inventory systems. Thirumalai [123] modeled a supply chain system with three companies arranged in series. He studied the cases of deterministic and stochastic demands and developed an optimization algorithm to help companies achieve supply
chain efficiency. DiFillipo [106] extended the one-warehouse multi-retailer system
using a multi-criteria approach that explicitly considered freight rate continuous
functions to emulate actual freight rates for both centralized and decentralized
cases. Natarajan [19] studied the one-warehouse multi-retailer system under decentralized control. The multiple criteria models are solved to generate several
efficient solutions and the value path method is used to display tradeoffs associated with the efficient solutions to the decision maker of each location in the
system.
2.6
Conclusions
Several authors have pointed out the importance of supplier selection by emphasizing the impact that decisions throughout the entire supply chain have, from
procurement of raw materials to delivery of finished products to final customers.
In order to help decision makers or purchasers make sound decisions with respect to
supplier selection, researchers have developed decision methods and models dealing with different aspects of the supplier selection process. This chapter presents
an extensive review and analysis of these decision methods and models as well
as research related to the subsequent chapters in this research (inventory models
with transportation costs and inventory models for multi-stage inventory systems).
This literature review has helped identify some opportunities and limitations in
the area of supplier selection research.
35
Several mathematical programming models have been proposed to solve the
supplier selection problem. Most of these models include approaches with a single
objective such as cost minimization or profit maximization. Despite the multiple
criteria nature of the supplier selection problem, very little work has been devoted
to this problem using multi-criteria techniques. In addition to the lack of models
considering the multi-criteria nature of the problem, most of the existing models for
supplier selection are concerned only with the ‘final selection’ step in the supplier
selection process. However, the quality of the final choice largely depends on the
quality of the steps prior to it. For instance, the quality of the final set of selected
suppliers depends on the quality of the screening process prior to arriving at the
final selection step.
Inventory management is an often overlooked area in supplier selection research. Currently, order quantity allocation continues to be considered part of the
final selection step in the supplier selection process and most mathematical models
in the literature assume the size of the order quantity to allocate to suppliers is
predetermined. This is equivalent to solving a single-period problem (short-term
planning) that does not consider inventory management over time and is generally used to make a one-time decision. However, when a planning horizon covers
multiple periods, problems have the potential to yield much better procurement
decisions by incorporating inventory management.
Another particular characteristic among the existing order quantity allocation
models is that they only study the impact of inventory on the direct purchaser
(single-stage systems). Given the prevalence of both supplier selection and inventory control problems in supply chain management, single-stage systems must be
extended to consider different supply chain configurations. In response, the problem will be to evaluate this trade-off to determine the appropriate level of inventory
throughout the different stages of the supply chain while properly allocating orders
to the selected suppliers.
Finally, existing literature in supplier selection considering transportation costs
in its analysis assumes that the transportation cost is part of the unit purchasing
price implying a fixed rate per unit. These assumptions are unrealistic because
freight rates usually decrease as shipping weights increase. In order to incorporate
transportation costs into the order allocation models, one should first be able
36
to identify transportation cost functions emulating actual freight rates without
increasing the complexity of the decision model.
Chapter
3
A Three-Phase Multi-Criteria
Methodology for Supplier Selection
3.1
Introduction
The supplier selection problem is complicated and risky, owing to a variety of
qualitative and quantitative factors affecting the decision-making process. Despite
the multiple criteria nature of the problem, very little work has been devoted to
the study of the supplier selection problem by using multi-criteria techniques such
as goal programming, multi-objective programming, or other similar approaches.
In addition, as noted in the literature review, most attention has been paid
to the final choice phase in the supplier selection process. However, the quality
of the final choice largely depends on the quality of the steps prior to that final
choice step. In this regards, there has not been an integrated approach involving
all the steps in supplier selection process. The importance of the methodology
presented in this chapter is that it considers the various phases of the supplier
selection process and presents an efficient methodology that integrates them.
First, L2 metric is used to screen an initial list of suppliers; then, the Analytical
Hierarchy Process (AHP) is utilized to determine the weights of both, qualitative
and quantitative criteria in a very powerful and easy way. Another important tool
implemented in our approach is Goal Programming (GP). Unlike most mathematical programming models, goal programming provides the decision maker (DM)
with enough flexibility to set target levels on the different criteria and obtain the
best compromise solution that comes as close as possible to each one of the defined
38
targets.
In general, this methodology can be applied to any kind of company. For
illustrative purposes, it has been applied to a manufacturing facility located in
Tijuana, Mexico. Because of confidentiality issues, the data used in this paper
have been disguised. Nevertheless, the criteria and goals shown do reflect the
actual procedure developed jointly with the Purchasing Manager of this company.
The chapter is divided as follows. Section 3.2 presents the proposed methodology for supplier selection along with a numerical example that illustrates each
one of the phases involved. Section 3.3 presents the development and application
of the goal programming model (Phase 3) along with an analysis of the results.
Sections 3.4 presents some important managerial implications and conclusions.
3.2
3.2.1
The Three-Phase Multi-criteria Methodology for Supplier Selection
Phase 1: Screening Process with an Lp Metric
The first phase in the methodology requires that the company define the criteria
that will be used to select their suppliers. The set of criteria chosen is unique to
every company and component/product, though they all reflect several similarities.
The purpose of using an Lp metric in this phase is to reduce the initial list of
suppliers with minimal effort. A short manageable list is not only easy to handle
but will help the DM to efficiently collect detailed data on the suppliers and apply
AHP in Phase 2. The technical details on how to implement the Lp metric (Phase
1) are described next and summarized in Figure 3.1.
The Lp metric represents the distance between two vectors x, y with the same
number of elements. One of the most commonly used Lp metrics is the L2 metric,
which measures the Euclidean distance between vectors. The ranking of alternatives is done by calculating the L2 metric between the ideal solution and each
vector representing the supplier’s ratings for the criteria. Mathematically, this is
computed as follows:
v
u n
uX
kx − yk2 = t
|xi − yi |2
i=1
(3.1)
39
List of
Potential
Suppliers
SUPPLIER
SELECTION
Delete Dominated
Suppliers
Use the L2 Metric
to Screen the
List
1. Define the Ideal
Value for Each
Criterion
2. With these
Values, Form the
Vector y
3. Compute the L2
Norm for Every
Supplier
4. Rank Suppliers
in Ascending
Order
M k a List
Make
Li t
With the Top
Suppliers
Small Base
of Suppliers
1
Figure 3.1. Phase 1 – Screening the Initial List of Suppliers
The algorithm for this phase is described next:
STEP 1. Define the ideal value for each criterion and sub-criterion. The ideal
value represents the best value attainable for each criterion/subcriterion from
the list of potential suppliers.
STEP 2. Use these values to form the ideal vector (denoted by y) as in Table 3.1.
Table 3.1. Ideal Values for Each Criterion
Ideal Values
Ideal Vector y
Price
($)
Cpk
(index)
Defective
Parts (ppm)
Flexibility
(%)
Service
(%)
Distance
(km)
Leadtime
(hrs/part)
40
2
3.4
25
100
5
0.05
40
STEP 3. Use the L2 metric to measure how “close” the rating vector xi for each
supplier matches the ideal supplier vector y (Supplier’s data is provided in
Table 3.2).
Table 3.2. Initial Suppliers’ Data
Criteria
Supplier
Price
($)
Cpk
(index)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
50
80
45
60
40
60
65
70
45
70
75
65
80
75
70
70
85
65
55
80
85
0.95
2
0.83
1
1.17
1.5
1.33
1.5
1
1.25
0.83
1
1.33
1.15
1.33
1.05
1.25
0.95
0.83
1.25
0.83
Defective
Parts(ppm)
105,650
3.4
158,650
66,800
22,750
1,350
6,200
1,350
66,800
12,225
158,650
66,800
6,200
22,750
6,200
44,500
12,225
105,650
158,650
12,225
158,650
Flexibility
(%)
Service
(%)
10
0
25
15
18
5
0
0
5
10
15
0
0
2
5
0
5
0
10
10
0
75
100
65
85
90
99
100
50
80
85
75
80
85
87
86
65
70
77
89
85
50
Distance
(km)
Leadtime
(hrs/part)
500
1,500
50
5,000
9,500
7,250
10
15,000
7,500
12,500
1,345
6,680
5,000
16,000
17,000
1,860
1,789
1,775
2,500
12,500
17,500
0.25
0.60
0.20
0.80
0.95
0.50
0.10
1.50
1.75
2.00
1.25
1.15
1.00
0.90
0.95
1.50
1.45
0.90
0.75
1.50
2.00
In case the different criteria and sub-criteria chosen are not measured using
the same scale, i.e. 0–1, 0–10, 0–100, the initial list of criteria values of the
suppliers must be normalized before computing the L2 norm.
To normalize the data it must be recognized whether each criterion is improved when minimized or maximized. Once this is established, one of the
following two equations is used to normalize the data:
If Minimizing, use
Hj − fij
fij − Lj
; otherwise, use
,
Rj
Rj
where Hj is the maximum value, and Lj is the minimum value for the j th
criterion, fij is the score of the ith supplier for the j th criterion and Rj
represents the corresponding range, Hj − Lj . Scores that represent or match
41
the ideal value get a normalized value of one, while the lowest scores get a
normalized value of zero. Table 3.3 shows the normalized data for Table 3.2.
Note that this normalization method converts all criteria to maximization.
Hence the ideal values are all ones.
Table 3.3. Normalized Suppliers’ Data
Criteria
Supplier
Price
($)
Cpk
(index)
Defective
Parts(ppm)
Flexibility
(%)
Service
(%)
Distance
(km)
Leadtime
(hrs/part)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0.78
0.11
0.89
0.56
1.00
0.56
0.44
0.33
0.89
0.33
0.22
0.44
0.11
0.22
0.33
0.33
0.00
0.44
0.67
0.11
0.00
0.10
1.00
0.00
0.15
0.29
0.57
0.43
0.57
0.15
0.36
0.00
0.15
0.43
0.27
0.43
0.19
0.36
0.10
0.00
0.36
0.00
0.33
1.00
0.00
0.58
0.86
0.99
0.96
0.99
0.58
0.92
0.00
0.58
0.96
0.86
0.96
0.72
0.92
0.33
0.00
0.92
0.00
0.40
0.00
1.00
0.60
0.72
0.2
0.00
0.00
0.20
0.40
0.60
0.00
0.00
0.08
0.20
0.00
0.20
0.00
0.40
0.40
0.00
0.50
1.00
0.30
0.70
0.80
0.98
1.00
0.00
0.60
0.70
0.50
0.60
0.70
0.74
0.72
0.30
0.40
0.54
0.78
0.70
0.00
0.97
0.91
1.00
0.71
0.46
0.59
1.00
0.14
0.57
0.29
0.92
0.62
0.71
0.09
0.03
0.89
0.90
0.9
0.86
0.29
0.00
0.90
0.72
0.92
0.62
0.54
0.77
0.97
0.26
0.13
0.00
0.38
0.44
0.51
0.56
0.54
0.26
0.28
0.56
0.64
0.26
0.00
Sometimes it is easy to identify dominated alternatives, i.e. alternatives
(suppliers) whose individual scores are less than or equal to the criterion
scores for another alternative (supplier). The dominated alternatives are
obviously not good choices; hence they can be eliminated from the analysis.
To compute the L2 metric use Equation (3.1).
STEP 4. Rank the suppliers by ordering them in ascending order; i.e., the
supplier with the smallest L2 value should be ranked as # 1 and so on
(See Table 3.4). Pre-select the list of suppliers to a short list for further
consideration based on their ranking (e.g. the top 5, top 10, etc).
For illustration, the first seven suppliers are chosen for further consideration.
The number of selected suppliers is up to the decision maker (DM), but
42
Table 3.4. Ranking Ordering of Suppliers Based on L2 Value
Supplier
L2 value
Rank
Supplier
L2 value
Rank
1
2
3
4
5
6
7
8
9
10
1.92
1.88
2.51
1.58
1.15
1.25
1.64
3.91
2.66
2.82
#6
#5
#7
#3
#1
#2
#4
#20
#10
#13
11
12
13
14
15
16
17
18
19
20
3.4
2.84
2.53
3.09
2.65
3.24
2.94
2.97
2.67
2.72
#19
#14
#8
#17
#9
#18
#15
#16
#11
#12
generally this number should be less than 10. The data for the top ranked
suppliers will be used in later sections in Phases 2 and 3.
3.2.2
Phase 2: Criteria Weights and Ranking of Suppliers
with AHP
The relevance of using AHP in this phase relies on the fact that this technique
allows a company to involve the decision maker (DM) in the assessment of not only
numerical but also intangible factors (e.g., supplier’s prestige, financial stability,
or the matureness of their quality management system).
Figure 3.2 shows a typical example of the criteria used for supplier selection.
The structure given by Figure 3.2 will be shown to be very useful when performing
AHP to compute the criteria weights.
The value of Phase 1 becomes obvious when AHP is implemented in Phase
2 because AHP can be a tedious and inefficient process for ranking more than
10 suppliers. AHP requires a number of pairwise comparison questions between
criteria/subcriteria and between alternatives. Figure 3.3 shows how the number
of questions to be answered by the DM increases when using AHP; this number
exceeds 500 questions for more than 10 alternatives (suppliers) and nine criteria.
Figure 3.4 summarizes the steps for Phase 2. The two outputs from this phase
consist of the weights for the criteria and a list of suppliers with their respective
total scores. This output will be used in Phase 3, during the formulation of the
43
Supplier
pp
Selection
Main Criteria
Fl ibilit
Flexibility
Q lit
Quality
Process
Capability
Supplier 1
Pi
Price
QMS
Maturity
S i
Service
Defective
Parts
Supplier 2
…Sub-Criteria…
Supplier 3
D li
Delivery
Direct
Distance
Leadtime
…
Supplier 4
Supplier N
Figure 3.2. AHP Structure
# of questions asked
30000
25000
20000
15000
10000
5000
0
0
10
20
30
40
50
60
70
80
# Alternatives to compare
Figure 3.3. Growth in the Number of Questions
GP model.
3.2.2.1
AHP Algorithm
This section summarizes the basic blocks in the AHP algorithm. The figures and
tables shown were used to develop the example in this research. AHP uses the
rating scale shown in Table 3.5 for the pairwise comparison questions.
STEP 1. Do a pairwise comparison of the main criteria using the scale in
Table 3.5. Form the matrix Anxn = [aij ] , where the aij entry represents
the relative importance of criterion ‘i ’ with respect to criterion ‘j ’. That is,
how much more important the ith criterion is relative to the j th criterion. To
illustrate this concept, a decision maker (e.g. purchasing manager) may be
44
1
Phase 2
Criteria and
Sub-Criteria
1. Perform the Pairwise
i C
Comparison
i
of Criteria
2. Compute Normalized Weights for
Criteria (w)
3. Check for
Consistency
Weights for
Criteria
For Each Criterion,
Compare all
Suppliers
1
Perform Steps
1, 2, 3 of AHP
Methodology
For each criterion,
The weights from
A Column of the
Score Matrix (S)
Compute the
Total Scores
1
Total Scores
(TS)
2
Figure 3.4. Phase 2 – Defining the Weights with AHP and Supplier Screening
asked how much more important is quality compared to price in selecting a
supplier? The answer should then be coded based on the scale.
Let aii = 1, ∀i, and aji = 1/aij . The matrix A for the numerical example
is shown in Table 3.6.
STEP 2. Compute the normalized weights for the main criteria from matrix A.
The most common way to do this is by normalizing each column using L1
norm, as follows:
aij
Compute rij = Pn
, then average the rij values to get the weights,
i=1 aij
P
j rij
wi =
.
n
The normalized weights that correspond to the pairwise comparison matrix
(Table 3.6) are shown in Table 3.7.
45
Table 3.5. Rating Scale for Pairwise Comparison
Degree of Importance
Definition
1
3
5
7
9
2, 4, 6, 8
Equal importance
Weak importance of one over another
Essential or strong importance
Demonstrated importance
Absolute importance
Intermediate values between the two
adjacent judgments
Table 3.6. Pairwise Comparison Matrix
Criteria
Quality
Delivery
Flexibility
Service
Price
Quality
Delivery
Flexibility
Service
Price
1
0.333333
0.333333
0.2
1
3
1
1
0.333333
1
3
1
1
0.333333
3
5
3
3
1
5
1
1
0.333333
0.2
1
Steps 1 and 2 are continuously performed throughout every sub-level of criteria and sub-criteria. In the current example, the weights for the five main
criteria are determined first, and then the two sub-levels of quality and delivery are compared separately. The final weight of a sub-criterion is the
product of the weights along the corresponding branch. Figure 3.5 shows the
numerical values.
Table 3.7. Normalized Matrix
Criteria
Quality
Delivery
Flexibility
Service
Price
Quality
Delivery
Flexibility
Service
Price
Weights
0.348837
0.116279
0.116279
0.069767
0.348837
0.473684
0.157895
0.157895
0.052632
0.157895
0.36
0.12
0.12
0.04
0.36
0.294118
0.176471
0.176471
0.058824
0.294118
0.283019
0.283019
0.09434
0.056604
0.283019
0.351932
0.170733
0.132997
0.055565
0.288774
46
Supplier
pp
Selection
Main Criteria
W1 = 0.133
Fl ibilit
Flexibility
W2 = 0.352
Q lit
Quality
W21 = 0.25
Process
Capability
W21 = (0.25)(0.352)=0.088
Supplier 1
W22 = 0.5
QMS
Maturity
W22 = 0.176
Supplier 2
W3 = 0.289
Pi
Price
W4 = 0.055
S i
Service
W23 = 0.25
W5 = 0.171
D li
Delivery
W32 = 0.857
W31 = 0.143
Defective
Parts
…Sub-Criteria…
W23= 0.088
Direct
Distance
Leadtime
W31 = 0.024
Supplier 3
Supplier 4
…
W32 = 0.147
Supplier N
Figure 3.5. Supplier Selection Criteria Weights
STEP 3. Test consistency of the decision maker’s responses via the pairwise
comparison matrix A. If the DM is perfectly consistent then, A (before
W1 = 0.133
normalization) has the following
property:

  
1
w1 /w2 · · · w1 /wn
w1

  
 w /w
 
1
· · · w2 /wn 
 2 1
  w2 
  
A·w
~ =
~
 ..
..
..  ·  ..  = λmax w.
..
 .



.
.
.
.

  
wn /w1 wn /w2 · · ·
1
wn
where w
~ is the eigen vector corresponding to the eigen value λmax . Saaty [52]
has proven that if A is perfectly consistent, then λmax = n and that λmax ≥ n.
The difference between λmax and n is used as a measure of DM’s consistency.
In particular, (λmax − n)/(n − 1) is the variance of the error incurred in
estimating aij , and is called Consistency Index (CI ):
CI =
λmax − n
n−1
where the eigen value (λmax ) is calculated as follows:
A1• · w A2• · w
Am• · w
λmax = Avg
,
,··· ,
.
w1
w2
wm
(3.2)
47
Finally, the Consistency Ratio (CR) is given by:
CR =
CI
RI
where RI is obtained from Table 3.8 as a function of the number of criteria compared (n). The set of numbers in Table 3.8 is an average random
consistency index derived from a sample of randomly generated reciprocal
matrices using the scale 1/9, 1/8, . . . , 8, 9 (Saaty and Vargas [124]). If CR<
0.1, accept the pairwise comparison matrix. An inconsistency of 10 percent
or less implies that the adjustment is small compared to the actual values of
the eigenvector entries.
Table 3.8. Random Index (RI) Values (Saaty [51])
n
RI
2
0
3
0.52
4
0.89
5
1.11
6
1.25
7
1.35
8
1.4
9
1.45
10
1.49
The respective computations for the example lead to the results shown in
Figure 3.6.
Axw
(A x w)/wi
1.829720
5.1990786
0.876509
5.1338128
0.683994
5.1429332
0.284949
5.1281943
1.488255
5.1537058
λmax
5.1515449
Consistency Index:
0.037886
Consistency Ratio:
0.034132
Figure 3.6. Consistency Test Results for the Pairwise Comparison Matrix
Figure 9: Consistency Ratio and Consistency Index
48
Since the consistency test is good, the DM should proceed to rank all the
suppliers by comparing them with regard to each criterion using AHP. The
weights computed for each criterion form a column. All the columns form
the so-called Score Matrix S. Finally, a total score (T S) for each supplier is
determined by using Eq. (3.3).
T S = [S × w] .
(3.3)
where w corresponds to the criteria weights computed in previous steps. The
suppliers are ranked based on their TS values (higher the better).
3.2.3
Phase 3: Order Quantity Allocation with a Preemptive GP Model
The model described in this phase is used to allocate the required demand to suppliers. Therefore, model variables are the planned purchases from each vendor.
It is assumed that the required demand corresponds to an order quantity that is
determined in advance. Therefore, Phase 3 is equivalent to solving a single-period
problem in which no inventory management is considered (one-time decision).
To allocate this order quantity, goal programming (GP) is used as an appropriate technique. In GP, all the objectives are assigned target levels for achievement
and a relative priority on achieving these levels. GP treats these targets as goals
to aspire for and not as absolute constraints (Ravindran et al. [125]). There are
two types of goal programming: preemptive and non-preemptive. In the preemptive case, goals at higher priority must be satisfied as far as possible before lower
priority goals are even considered. Therefore, the problem reduces to a sequence
of single-objective optimization problems. In the non-preemptive case, different
weights are assigned to each goal turning the problem into a single-objective optimization problem, consequently assuming a linear utility function (Goicoechea
et al. [126]). Since the nature of the Supplier Selection problem suggests that the
utility function is nonlinear, implementing a non-preemptive GP model might not
be very realistic; therefore a preemptive GP model is proposed to emulate the
behavior of such utility functions.
The advantages of using goal programming are that (1) it allows the firm
49
to set planning goals related to the supplier selection criteria and policies, (2)
GP also lets the company assign priorities on these goals, reflecting their relative
importance, and (3) setting goals allows a company to control the deviation from
targets and achieve tradeoffs for goals in conflict.
3.2.3.1
Goal Constraints
Goal constraints must be developed together with management and must be defined according to the company’s main goals. In this case, the constraints were
derived from the Scorecard used in the Supplier’s Evaluation process. Some constraints had to be redefined or changed to meet the model’s specific needs. Table 3.9
presents the notation and terminology used.
Table 3.9. Problem Notation
n
Xi
D
Ci
T Si
Li
li
C̄pk
Cpi
qi
SL
Si
F
∆i
P Ri
Zi
Yi
d+
d−
Number of suppliers
Ordered quantity from ith supplier
Demand (predetermined order quantity)
Capacity of ith supplier
Total score of ith supplier
Company’s required leadtime for the ith supplier
Time required by ith supplier to procure one unit of product
Company’s required level of Cpk
Cpk of ith supplier
Defects of ith supplier (in parts per million)
Service level required
Service level of ith supplier
Level of flexibility required
Flexibility level of ith supplier
Price of ith supplier
Distance from ith supplier to buyer
1, if an order is allocated to ith supplier; 0, otherwise
Amount of deviation above the goal
Amount of deviation below the goal
The goal constraints included in the model are introduced next.
Weighted Value of Purchase – WVP. In this goal constraint, the total
scores obtained in Phase 2 form the coefficients T Si for each supplier. The aim is
to maximize the total WVP. In other words, the total scores indicate particular
preferences of the DM when comparing the suppliers with respect to the criteria.
It is then tried to maximize the number of units allocated to suppliers with higher
50
total scores. In general, WVP is maximized by setting an ideal value (M ) to
the goal constraint and trying to minimize the underachievement d−
1 as much as
possible.
n
X
+
T Si Xi + d−
1 − d1 = M.
(3.4)
i=1
Distance goal. Globalization seems to be changing paradigms in industry
with international suppliers. Unfortunately there is still a strong negative correlation between quick delivery and distance. JIT requires that ideally suppliers
should be close to the buyer; as a matter of fact several companies keep as many
suppliers as possible to a distance where they can supply any order within minutes. The following goal constraint minimizes the total distance to the suppliers
selected. The distance is minimized by setting an ideal goal of zero. The objective
is to try to minimize the overachievement d+
2 .
n
X
+
Zi Yi + d−
2 − d2 = 0.
(3.5)
i=1
Process Capability (Cpk ). Current Six Sigma trends motivate companies
to ensure certain quality level throughout the value stream. Consequently, it is
logical to avoid as much as possible, suppliers that do not meet a specific quality
level. This constraint is strictly on the average, hence the restriction does not
discriminate any supplier for not achieving this goal, but it does select a group
of suppliers satisfying such constraint. For the current example, this index represents the supplier’s sigma level with respect to a critical quality feature, given the
respective LSL (Lower Specification Limit) and USL (Upper Specification Limit)
provided by the company. The objective is established as to minimize d−
3 , the
underachievement of Cpk .
n
X
i=1
Cpi Yi +
d−
3
−
d+
3
= C̄pk
n
X
Yi .
(3.6)
i=1
Flexibility goal. One of the most important competitive advantages of world
class companies is their ability to satisfy a dynamic demand. Flexibility allows
a company to expand its capacity and respond to changes in demand. Hence,
companies must try to select suppliers that maximize the company’s flexibility.
51
The objective of this goal is to minimize d−
4 , the underachievement of a flexibility
level required by the purchaser.
n
X
+
∆i Yi + d−
4 − d4 = F
n
X
i=1
Yi .
(3.7)
i=1
Quality – Defective parts per million (ppm). This goal constraint was
chosen to minimize the defective percentage rate of our suppliers. It is known
that there is a direct relationship between Cpk and ppm, but herein they are
differentiated by considering ppm in a more general sense; i.e., considering not
only as defective products, those who do not meet the company’s specifications
for a certain critical quality feature, but for any non-conformance issue that may
appear. The objective of this goal is set to minimize d+
5 , the overachievement of
defective parts.
n
X
+
qi Yi + d−
5 − d5 = 0.
(3.8)
i=1
Service level goal. With the increasing importance in keeping a performance
indicator to monitor service satisfaction, most of the companies keep track of their
supplier service level. It is a prudent choice to keep suppliers that provide an
average satisfaction level (SL). The service level required is kept at an optimal
value by minimizing d−
6 .
n
X
+
Si Yi + d−
6 − d6 = SL
i=1
n
X
Yi .
(3.9)
i=1
Purchasing expenses. Purchasing expenses reflects the total cost in the
buyer’s location warehouse, including cost of distance for freight, and broker costs
as well. This constraint minimizes the purchasing expenses made by the company,
according to the orders placed and the individual price (total cost) offered by every
supplier. The objective in this case, is to minimize the overachievement (d+
7 ) of an
unrealistic target of zero cost.
n
X
+
P Ri Xi + d−
7 − d7 = 0.
(3.10)
i=1
Leadtime goal. Take li to be the production rate at which an order can be
52
satisfied by the ith supplier. Therefore, the time it takes the supplier to fulfill
an order is directly proportional to this variable. The company usually has a
maximum allowed leadtime for every single supplier (Li ), usually being more strict
with local suppliers. There will be at most ‘n’ constraints of this type. The
objective is established as to minimize d+
8 , the overachievement of Li .
+
l i X i + d−
8 − d8 = Li , i = 1, 2, ..., n.
3.2.3.2
(3.11)
Real Constraints
The following two constraints must be always satisfied:
n
X
Xi = D,
(3.12)
i=1
X i ≤ Ci ,
i = 1, 2, ..., n.
(3.13)
Equation (3.12) implies that the orders placed over a given period must satisfy
the demand. Equation (3.13) refers to the fact that a particular order can not
exceed the corresponding capacity of that supplier.
Figure 3.7 summarizes the steps for Phase 3.
2
Phase 3
Purchasing
Department
Goals
Define the
Goal Priorities
Develop the
Mathematical GP
Model
Solve the
Model
Perform a
Sensitivity
Analysis
Display
Results
Figure 3.7. Phase 3 – Goal Programming
53
3.3
Application and Analysis
This section presents the application of the GP model along with the analysis of
results. It is important to note that this analysis is performed on the top seven
suppliers obtained in Phase 1. For this application a preemptive GP model is
considered, as explained before. Before deciding how to allocate the order quantity,
the specific goal priorities used in this model are presented in Table 3.10. This
priority structure was defined by the company and reflects the importance given
(by the DM) to the different criteria considered in the supplier selection process.
Table 3.10. GP Model Priorities
Priority
1
2
3
4
5
6
7
8
Goal Constraint
Deviational Variables
Weighted value of purchase
Purchasing expenses
Quality (ppm)
Flexibility
Leadtime
Service Level
Process Capability
Distance
d−
1
d+
2
d+
3
d−
4
+ + + + +
+
d+
5 , d6 , d7 , d8 , d9 , d10 , d11
−
d12
d−
13
d+
14
Based on this priority structure, the objective function is as follows:
+
+
−
+
+
+
Min Z = P1 (d−
1 ) + P2 (d2 ) + P3 (d3 ) + P4 (d4 ) + P5 (d5 + d6 + d7
+
+
+
−
−
+
+ d+
+
d
+
d
+
d
8
9
10
11 + P6 (d12 ) + P7 (d13 ) + P8 (d14 ). (3.14)
where Pk , k = 1, . . . , 8, is called a priority factor. These factors represent a
priority ‘k’ with the assumption that Pk is much larger than Pk+1 (Pk >> Pk+1 ).
This is equivalent to stating that goal ‘k’ has absolute priority over goal ‘k + 1’.
Furthermore, these factors are conceptually different from weights. Recall that
preemptive goal programming is essentially a sequential optimization process, in
which successive optimizations are carried out on the alternate optimal solutions
of the previously optimized goals at higher priorities (Ravindran et al. [125]).
In order to test the model, different profiles (characterizations) for each supplier
are proposed. These profiles represent characteristics of each supplier with respect
to each criterion.
54
Supplier 1: supplier 1 offers a low price for the product and a relatively bad
performance in all the remaining criteria.
Supplier 2: supplier 2 provides an excellent service. It also offers products
with superior quality but at a high price.
Supplier 3: supplier 3 presents an excellent flexibility but at the expense of
low quality.
Supplier 4: supplier 4 offers an average performance in all criteria.
Supplier 5: supplier 5 stands out for its very low price, although it is far away
in terms of travel distance.
Supplier 6: supplier 6 also offers an average performance but, unlike supplier
4, its service level is nearly perfect. Also, in terms of quality level (ppm), supplier
6 offers a higher level than supplier 4.
Supplier 7: supplier 7 maintains the shortest leadtime of all suppliers (given
its proximity to the purchasing company); it also provides an excellent service;
however, it offers poor technical capability.
The numerical data for the illustrative example is provided in Table 3.11. In
Table 3.11. Input Data for the GP Model
Criteria
Supplier’s
Profile
Supplier
Supplier
Supplier
Supplier
Supplier
Supplier
Supplier
1
2
3
4
5
6
7
Price
($)
Cpk
(index)
Defective
Parts(ppm)
Flexibility
(%)
Service
(%)
Distance
(km)
Leadtime
(hrs/part)
50
80
45
60
40
60
65
0.95
2.00
0.83
1.00
1.17
1.50
1.33
105,650
3.4
158,650
66,800
22,750
1,350
6,200
10
0
25
15
18
5
0
75
100
65
85
90
99
100
500
1,500
50
5,000
9,500
7,250
10
0.25
0.60
0.20
0.80
0.95
0.50
0.10
addition, a demand (D) of 13,000 units is considered. Recall that this demand
represents the predetermined order quantity to allocate to the selected suppliers.
That is, one supplier or a combination of them must satisfy this demand in its
entirety.
55
3.3.1
Computational Results
On this final stage, the results obtained with the preemptive GP model are presented. All results were generated using the optimization software LINDO. In particular, the ‘preemptive goal ’ option available in this software is applied in solving
the model. This option solves preemptive (lexicographic) goal programs sequentially by priority. Table 3.12 shows the final allocation of the demand to each
supplier. Notice that suppliers 2 and 4 were not chosen. In particular, they both
Table 3.12. Orders Allocated to Each Supplier
Supplier
1
2
3
4
5
6
7
Total Cost
Quantity (units)
2,200
3,000
3,200
1,500
3,100
$665,500.00
possess the lowest Total Score values (T Si ) for the first priority (WVP ). Moreover,
Supplier 2 offers the highest price among all suppliers. This makes it less likely to
be chosen given the priority structure, on which ‘Purchasing Expenses’ is defined
as the second most important criterion to consider. In the case of Supplier 4, although it offers an average performance on all criteria, its performance is surpassed
by other suppliers.
Another important result is the achieved levels for each criterion with respect
to the desired goals. These results are summarized in Table 3.13.
Based on the results, only the leadtime goal was fully achieved. That is, suppliers 1, 3, 5, 6, and 7 loosely fulfilled the levels set by the company as goals in terms
of total leadtime (hrs). The rest of the goals are partially achieved with respect to
the corresponding deviational variables and target levels initially set by the DM.
56
Table 3.13. Goal Achievement
Criteria
Achievements
Weighted value of purchase
Purchasing expenses ($)
Quality level (ppm)
Flexibility achieved (%)
Leadtime underachievement (hrs)
Service Level achieved (%)
Process Capability achieved (Cpk)
Average distance (km)
3.3.2
7,719.00
665,500.00
73,650.00
11.60
200.00
85.80
1.15
3,462.00
Sensitivity Analysis
As part of the analysis performed, several scenarios were analyzed. Each scenario
defines a different priority structure with respect to the criteria. Scenarios are
evaluated to check the robustness of the response for the GP model. The scenarios
are described in Table 3.14. The first scenario corresponds to the priority structure
originally defined by the DM, while the rest of them reflect situations where price
may not be as important and leadtime or distance are crucial, etc.
Table 3.14. Analysis of Scenarios
Priorities
Scenario
P1
P2
P3
P4
P5
P6
P7
P8
1
2
3
4
5
6
7
8
WVP
P.Exp.
P.Exp.
Flexib.
Service
Distance
Quality
Leadtime
P.Exp.
Quality
Quality
Leadtime
WVP
P.Cap.
Flexib.
Distance
Quality
Flexib.
WVP
Service
Quality
Service
Leadtime
Flexib.
Flexib.
Leadtime
Leadtime
Quality
Distance
Leadtime
P.Cap.
P.Exp.
Leadtime
Service
P.Cap.
Distance
Leadtime
Flexib.
P.Exp.
Quality
Service
P.Cap.
Distance
P.Exp.
Flexib.
Quality
WVP
Service
P.Cap.
WVP
Flexib.
P.Cap.
P.Cap.
P.Exp.
Service
WVP
Distance
Distance
Service
WVP
P.Exp.
WVP
Distance
P.Cap.
It is worthwhile to mention that there are a total of 8!, or equivalently 40,320
different scenarios, many of them providing the exact same answer. Only a few
of them were chosen, for being considered as representative of actual scenarios in
industry. The results displayed in Table 3.15 show the allocation of orders under
each scenario.
Notice that most solutions are in the same form as the original solution for
57
Table 3.15. Allocation for the Different Scenarios
Supplier
Scenario
X1
X2
X3
X4
X5
X6
X7
1
2
3
4
5
6
7
8
2,200
2,200
2,200
2,200
2,200
2,200
3,200
2,900
1,700
3,000
3,000
3,000
2,400
3,000
3,000
2,400
400
400
1,240
1,900
-
3,200
3,200
3,200
3,160
3,200
3,200
-
1,500
4,200
4,200
4,000
3,700
4,200
4,000
3,100
3,100
2,700
2,700
2,700
Scenario 1. In general, there seems to be a tendency to choose Suppliers 1, 3, 5, 6
and 7. Order quantities don’t seem to vary much and a more careful analysis on
the deviational variables shows that the priorities are optimized to similar values
for all solutions.
The solutions presented in Table 3.12 could be shown to the DM along with
information regarding the achieved values for each priority (as in Table 3.13).
This should provide the DM with a good vision of possible alternatives for the
final decision.
3.4
Conclusions
The Three-Phase integrated methodology presented herein allows managers to
make sound decisions with respect to supplier selection. In particular, Phase 1
offers an easy way to screen a large number of potential suppliers to a manageable
number. Then, the advantage of AHP (in Phase 2) is that it can help managers
in formulating decisions concerning the impact of alternative suppliers based on
the multiple criteria of the organization. It also provides a strategic approach to
evaluate alternatives. AHP is very useful for managerial decision making because it
is flexible enough to accommodate a larger set of evaluation criteria. This enables
managers to make sound selections based on both qualitative and quantitative
criteria.
In Phase 3, managers can evaluate the impact of changing business conditions
58
(e.g., increase service level, change the required flexibility, leadtime, etc.) and obtain the proper allocation of demand to each supplier by means of goal programming, which unlike other mathematical programming approaches, allows managers
to consider different criteria levels of achievement and give their respective priority
with certain flexibility. Different criteria and goal constraints can be introduced
to account for specific needs of a company. In summary, use of this methodology
can facilitate the supplier selection and the purchasing problems.
Given the assumption in Phase 3 that the order quantity to allocate to suppliers
is predetermined. Mathematical models are developed in later chapters to help DM
find the optimal order quantity by considering inventory management costs into
the analysis.
Chapter
4
Analytical Models for Supplier
Selection and Order Quantity
Allocation
4.1
Introduction
Chapter 3 presented an integrated approach involving all steps in the supplier selection process. However, the amount of the order quantity to allocate to suppliers
is assumed to be determined in advance. This is equivalent to solving a singleperiod problem in which no inventory management costs are considered over time.
Consequently, a single-period problem does not lead to an inventory policy for continuous replenishment over an infinite planning horizon. This chapter addresses
the order quantity allocation problem in supplier selection while considering inventory costs in the analysis. The objective is to allocate the corresponding order
quantities over time to the selected suppliers. The result is an optimal inventory
policy with minimum cost per time unit. Additional criteria such as quality and
capacity are also considered. When a planning horizon covers multiple periods,
inventory policies have the potential to yield much better procurement decisions
providing an opportunity for companies to reduce costs, further affecting the entire
supply chain.
The remainder of this chapter is organized as follows: Section 4.2 presents the
description of the problem and the assumptions and notations to be used throughout the chapter. Sections 4.3, 4.4, and 4.5 present the proposed order quantity
60
allocation models for the problem under consideration. Section 4.6 provides a
summary of the analysis of the models presented.
4.2
Problem Description and Assumptions
The problem considered in this chapter is a single-stage system with multiple
suppliers, see Figure 4.1.
Supplier
1
Supplier
2
Manufacturer
…
Supplier
r
Figure 4.1. System Under Consideration
The model considers a single product. A constant demand rate for this product
is assumed and must be satisfied without shortages. The goal is to determine how
much, how often, and the number of orders allocated to selected suppliers, while
minimizing the total cost per time unit. The total cost per time unit includes
setup, holding, and purchasing costs. The notations used throughout this chapter
are:
Data
r – number of available suppliers
d – demand per time unit
a – inventory holding cost rate
h – inventory holding cost per unit and time unit
ki – setup cost of ith supplier
pi – unit price of ith supplier
ci – capacity of ith supplier per time unit
qi – perfect rate of ith supplier
61
qa – minimum acceptable perfect rate of parts
li – leadtime of ith supplier
zi – reorder point of ith supplier
Variables
Ji – number of orders of ith supplier per order cycle
M – total number of orders allocated to all selected suppliers
Qi – order quantity to ith supplier
Ti – reorder interval given that an order from ith supplier has just arrived
Tc – (repeating) order cycle time
This chapter presents three single-stage order quantity allocation models. The
first model is a generalization of the work by Ghodsypour and O’Brien [36]. The
second and third models are extensions of the first one. Two key assumptions in
the original formulation by Ghodsypour and O’Brien [36] are:
1. Only one order per supplier is allowed in each order cycle.
2. The order quantities placed to the selected suppliers can be of different size
P
(namely Qi , and Q = ri=1 Qi ).
The first assumption is an unnecessary restriction to the problem and is relaxed
in the three proposed models in this chapter. This is done by allowing the buyer
to order more than one time within a repeating order cycle. Since the demand rate
is constant, the following can be stated: Ti = Qi /d. In one order cycle, Tc , there
P
will be ri=1 Ji orders placed to the selected suppliers. This implies that multiple
orders to one supplier are allowed within one order cycle. After all orders in one
order cycle have been placed, the cycle is repeated. For this reason, Tc is defined
as ‘repeating cycle time’. In order to avoid confusion, from now on this concept is
simply referred to as order cycle time.
The length of an order cycle becomes Tc =
Pr
Ji Ti = (
Pr
Ji Qi )/d, and
P
the total number of order cycles per time unit is given by 1/Tc = 1/( ri=1 Ji Ti ) =
P
d/ ri=1 Ji Qi ). Figure 4.2 illustrates the order cycle concept for three selected
i=1
i=1
suppliers.
In this example, one order cycle includes four orders allocated to three suppliers
(one to supplier 1, two to supplier 2, and one to supplier 3). Once the inventory
Order Cycle Concept for Different Qi’s
1 Order from
Supplier
pp
1
62
1 Order from
Supplier
pp
3
2 Orders from
Supplier 2
Units
Q2
l2
Q1
Q3
l3
l1
z1
T1
z2
z3
T2
T2
Tc
=
T3
Time
∑ i =1 J T
3
i
i
Figure 4.2. Order Cycle for Three Selected Suppliers
from the fourth order is depleted, the manufacturer starts a new order cycle by
again placing an order to supplier 1, two to supplier 2 (one at a time), and one to
supplier 3. Notice that leadtimes are positive (li > 0) and reorder points can be
calculated as zi = d × li . As long as leadtimes are essentially fixed, they become
irrelevant and can be assumed to be zero.
Regarding the second assumption, the first proposed model (Section 4.3) still
assumes that the order quantities placed to selected suppliers within one order cycle
are of different sizes. The second (Section 4.4) and third (Section 4.5) proposed
models relax the second assumption and assume that the order quantities allocated
to each supplier per cycle are the same size for all suppliers (Q). When the order
P
P
quantity is Q, the order cycle time is defined as Tc = T · ri=1 Ji = (Q/d) · ri=1 Ji ,
and the total number of order cycles per time unit is given by 1/Tc = 1/(T ·
Pr
Pr
J
)
=
d/(Q
·
i
i=1
i=1 Ji ).
4.3
Different-Size Order Quantities and Dependent Holding Costs
As mentioned earlier, the objective is to minimize the total cost per time unit which
includes setup, holding, and purchasing costs. The development of the objective
function follows:
63
Setup Cost Per Time Unit. Since a product is ordered ‘Ji ’ times from the
P
ith supplier, the total setup cost per order cycle is ri=1 Ji ki . By dividing the total
P
setup cost by the length of an order cycle Tc = ( ri=1 Ji Qi )/d, the setup cost per
time unit becomes,
r
P
i=1
r
P
d·
Ji ki
.
(4.1)
Ji Qi
i=1
Holding Cost Per Time Unit. The holding cost due to ith supplier is the
product of its unit holding cost, api , the average inventory level, Qi /2, and the
P
fraction of demand replenished by this supplier, Ji Qi / ri=1 Ji Qi . Hence, the total
holding cost per time unit is given by,
r
P
Ji Q2i pi
a i=1
· r
.
2 P
Ji Qi
(4.2)
i=1
Purchasing Cost Per Time Unit. This cost is expressed as,
r
P
d·
Ji Qi pi
i=1
r
P
,
(4.3)
Ji Qi
i=1
where
Pr
i=1
Ji Qi pi /
Pr
i=1
Ji Qi indicates an average price of a purchased unit.
Two types of constraints are considered in the problem: capacity and quality.
The use of capacity and quality constraints for supplier selection has extensively
been suggested in the literature. For instance, [127] and [13] highlighted the importance of considering supplier’s capacity as a response to order inquiries. [1]
concluded that quality is one criterion that is used in most practical situations.
Furthermore, an empirical analysis of the supplier selection process by [128] has
shown that quality is the most important factor along with cost.
The capacity constraints are as follows:
Jj Qj
d· P
≤ cj , for j = 1, . . . , r,
r
Ji Qi
i=1
(4.4)
64
where the left-hand side represents the proportion of demand per time unit that is
assigned to the ith supplier, which is limited by its offered capacity per time unit
(right-hand side). The quality constraint is,
r
P
Ji Qi qi
i=1
r
P
≥ qa ,
(4.5)
Ji Qi
i=1
where the left-hand side represents the average perfect rate offered by suppliers.
This average needs to meet the minimum acceptable perfect rate (qa ) imposed by
the purchaser.
Notice that the term Ji Qi makes the terms of the objective function (Eqs. (4.1),
(4.2), (4.3)), as well as the constraints Eqs. (4.4) and (4.8) nonlinear. In order to
simplify the optimization model (e.g. linearize the constraints) the following is
defined, Ri = Ji Qi . The final mixed integer nonlinear programming model after
substituting Ri and rearranging terms becomes,
r R2
r
P
P
i
pi
Ri pi
a i=1 Ji
i=1
i=1
ZS = d · P
+ · P
+d· P
,
r
r
r
2
Ri
Ri
Ri
r
P
(P4.1)
minimize
Ji ki
i=1
subject to
dRj − cj
r
X
i=1
i=1
Ri ≤ 0, j = 1, . . . , r,
i=1
r
X
i=1
r
X
Ri qi − qa ≥ 0,
Ji = M,
i=1
M ≥ 1, integer,
Ji ≥ 0, integer, i = 1, . . . , r,
Ri ≥ 0, i = 1, . . . , r.
Observe that the constraints are now linear. It can be shown that Problem
(P4.1) is convex for fixed Ji ’s. In addition, since all constraints are linear, it can be
efficiently solved by commercial nonlinear optimization software (e.g. LINGO [129]
P
or GAMS [130]). Notice that the constraint ri=1 Ji = M has been added to the
65
formulation. This constraint represents the total number of orders allocated to
all selected suppliers in order cycle. The implications are as follows. The optimal
value of M that minimizes ZS may result in an excessively large order cycle time.
In this case, a company may be interested in restricting M to a reasonable small
value to reduce the entire order cycle time. Short cycle times facilitate interaction
with suppliers and simplify the inventory control process. In order to do so, an
upper bound on M can be added or M can be fixed to a small integer value.
Importantly, Problem (P4.1) represents a generalized form of the model proposed by Ghodsypour and O’Brien [36]. An optimal solution given by Ghodsypour
and O’Brien’s represents a feasible solution to Problem (P4.1) and therefore that
solution can be improved. This is achieved by allowing allocation of multiple orders to a selected supplier within an order cycle. In Ghodsypour and O’Brien’s,
they restrict each supplier to be allocated at most one order in each order cycle.
4.3.1
Illustrative Example
The following example is used to illustrate the advantages of the proposed Problem
(P4.1). Table 4.3 shows the data for the three suppliers to be considered.
Table 4.1. Supplier’s Data for the Illustrative Example
Supplier
i
Price (pi )
($)
Setup Cost (ki )
($)
Perf. Rate
(qi )
Capacity (ci )
(units/month)
1
2
3
7.2
12.8
25.6
1,000
500
900
0.92
0.95
0.98
600
700
500
In addition, d = 1,000 units/month, a = 0.3/unit/month, and the minimum
acceptable perfect rate is qa = 0.95. First, Problem (P4.1) is solved (using LINGO)
to obtain the absolute optimal solution. Then, the model is solved for different
fixed values of M . Table 4.2 shows the detailed solutions for M = 3 to 25, and
M = 47. These solutions include the number of orders assigned to each supplier
in an order cycle, the size of these orders, the corresponding total monthly cost,
the length of the order cycle, and the percentage deviation from the optimal total
monthly cost.
66
Table 4.2. Detailed Solutions for the Illustrative Example
M
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
47
Number of Orders
Zs
Order Cycle’s
% Dev.from
J1
J2
J3
Order Quantity (units)
Q1
Q2
Q3
($/month)
Length (months)
Optimal Solution
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
4
11
1
2
3
4
5
6
6
7
8
9
10
11
11
12
12
13
14
15
15
16
17
18
19
35
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
8
227
335
425
504
576
642
763
838
909
977
1042
1105
1222
1288
710
746
780
814
874
909
944
977
1010
960
1058
782
661
588
537
499
593
559
530
507
486
469
518
501
552
535
520
506
544
530
518
507
496
512
227
335
425
504
576
642
381
419
455
489
521
552
407
429
473
497
520
542
437
455
472
489
505
480
17,057.17
16,476.30
16,282.18
16,202.94
16,173.46
16,169.76
16,161.18
16,135.63
16,123.82
16,121.29
16,125.28
16,134.00
16,138.65
16,138.80
16,140.13
16,133.15
16,129.76
16,129.24
16,128.34
16,123.82
16,121.59
16,121.29
16,122.60
16,121.22
1.5
2.2
2.8
3.4
3.8
4.3
5.1
5.6
6.1
6.5
6.9
7.4
8.1
8.6
9.5
9.9
10.4
10.8
11.7
12.1
12.6
13.0
13.5
25.6
5.81
2.20
1.00
5.1·10−1
3.2·10−1
3.0·10−1
2.5·10−1
8.9·10−2
1.6·10−2
4.3·10−4
2.5·10−2
7.9·10−2
1.1·10−1
1.1·10−1
1.2·10−1
7.4·10−2
5.3·10−2
4.9·10−2
4.4·10−2
1.6·10−2
2.3·10−3
4.3·10−4
8.5·10−3
0.00
The absolute minimum (optimal solution to the proposed model) is given by
M = 47 with a corresponding Zs∗ =$16,121.22/month. Figure 4.3 shows a graphical
representation of the total monthly cost for the M values displayed in Table 3,
except that the optimal solution (M = 47) is represented by a line.
In Table 4.2, the optimal solution obtained when M = 3 corresponds to the
optimal solution obtained solving the model by Ghodsypour and O’Brien [36]. This
solution corresponds to an allocation of one order per supplier in an order cycle.
In their work they assumed that at most one order per supplier is allowed within
an order cycle, which is an unnecessary restriction. In this research, their solution
is improved by 5.81% by allowing multiple orders within an order cycle.
In this example, the optimal value of M that minimized Zs results in a large
cycle time (25.6 months), as displayed in Table 4.2. This order cycle time may
be too large for the company. However, one of the advantages of Problem (P4.1)
is that the values of M can be controlled by the decision maker. For instance, a
company can set M = 12 and reduce the order cycle’s length from 25.6 months
67
Total Co
ost ($/month)
17,000.00
16,800.00
16,600.00
16,400.00
Absolute Minimum $16,121.22 (M=47)
16,200.00
16,000.00
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
M
Figure 4.3. Total Monthly Cost for Different M Values
to 6.5 months. Shortening the cycle time will facilitate the inventory management
process as well as give the company the opportunity to re-evaluate suppliers in
a short-term period. Moreover, reducing M from the optimal solution of 47 to
12, would only penalize Zs by (an increment of) $0.07/month (4.3 · 10−4 %). This
difference is clearly marginal and therefore the proposed model allows decision
makers to control M while maintaining the percentage increase in cost per time
unit, in comparison to the optimal solution, to a minimum.
4.4
Equal-Size Order Quantities and Dependent
Holding Costs
One of the reasons why the analysis on single-stage models is important is because
these models provide a strong foundation for subsequent analysis of multi-stage
supply chain systems. One important issue to address in multi-stage systems is
coordination of the inventory being transferred from one stage to another. For
example, Roundy [112] and Muckstadt and Roundy [119] have shown that for
serial supply chain system to achieve coordination, the order quantity placed at
one stage needs to be a multiple of the order quantity placed at the immediate
subsequent stage; and that applies to all stages of the serial system. In an effort
68
to address this issue of an order quantity being multiple of one another, Problem
(P4.1) is modified to the case where the order allocated to all selected suppliers is
of the same size (Q). All other assumptions and notation remain the same as in
Section 4.3. The total cost per time unit (Zs ) is as follows:
r
P
Ji ki
d
Zs = · i=1
r
Q P
Ji
r
P
aQ i=1
+
· P
r
2
i=1
r
P
Ji pi
+d·
Ji
i=1
J i pi
i=1
r
P
,
(4.6)
Ji
i=1
where the first term represents the setup cost, which is obtained by dividing
P
the total setup cost per order cycle, ri=1 Ji ki , by the length of the order cycle,
P
Tc = (Q/d) · ri=1 Ji . The second term accounts for the holding cost. Since the
order quantity (Q) is the same for all suppliers, the average inventory on-hand is
P
P
Q/2, and a · ri=1 Ji pi / ri=1 Ji indicates the average holding cost of a purchased
P
P
unit. The last term corresponds to the purchasing cost, where ri=1 Ji pi / ri=1 Ji
denotes the average price of a purchased unit. The capacity and quality are still
considered in the model. The capacity constraints are as follows:
Jj
d· P
≤ cj , for j = 1, . . . , r,
r
Ji
(4.7)
i=1
where the left-hand side represents the proportion of demand per time unit that is
assigned to the ith supplier, which is limited by its offered capacity per time unit
(right-hand side). The quality constraint is,
r
P
Ji qi
i=1
r
P
≥ qa ,
(4.8)
Ji
i=1
where the left-hand side represents the average perfect rate offered by suppliers.
This average needs to meet the minimum acceptable perfect rate (qa ) imposed by
the manufacturer. Since all the constraints are independent of Q, the optimal order
quantity can be obtained by taking the first derivative of the objective function
(Eq. 4.6) with respect to Q, setting it to zero and solving for Q. The optimum is
69
given by the following expression:
v
u
r
P
u
Ji ki
u
u 2d i=1
∗
Q =u · P
,
r
ta
Ji pi
(4.9)
i=1
The complete mixed integer nonlinear programming model, after substituting
Q in Eq. (4.6) by its optimal value provided by Eq. (4.9) and rearranging terms,
is as follows:
(P4.2)
minimize
1
Zs = P
r
Ji
v

!
!
u
r
r
r
X
X
X
u
t2ad
Ji ki
Ji pi + d
Ji pi  ,
i=1
i=1
i=1
i=1
subject to
dJj − cj
n
X
Ji ≤ 0, j = 1, . . . , r,
i=1
r
X
Ji (qi − qa ) ≥ 0,
i=1
r
X
Ji = M,
i=1
M ≥ 1, integer,
Ji ≥ 0, integer, i = 1, . . . , r.
Notice the constraint
Pr
i=1
Ji = M has also been added to the formulation of
Problem (P4.2), just as in Problem (P4.1). Controlling M brings about the same
advantages as for Problem (P4.1). In addition to controlling the length of the order
cycle, setting M to a fixed value simplifies the mathematical model in two ways:
(1) the variables in the objective function’s denominator are eliminated, and (2)
the feasible region is restricted as the value of M creates implicit upper bounds on
the Ji variables.
Since all constraints are linear, Problem (P4.2) can be easily solved using a
commercial optimization package such as LINGO [129] or GAMS [130]. The following
theorem follows for an optimal order allocation.
Theorem 4.1. If (J1 , J2 , . . . , Jr ) is an optimal solution to Problem (P4.2) that
minimizes the order cycle time, then (J1 , J2 , . . . , Jr ) must be relative prime num-
70
bers.
Proof: (By contradiction) Let us assume that (J1 , J2 , . . . Jr ) is an optimal solution with minimum order cycle time Tc , where (J1 , J2 , . . . , Jr ) are not relative
prime numbers. Note that the greatest common denominator (g.c.d.) is an integer
number greater than 1. Then, the solution (J1 /g.c.d., J2 /g.c.d., . . . , Jr /g.c.d.) is
an equivalent solution where the (J1 , J2 , . . . , Jr ) are relative prime numbers. Let
P
Tc = ri=1 Ji T be the order cycle time for the first solution. Then the order cycle
time for the second solution would be Tc /g.c.d., which is smaller than Tc . This is
a contradiction.
4.4.1
Illustrative Example
The following example is used to illustrate the advantages of the proposed Problem
(P4.1). Table 4.3 shows the data for the three suppliers to be considered.
Table 4.3. Supplier’s Data for the Illustrative Example
Supplier
i
Price (pi )
($)
Setup Cost (ki )
($)
Perf. Rate
(qi )
Capacity (ci )
(units/month)
1
2
3
9
16
32
9
4
8
0.92
0.95
0.98
600
700
500
In addition, d = 1,000units/month, a = 0.2/unit/month, and the minimum
acceptable perfect rate is qa = 0.95. First, Problem (P4.1) is solved (using LINGO)
to obtain the absolute optimal solution. Figure 4.4 shows the total cost per time
unit versus M (the maximum number of orders allowed within an order cycle).
Notice that the total annual cost decreases as M increases within certain ranges.
This is because by restricting the order quantity allocated to suppliers to be of equal
size, there are points at which the holding cost cannot be further reduced and the
total monthly cost increases again. This produces the “sawtooth” type of behavior
exhibited in Figure 4.4. The optimal solution to Problem (P4.2) is attained at
M = 40 (J1 = 6, J2 = 28, J3 = 6) with a corresponding total monthly cost of
$17,542/month. From Theorem 4.1, it is easy to see that if M = 40 attains the
71
19,100.00
Total Cost ($/mo
onths)
18,900.00
18,700.00
18,500.00
18,300.00
18,100.00
17,900.00
17,700.00
17,500.00
3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
20
40
M
Figure 4.4. Total Monthly Cost Versus M Values for Multiple Equal-Size Orders
minimum cost per time unit and the Ji values are not relative primer numbers,
then same total cost per time unit is attained at M = 40 with the order allocation
J1 = 6, J2 = 28, and J3 = 6. This Ji values are relative prime numbers. This
property will be useful in the development of a closed-form solution for the case of
r = 2 suppliers in Section 4.5.1.
4.5
Equal-Size Order Quantities and Constant
Holding Costs
In this section, Problem (P4.2) is modified to consider holding costs that are not
dependent on the purchasing price (h). Such assumption along with the one of
equal-size order quantities (Q) are important as the model in this section serves as
a basis for the multi-stage model presented in Chapter 6. The objective function
becomes,
r
P
Ji ki
d
ZS = · i=1
r
Q P
i=1
Ji
r
P
J i pi
hQ
i=1
+d· P
+
,
r
2
Ji
(4.10)
i=1
where the first and third terms represent the setup and purchasing costs, respectively, as introduced in Section 4.4. Since the holding cost rate (h) is constant, the
72
holding cost per time unit is simply expressed as the unit holding cost times the
average inventory on-hand, Q/2. The capacity and quality constraints are given by
Eq. (4.7) and (4.8), respectively. Since the constraints are independent of Q, the
optimal order quantity can be obtained by taking the first derivative of Eq. 4.10
with respect to Q, setting it to zero and solving for Q. The optimum is provided
by,
v
u
r
u 2d X
∗
u
Q =t P
Ji ki .
r
h Ji i=1
(4.11)
i=1
The complete mixed integer nonlinear programming model, after substituting
Q in Eq. (4.10) by its optimal value given by Eq. (4.11) and adding the constraints,
is as follows:
(P4.3)
minimize
subject to
v
u
r
r
u 2dh X
d X
ZS =t
·
J i pi ,
Ji ki +
M i=1
M i=1
dJi ≤ ci M, i = 1, . . . , r,
r
X
Ji qi ≥ qa M,
i=1
r
X
Ji = M,
i=1
M ≥ 1, integer,
Ji ≥ 0, integer, i = 1, . . . , r.
4.5.1
Closed-Form Solution Analysis for Two Suppliers
This section presents a closed form solution analysis for a particular case of Problem
(P4.3), where only two potential suppliers are considered (r = 2). The purpose
is to determine the optimal order quantity and the number of orders per order
cycle allocated to each supplier. The mathematical formulation for this particular
analysis is as follows:
(P4.3’)
minimize
subject to
" #
k1
k2
d
ZS =
J1
+ p1 + J 2
+ p2 ,
(J1 + J2 )
Q
Q
dJ1 ≤ c1 (J1 + J2 ),
(4.12)
73
dJ2 ≤ c2 (J1 + J2 ),
(4.13)
J1 q1 + J2 q2 ≥ qa (J1 + J2 ),
(4.14)
Q ≥ 0,
J1 , J2 ≥ 0, integer.
where (4.12) and (4.13) represent the capacity constraints, and (4.14) is the quality
constraint. From Eq. (4.11), the following represents the optimal order quantity
for this particular case:
s
Q∗ =
2d(J1 k1 + J2 k2 )
.
h(J1 + J2 )
(4.15)
The following property follows for an optimal order allocation.
Property 4.1. If (J1 , J2 ) represents the optimal number of orders placed to suppliers 1 and 2, respectively, then (nJ1 , nJ2 ) is also an optimal allocation for n > 0,
integer.
Proof: From the objective function in Problem (P4.3’), it is easy to see that if
J1 and J2 are multiplied by n, then n can be factorized and canceled out. The
same applies for the constraints (4.12), (4.13), and (4.14). Consequently, the model
remains the same and the optimal order quantity does not change.
Property 4.1 provides a way to find the shortest cycle time for a given solution.
This is done by finding the relative prime numbers of J1 and J2 . Using the notation
(J1 , J2 ), to denote the greatest common divisor, J1 and J2 are relative prime if
(J1 , J2 ) = 1.
The following property also provides useful relationships for the closed-form
analysis.
Property 4.2. Let J1 and J2 be positive integers and 0 < c < d, where c represents
the capacity of a given supplier. Then, the following holds:
If
c
(d − c)
c
(d − c)
d
.
=
, then
=
=
J1
J2
J1
J2
(J1 + J2 )
Proof: c/J1 = (d − c)/J2 can be rearranged as follows:
cJ2 = (d − c)J1 = dJ1 − cJ1
74
c(J1 + J2 ) = dJ1 =⇒
c
d
=
.
J1
J1 + J2
In order to determine Q∗ , J1 , and J2 from the known parameters of the problem,
all possible combinations of quality, capacity, setup cost, and price need to be
considered. First, quality and capacity combinations are presented. The feasibility
conditions for Problem (P4.3’) are derived from these combinations.
Without loss of generality, let us assume q1 ≤ q2 . Therefore, the following
represents the different possible combinations of quality rates:
q1 < qa ≤ q2 ,
(4.16)
q1 ≥ q a ,
(4.17)
q2 < qa .
(4.18)
In terms of capacity, the following combinations need to be considered:
c1 < d, c2 < d, and c1 + c2 ≥ d,
(4.19)
c1 < d and c2 ≥ d,
(4.20)
c1 ≥ d and c2 ≥ d,
(4.21)
c1 ≥ d and c2 < d,
(4.22)
c1 + c2 < d.
(4.23)
The feasibility conditions for Problem (P4.3’) are stated in the following theorem.
Theorem 4.2. (Feasibility Conditions) The problem is feasible if and only if one
of the following conditions holds:
(i) q1 ≥ qa and c1 + c2 ≥ d.
(ii) q1 < qa ≤ q2 , c2 ≥ d(qa − q1 )/(q2 − q1 ), and c1 + c2 ≥ d.
Proof: It is easy to see that if either (4.18) or (4.23) is true, the problem is
infeasible. First, if (4.18) is true, then q2 < qa , which implies q1 < qa . Hence,
J1 q1 + J2 q2 < qa (J1 + J2 ). This implies that the quality constraint (4.14) in
Problem (P4.3’) is not satisfied and, therefore, the problem is infeasible. Second,
75
if (4.23) is true, the following is obtained by adding capacity constraints (4.12) and
(4.13):
d(J1 + J2 ) ≤ (c1 + c2 )(J1 + J2 ) =⇒ d ≤ c1 + c2 ,
which implies that if c1 + c2 < d, the problem is infeasible. Consequently, if
c1 + c2 ≥ d, both capacity constraints, (4.12) and (4.13), are satisfied. Therefore,
parts (i) and (ii) need to be checked for feasibility only with respect to the quality
constraint (4.14).
For part (i), since q1 ≥ qa , then the quality constraint (4.14) is automatically
satisfied because the two suppliers exceeds the quality required.
For part (ii), since q1 < qa ≤ q2 , when c2 ≥ d (irrespective of c1 ) the problem
is automatically feasible (e.g. by using only supplier 2). However, when c2 < d,
it must be proved that the quality constraint (4.14) is satisfied if and only if
c2 ≥ d(qa − q1 )/(q2 − q1 ).
First, it will be proved that if q1 < qa ≤ q2 and c2 ≥ d(qa − q1 )/(q2 − q1 ), then
constraint (4.14) is satisfied. To do so, it will be shown that the point (J1 , J2 ),
J1 , J2 > 0, integer, such that it satisfies the following relation is feasible:
(d − c2 )/J1 = c2 /J2 .
The term c2 ≥ d(qa − q1 )/(q2 − q1 ), may be rewritten as follows:
c2 q2 − c2 q1 ≥ dqa − dq1 ,
q1 (d − c2 ) + c2 q2 ≥ dqa ,
(4.24)
From Property 4.2, (d−c2 ) = dJ1 /(J1 +J2 ) and c2 = dJ2 /(J1 +J2 ). Substituting
(d − c2 ) and c2 into Eq. (4.24) and multiplying both sides by (J1 + J2 )/d, the
following is obtained:
J1 q1 + J2 q2 ≥ qa (J1 + J2 ),
which implies that the quality constraint (4.14) is also satisfied. Consequently, if
q1 < qa ≤ q2 , c2 ≥ d(qa − q1 )/(q2 − q1 ), and c1 + c2 ≥ d, the problem is feasible.
Second, it will be proved that if the problem is feasible, then c2 ≥ d(qa −
q1 )/(q2 − q1 ) must hold. (By contradiction) Assume that the point (J1 , J2 ),
J1 , J2 > 0, integer, such that it satisfies the relation, c1 /J1 = (d − c1 )/J2 , is
feasible, and that c2 < d(qa − q1 )/(q2 − q1 ). From, c1 + c2 ≥ d =⇒ c2 ≥ d − c1 .
76
Consequently, c2 < d(qa − q1 )/(q2 − q1 ) may be stated as follows:
d − c1 ≤ c2 <
d(qa − q1 )
,
(q2 − q1 )
and this may be rewritten as,
(d − c1 )(q2 − q1 ) < d(qa − q1 ) =⇒ c1 q1 + q2 (d − c1 ) < dqa .
(4.25)
From Property 4.2, c1 = J1 d/(J1 +J2 ) and (d−c1 ) = dJ2 /(J1 +J2 ). Substituting
c1 and (d − c1 ) into Eq. (4.25) and multiplying both sides by (J1 + J2 )/d, the
following is obtained,
J1 q1 + J2 q2 < qa (J1 + J2 ).
This implies that the quality constraint (4.14) is not satisfied. This contradicts
the feasibility assumption. Hence, when c2 < d, if q1 < qa ≤ q2 , then c2 ≥
d(qa − q1 )/(q2 − q1 ) must hold for the problem to be feasible.
Now, in addition to capacity and quality combinations, setup cost (k1 , k2 )
and purchasing price (p1 , p2 ) combinations need also need to be considered as
they directly affect the determination of Q∗ , J1 , and J2 . Table 4.4 shows a summary of the feasible cases derived from the different combinations of quality rates,
capacities, setup costs, and purchasing prices.
Table 4.4. Cases Considered in the Closed-Form Solution
k1 k2
<
≤
p1 p2
<
≥
≥
>
≤
>
c1 < d
c2 < d
q1 < qa ≤ q2
c1 < d c1 ≥ d
c2 ≥ d c2 ≥ d
c1 ≥ d
c2 < d
c1 < d
c2 < d
q1 ≥ qa
c1 < d c1 ≥ d
c2 ≥ d c2 ≥ d
c1 ≥ d
c2 < d
Note that for ease of analysis, the cases where k1 ≤ k2 , p1 ≥ p2 and k1 ≥
k2 , p1 ≤ p2 are analyzed together as they represent the combinations where no
supplier dominates the other with respect to both setup cost and purchasing price.
77
4.5.1.1
Development of the Closed-Form Solution
The following lemmas provide some useful properties for the analysis of the combinations presented in Table 4.4. Each lemma corresponds to one column in the
table and derives the optimal order quantity and number of orders per order cycle in closed form under the combinations indicated in its corresponding row and
column.
Lemmas 4.1 – 4.4 represent the first four columns in Table 4.4. Recall that by
Theorem 4.2 (ii), c1 + c2 ≥ d must apply for feasibility.
Lemma 4.1. If q1 < qa ≤ q2 , and c1 < d, c2 < d, then the optimal order quantity
is given by one of the following:
s d(q2 − qa )
2
∗
(i) If k1 < k2 , p1 < p2 , and c1 ≤
, then Q =
c1 k1 + (d − c1 )k2 ,
(q2 − q1 )
h
s
2d
∗
otherwise, Q =
(q2 − qa )k1 + (qa − q1 )k2 .
h(q2 − q1 )
s 2
∗
(d − c2 )k1 + c2 k2 .
(ii) If k1 > k2 and p1 > p2 , then Q =
h
(iii) If k1 ≤ k2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥ p2 , then:
s 2
d(q2 − qa )
∗
, then Q =
c1 k1 + (d − c1 )k2 , or
(a) If c1 ≤
(q2 − q1 )
h
s 2
∗
Q =
(d − c2 )k1 + c2 k2 , the one which results in a minimum ZS .
h
s 2
∗
(d − c2 )k1 + c2 k2 or
(b) Otherwise, Q =
h
s
2d
∗
Q =
(q2 − qa )k1 + (qa − q1 )k2 , the one which results in
h(q2 − q1 )
a minimum ZS .
Proof: For part (i), since k1 < k2 and p1 < p2 , then it is desirable to allocate
as many units as possible to supplier 1 in order to minimize the ZS . However,
since q1 < qa , the quality constraint (4.14) may become infeasible using the entire
78
capacity of supplier 1. By Theorem 4.2 (ii), c1 +c2 ≥ d and c2 ≥ d(qa −q1 )/(q2 −q1 ).
It follows, that c2 ≥ d − c1 , so c2 ≥ d(qa − q1 )/(q2 − q1 ) may be restated as follows:
d − c1 ≥
d(qa − q1 )
d(q2 − qa )
=⇒ c1 ≤
.
(q2 − q1 )
(q2 − q1 )
(4.26)
First, from Eq. (4.26), if c1 ≤ d(q2 − qa )/(q2 − q1 ), the quality constraint (4.14)
is satisfied, and therefore the entire capacity of supplier 1 can be used. Then,
supplier’s 1 purchasing rate can be stated as c1 /J1 . From Property 4.2, c1 /J1 =
(d − c1 )/J2 = d/(J1 + J2 ). And so, J1 = c1 (J1 + J2 )/d, and J2 = (d − c1 )(J1 + J2 )/d.
Substituting J1 and J2 into the optimal order quantity provided by Eq. (4.15), Q∗
is obtained as follows:
s
Q∗ =
2d(J1 k1 + J2 k2 )
=
h(J1 + J2 )
r 2
c1 k1 + (d − c1 )k2 .
h
(4.27)
In order to obtain the optimal number of orders to place to each supplier, J1
and J2 , from the above analysis the following is known:
J1 /(J1 + J2 ) = c1 /d and J2 /(J1 + J2 ) = (d − c1 )/d,
since they share a common denominator, using Property 4.1, the optimum minimum number of orders placed to each supplier can be obtained as (J1 , J2 ) =
(c1 , d − c1 ) = 1, where (J1 , J2 ) denotes the greatest common divisor. Note that in
order to guarantee an optimal solution such that J1 , J2 are integers, c1 and d must
either be integers, or rational numbers.
Second, from Eq. (4.26), if c1 > d(q2 − qa )/(q2 − q1 ), since q1 < qa , using all
capacity of supplier 1 makes the quality constraint (4.14) infeasible. In such a case,
it is optimal to satisfy (4.14) as equality,
J1 q1 + J2 q2 = qa (J1 + J2 ),
(4.28)
By solving Eq. (4.28) for J1 and substituting it into Eq. (4.15), the following
Q∗ is obtained:
s
s
2d(J
k
+
J
k
)
2d
1 1
2 2
∗
=
(q2 − qa )k1 + (qa − q1 )k2 .
Q =
h(J1 + J2 )
h(q2 − q1 )
(4.29)
79
In order to obtain J1 and J2 from Eq. (4.29), it is easy to see that:
J1 /(J1 + J2 ) = (q2 − qa )/(q2 − q1 ), and J2 /(J1 + J2 ) = (qa − q1 )/(q2 − q1 ),
since both terms share a common denominator, using Property 4.1, the optimum
can be obtained by finding n, n > 0, integer, such that J1 and J2 become the
smallest possible integers (e.g. J1 = (q2 − qa )n, J2 = (qa − q1 )n).
For part (ii), since k1 > k2 , p1 > p2 , and q2 ≥ qa , then in order to minimize the
ZS it is desirable to use all capacity from supplier 2 ( its purchasing rate = c2 /J2 ).
From Property 4.2, c2 /J2 = (d−c2 )/J1 = d/(J1 +J2 ). So, J1 = ((d−c2 )/d)(J1 +J2 ),
c2
and J2 = (J1 + J2 ). Substituting J1 and J2 into Eq. (4.15), Q∗ is obtained:
d
s
r 2d(J
k
+
J
k
)
2
1
1
2
2
(4.30)
Q∗ =
=
(d − c2 )k1 + c2 k2 .
h(J1 + J2 )
h
It is easy to see that
J1 /(J1 + J2 ) = (d − c2 )/d, and J2 /(J1 + J2 ) = c2 /d,
and since both terms share a common denominator, to calculate the optimal
J1 and J2 , using Property 4.1, (J1 , J2 ) = (d − c2 , c2 ) = 1, where (J1 , J2 ) denotes
the greatest common divisor.
For part (iii), given that no one supplier dominates the other with respect to
both setup cost and purchasing price, Q∗ is given by a combination of parts (i)
and (ii). Specifically, from part (i) if c1 ≤ d(q2 − qa )/(q2 − q1 ), then Q∗ is given
by either either Eq. (4.27) or Eq. (4.29), the one which results in the lower ZS .
However, if c1 > d(q2 − qa )/(q2 − q1 ), then Q∗ is given by either either Eq. (4.29)
or Eq. (4.30), the one which results in the lower ZS . The number of orders is
calculated accordingly, as per parts (i) and (ii).
Lemma 4.2. If q1 < qa ≤ q2 and c1 < d, c2 ≥ d, then the optimal order quantity
is given by one of the following:
s d(q2 − qa )
2
∗
(i) If k1 < k2 , p1 < p2 , and c1 ≤
, then Q =
c1 k1 + (d − c1 )k2 ;
(q2 − q1 )
h
s
2d
∗
otherwise, Q =
(q2 − qa )k1 + (qa − q1 )k2 .
h(q2 − q1 )
80
r
(ii) If k1 > k2 and p1 > p2 , then Q∗ =
2k2 d
.
h
(iii) If k1 ≤ k2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥ p2 , then:
s r
d(q2 − qa )
2
2k2 d
∗
∗
(a) If c1 ≤
,Q =
c1 k1 + (d − c1 )k2 or Q =
, the
(q2 − q1 )
h
h
one which results in a minimum ZS .
s
2d
(b) Otherwise, Q∗ =
(q2 − qa )k1 + (qa − q1 )k2 or
h(q2 − q1 )
r
2k2 d
∗
, the one which results in a minimum ZS .
Q =
h
Proof: For part (i), since k1 < k2 , p1 < p2 , and c1 < d, then Lemma 4.1 (i)
provides Q∗ and the optimal order allocation for both suppliers.
For part (ii), since k1 > k2 , p1 > p2 , q2 ≥ qa , and c2 ≥ d, then it is optimal to
use only supplier 2 to satisfy the entire demand. This implies that J1 = 0, and by
Property 4.1, J2 = 1. Therefore, from Eq. (4.15), Q∗ is obtained as follows:
s
r
2d(J
k
+
J
k
)
2k2 d
1
1
2
2
Q∗ =
=
.
(4.31)
h(J1 + J2 )
h
For part (iii), given that no supplier dominates the other with respect to
both setup cost and purchasing price, the optimal order quantity is given by a
combination of parts (i) and (ii). In particular, since the optimum for part (i)
is given by Lemma 4.1(i), if c1 ≤ d(q2 − qa )/(q2 − q1 ), the optimum is given
by either Eq. (4.27) or Eq. (4.31), the one which results in the lower ZS . If
c1 > d(q2 −qa )/(q2 −q1 ), the optimum is given by by either Eq. (4.29) or Eq. (4.31),
the one which results in the lower ZS . J1 and J2 are calculated as per parts (i)
and (ii).
Lemma 4.3. If q1 < qa ≤ q2 and c1 ≥ d, c2 ≥ d, then the optimal order quantity
is given by one of the following:
s
(i) If k1 < k2 , p1 < p2 , then Q∗ =
2d
(q2 − qa )k1 + (qa − q1 )k2 .
h(q2 − q1 )
r
(ii) If k1 > k2 and p1 > p2 , then Q∗ =
2k2 d
.
h
81
r
2k2 d
or
(iii) If k1 ≤ k2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥ p2 , then Q∗ =
h
s
2d
∗
Q
(q2 − qa )k1 + (qa − q1 )k2 , the one which results in a minh(q2 − q1 )
imum ZS .
Proof: For part (i), given that k1 < k2 , p1 < p2 , and c1 ≥ d, this represents
the particular case from Lemma 4.1(i) where c1 > d(q2 − qa )/(q2 − q1 ). Therefore,
Q∗ is obtained by Eq. (4.29). In addition, J1 and J2 are obtained by finding n,
n > 0, integer, such that J1 and J2 become the smallest possible integers (e.g.
J1 = (q2 − qa )n, J2 = (qa − q1 )n).
For part (ii), since k1 > k2 , p1 > p2 , q2 ≥ qa , and c2 ≥ d, by Lemma 4.2 (ii),
Q∗ is given by Eq. (4.31) and J1 = 0, J2 = 1.
For part (iii), since no one supplier dominates the other with respect to both
setup cost and purchasing price, Q∗ is given by either Eq. (4.29) or Eq. (4.31), the
one that results in the lower ZS . The number of orders is found accordingly, as
per parts (i) and (ii).
Lemma 4.4. If q1 < qa ≤ q2 and c1 ≥ d, c2 < d, then the optimal order quantity
is given by one of the following:
s
∗
(i) If k1 < k2 , p1 < p2 , then Q =
2d
(q2 − qa )k1 + (qa − q1 )k2 .
h(q2 − q1 )
s 2
(ii) If k1 > k2 and p1 > p2 , then Q∗ =
(d − c2 )k1 + c2 k2 .
h
(iii) If k1 ≤sk2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥ p2 , then
2d
Q∗ =
(q2 − qa )k1 + (qa − q1 )k2 or
h(q2 − q1 )
s 2
∗
Q =
(d − c2 )k1 + c2 k2 , the one which results in a minimum ZS .
h
Proof: For part (i), given that k1 < k2 , p1 < p2 , and c1 ≥ d, Lemma 4.3 (i)
provides Q∗ using Eq.(4.29), and the optimal values of J1 and J2 .
For part (ii), since k1 > k2 , p1 > p2 , and c2 < d, by Lemma 4.1 (ii) Q∗ is
obtained using Eq.(4.30) and the optimal allocation J1 and J2 .
82
For part (iii), since no one supplier dominates the other with respect to both
setup cost and purchasing price, the optimal order quantity is given by either
Eq. (4.29) or Eq. (4.30), the one that results in a lower ZS . The number of orders
is found accordingly, as per parts (i) and (ii).
Now, Lemmas 4.5 – 4.8 are presented. These lemmas provide the optimal order
quantity and order allocation to the last four columns in Table 4.4.
Lemma 4.5. If q1 ≥ qa and c1 < d, c2 < d, then the optimal order quantity is
given by one of the following:
s 2
∗
(i) If k1 < k2 , p1 < p2 , then Q =
c1 k1 + (d − c1 )k2 .
h
s 2
∗
(d − c2 )k1 + c2 k2 .
(ii) If k1 > k2 and p1 > p2 , then Q =
h
(iii) If k1 ≤ s
k2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥
sp2 ,then
2
2
∗
∗
Q =
c1 k1 + (d − c1 )k2 or Q =
(d − c2 )k1 + c2 k2 , the one
h
h
which results in a minimum ZS .
Proof: For part (i), since k1 < k2 , p1 < p2 , and q1 ≥ qa , then all capacity from
supplier 1 can be used in the optimal solution (purchasing rate = c1 /J1 ). From
Property 4.2, c1 /J1 = (d − c1 )/J2 = d/(J1 + J2 ). And so, J1 = c1 (J1 + J2 )/d, and
J2 = (d − c1 )(J1 + J2 )/d. Substituting J1 and J2 into Eq. (4.15), Q∗ is obtained:
s
r 2d(J
k
+
J
k
)
2
1
1
2
2
=
c1 k1 + (d − c1 )k2 .
Q∗ =
h(J1 + J2 )
h
Notice that Q∗ is same as Eq. (4.27). The optimal number of orders is given
by (J1 , J2 ) = (c1 , d − c1 ) = 1, where (J1 , J2 ) denotes the greatest common divisor.
For part (ii), since k1 > k2 , p1 > p2 , c2 < d, and q2 ≥ qa , by Lemma 4.4 (ii)
Q∗ is giving by Eq. (4.30). (J1 , J2 ) = (d − c2 , c2 ) = 1, where (J1 , J2 ) denotes the
greatest common divisor.
For part (iii), since no one supplier dominates the other with respect to both
setup cost and purchasing price, the optimal order quantity is given by either
Eq. (4.27) or Eq. (4.30), the one that results in a lower ZS . The number of orders
is found accordingly, as per parts (i) and (ii).
83
Lemma 4.6.
If q1 ≥ qa and c1 < d, c2 ≥ d, then the optimal order quantity is
given by one of the following:
s 2
∗
(i) If k1 < k2 , p1 < p2 , then Q =
c1 k1 + (d − c1 )k2 .
h
r
2k2 d
∗
(ii) If k1 > k2 and p1 > p2 , then Q =
.
h
(iii) If k1 ≤s
k2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥ p2 , then
r
2
2k2 d
Q∗ =
c1 k1 + (d − c1 )k2 or Q∗ =
, the one which results in a
h
h
minimum ZS .
Proof: For part (i), since k1 < k2 , p1 < p2 , and c1 < d, by Lemma 4.5 (i), Q∗
is given by Eq. (4.27) and the optimal minimum number of orders placed to each
supplier is (J1 , J2 ) = (c1 , d − c1 ) = 1, where (J1 , J2 ) denotes the greatest common
divisor.
For part (ii), since k1 > k2 , p1 > p2 , q2 ≥ qa , and c2 ≥ d, by Lemma 4.2 (ii),
Q∗ is given by Eq. (4.31), and J1 = 0, J2 = 1.
For part (iii), since no one supplier dominates the other with respect to both
setup cost and purchasing price, the optimal order quantity is given by either
(4.27) or (4.31), the one that results in a lower ZS . The number of orders is found
accordingly, as per parts (i) and (ii).
Lemma 4.7. If q1 ≥ qa and c1 ≥ d, c2 ≥ d, then the optimal order quantity is
given by one of the following:
r
2k1 d
.
h
r
2k2 d
(ii) If k1 > k2 and p1 > p2 , then Q∗ =
.
h
(i) If k1 < k2 , p1 < p2 , then Q∗ =
r
(iii) If k1 ≤ k2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥ p2 , then Q∗ =
r
2k2 d
∗
Q =
, the one which results in a minimum ZS .
h
2k1 d
or
h
Proof: For part (i), since k1 < k2 , p1 < p2 , q1 ≥ qa , and c1 ≥ d then the minimum
ZS is attained when using only supplier 1 to satisfy the required demand, which
84
implies that supplier 2 is not used and J2 = 0. Moreover, by Property 4.1, J1 = 1.
Then Q∗ is:
s
Q∗ =
2d(J1 k1 + J2 k2 )
=
h(J1 + J2 )
r
2K1 d
.
h
(4.32)
For part (ii), since k1 > k2 , p1 > p2 , q1 ≥ qa , and c2 ≥ d, by Lemma 4.6 (ii),
∗
Q is given by Eq. (4.31) and J1 = 0, J2 = 1.
For part (iii), since no one supplier dominates the other with respect to both
setup cost and purchasing price, the optimal order quantity is given by either
Eq. (4.31) or Eq. (4.32), the one that results in a lower ZS . The number of orders
is found accordingly, as per parts (i) and (ii).
Lemma 4.8. If q1 ≥ qa and c1 ≥ d, c2 < d, then the optimal order quantity is
given by one of the following:
r
2k1 d
.
h
s 2
∗
(ii) If k1 > k2 and p1 > p2 , then Q =
(d − c2 )k1 + c2 k2 .
h
(i) If k1 < k2 , p1 < p2 , then Q∗ =
r
2k1 d
or
(iii) If k1 ≤ k2 and p1 ≥ p2 , or k1 ≥ k2 and p1 ≥ p2 , then Q =
h
s 2
Q∗ =
(d − c2 )k1 + c2 k2 , the one which results in a minimum ZS .
h
∗
Proof: For part (i), since k1 < k2 , p1 < p2 , q2 ≥ qa , and c1 ≥ d, by Lemma
4.1 (i), Q∗ is given by Eq. (4.32), and J1 = 1, and J2 = 0.
For part (ii), since k1 < k2 , p1 < p2 , q2 ≥ qa , and c2 < d, by Lemma 4.5 (ii),
Q∗ is given by Eq. (4.30). (J1 , J2 ) = (d − c2 , c2 ) = 1, where (J1 , J2 ) denotes the
greatest common divisor.
For part (iii), since no one supplier dominates the other with respect to both
setup cost and purchasing price, the optimal order quantity is given by either
Eq. (4.30) or Eq. (4.32), the one that results in a lower ZS . The number of orders
is found accordingly, as in parts (i) and (ii).
As shown in Lemmas 4.1 – 4.8, Q∗ is given by one of the following equations:
(4.27), (4.29), (4.30), (4.31), or (4.32). For ease of analysis their corresponding
total cost per time unit can be computed as follows:
85
• When Q is given by Eq. (4.27),
k1
k2
hQ
S1
ZS (Q) = c1
+ p1 + (d − c1 )
+ p2 +
.
Q
Q
2
• When Q is given by Eq. (4.29),
d
k1
k2
hQ
B
ZS (Q) =
(q2 − qa )
+ p1 + (qa − q1 )
+ p2
+
.
(q2 − q1 )
Q
Q
2
• When Q is given by Eq. (4.30),
k1
k2
hQ
S2
ZS (Q) = (d − c2 )
+ p 1 + c2
+ p2 +
.
Q
Q
2
• When Q is given by Eq. (4.31),
ZSO2 (Q)
k2
=d
+ p2
Q
+
hQ
.
2
+
hQ
.
2
• When Q is given by Eq. (4.32),
ZSO1 (Q)
k1
=d
+ p1
Q
These functions are used in Table 4.5. Table 4.5 provides the optimal order
quantity (Q∗ ) and the optimal order allocation (J1 and J2 ) in closed form for
each one of the cases presented in Table 4.4. The same column and row order is
preserved.
<
≥
≤
>
<
≤
≥
>
B
ZS
h(q2 − q1 )
h
(c1 k1 + (d − c1 )k2 ) ,
(q2 − qa )k1 + (qa − q1 )k2
(d − c2 )k1 + c2 k2
r 2
2d
h
, then
B
ZS
O2
ZS
S1
ZS
s
Q∗ =
h(q2 − q1 )
2d
h
h
,
r 2k d 2
h
2k2 d
(q2 − qa )k1 + (qa − q1 )k2
r
O2
ZS
else
(c1 k1 + (d − c1 )k2 ) ,
, then
r 2k d 2
h
2
(q2 − q1 )
r
Q∗ = argmin
Q∗ = argmin
d(q2 − qa )
c1 < d, c2 ≥ d
If c1 ≤
(q2 − qa )k1 + (qa − q1 )k2
otherwise,
c1 k1 + (d − c1 )k2
h(q2 − q1 )
((d − c2 )k1 + c2 k2 )
h
2
, then
Q∗ =
s
Q∗ =
d(q2 − qa )
r (q2 − q1 )
2
If c1 ≤
h
else
2
((d − c2 )k1 + c2 k2 ) ,
h
2
r
2d
S2
ZS
d(q2 − qa )
(q2 − q1 )
r
r
S1
ZS
S2
ZS
Q∗ =
s
Q∗ = argmin
Q∗ = argmin
If c1 ≤
c1 < d, c2 < d
Optimal values of J1 and J2 :
q
– If Q∗ = 2/h c1 k1 + (d − c1 )k2 , then (J1 , J2 ) = (c1 , d − c1 ) = 1, where (J1 , J2 ) denotes the common greatest denominator.
q
– If Q∗ = (2d/h(q2 − q1 )) (q2 − qa )k1 + (qa − q1 )k2 , then J1 = n(q2 − qa ), J2 = n(qa − q1 ), where n > 0, integer,
such that
qJ1 and J2 becomes the smallest possible integer.
– If Q∗ = 2/h (d − c2 )k1 + c2 k2 , then (J1 , J2 ) = (d − c2 , c2 ) = 1, where (J1 , J2 ) denotes the common greatest denominator.
p
– If Q∗ = 2k2 d/h, then J1 = 0 and J2 = 1.
p1 p2
k1 k2
q1 < qa ≤ q2
Table 4.5. Closed-Form Solution of Feasible Cases (Part 1)
;
86
<
≥
≤
>
<
≤
≥
>
B
ZS
s
2d
h(q2 − q1 )
O2
ZS
h
,
r 2k d 2
Q∗ =
s
Q∗ =
r
h
2k2 d
2d
h(q2 − q1 )
(q2 − qa )k1 + (qa − q1 )k2
Q∗ = argmin
c1 ≥ d, c2 ≥ d
Q∗ = argmin
2d
2
h
(q2 − qa )k1 + (qa − q1 )k2
((d − c2 )k1 + c2 k2 )
(d − c2 )k1 + c2 k2
h
r 2
r
h(q2 − q1 )
S2
ZS
s
Q∗ =
B
ZS
(q2 − qa )k1 + (qa − q1 )k2
c1 ≥ d, c2 < d
,
Optimal Values of J1 and J2 :
q
– If Q∗ = (2d/h(q2 − q1 )) (q2 − qa )k1 + (qa − q1 )k2 , then J1 = n(q2 − qa ), J2 = n(qa − q1 ), where n > 0, integer,
such that
pJ1 and J2 becomes the smallest possible integer.
– If Q∗ = 2k2 d/h, then J1 = 0 and J2 = 1.
q
– If Q∗ = 2/h (d − c2 )k1 + c2 k2 , then (J1 , J2 ) = (d − c2 , c2 ) = 1, where (J1 , J2 ) denotes the common greatest denominator.
p1 p2
k1 k2
q1 < qa ≤ q2
Table 4.5: Closed-Form Solution of Feasible Cases (Part 2)
87
<
≥
≤
>
<
≤
≥
>
h
r 2
h
r 2
(d − c2 )k1 + c2 k2
h
,
,
Q∗ = argmin
Q∗ =
r
h
2k2 d
h
r 2k d 2
c1 k1 + (d − c1 )k2
O2
ZS
h
r 2
c1 < d, c2 ≥ d
S1
ZS
c1 k1 + (d − c1 )k2
h
r 2
c1 k1 + (d − c1 )k2
Q∗ =
(d − c2 )k1 + c2 k2
r 2
S1
ZS
S2
ZS
Q∗ =
Q∗ = argmin
c1 < d, c2 < d
,
Optimal Values of J1 and J2 :
q
– If Q∗ = 2/h c1 k1 + (d − c1 )k2 , then (J1 , J2 ) = (c1 , d − c1 ) = 1, where (J1 , J2 ) denotes the common greatest denominator.
q
– If Q∗ = 2/h (d − c2 )k1 + c2 k2 , then (J1 , J2 ) = (d − c2 , c2 ) = 1, where (J1 , J2 ) denotes the common greatest denominator.
p
– If Q∗ = 2k2 d/h, then J1 = 0 and J2 = 1.
p1 p2
k1 k2
q1 ≥ qa
Table 4.5: Closed-Form Solution of Feasible Cases (Part 3)
88
<
≥
≤
>
<
≤
≥
>
r
,
h
r 2k d 1
h
2k2 d
h
r 2k d 2
O1
ZS
O2
ZS
Q∗ =
Q∗ = argmin
c1 ≥ d, c2 ≥ d
r
h
Q∗ =
h
((d − c2 )k1 + c2 k2 ) ,
(d − c2 )k1 + c2 k2
r 2
h
h
2
r 2k d 1
r
O1
ZS
S2
ZS
2k1 d
Q∗ = argmin
Q∗ =
c1 ≥ d, c2 < d
Optimal Values of J1 and J2 :
p
– If Q∗ = 2k1 d/h, then J1 = 1 and J2 = 0.
p
– If Q∗ = 2k2 d/h, then J1 = 0 and J2 = 1.
q
– If Q∗ = 2/h (d − c2 )k1 + c2 k2 , then (J1 , J2 ) = (d − c2 , c2 ) = 1, where (J1 , J2 ) denotes the common
greatest denominator.
p1 p2
k1 k2
q1 ≥ qa
Table 4.5: Closed-Form Solution of Feasible Cases (Part 4)
89
90
Theorem 4.3. (Optimality) If the problem satisfies the feasibility conditions
provided by Theorem 4.2, Table 4.5 provides the optimal order quantity (Q∗ ) under
the various costs, quality, and capacity conditions specified.
Proof: The results of column 1 are proved in Lemma 4.1. Specifically, part (i)
proves the results when k1 < k2 and p1 < p2 , part (ii) proves the results when
k1 > k2 and p1 > p2 , and part (iii) proves the results when k1 ≤ k2 and p1 ≥ p2 , or
k1 ≥ k2 and p1 ≥ p2 . Similarly, the results of column 2 and its corresponding rows
are proved in Lemma 4.2. The results of column 3 and its corresponding rows are
proved in Lemma 4.3. Finally, the results of column 4 and its corresponding rows
are proved in Lemma 4.4.
4.6
Conclusions
Supplier selection literature has not often considered the inventory management
of the parts being purchased in the final selection of suppliers. In this chapter,
three single-stage models are presented to illustrate the order quantity allocation
concept in supplier selection decisions. The first of these models is an extension
of the work by Ghodsypour and O’Brien [36]. By allowing multiple orders to be
allocated to the selected suppliers within a replenishment cycle, the original results
were ouperformed. In this model, sometimes the optimal value of M (number of
orders allowed within an order cycle) that minimizes ZS may result in a large cycle
time. A very practical solution to this problem is that a company can restrict M
to a reasonable small value. It is shown that by doing so, the increase in total
cost per time unit can be justified by the advantages of having a shorter cycle
time. Short cycle times facilitate the interaction with suppliers and simplify the
inventory management process.
Although the first model provides optimal solutions, the fact that order quantities for different selected suppliers are unequal may prevent the application of these
results to multi-stage supply chain systems where inventory coordination between
stages is necessary in order to avoid shortages. For this reason, the second proposed model restricts all order quantities to be of equal size for the final selected
suppliers. In addition, by restricting M to a reasonable small value, the same
advantages as for the first model are obtained. In addition to these advantages,
91
the objective function of the second model is simplified, which makes the mathematical model considerably simpler and thus easier to solve. These advantages
make the models proposed in this chapter very practical as well as easy to solve
and implement in different practical situations.
The third model modifies the second model to consider holding costs that are
not dependent on the purchasing price. This model will serve as a basis for the
problems analyzed in Chapters 5 and 6. A closed-form solution for the particular
case of two suppliers is developed.
Chapter
5
Incorporating Transportation Costs
into Supplier Selection and Order
Quantity Allocation
5.1
Introduction
Chapter 4 addressed the order quantity allocation in the supplier selection problem.
To derive optimal inventory policies that simultaneously determine how much, how
often, and from which suppliers to order, typical inventory costs (holding, setup,
and purchasing) were considered.
In this chapter, Problem (P4.3) from Chapter 4 is extended to consider transportation costs in addition to inventory costs. The relevance of incorporating
transportation costs into replenishment decisions has been highlighted by several
authors in the literature (e.g., Langley [85], Hall [86], Buffa [88], Carter and Ferrin [87], and Swenseth and Godfrey [103]).
Despite the importance of transportation costs to determine supplier selection
and order quantity allocation, existing models have typically assumed that: (1)
transportation costs are managed by suppliers and, therefore, considered a part of
the unit price; or (2) transportation costs are managed by the buyer and therefore,
included as part of the setup cost. These models are insensitive to the effect of the
shipment quantity on the per-shipment cost of transportation and seem unrealistic
for situations where goods are moved in smaller-size, less-than-truckload shipments
(Warsing [21]).
93
This chapter focuses on the usage of trucks as a means of transporting goods
and incorporates the transportation cost as a function of the shipment quantity.
This is the mode for which most data is available and for which the full-truckload (TL) versus less-than-truckload (LTL) is most interesting. The goal is to
determine if it is more cost effective to order smaller shipments from selected
suppliers more frequently at a higher cost per-unit shipping cost or to order larger,
but less expensive, shipments less frequently.
5.2
Actual Transportation Freight Rates
In practice, freight can be transported using TL or LTL. According to Swenseth
and Godfrey [103], TL rates are usually stated on a per-mile basis and LTL rates are
generally stated per hundredweight (CWT). Table 5.1 shows an example of freight
rates for a particular shipping route (this data has been extracted for illustrative
purposes from Swenseth and Godfrey [103]).
Table 5.1. Example of Nominal Freight Rates
Weight Break (lbs)
Freight Rate
Minimum Charge
1–499
500–999
1,000–1,999
2,000–4,999
5,000–9,999
10,000–19,999
20,000–46,000 *
$40.00
$17.60/CWT
$14.80/CWT
$13.80/CWT
$12.80/CWT
$12.40/CWT
$6.08/CWT
$1,110.00
* TL Capacity
Figure 5.1 shows a graphical representation of the freight rates versus the weight
shipped using the data from Table 5.1.
Notice that freight rates take the form of a step function with a decreasing
rate as shipping weights increase. This reflects the economies of scale for larger
shipping weights.
94
45
40
Freightt Rate ($/CWT)
35
30
25
20
15
10
5
0
Shipment Weight (lbs)
Figure 5.1. Freight Rate Vs. Weight Shipped
Now, Figure 5.2 graphically represents the weight shipped (lbs) with its corresponding total transportation cost ($) for the rates given in Table 5.1 (weight is
only shown up to 1,010 lbs).
Total TTransportation Cost ($)
160
140
138
120
100
80
74
60
40
weight breakpoints
20
0
10
110
210
310
421
410
500 510
610
710
810
933 1,000 910
1,010
indifference points
Shipment Weight (lbs)
Figure 5.2. Total Transportation Cost Structure as Typically Stated
In Figure 5.2, there exist some weights that when multiplied by its corresponding freight rate will yield the same total cost as that for the next weight break-
95
point. These points are called indifference points and give rise to the concept
of ‘over-declare’. Over-declared shipments are used by shippers to achieve a
lower total transportation cost. This is accomplished by artificially inflating the
weight to a higher weight breakpoint resulting in a lower total cost (Swenseth and
Godfrey [102]). For example, consider Figure 5.2. The first indifference point (421
lbs) is calculated as follows: (500 lbs · 14.80 $/CWT)/17.60 $/CWT. If the weight
shipped is between 421 and 500 lbs, then the shipment can be over-declared to 500
lbs. In this way, the company is charged a fixed amount of $74. The effective
rate for a given shipment in this range is calculated as the fixed amount of $74
divided by the weight shipped. For this example, the effective rate for a shipment
of 450 lbs is 74/450 = $0.164/lb or $16.4/CWT. Likewise, the second indifference
point (933 lbs) is obtained as follows: (1, 000 lbs·13.80 $/CWT)/14.80.60 $/CWT.
Therefore, any shipping weight between 933 and 1,000 lbs can be over-declared to
1,000 lbs to obtain a reduced total transportation cost.
Once all the indifference points for the rates in Table 5.1 are found, a schedule
of actual freight rates is created that alternates between ranges of a constant charge
per CWT followed by a fixed charge. The fixed charge is the result of over-declaring
a LTL shipment to the next LTL weight break or the TL shipment (See Table 5.2).
Figure 5.3 shows a pictorial representation of the rates shown in Table 5.2.
Notice that all shipments between the weight breakpoints and the indifference
points are charged the same amount.
Although the function shown in Figure 5.3 is continuous, it is non-differentiable
due to the indifference points. Hence, it becomes difficult to incorporate actual
freight rates into analytical models. Natarajan [19] identified two main problems
when trying to incorporate actual freight rates into analytical models:
1. Determining the exact rates between every origin and destination is time
consuming and expensive.
96
Table 5.2. Actual Freight Rate Schedule
Weight Break (lbs)
Freight Rate
Min charge (up to 227 lbs)
228–420
421–499
500–932
933–999
1,000–1,855
1,856–1,999
2,000–4,749
4,750–9,999
10,000–18,256
18,257–46,000
$40.00
$17.60/CWT
$74.00
$14.80/CWT
$138.00
$13.80/CWT
$256.00
$12.80/CWT
$608.00
$6.08/CWT
$1,110.00
Totall Transportation Cost ($)
160
140
138
120
100
80
74
60
40
weight breakpoints
20
0
10
110
210
310
421
410
500 510
610
710
810
933 1,000 910
1,010
indifference points
Shipment Weight (lbs)
Figure 5.3. Total Transportation Cost Function as Typically Charged
2. The freight rate is a function of total weight shipped, making its representation a step function (as in Figure 5.2).
Because of the difficulty that arises when working with actual freight rates,
several researchers have proposed the use of continuous functions to properly
estimate actual freight rates. Section 5.4 presents two continuous functions that
estimate the freight rates used to determine the total transportation cost in the
proposed supplier selection and order quantity allocation models.
97
5.3
Problem Description and Assumptions
As in Chapter 4, a single-stage system is studied here. The system consists of a
manufacturing facility operating in a centralized framework that procures items
from different suppliers. The demand placed on the manufacturer must be satisfied
by the different selected suppliers without shortages. The problem is determining
the order quantity and the number of orders per order cycle allocated to each
selected supplier, while minimizing the total cost per time unit of the system.
The total cost includes setup, holding, purchasing, and transportation costs. The
inbound transportation cost of the manufacturer is initially modeled using LTL
rates. As noted by Spiegel [131], this assumption is reasonable given the fact that
in today’s market many factors are driving the use of small shipment sizes (LTL).
These include: an increasing number of stock-keeping-units driven by customers’
demands for greater customization, lean philosophies and the resulting push to
more frequent shipments, and customer service considerations driving more decentralized distribution networks serving fewer customers per distribution center.
Nonetheless, in Section 5.5.7 the case where more than one TL might be needed
to transport items from suppliers is considered.
In addition the to the EOQ assumptions used in Chapter 4 for Problem (P4.3),
the following assumptions are included to study the problem in this chapter:
• Lead times from suppliers to the manufacturer are fixed.
• A continuous review policy is followed. This implies that orders are placed
when the reorder point is reached. The reorder point for suppliers will be
different as a consequence of suppliers’ leadtimes being different.
• Free-on-board (FOB) origin, freight collect is assumed. Following the description presented in Hughes Networks Systems [132], this implies that the
buyer (manufacturer) pays the freight charges and also owns the goods intransit (in-transit inventory).
98
The following notation is used throughout this chapter:
Data
r – number of available suppliers
d – demand per time unit
w – weight of an item
h – inventory holding cost per item and time unit
ki – setup cost of ith supplier
pi – unit price of ith supplier
ci – capacity of ith supplier per time unit
qi – perfect rate of ith supplier
qa – minimum acceptable perfect rate of parts
li – leadtime of ith supplier
Variables
Ji – number of orders of ith supplier per order cycle
Q – ordered quantity from selected suppliers
T – time between consecutive orders
Tc – (repeating) order cycle time
P
Recall from Chapter 4 that the length of an order cycle is Tc = T · ri=1 Ji =
P
(Q/d) · ri=1 Ji . The total number of order cycles per time unit is given by 1/Tc =
P
P
1/(T · ri=1 Ji ) = d/(Q · ri=1 Ji ).
5.4
Freight Rate Continuous Functions
In this section, two continuous functions used to fit the actual freight rates are
introduced.
Swenseth and Godfrey [102] proposed the use of the proportional function to
99
model LTL freight rates. The function is as follows:
Fy = Fx + α(Wx − Wy ),
(5.1)
where Fy is the freight rate for shipping a given load ($/CWT), Fx is the TL
rate per CWT, Wx is the TL weight (lbs), Wy is the shipping weight (lbs), and
α represents the rate at which the freight rate increases per 100 lbs decrease in
shipping weight. Notice that the terms Fx and αWx are constants and can be
substituted by another constant (say, A), and that Wy can also be expressed as
Wy = Qw, where Q is the order quantity and w is the weight of the item under
consideration. Hence, Eq. (5.1) can be rewritten as:
Fy = A − αQw.
(5.2)
Eq. (5.2) is the proportional function proposed by Langley [85]. It is easy to
see that the freight rate decreases (at a rate α), for every unit increase in Q. The
value of α can be obtained in two ways (Natajaran [19]): (1) from Eq. (5.1) by
minimizing the mean squared error between actual and estimated LTL freight rates
for each route. In this case, rates are generated over a realistic range of shipment
quantities (Q) for a lane and then a curve is fitted to the rate data; (2) from
Eq. (5.2) by fitting a simple linear regression model between the freight rate and
order quantity.
The proportional function by Swenseth and Godfrey [102] has basically been
used to develop analytical results in problems that incorporate transportation costs
in the analysis. However, as in the case of Natajaran [19] and DiFillipo [106],
when the transportation freight rates are known, the proportional function by
Langley [85], Eq. (5.2), is used in actual implementations and α is obtained by
fitting a linear regression. In this research, since it is assumed that the freight
rates from the potential suppliers are known , Langley’s function is fit between the
100
freight rate and the weight shipped to estimate freight rates.
In addition to Langley’s function (Eq. (5.2)), Tyworth and Ruiz-Torres [105]
proposed a method to generate a power continuous function. The general form of
this estimate as a function of the weight shipped is as follows:
Fy = a(Qw)b ,
(5.3)
where a and b are the corresponding coefficients. These coefficients can be found
using non-linear regression analysis. However, notice that Eq. (5.3) can also be
expressed as follows:
ln(Fy ) = ln[a(Qw)b ] = ln(a) + b ln(Qw).
(5.4)
In this way, the coefficients can also be found by performing a simple linear
regression analysis.
To generate the rate functions, effective rates need to be computed. Figure 5.4
shows the continuous functions generated using Eq. (5.2) and Eq. (5.3) to fit the
freight rates for the data in Table 5.1.
80
Effective Rate ($/CWT)
70
Effective Rate
Power Estimate
Langley's Estimate
60
50
40
30
20
10
0
50
227
400
499
800
999
1,600 1,950 3,000 5,000 9,000 14,000 25,000 46,000
Shipment Weight (lbs)
Figure 5.4. Langley’s and Power Function Estimates
101
5.5
Transportation-Inclusive Models with
Equal-Size Order Quantities
5.5.1
Estimating Transportation Costs
Let Fy i , i = 1, . . . , r, be the freight rate function for shipping a given load from
supplier i ($/CWT). Thus, the transportation freight rate from supplier ‘i’ using
Langley’s proportional function (Eq. (5.2)) is,
Fy i = Ai + αi Qw,
(5.5)
and the transportation cost for shipping an order quantity (Q) from supplier ‘i’
using (Eq. (5.5)) is,
Ai + αi Qw
Qw
.
100
(5.6)
Since freight rates are given in $/CWT, the order quantity to be shipped is
multiplied by the weight of the item (w) and divided by 100 in order to express
the weight shipped in CWT. The transportation cost per time unit is obtained by
multiplying Eq. (5.6) by the total number of orders allocated to all suppliers in
P
one order cycle ( ri=1 Ji ) and by the total number of order cycles per time unit
P
d/(Q · ri=1 Ji ),
(
)
r
X
Ji Qw d
1
Ai + αi Qw
· · Pr
100 Q
i=1 Ji
i=1
(
)
r
Ji
dw X
=
Ai + αi Qw Pr
.
(5.7)
100 i=1
i=1 Ji
Similarly, the transportation freight rate from supplier ‘i’ using the power function (Eq. (5.3)) is,
Fy i = ai (Qw)bi ,
(5.8)
and the total transportation cost for shipping an order quantity (Q) from supplier
102
‘i’ using Eq. (5.8)) is,
ai (Qw)bi
Qw
.
100
(5.9)
Finally, its corresponding transportation cost per time unit is,
r
dw X
100 i=1
(
)
J
i
ai (Qw)bi P
.
r
Ji
(5.10)
i=1
5.5.2
In-Transit Inventory
Since FOB origin, freight collect is assumed, the manufacturer not only pays for
freight charges but is also responsible for goods in transit. Therefore, the in-transit
inventory should also be reflected in the total inventory per time unit held by the
manufacturer. The in-transit inventory per time unit for each supplier is,
li dJi
· r
· h,
Y P
Ji
(5.11)
i=1
where Y is the number of days per time unit. The first term (li /Y ) represents
the fraction of time that an order of size Q spends in transit. The second term
represents the fraction of the total demand procured from supplier i, and h is the
holding cost rate (herein assumed to be the same as the regular holding inventory
cost). The final expression for in–transit inventory per time unit considering all
suppliers is,
r
P
li Ji
dh i=1
· P
,
r
Y
Ji
(5.12)
i=1
where
Pr
i=1 li Ji /
Pr
i=1
Ji indicates an average leadtime. Note that the in-transit
inventory cost does not depend on the order quantity Q. While this cost does not
directly affect the size of the order, it does affect the number of orders allocated
to suppliers (Ji ’s).
103
5.5.3
Model Considering Continuous Functions
The total cost per time unit considering continuous functions to estimate the transportation freight rates is the following:
r
P
ZF Q
Ji ki
d
= · i=1
r
Q P
Ji
r
P
Ji pi
hQ
+
+ d · i=1
r
P
2
i=1
Ji
r
P
dw
+
·
100
i=1
i=1
Ji · Fy i
r
P
i=1
Ji
r
P
Ji li
dh i=1
+
· P
, (5.13)
r
Y
Ji
i=1
where the first term represents the setup cost, the second term denotes the holding cost, the third term is the purchasing cost, the fourth term accounts for
the transportation cost, and the fifth term denotes the cost corresponding to
the in-transit inventory. Eq. (5.5) replaces Fy i when Langley’s function is used
to estimate the actual freight rate from supplier i. Likewise, Eq. (5.8) replaces Fy i
when the power function is employed to estimate the freight rate from supplier
i. This is equivalent to replacing the fourth term in Eq. (5.13) with Eqs. (5.7) or
(5.10).
By including capacity and quality constraints (from Problem (P4.3)), and rearranging terms of Eq. (5.13), the complete formulation for the supplier selection and
order quantity allocation problem considering transportation costs is the following:
(P5.1)
"
r
r
r
X
d 1 X
w X
=
·
Ji ki +
Ji pi +
·
Ji Fy i
M Q i=1
100
i=1
# i=1
r
X
h
hQ
Ji li +
,
+ ·
Y i=1
2
minimize
ZF Q
subject to
dJi ≤ ci M, i = 1, . . . , r,
r
X
Ji qi ≥ qa M,
i=1
r
X
Ji = M,
i=1
Q ≥ 0,
104
Ji ≥ 0, integer, i = 1, . . . , r,
M ≥ 1, integer,
where the total number of orders allocated to all selected suppliers in one order
P
cycle, ri=1 Ji , has been defined as M . Recall from Chapter 4 that if the optimal
value of M that minimizes the total cost per time unit results in an excessively
large order cycle time, then the manufacturer may restrict M to a reasonably small
value to reduce the entire order cycle. Short cycle times facilitate interaction with
suppliers and simplify the inventory control process.
5.5.4
Linearizing Actual LTL Freight Rates
So far, two functions that estimate the actual transportation freight rates have
been introduced. Since it is of interest to study the performance of such estimates, a mathematical model to obtain the optimal order quantity allocation to
suppliers considering the actual freight rate structure is proposed. This is done
by representing transportation costs as a continuous piecewise linear function (of
the weight shipped) using binary variables. The logic is as follows. Suppose that
the transportation cost function f (x) has breakpoints b1 , b2 , . . . , bm , as shown in
Figure 5.5.
A breakpoint indicates a point at which a change in freight rate occurs. Suppose that a given shipment of size x is to be transported such that for some k,
bk ≤ x ≤ bk+1 . Then, for some λk (0 ≤ λk ≤ 1), x may be written as,
x = λk bk + (1 − λk )bk+1 ,
and because f (x) is linear for bk ≤ x ≤ bk+1 , its corresponding transportation cost
may be written as,
f (x) = λk f (bk ) + (1 − λk )f (bk+1 ).
105
f (x )
f (bk +1 )
f (bk )
bk −1
bk
bk +1
x
Figure 5.5. LTL Rate Structure
Observe from Figure 5.5 that for a shipment in the range bk−1 ≤ x ≤ bk , its
corresponding cost f (x) will always be the same, irrespective of the values of λk−1 ,
given that a flat transportation cost is charged for the shipments falling within
this range.
Applying the above logic to the specific problem under consideration, the total
weight shipped from supplier ‘i’ (in one order) can be defined as follows:
Q·w =
u
i +1
X
bi,k · λi,k ,
(5.14)
k=1
where bi,k represents a breakpoint (lbs) that can be obtained from the actual LTL
rate structure and ui is the total number of breakpoints in the actual LTL rate
structure. Further, bi,1 = 0 and bi,ui +1 equals the capacity of a truckload (lbs).
The total transportation cost charged to supplier i for the weight (Qw) shipped
may be written as,
T Ci (Qw) =
u
i +1
X
gi,k · λi,k ,
(5.15)
k=1
where gi,k is the total transportation cost ($) obtained by evaluating the corresponding breakpoint bi,k into the actual LTL rate structure. As per the definition
of bi,ui +1 , gi,ui +1 is the cost of a full truckload.
106
In order to illustrate these concepts consider the nominal and actual freight
rates in Table 5.3. The nominal freight rates were obtained from Ballou [133] and
the actual freight rates were calculated as explained in Section 5.2. The capacity
of the truck is 40,000 lbs and the TL rate is $18.8125/CWT, which corresponds to
a cost of $7,525/TL.
Table 5.3. Nominal and Actual Freight Rates for Supplier i
Nominal Freight Rate
Weight Break (lbs)
1–499
500–999
1,000–1,999
2,000–4,999
5,000–9,999
10,000–19,999
20,000–29,999
30,000–40,000
Freight Rate
$107.75/CWT
$92.26/CWT
$71.14/CWT
$64.14/CWT
$52.21/CWT
$40.11/CWT
$27.48/CWT
$7,525
Actual Freight Rate
Weight Break (lbs)
1–428
429–499
500–771
772–999
1,000–1,803
1,804–1,999
2,000–4,070
4,071–4,999
5,000–7,682
7,683–9,999
10,000–13,702
13,703–19,999
20,000–27,383
27,384–40,000
Freight Rate
$107.75/CWT
$461.3
$92.26/CWT
$711.4
$71.14/CWT
$1,282.8
$64.14/CWT
$2,610.5
$52.21/CWT
$4,011
$40.11/CWT
$5,496
$27.48/CWT
$7,525
In this particular case, the total number of breakpoints is ui = 14. The total
weight shipped, using Eq. (5.14), is expressed as follows:
Qw = 0 · λi,1 + 429 · λi,2 + 500 · λi,3 + 772 · λi,4 + 1, 000 · λi,5 + 1, 804 · λi,6
+ 2, 000 · λi,7 + 4, 071 · λi,8 + 5, 000 · λi,9 + 7, 683 · λi,10 + 10, 000 · λi,11
+ 13, 703 · λi,12 + 20, 000 · λi,13 + 27, 384 · λi,14 + 40, 000 · λi,15 ,
107
and the corresponding total transportation cost using Eq. (5.15), is,
T Ci (Qw) = 0 · λi,1 + 461.3 · λi,2 + 461.3 · λi,3 + 711.4 · λi,4 + 711.4 · λi,5
+ 1, 282.8 · λi,6 + 1, 282.8 · λi,7 + 2, 610.5 · λi,8 + 2, 610.5 · λi,9
+ 4, 011.0 · λi,10 + 4, 011.0 · λi,11 + 5, 496.0 · λi,12 + 5, 496.0 · λi,13
+ 7, 525 · λi,14 + 7, 525 · λi,15 ,
5.5.5
Model Considering Actual Freight Rates
The total cost per time unit considering actual transportation costs for the system
under consideration is the following:
r
P
Ji ki
d
ZA = · i=1
r
Q P
Ji
r
P
Ji pi
hQ
+
+ d · i=1
r
P
2
i=1
Ji
r
P
d
+ · i=1
Q
i=1
Ji · T Ci (Qw)
n
P
Ji
i=1
r
P
Ji li
dh i=1
+
· P
, (5.16)
r
Y
Ji
i=1
where the first term represents the setup cost, the second term denotes the holding cost, the third term is the purchasing cost, the fourth term accounts for the
transportation cost, and the fifth term denotes the cost corresponding to the
in-transit inventory costs.
The complete mathematical model for supplier selection and order quantity
allocation using actual transportation freight rates, after rearranging Eq. (5.16)
and including capacity and quality constraints, is as follows:
(P5.2)
minimize
subject to
"
r
r
r
X
d 1 X
1 X
ZA =
Ji ki +
Ji pi +
Ji · T Ci (Qw)
M Q i=1
Q
i=1
i=1
#
r
X
h
hQ
+ ·
Ji li +
,
Y i=1
2
Ji d ≤ ci M, i = 1, . . . , r,
r
X
i=1
Ji qi ≥ qa M,
(5.17)
(5.18)
108
r
X
Ji = M,
(5.19)
i=1
Q·w =
u
i +1
X
bi,k · λi,k , i = 1, . . . , r,
(5.20)
k=1
T Ci (Qw) =
u
i +1
X
gi,k · λi,k , i = 1, . . . , r,
(5.21)
k=1
λi,k ≤ Zi,k , i = 1, . . . , r; k = 1,
(5.22)
λi,k ≤ Zi,k−1 + Zi,k , i = 1, . . . , r; k = 2, . . . , ui , (5.23)
λi,k ≤ Zi,k−1 , i = 1, . . . , r; k = ui + 1,
u
i +1
X
k=1
u
i +1
X
(5.24)
λi,k = 1, i = 1, . . . , r,
(5.25)
Zi,k = 1, i = 1, . . . , r,
(5.26)
k=1
Zi,k ∈ {0, 1}, i = 1, . . . , r; k = 1, . . . , ui ,
(5.27)
Q ≥ 0,
(5.28)
Ji ≥ 0, integer, i = 1, . . . , r,
(5.29)
M ≥ 1, integer.
(5.30)
Each binary variable, Zi,k , represents one linear segment of the freight rate
function. By constraint (5.26), only one Zi,k per supplier can get a value of ‘1’.
Then, the specific segment chosen contains the weight shipped (Qw) and its corresponding total transportation cost T Ci (Qw) is expressed as the linear combination
of λi,k and λi,k+1 .
5.5.6
Illustrative Example
In this section, a numerical example is presented to analyze the impact of transportation costs on supplier selection and order quantity allocation decisions. It is
important to compare the estimated solutions obtained from using Langley’s and
109
the power functions to the absolute optimal solution obtained by solving Problem
(P5.2). Important properties and conclusions are derived from this analysis.
5.5.6.1
Data and Parameters
The example problem consists of one manufacturer and three potential suppliers.
The manufacturer needs to decide its inventory policy with respect to a component
part needed in the assembly process. The weight of the component part is w = 16
lbs and its demand has been estimated at d = 1000 units/month with a corresponding holding cost of h = $10/unit/month. The manufacturer has specified its
minimum acceptable perfect rate as qa = 0.95. Table 5.4 shows additional data for
the three potential suppliers.
Table 5.4. Supplier’s Data
Supplier
i
Price (pi )
($)
Setup Cost (ki )
($)
Perf. Rate (qi )
Capacity (ci )
(units/month)
Leadtime (li )
(days)
1
2
3
20
24
30
160
140
130
0.93
0.95
0.98
700
800
750
1
3
2
The suppliers are located in different geographical areas, and therefore, the
corresponding freight rates are different. The capacity of the trucks is Wx =
40,000 lbs. Tables 5.5, 5.6, and 5.7, respectively, show the nominal (Ballou [133])
and actual freight rates for suppliers 1, 2, and 3. The actual freight rates were
calculated as explained in Section 5.2. Additionally, their corresponding TL rates
are: $18.8125/CWT ($7,525/TL), $33/CWT ($13,200/TL), and $12.575/CWT
($5,030/TL).
110
Table 5.5. Nominal and Actual Freight Rates for Supplier 1
Nominal Freight Rate
Weight Break (lbs)
1–499
500–999
1,000–1,999
2,000–4,999
5,000–9,999
10,000–19,999
20,000–29,999
30,000–40,000
Freight Rate
$107.75/CWT
$92.26/CWT
$71.14/CWT
$64.14/CWT
$52.21/CWT
$40.11/CWT
$27.48/CWT
$7,525
Actual Freight Rate
Weight Break (lbs)
1–428
429–499
500–771
772–999
1,000–1,803
1,804–1,999
2,000–4,070
4,071–4,999
5,000–7,682
7,683–9,999
10,000–13,702
13,703–19,999
20,000–27,383
27,384–40,000
Freight Rate
$107.75/CWT
$461.3
$92.26/CWT
$711.4
$71.14/CWT
$1,282.8
$64.14/CWT
$2,610.5
$52.21/CWT
$4,011
$40.11/CWT
$5,496
$27.48/CWT
$7,525
The functions generated from fitting Eqs. (5.5) and (5.8) to the effective rates
of each supplier are summarized in Table 5.8 along with their corresponding coefficient of determination (R2 , 0 ≤ R2 ≤ 1). The analysis was performed using
Minitab [134]. In general, the higher the R2 , the better the model fits the freight
rates.
5.5.6.2
Analysis of Results
The following results are labeled for the purpose of simplifying the analysis:
1. WTA (Problem (P4.3) + actual transportation cost): results are obtained in
three steps. First, Problem (P4.3), which neither considers transportation
nor in-transit inventory costs, is solved to find the total cost per time unit
(ZS∗ ), the order allocation (Ji ’s), and the order quantity allocated to selected
suppliers (Q∗ ). Second, the transportation and in-transit inventory costs
are computed. The freight rates for the three suppliers are obtained from
Tables 5.5–5.7 considering the shipping weight (Q∗ · w). The transportation
111
Table 5.6. Nominal and Actual Freight Rates for Supplier 2
Nominal Freight Rate
Weight Break (lbs)
1–499
500–999
1,000–1,999
2,000–4,999
5,000–9,999
10,000–19,999
20,000–29,999
30,000–40,000
Actual Freight Rate
Freight Rate
$136.26/CWT
$109.87/CWT
$91.61/CWT
$79.45/CWT
$69.91/CWT
$54.61/CWT
$48.12/CWT
$13,200
Weight Break (lbs)
1–403
404–499
500–833
834–999
1,000–1,734
1,735–1,999
2,000–4,399
4,400–4,999
5,000–7,811
7,812–9,999
10,000–17,623
17,624–19,999
20,000–27,431
27,432–40,000
Freight Rate
$136.26/CWT
$549.35
$109.87/CWT
$916.1
$91.61/CWT
$1,589
$79.45/CWT
$3,495.5
$69.91/CWT
$5,461
$54.61/CWT
$9,624
$48.12/CWT
$13,200
Table 5.7. Nominal and Actual Freight Rates for Supplier 3
Nominal Freight Rate
Weight Break (lbs)
1–499
500–999
1,000–1,999
2,000–4,999
5,000–9,999
10,000–19,999
20,000–29,999
30,000–40,000
Actual Freight Rate
Freight Rate
$81.96/CWT
$74.94/CWT
$61.14/CWT
$49.65/CWT
$39.73/CWT
$33.44/CWT
$18.36/CWT
$5,030
Weight Break (lbs)
1–428
429–499
500–771
772–999
1,000–1,803
1,804–1,999
2,000–4,070
4,071–4,999
5,000–7,682
7,683–9,999
10,000–13,702
13,703–19,999
20,000–27,383
27,384–40,000
Freight Rate
$81.96/CWT
$374.7
$74.94/CWT
$611.4
$61.14/CWT
$993
$49.65/CWT
$1,986.5
$39.73/CWT
$3,344
$33.44/CWT
$3,672
$18.36/CWT
$5,030
cost per time unit is given by
r
P
dw
·
100
Ji FiA
i=1
r
P
i=1
,
Ji
(5.31)
112
Table 5.8. Summary of Freight Rate Continuous Estimates
Supplier
Langley’s Fn ($/CWT)
R2 value
Power’s Fn ($/CWT)
R2 value
1
2
3
Fy 1 = 61.7-0.00127 (Qw)
Fy 2 = 80.3-0.00129 (Qw)
Fy 3 = 48.2-0.00109 (Qw)
0.763
0.746
0.758
Fy 1 = 1586.21 (Qw)−0.4028
Fy 2 = 789.97 (Qw)−0.2831
Fy 3 = 2247.57 (Qw)−0.4757
0.947
0.935
0.938
where FiA indicates the actual freight rates obtained ($/CWT). The in-transit
transportation cost per time unit is calculated as follows:
r
P
Ji li
dh i=1
.
r
Y P
Ji
(5.32)
i=1
The resulting costs from Eqs. (5.31) and (5.32) are added to the cost found
in step one, ZS∗ .
2. LF (Langley’s function): results are obtained by solving Problem (P5.1) using
the Langley’s functions (Fy 1 , Fy 2 , and Fy 3 ) provided in the second column
of Table 5.8. These results consider estimated transportation and inventory
costs simultaneously.
3. LFA (LF with actual transportation costs): these results are calculated in
two steps. First, using the order quantity obtained in LF, the corresponding
actual freight rates for the selected suppliers are determined from Tables 5.5–
5.7. Second, the total transportation cost is recalculated using these actual
freight rates in Eq. (5.31).
4. PF (Power function): results are obtained by solving Problem (P5.1) using
the Power functions (Fy 1 , Fy 2 , and Fy 3 ) provided in the fourth column of
Table 5.8. These results consider estimated transportation and inventory
costs simultaneously.
113
5. PFA (PF with actual transportation costs): these results are found in two
steps. First, using the order quantity from PF, the corresponding actual
freight rates for the selected suppliers are determined from Tables 5.5–5.7.
Second, the total transportation cost is recalculated using these actual freight
rates in Eq. (5.31).
6. AO (Absolute Optimal solution): results are obtained by solving Problem
(P5.2), which considers actual transportation costs.
LFA and PFA are calculated to compare the continuous functions with the
absolute optimal (AO) solution. The primary concern is the comparison of actual
costs and not the estimated costs represented in LF and PF. Table 5.9 shows the
results obtained and studied for cases 1–6.
Table 5.9. Solutions to Illustrative Example (Same Q’s)
Order Allocation
J1
J2
J3
WTA
LFA
PFA
AO
3
3
3
3
20
0
0
0
2
2
2
2
Ordering Quantity
Q
Cycle’s Length
Tc (month)
Total Cost
($/month)
% Deviation
(from AO)
168
277
551
625
4.2
1.4
2.8
3.1
38,346.1
34,917.5
34,283.3
33,819.1
13.4
3.4
1.4
–
Conclusions from the results summarized in Table 5.9:
• By incorporating transportation costs and inventory costs simultaneously, as
in LFA, PFA, and AO, the manufacturer can take advantage of economies of
scale in shipping. The order quantity for WTA is based on a model that only
optimizes inventory costs and does not take advantage of economies of scale
in transportation. For this reason, the order quantity is smaller than those
of LFA, PFA, and AO. Order quantities obtained considering transportation
and inventory costs simultaneously are larger and in less frequent shipments.
This makes the most impact on freight rates. After adding the transportation
114
and in-transit inventory costs to WTA, it provides the worst total cost per
time unit (13.4% greater than that of AO).
• Considering transportation and inventory costs simultaneously, as in LFA,
PFA, and AO, changes the order allocation solution. In contrast to WTA,
which does not consider transportation and inventory costs simultaneously,
LFA, PFA, and AO have obtained different allocation solutions. In this
particular case, LFA, PFA, and AO have eliminated supplier 2 altogether.
This is mainly due to the high average actual freight rates offered by supplier
2.
• The order allocations for LFA, PFA, and AO are the same. This implies that
by solving Problem (P5.1), one can obtain the number of orders allocated
to each selected supplier. Consequently, this solution (for Ji ’s) can be introduced to solve Problem (P5.2). Problem (P5.2) will be easier to solve once
the Ji ’s are known. Specifically, Problem (P5.2) is simplified in the following
manner: constraints (5.36), (5.37), (5.38) , and (5.48) are no longer necessary
and the objective function will be dependent only on Q.
An analysis of transportation costs for different solutions was performed. Results were obtained for different fixed values of M . Table 5.10 shows the transportation costs for values of M from 2 to 25.
The impact of not considering inventory and transportation costs simultaneously results in an average deviation of 87% from the optimal solution (AO). Essentially, this translates to higher shipping costs. Modeling of freight rates using
Langley’s function results in transportation costs that are 43% higher than AO. In
contrast, using the power function results in a 14% deviation from AO. Therefore,
a power function is a better estimate of the actual freight rates.
The use of continuous functions is recommended when the number of potential
suppliers is large or when no optimization software is available to solve Problem
115
Table 5.10. Analysis of Transportation Costs
M
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Average
% Deviation
WTA
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
9,103.2
10,306.1
10,907.6
9,335.0
11,509.0
11,680.9
10,601.4
10,835.9
11,023.5
11,177.0
11,304.9
11,413.1
11,505.9
11,586.3
11,656.7
11,718.7
11,773.9
11,823.3
11,867.7
11,907.9
11,944.5
11,977.8
12,008.4
12,036.6
% Dev
from AO
55%
51%
85%
56%
96%
96%
68%
82%
84%
79%
89%
85%
93%
93%
89%
96%
92%
98%
98%
95%
100%
96%
101%
101%
H
87%
LFA
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
8,491.9
9,783.2
8,491.9
8,532.5
9,163.2
8,521.5
9,000.2
8,515.2
8,532.5
8,877.6
8,526.1
8,820.2
8,521.5
8,532.5
8,770.0
8,528.0
8,740.2
8,524.4
8,532.5
8,713.6
8,529.0
8,695.0
8,526.1
8,532.5
% Dev
from AO
44%
43%
44%
42%
56%
43%
43%
43%
42%
42%
43%
43%
43%
42%
42%
43%
43%
43%
42%
42%
43%
43%
43%
42%
H
43%
PFA
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
6,739.1
7,654.3
6,739.1
6,791.8
7,241.7
6,776.9
7,118.4
6,768.6
6,791.8
7,038.5
6,783.1
6,993.2
6,776.9
6,791.8
6,961.8
6,785.7
6,937.4
6,780.8
6,791.8
6,921.5
6,787.1
6,905.8
6,783.1
6,791.8
% Dev
from AO
15%
12%
15%
13%
23%
14%
13%
14%
13%
13%
14%
13%
14%
13%
13%
14%
13%
14%
13%
13%
13%
13%
14%
13%
AO
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
5,884.0
6,835.2
5,884.0
5,990.7
5,884.0
5,960.2
6,307.4
5,943.2
5,990.7
6,240.4
5,972.9
6,185.6
5,960.2
5,990.7
6,162.4
5,978.1
6,131.4
5,968.2
5,990.7
6,121.5
5,981.0
6,100.8
5,972.9
5,990.7
H
14%
(P5.2). Continuous functions do not require specification of rate breakpoints or any
embedded analysis to determine when to over-declare a given shipment. Further,
fitting continuous functions and solving Problem (P5.1) can easily be done using
Excel.
In solving Problem (P5.1), LTL is assumed for all shipments from suppliers and
either Langley’s or the power function is used to estimate the actual freight rates. If
after solving Problem (P5.1), the shipping weight can be over-declared as a TL, the
function used to estimate actual freight rates is overestimating the transportation
cost per time unit. This can be corrected by recalculating the transportation cost
per time unit with the [lower] freight rate corresponding to a full TL. This results
in a lower total transportation cost but the order quantity remains unchanged.
116
5.5.7
Use of Multiple Trucks
The analysis of the previous section was focused on the use of LTL for all shipments
from suppliers. This included the cases where it was appropriate to over-declare
LTL shipments as TL shipments. In this section, the case where more than one
TL might be needed to transport items from suppliers is considered. A procedure
is provided to determine the number of TL’s or the combination of TL and LTL
needed to ship the orders from suppliers.
The solution procedure consists of three main steps. First, the order quantity
is calculated solving Problem (P4.3) with the price of an item redefined as,
p0i = pi + τi ,
(5.33)
where pi is the unit price of ith supplier and τi is the TL rate per unit shipped and
can be calculated from the actual freight rates as follows:
!
Fxi w
,
100
τi =
(5.34)
where Fxi is the TL rate per CWT from supplier i. Denote the order quantity
obtained from solving Problem (P4.3) as Q 0 . This order quantity takes advantage
of the largest discount available (e.g. the discount of shipping the full TL).
Second, the smallest number of trucks where all items take advantage of the
largest discount is obtained,
$
η=
%
Q 0w
.
Wx
(5.35)
where Wx is the TL capacity in lbs.
Finally, the optimal solution is found by solving the problem of using η trucks,
η + 1 trucks, or η trucks + LTL. The following mathematical model is used to solve
117
these cases:
(P5.3)
minimize
subject to
"
r
r
r
X
1 X
d 1 X
Ji ki +
Ji pi +
Ji · T Ci (Qw)
ZA =
M Q i=1
Q i=1
i=1
#
r
r
η Wx X
h X
hQ
+
·
Ji Fxi + ·
Ji li +
,
100 Q i=1
Y i=1
2
Ji d ≤ ci M, i = 1, . . . , r,
r
X
i=1
r
X
(5.36)
Ji qi ≥ qa M,
(5.37)
Ji = M,
(5.38)
i=1
Q · w − ηWx =
T Ci (Qw) =
u
i +1
X
bi,k · λi,k , i = 1, . . . , r,
k=1
u
+1
i
X
gi,k · λi,k , i = 1, . . . , r,
(5.39)
(5.40)
k=1
λi,k ≤ Zi,k , i = 1, . . . , r; k = 1,
(5.41)
λi,k ≤ Zi,k−1 + Zi,k , i = 1, . . . , r; k = 2, . . . , ui , (5.42)
λi,k ≤ Zi,k−1 , i = 1, . . . , r; k = ui + 1,
u
i +1
X
k=1
u
i +1
X
(5.43)
λi,k = 1, i = 1, . . . , r,
(5.44)
Zi,k = 1, i = 1, . . . , r,
(5.45)
k=1
Zi,k ∈ {0, 1}, i = 1, . . . , n; k = 1, . . . , ui ,
(5.46)
η(Wx /w) ≤ Q ≤ (η + 1)(Wx /w),
(5.47)
Ji ≥ 0, integer, i = 1, . . . , r,
(5.48)
M ≥ 1, integer.
(5.49)
Constraint (5.47) bounds the optimal order quantity to be between η and η +
1 trucks. Therefore, constraint (5.39) ensures that the weight shipped as LTL
118
remains between η and η + 1 trucks.
Problem (P5.3) is an extension of Problem (P5.2). The original transportation
cost used in Problem (P5.2) was divided in two: the transportation cost due to TL’s
and the transportation cost due to the LTL portions. These costs were obtained
as follows. The cost of shipping multiple trucks from ith supplier is given by
ηJi
!
Fxi Wx
,
100
(5.50)
where the term within parenthesis represents the cost of one truck from supplier i.
The total transportation cost per time unit due to TL shipments from all suppliers
is obtained by multiplying Eq. (5.50) by the total number of orders allocated to
all selected suppliers in one order cycle and by the total number of order cycles
per time unit,
Q
d
r
P
·
Ji
r
X
ηJi
i=1
Fxi Wx
100
!
r
P
d η Wx
·
=
100 Q
i=1
Ji Fxi
i=1
r
P
.
(5.51)
Ji
i=1
The total transportation cost per time unit corresponding to the LTL shipments
was obtained from Eq.(5.16),
r
P
d
·
Q
Ji · T Ci (Qw)
i=1
r
P
.
(5.52)
Ji
i=1
The procedure to determine the number of TL’s or the combination of TL and
LTL needed to ship orders from suppliers is outlined below:
1. For i = 1, . . . , r, calculate τi = (Fxi w)/100, and set p0i = pi + τi . Use these
p0i values in place of the suppliers’ unit prices and solve Problem (P4.3) to
obtain Q 0 and Ji0 .
119
2. Compute η = b(Q 0 w)/Wx c.
3. Solve Problem (P5.3) to obtain Q∗ and Ji∗ , i = 1, . . . , r, and calculate the
optimal number of units shipped by LTL, QLT L∗ = Q∗ − η(Wx /w).
5.5.7.1
Illustrative Example
In this section the procedure introduced above is illustrated. The same data as in
Section 5.5.6.1 are used, except that the demand rate is increased to d = 1, 000, 000
units/month and the capacities from suppliers are now: c1 =700,000 units/month,
c2 =800,000 units/month, and c3 = 750,000 units/month.
First, Problem (P4.3) is solved using p01 = $23.01/unit, p2 =0 $29.28/unit, and
p03 = $32.012/unit in place of p1 , p2 , and p3 , respectively. The order quantity obtained is Q 0 = 5,440 units and the number of orders allocated to each supplier is
J1 = 3 orders, J2 = 0 orders, and J3 = 2 orders. Next, using Q 0 , the smallest number of trucks where all items take advantage of economies of scale in transportation
is calculated using Eq. (5.35) as follows:
$
η=
5, 440 · 16
40, 000
%
$
%
= 2.1 = 2 trucks.
This implies that the optimal order quantity will be between 2 and 3 trucks.
The order quantity is therefore bounded as,
5, 000 ≤ Q ≤ 7, 500,
and the constraint on the maximum number of LTL units becomes,
Q − 5, 000.
Finally, Problem (P5.3) is solved. The optimal order quantity is Q∗ = 5, 000
units, and J1∗ = 3, J2∗ = 0, J3∗ = 2. This implies that only 2 TLs are needed to
120
transport the orders from suppliers 1 and 3 (QLT L∗ = 0). The corresponding costs
for this optimal solution are highlighted in Table 5.11. Additionally, this table
shows other feasible solutions that make use of LTL portions of a third truck. All
of the solutions use an order allocation of J1 = 3, J2 = 0, J3 = 2 and, therefore,
the purchasing and in-transit inventory costs are the same for all solutions and
have been omitted from the table.
Table 5.11. Analysis of Variable Costs ($/month)
Scenario
2
2
2
2
3
TL
TL+750 units
TL+1250 units
TL+1685 units
TL (2 TL+2500 units)
5.6
TL
Cost
LTL
Cost
Setup
Cost
Holding
Cost
Sum of
Var.Costs
Total
Cost
2,610,800.0
2,270,260.8
2,088,640.0
1,952,729.9
1,740,533.3
781,398.2
762,624.0
923,052.3
870,266.6
29,600.0
25,739.1
23,680.0
22,139.1
19,733.3
25,000.0
28,750.0
31,250.0
33,425.0
37,500.0
3,132,066.6
3,572,814.9
3,372,860.6
3,398,013.1
3,134,700.0
27,132,066.6
27,572,814.9
27,372,860.6
27,398,013.1
27,134,700.0
Transportation-Inclusive Models with
Different–Size Order Quantities
The advantage of the proposed models, which consider order quantities of the same
size (Q), is that they can be extended into multi-stage supply chain systems where
inventory coordination between stages is necessary in order to avoid shortages.
However, the fact that these models also consider transportation costs implies
that the fixed order quantity will force some suppliers to sacrifice economies of
scale and allow others to take advantage of them. Appendix A analyzes the case
where order quantities allocated to selected suppliers may be of different sizes, Qi .
121
5.7
Conclusions
In this chapter, the relevance of incorporating transportation costs into replenishment decisions has been highlighted. Problem (P4.3) from Chapter 4 was extended
to consider transportation and inventory costs simultaneously in the determination
of the order quantities to allocate to selected suppliers.
Under the assumption that shipments from suppliers are LTL the order quantities were assumed to be of the same size for all selected suppliers. Two continuous
functions were used to determine the actual freight rates from different suppliers.
Additionally, a model was presented considering actual freight rates in order to
provide optimal solutions. In this model, the transportation costs are represented
as a continuous piecewise linear function (of the weight shipped) using binary variables. The model was used to determine the effectiveness of continuous functions
in estimating transportation freight rates. It is worth noting that this model can
be easily extended to consider piecewise linear functions that are not continuous.
This is the case when the capacity of a TL is exceeded and an extra truck might be
needed. In such situations, there exists a sudden increase in total transportation
cost, which may make the function discontinuous.
The LTL assumptions were also extended to consider the case where more
than one TL might be needed to transport items from suppliers. A procedure
was provided to determine the number of TL’s or the combination of TL and LTL
needed to ship the orders from suppliers.
It has been shown that incorporating transportation costs into inventory decisions not only affects the order quantity shipped from selected suppliers but also
the actual selection of suppliers. This can produce a significant impact on supply
chain configurations.
Chapter
6
A Serial Inventory System with
Supplier Selection and Order
Quantity Allocation
6.1
Introduction
Chapters 4 and 5 addressed the supplier selection and the order quantity allocation
problem for a single-stage system. However, the focus will now shift to a multistage system where the inventory impacts the direct purchaser and the subsequent
stages of a supply chain. This is important when you consider that in today’s global
market, many factors are encouraging companies to gain a competitive advantage
by focusing attention on their entire supply chain.
Given the prevalence of both supplier selection and inventory control problems
in supply chain management, this chapter addresses these problems simultaneously
by analyzing a serial supply chain system that effectively ties these interrelated
issues together by incorporating the supplier selection into a multi-stage context.
First, the supplier selection problem is considered at Stage 1 of the serial supply
chain system. Problem (P4.3) is integrated into the multi-stage system and used to
123
determine the selection of suppliers and the order quantity to allocate to selected
suppliers. Next, the item procured at Stage 1 is assembled into a final product
that is moved throughout the serial supply chain system until it reaches the end
customer. For the inventory control problem, an inventory policy is developed to
determine the inventory held at each stage of the supply chain system to replenish
subsequent stages accordingly. The objective of the system is to coordinate the
inventory at the various stages to minimize the total cost associated with the entire
supply chain system while selecting a set of suppliers which best meet the capacity
and quality required by Stage 1.
The remainder of this chapter is organized as follows. In Section 6.2, the
assumptions of the system under consideration are stated. In Section 6.3, the
development of the proposed multi-echelon inventory model is presented and a
power-of-two inventory policy for the system is introduced. To prove the effectiveness of this policy, a lower bound on the optimal total cost per time unit is
obtained. In Section 6.4 an illustrative example is provided to show the application of both the proposed mathematical model and the power-of-two procedure.
Finally, some conclusions are summarized in Section 6.5.
6.2
Problem Description and Assumptions
In a serial system, raw materials and products flow sequentially through a chain of
stages to satisfy the demand of a customer. It is assumed that the entire system
belongs to a single firm. Hence, inventory decisions are made by a single decision
maker (e.g. centralized control) whose objective is to minimize the total cost of the
system. Stage 1 has its inventory replenished periodically from a set of selected
suppliers. Stages 2 through N replenish their inventory from their immediate
predecessor. Demand occurs at Stage N at a constant rate per time unit and must
be met without shortages. In this way, the supplier selection only occurs at the
124
first stage in the system. Therefore, purchasing costs are only incurred at Stage
1 while the product is transferred internally through the company in subsequent
stages. Figure 6.1 depicts the system under consideration for three stages.
Supplier
1
Supplier
2
…
Manufacturer
Assembler
Distribution
Center
Stage 1
Stage 2
Stage 3
Retailer
Supplier
r
Figure 6.1. Serial System with Three Stages and Multiple Potential Suppliers
Some or all the stages of the system might be processing centers that process
the items received from the preceding stage and transform them into something
closer to the finished product. Stages are also used to store items until they are
ready to be moved to the next processing center or the next storage facility bringing
the item closer to the customer. Finally, Stage N might perform any needed final
processing and/or store the final product at a location where it can be immediately
used to meet the demand for that product on a continuous basis.
As an item is being processed into something closer to the end product, the
item can be referred to as item 1 at Stage 1, item 2 at Stage 2, and so on. A
unit of item 1 at Stage 1 will result in one unit of item 2 at Stage 2, and so on.
For example, consider the case in Figure 6.1. Suppose Stage 1 is a manufacturing
facility producing item 1, which in turn requires raw materials for processing.
These raw materials are procured from different suppliers. The final item at Stage
1 is then used to replenish Stage 2 (assembler). In Stage 2 some final assembly is
performed before the item is used to replenish Stage 3 (distribution center). The
distribution center satisfies the demand of the end product.
125
Under the above assumptions, an inventory policy based on the so-called echelon inventory is an effective way to manage the system. Hillier and Lieberman [135]
define an echelon of an inventory system as “each stage at which inventory is held
in the progression through a multi-stage inventory system”.
In addition to determining the inventory policy for all the stages involved in
the system, supplier selection takes place at the first stage of the system. This
allows Stage 1 to replenish its inventory from different suppliers by performing a
selection process whose output is the number of orders that are to be procured
from selected suppliers, the size of these orders, and the frequency with which
orders are to be received. In particular, Problem (P4.3) from Chapter 4 is used
to perform the supplier selection and order quantity allocation at Stage 1. Recall
that Problem (P4.3) considers different criteria into the analysis, namely price,
capacity, and quality.
The objective for the proposed model is to coordinate the inventory from the
point of supply to the point of consumption to minimize the total cost per time
unit associated with the entire system while also allocating orders to selected suppliers at Stage 1. According to Hillier and Lieberman [135], serial supply chain
systems might often lead to developing partnership relationships with suppliers as
well as mutually beneficial supply contracts that enable reducing the total cost of
operating a jointly managed multi-echelon inventory system.
6.3
Multi-Stage Serial Inventory Model
In this section the proposed multi-stage serial inventory model with supplier selection at the first stage is derived. Problem (P4.3) from Chapter 4 is integrated
into the multi-stage system to allow Stage 1 to select the proper set of suppliers
while allocating their corresponding order quantities over time. The notation used
throughout this chapter is as follows:
126
Data
N – number of stages
r – number of available suppliers
d – demand per time unit
k1i – setup cost for placing an order to ith supplier at Stage 1, for i = 1, . . . , r
kj – setup cost at Stage j, for j = 2, . . . , N
pi – unit price of ith supplier, for i = 1, . . . , r
ci – capacity of ith supplier per time unit
qi – perfect rate of ith supplier
qa – minimum acceptable perfect rate of parts
hj – conventional (unit) holding cost per time unit at Stage j, for j = 1, . . . , N
Variables
Ji – number of orders of ith supplier per order cycle at Stage 1
Qj – order quantity at Stage j, j = 1, . . . , N
T – time between consecutive orders
Tc – (repeating) order cycle time
Note that the order quantity at Stage 1, Q1 , is equivalent to Q in Problem
(P4.3).
The assumptions of the EOQ model hold at Stage N and transfer times of
items between stages are instantaneous (e.g. leadtimes are assumed to be zero).
Units increase in value as they move forward in the supply chain (e.g. each time
they reach the next stage, replenishment and processing costs increase the value
of the units); thus, h1 < h2 < . . . < hN . The echelon (unit) holding cost (ej ) is
defined as the increase in unit holding cost between stages j − 1 and j. That is,
e1 = h1 , e2 = h2 − h1 , e3 = h3 − h2 , . . . , eN = hN − hN −1 . Our calculations utilize
the echelon holding cost instead of the conventional holding costs.
Before developing the model, Figure 6.2 is used to discuss why echelon holding
127
costs are used rather than conventional holding costs and how the inventory levels need to be coordinated in a multi-echelon inventory system in order to avoid
shortages in the system. Figure 6.2 depicts the inventory levels for a serial system
where N = 3.
Inventory Levels
at Installation 1
Q1
Echelon Inventory
Stage Inventory
Q1 - Q2
Q1 - 2Q2
T1
T1
Time
Stage Inventory
Echelon Inventory
Inventory Levels
at Installation 2
Q2
Q2 – Q3
T2
T2
T2
Time
Inventory Levels
at Installation 3
Stage Inventory = Echelon Inventory
Q3
T3
T3
T3
T3 T3
T3
Time
Figure 6.2. Synchronized Inventory Levels at Three Stages
The shaded areas represent the actual inventory levels at each stage. First,
consider the stage inventory. The graph for Stage 3 is the usual saw-tooth pattern.
However, the actual stage inventory at Stage 2 is not of this form. In fact, the
128
inventory at Stage 2 is a function of both Q2 and Q3 , whereas the average inventory
at Stage 3 is always 12 Q3 , independent of the choice of Q2 . The fact that the average
inventory at Stage 2 depends on both Q2 and Q3 makes it undesirable to calculate
the holding costs using the on-hand inventories.
Now consider the echelon inventories. The echelon inventory for Stage 2 consists
of the inventory on hand at Stage 2 plus the amount on hand at Stage 3. When
echelon inventories are considered, the graph of echelon inventory in both stages
follows a saw-tooth pattern. More importantly, the average echelon inventory level
at Stage 2 depends only on Q2 . Similarly, the average echelon inventory level at
Stage 3 depends only on Q3 . Thus, if inventory costs are charged proportional to
echelon inventory levels, using the corresponding echelon holding cost, then the
average costs are e2 · 12 Q2 and e3 · 12 Q3 , for stages 2 and 3, respectively. The same
logic applies to the echelon inventory levels at Stage 1. It can easily be shown that
the total inventory holding cost per time unit is the same whether using echelon or
actual stage inventories. Using echelon inventories to calculate the total holding
cost per time unit, however, simplifies the analysis. It is important to point out also
that the order quantities at Stages 1 and 2 are multiples of each other (Q1 = 3Q2 ),
and so are the order quantities at installations 2 and 3 (Q2 = 2Q3 ).
Schwarz and Schrage [117], and Love [118] introduced the first results for a serial
system. They proved that for an N -stage serial system, an optimal policy must be
nested and inventory replenished only when the inventory level is zero. A policy is
said to be nested provided that, if a stage orders at a given time, every downstream
stage must order at this time as well, i.e., Qj = nj Q(j+1) , j = 1, . . . , N −1, where nj
is a fixed positive integer (as in the case shown in Figure 6.2); otherwise, shortages
might occur. The zero-inventory ordering property implies that Stage j should
replenish its inventory (Qj units) only when its inventory level drops to zero and
it is time to supply Stage j + 1 with an order quantity of size Qj+1 .
In the system under consideration, although multiple suppliers may be used to
129
replenish the inventory in Stage 1, only one supplier at a time is asked to replenish
it (e.g. no order splitting is considered every time Stage 1 places an order).
The total cost of the system per time unit (ZN ) includes holding, setup, and
purchasing costs. Because of the nested and zero-inventory ordering properties,
the system under consideration can be formulated as an extension of the serial
system formulated by Schwarz and Schrage [117], except that the supplier selection
problem for Stage 1 is included:
(P6.1)
minimize
r
N
N
X
X
d
1 X
kj
ZN =
+ ·
·
Ji k1i + d ·
ej Qj
Q1 M i=1
Q
2
j
j=2
j=1
r
d X
+
·
Ji pi ,
M i=1
subject to
dJi ≤ ci M,
r
X
i = 1, . . . , r,
(6.1)
Ji qi ≥ qa M,
(6.2)
Ji = M,
(6.3)
i=1
r
X
i=1
Qj = nj Q(j+1) , j = 1, . . . , N − 1,
(6.4)
nj ≥ 1, integer, j = 1, . . . , N − 1,
(6.5)
Qj ≥ 0, j = 1, . . . , N,
(6.6)
Ji ≥ 0, integer, i = 1, . . . , r,
(6.7)
M ≥ 1, integer,
(6.8)
where the first term in the objective function represents the setup cost at Stage
1. The second term represents the setup cost for installations j = 2, . . . , N .
The third term accounts for the total holding cost per time unit. The last term
corresponds to the purchasing cost incurred for all the units purchased at Stage
1. Similar to Problem (P4.3), an upper bound on M can be added or M can be
fixed to a small integer value. From a mathematical standpoint, the advantages of
130
doing so are the same as the ones discussed for Problem (P4.3).
Multi-stage inventory problems, like Problem (P6.1), become surprisingly difficult as the number of stages increases. Because of this complexity, many researchers have proposed heuristics that can be shown to be effective with respect
to theoretical lower bounds. For example, power-of-two policies have been used
as a practical approach to determine inventory policies in supply chain systems
(Roundy [112], [114], and Muckstadt and Roundy [119]). The implementation
of such policies is simple and computationally efficient. Section 6.3.1 presents a
power-of-two policy for the N -stage serial system under consideration. The effectiveness of this policy is calculated with respect to a theoretical lower bound.
A lower bound for Problem (P6.1) can be obtained by dropping the coordination
requirement. The following relaxation Problem (R6.1) is obtained,
(R6.1)
minimize
r
X
d
ZN =
·
Ji k1i +
Q1 M i=1
|
stage 1
+
N
N
X
1 X
kj
·
ej Qj + d ·
,
2 j=2
Q
j
j=2
|
{z
}
stages 2,...,N
subject to
dJi ≤ ci M,
r
X
i = 1, . . . , r,
Ji qi ≥ qa M,
i=1
r
X
Ji = M,
i=1
Qj ≥ Q(j+1) , j = 1, . . . , N,
Qj ≥ 0, j = 1, . . . , N,
Ji ≥ 0, integer, i = 1, . . . , r,
M ≥ 1, integer.
r
1
d X
e1 Q1 +
·
J i pi
2
M i=1
{z
}
131
By replacing constraint (6.4) by Qj ≥ Qj+1 one can obtain an easily computed
lower bound on the cost of the optimal solution. To obtain a lower bound from
Problem (R6.1), the problem can be separated in stages. In this way, a solution can
be obtained by solving multiple independent subproblems, one for every stage: the
order quantity and the supplier order allocation for Stage 1 are obtained by solving
Problem (P4.3), and the order quantities for Stages 2 through N are obtained
by applying the standard EOQ formula at each stage. If after computing the
corresponding Qj values all constraints Qj ≥ Qj+1 , for j = 1, . . . , N − 1, are
satisfied, then such a solution is optimal for the relaxed Problem (R6.1) and its cost
represents a lower bound on the cost of the optimal solution for Problem (P6.1).
Otherwise, we know that in any feasible solution Qj must be at least as large as
Qj+1 . Therefore, if any Qj < Qj+1 , for any j = 1, . . . , N − 1, then the violated
constraint is forced to hold at equality. This is done by collapsing Stages j and j +1
into a single stage with setup cost kj +kj+1 and an echelon holding cost of ej +ej+1 .
The logic of this rule is derived from the EOQ formula. In particular, the only
non-constant terms in the EOQ formula are kj and ej . Hence, if kj /ej ≤ kj+1 /ej+1 ,
then based on Qj ≥ Qj+1 , the minimum required order quantity from Stage j to
be able to meet the required order quantity by Stage j + 1 is Qj = Qj+1 , in which
case, as mentioned above, both stages are collapsed into one stage. Schwarz and
Schrage [117], and Muckstadt and Roundy [119] applied this reduction rule to a
similar serial inventory model.
6.3.1
Power-of-Two Inventory Policy
The proposed algorithm can be used to find a power-of-two inventory policy for
Problem (P6.1). The assumption is that Qj = nj Qj+1 , nj = 2m , j = 1, . . . , N − 1,
where m is a fixed positive integer; so the values of nj are 1, 2, 4, 8, . . .. This assumption reduces the feasible region of Problem (P6.1). Additionally, Roundy [114] has
shown that an optimal solution using the power-of-two approximation is nearly
132
optimal for the original problem. In particular, he proved that the amount by
which the cost of an optimal solution for the approximation exceeds the cost of an
optimal solution for the original problem is within 2%.
The proposed algorithm consists of four main steps. In the first step, the
solution to the relaxed Problem (R6.1) is found. Using the setup costs kj and the
echelon costs ej , for j = 1, . . . , N , the special case when consecutive stages may be
merged is checked. Note that k1 represents the weighted average setup costs for
Stage 1 and is computed using the Ji values obtained from solving Problem (P4.3)
as follows,
r
P
k1 =
Ji k1i
i=1
r
P
.
(6.9)
Ji
i=1
Therefore, Step 1 begins by solving Problem (P4.3) to obtain the Ji values that
are used in computing k1 . Then, the EOQ formula is used to obtain the order
quantity at each stage.
The second step consists of coordinating the solutions of the relaxed Problem
(R6.1) to fit the assumption Qj = nj Qj+1 , nj = 2m , j = 1, . . . , N − 1, where m is a
fixed positive integer. Let QR
j , j = 1, . . . , N , be the solution of the relaxed problem
, j = 1, . . . , N , be a power-of-two solution. Hillier and Lieberman [135]
and QPOT
j
summarized Roundy’s [114] procedure to obtain a power-of-two solution from the
solutions of the relaxed problem. The procedure initially uses the value QPOT
to
N
determine the power-of-two solution for Stage N − 1, QPOT
N −1 . Since the value of
POT
QPOT
is initially unknown, the value QR
N
N is used as an approximation of QN .
The value of QPOT
to be used in subsequent iterations of our proposed algorithm
N
is derived later on (Eq. (6.11)). Once QPOT
N −1 is set, its value is used to determine
the power-of-two solution for Stage N − 2, QPOT
N −2 . The same procedure is repeated
until power-of-two solutions are determined for all the stages. Roundy’s procedure
is presented next:
133
Roundy’s Power-of-Two Rounding Procedure
1. If QPOT
is known, go to Step 2. Otherwise, set QPOT
to QR
N
N
N and go to Step 2.
2. For j = N − 1, N − 2, . . . , 1, determine the nonnegative value of m such that
R
m+1 POT
2m QPOT
Qj+1 . Go to Step 3.
j+1 ≤ Qj ≤ 2
2m+1 QPOT
QR
j+1
j
= nj QPOT
, set nj = 2m and QPOT
3. If m POT ≤
j+1 . Otherwise, set
j
R
2 Qj+1
Qj
nj = 2m+1 and QPOT
= nj QPOT
j
j+1 .
Once the QPOT
values have been determined according to the above procedure,
j
just obtained
Problem (P4.3) is resolved with the value of Q fixed to the value QPOT
1
from the rounding procedure. This is done mainly to check whether the order
allocation to suppliers, namely Ji , i = 1, . . . , r, has changed with respect to the
original set of Ji values obtained in the solution of the relaxed problem in the first
step.
After tentatively determining the nj and Ji values, the third step is to refine the
value of QN to attempt to obtain an overall optimal solution for Problem (P6.1).
The refined value is used as an input to the Roundy’s power-of-two rounding procedure (Step 2) where each Qj is expressed as a power-of-two of the refined QN value.
Therefore, we are interested in finding the value of QN that minimizes the total
cost per time unit (ZN ) when the order quantity at each stage is expressed in terms
of QN . By replacing Qj by sj QN in Problem (P6.1), where sj = nj nj+1 · · · nN −1 ,
for j = 1, . . . , N − 1, and SN = 1, the total cost per time unit becomes,

r
P

Ji k1i X
N
N
r
d 
QN X
kj 
d X
i=1


ZN =
+
+
·
ej sj +
·
Ji pi .
QN  s1 M
s 
2 j=1
M i=1
j=2 j
(6.10)
Hence, the value of QN that minimizes ZN is found by taking the first derivative
134
of Eq. (6.10) with respect to QN , setting it to zero, and solving for QN . This yields,
v 

u
r
P
u
u  Ji k1i
N k 
P
u  i=1
j
u 2d
+

u  s1 M
j=2 sj
u
u
Q∗N = u
.
u
N
P
t
e j sj
(6.11)
j=1
In the proposed algorithm QPOT
= Q∗N . Note that because this expression
N
requires knowing the nj values, the QR
N obtained from the solution to the relaxed
problem is initially used as an approximation of QPOT
in the Roundy’s power-of-two
N
rounding procedure introduced before.
Recall that the weighted average setup cost in Stage 1 is k1 =
Pr
i=1
Ji k1i /M .
Therefore, Eq. (6.11) can be rewritten as,
v
u
N k
P
u
j
u 2d ·
u
j=1 sj
.
Q∗N = u
u P
t N
e j sj
(6.12)
j=1
In the fourth step, the algorithm is terminated as the power-of-two inventory
policy has been determined. This implies that any of the nj or Ji values have
changed. Consequently, the QN value cannot be further refined.
Algorithm for finding a power-of-two inventory policy
1. Merge stages as necessary and solve the relaxed Problem (R6.1):
(a) Solve Problem (P4.3) to obtain Ji , for i = 1, . . . , r, and compute k1 using
Eq. (6.9).
(b) If kj /ej ≤ kj+1 /ej+1 for any j = 1, . . . , N − 1, collapse Stages j and
j + 1 with a setup cost of kj + kj+1 and an echelon cost of ej + ej+1 .
Renumber stages and reset the value of N accordingly.
135
(c) Set QR
j =
p
2kj d/ej , for j = 1, . . . , N.
(d) Compute the total cost per time unit for the relaxed problem,
ZNR
=d·
N
X
j=1
1
kj
+ ·
R
2
Qj
N
X
r
P
ej QR
j
+d·
j=1
Ji pi
i=1
r
P
.
Ji
i=1
2. Obtain a power-of-two solution to Problem (P6.1):
and nj using the Roundy’s power-of-two rounding
(a) Determine QPOT
j
procedure introduced earlier.
obtained in Step
(b) Solve Problem (P4.3) with Q fixed to the value QPOT
1
2a to obtain Ji , i = 1, . . . , r.
(c) If none of the nj (Step 2a) and Ji (Step 2b) values change (with respect
to the Ji values obtained in Step 1a), go to Step 4a. Otherwise, go to
Step 3a.
3. Refine the order quantity QN :
(a) Use the nj values obtained in Step 2a and compute sj = nj nj+1 · · · nN −1 ,
for j = 1, . . . , N − 1, SN = 1.
(b) Use the sj values obtained in Step 3a and the Ji values obtained in Step
2b and compute QPOT
using Eq. (6.12). Go to Step 2a.
N
4. Terminate the algorithm with the power-of-two inventory policy for Problem
(P6.1):
(a) The power-of-two inventory policy is (QPOT
, QPOT
, . . . , QPOT
1
2
N ) and, from
Step 2b, the order allocation at Stage 1 is (J1 , J2 , . . . Jr ).
136
(b) Calculate the corresponding total cost per time unit for the policy generated:
POT
ZN
=
d
QPOT
N
·
N
X
j=1
kj QPOT
+ N ·
sj
2
N
X
j=1
r
P
e j sj + d ·
Ji pi
i=1
r
P
.
Ji
i=1
The following observations about the proposed algorithm are in order. First,
the proposed algorithm is only guaranteed to converge, avoiding infinite loops from
P
Steps 2b and 3b, if an implicit upper bound on the Ji values, ri=1 Ji ≤ M , is added
to Problem (P4.3).
Second, because the relaxed Problem (R6.1) does not require any inventory
coordination, the cost that is calculated for its optimal solution (ZNR ), is a lower
bound on the cost of the optimal solution obtained by solving Problem (P6.1), say
ZN∗ . Since solving Problem (P6.1) becomes difficult as the number of stages (N )
increases, the total cost corresponding to the power-of-two policy (ZNPOT ) gives a
conservative estimate of how close ZNPOT must be to ZN∗ . That is, ZNR ≤ ZN∗ ≤ ZNPOT ,
which implies ZNPOT − ZN∗ ≤ ZNPOT − ZNR .
Third, although the solution obtained using this algorithm is not guaranteed
to be optimal, it provides a solution that is close to the optimal solution. Since
the power-of-two policy is an approximation of the original Problem (P6.1), the
solution obtained is adequate for practical purposes.
Fourth, setting QPOT
to the value QR
N
N obtained from the solution to the relaxed
problem and determining a power-of-two policy using Roundy’s procedure provides
a policy that is within 6% of the lower bound computed by solving the relaxed
problem (Muckstadt and Roundy [119]). That is, ZNPOT − ZNR ≤ 0.06 · ZNR . In
addition, refining the value QN , as in Step 3, guarantees a policy that is within
2% of the lower bound.
137
6.4
Illustrative Example and Analysis
Consider a serial system with four stages. Setup and unit holding costs for each
stage are shown in Table 6.1.
Table 6.1. Data for Stages
Stage
j
Setup Cost (kj )
($)
Holding Cost (hj )
($/unit/month)
1
2
3
4
9,000
4,000
1,500
8.00
12.50
28.90
59.20
Stage 4 expects to sell end products at a rate of d = 10, 000 units/month. Raw
materials needed at Stage 1 (e.g., manufacturer) are to be procured from different
suppliers. The manufacturer wants to maintain a minimum perfect rate qa = 0.95
of the materials procured from the selected suppliers. Table 6.2 shows additional
data for the potential suppliers.
Table 6.2. Suppliers’ Data
Supplier
i
Price (pi )
($)
Setup Cost (k1i )
($)
Perfect Rate (qi )
Capacity (ci )
(units/month)
1
2
3
28
40
46
2,900
3,500
1,500
0.96
0.93
0.97
4,400
7,000
6,600
From the data, the unit echelon costs can be computed as follows: e1 = h1 =
$8/unit/month, e2 = h2 −h1 = $4.5/unit/month, e3 = h3 −h2 = $16.4/unit/month,
and e4 = h4 − h3 = $30.3/unit/month.
The purpose of the numerical example is to calculate the inventory policy for the
4-echelon serial system using both the proposed Problem (P6.1) and the proposed
algorithm for finding a power-of-two inventory policy. The optimal solution for
138
Problem (P6.1) was obtained using LINGO [129]. The effectiveness of the power-oftwo policy is calculated by comparing its total cost to that of the optimal solution
and also to the cost corresponding to the lower bound (solution to the relaxed
problem).
First, Problem (P6.1) is solved. Table 6.3 shows the optimal inventory policy
for all stages and the supplier order allocation for Stage 1 along with the total cost
per time unit for this optimal policy.
Table 6.3. Optimal Solution to Problem (P6.1)
Stage j
sj
Q∗j (units)
1
4
4,280
2
3
4
4
2
1
4,280
2,140
1,070
Order Allocation
M = 100 (J1 = 44, J2 = 39, and J3 = 17)
Tc = 42.8 months (3.5 years)
∗
= 478,413.10
Total Cost ($/month) ZN
In this particular example, the optimal M value results in a very long order
cycle for Stage 1 (42.8 months). As previously discussed, one of the advantages
of Problem (P6.1) is that the number of orders allowed within an order cycle can
be controlled so as to shorten the entire order cycle at Stage 1. To do so, M is
fixed to different integer values in Problem (P6.1). In order to illustrate this idea,
Table 6.4 shows detailed solutions for M =2 to 25. These solutions include the
order quantity assigned to Stages 1 to 4. In addition, the optimal order allocation
to selected suppliers at Stage 1, the size of these orders, and the corresponding
length of the order cycle are also included. Finally, the total cost per time unit
for each solution is given along with its percentage deviation from the absolute
optimal solution at M = 100 (ZN∗ = $478,413/month).
Notice in Table 6.4, if M is set to 5, the length of the order cycle at Stage 1
is reduced from 42.8 months (optimal solution) to 2.14 months, and the increase
139
Table 6.4. Coordinated Inventory Policy for Different Values of M
M
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
J1
0
1
1
2
2
3
3
3
4
4
5
5
6
6
7
7
7
8
8
9
9
10
10
11
J2
1
1
1
2
2
2
3
3
4
4
4
5
5
6
6
6
7
7
8
8
8
9
9
9
J3
1
1
2
1
2
2
2
3
2
3
3
3
3
3
3
4
4
4
4
4
5
4
5
5
Order Cycle’s
Length at Stage 1
Order Quantity at Stage j
(units)
ZN
($/month)
(months)
Q1
Q2
Q3
Q4
0.85
1.28
1.69
2.14
2.55
2.98
3.42
3.83
4.28
4.69
5.12
5.55
5.98
6.42
6.84
7.26
7.69
8.12
8.55
8.98
9.40
9.84
10.26
10.69
4,247.01
4,258.10
4,234.51
4,276.89
4,258.10
4,261.27
4,269.85
4,258.10
4,276.89
4,266.65
4,267.78
4,272.56
4,273.11
4,276.89
4,277.10
4,270.46
4,273.76
4,274.10
4,276.89
4,277.05
4,271.92
4,279.48
4,274.68
4,274.90
4,247.01
4,258.10
4,234.51
4,276.89
4,258.10
4,261.27
4,269.85
4,258.10
4,276.89
4,266.65
4,267.78
4,272.56
4,273.11
4,276.89
4,277.10
4,270.46
4,273.76
4,274.10
4,276.89
4,277.05
4,271.92
4,279.48
4,274.68
4,274.90
2,123.51
2,129.05
2,117.25
2,138.44
2,129.05
2,130.63
2,134.93
2,129.05
2,138.44
2,133.33
2,133.89
2,136.28
2,136.55
2,138.44
2,138.55
2,135.23
2,136.88
2,137.05
2,138.44
2,138.52
2,135.96
2,139.74
2,137.34
2,137.45
1,061.75
1,064.53
1,058.63
1,069.22
1,064.53
1,065.32
1,067.46
1,064.53
1,069.22
1,066.66
1,066.95
1,068.14
1,068.28
1,069.22
1,069.27
1,067.62
1,068.44
1,068.53
1,069.22
1,069.26
1,067.98
1,069.87
1,068.67
1,068.73
$550,084.35
$500,397.88
$519,730.64
$484,929.03
$500,397.88
$486,201.60
$490,730.12
$500,397.88
$484,929.03
$493,366.87
$485,671.56
$488,498.97
$482,250.67
$484,929.03
$479,684.87
$485,453.22
$487,507.33
$482,955.51
$484,929.03
$480,933.48
$485,334.11
$479,263.07
$483,366.66
$480,072.90
in total cost with respect to the absolute optimal solution is only about 1%. This
indicates that a company may restrict M to a reasonable small value to reduce
the length of Stage 1’s order cycle without considerably increasing the total cost
of the solution in comparison to that of the optimal solution.
Next, the power-of-two inventory policy is calculated using the proposed algorithm. After solving Problem (P4.3), as indicated in Step 1a, the following order
allocation is obtained: J1 = 44, J2 = 39, and J3 = 17. These values are used to
compute the weighted average setup cost for Stage 1 using Eq. (6.9), k1 = $2, 896.
Then, since k1 /e1 < k2 /e2 , Stages 1 and 2 are treated as a single merged stage.
The conditions k2 /e2 ≤ k3 /e3 and k3 /e3 ≤ k4 /e4 do not hold. For this reason, all
other stages are treated as separate stages. Table 6.5 shows the data generated
after combining Stages 1 and 2.
Table 6.6 summarizes the results from applying the solution procedure.
140
Table 6.5. Adjusted Data for Stages
Stage
j
Setup Cost (kj )
($)
Echelon Cost (ej )
($/unit/month)
1+2
3
4
11,896
4,000
1,500
12.5
16.4
30.3
Table 6.6. Details of the Proposed Algorithm
Stage
Solution of
Relaxed Problem
Initial Solution of
Revised Problem
Final Solution of
Revised Problem
j
QR
j
(units)
Cost
($/month)
QPOT
j
(units)
Cost
($/month)
QPOT
j
(units)
Cost
($/month)
1+2
3
4
4,362
2,209
995
411,934.39
36,221.43
30,149.63
3,980
1,990
995
412,164.25
36,418.35
30,149.63
4,276
2,138
1,069
411,945.39
36,240.67
30,227.16
R
=478,305.45
ZN
ZN =478,732.23
POT
=478,413.22
ZN
The last two columns of the table summarize the results from completing the
algorithm. The refined QPOT
is calculated using n1+2 = 2, n3 = 2, J1 = 44,
4
J2 = 39, and J3 = 17. Repeating Step 2 of the proposed algorithm with this
= 1, 069 units) again yields the same nj and Ji values as above.
new value (QPOT
4
Because these values do not change, QPOT
cannot be further refined and the desired
4
power-of-two inventory policy has been obtained. The final Ji values correspond
to the original Ji values obtained in the solution of Problem (P4.3) in Step 1 of
the algorithm.
Notice that an inventory policy using the solutions of the relaxed problem
would lead to inventory shortages due to lack of coordination between the order
quantities. This is illustrated in Figure 6.3.
In Table 6.6, the total cost ZN =$478,732.23/month is 0.089% above the total
cost of the relaxed problem (ZNR ). The refinement procedure improves the cost
to ZNPOT =$478,413.22/month, only 0.023% above the lower bound ZNR . The rel-
141
Inventory Levels
at Installation 1+2
Q1+2=4,362
Q1+2
1 2-Q
Q3=2,153
=2 153
Q1+2-2*Q3=-56
Time
T1+2
Inventory Levels
att IInstallation
t ll ti
3
Q3= 2,209
Q3-Q4=1,214
Q3-2*Q4=219
Time
Q3-3*Q4=-776
T3
Inventory Levels
at Installation 4
Q4= 995
T4
T4
T4
T4
Time
Figure 6.3. Inventory Levels for All Stages (Separate Inventory Policies)
evance of this result is that if the optimal solution was unknown, the total cost
corresponding to the power-of-two policy (ZNPOT ) would be within 0.023% of the
optimal cost (ZN∗ ). In this particular case, ZNPOT is practically the same as ZN∗ .
142
6.5
Conclusions
The importance of supply chain management in today’s competitive environment
forces companies to focus their attention on the study and analysis of inventory
policies inclusive of their entire supply chain systems, rather than solving separate inventory policies for every stage involved in the system. In this chapter, a
model that ties both supplier selection and inventory control problems together
in a supply chain system is proposed. In particular, a serial multi-stage system is
analyzed. The first stage of the system performs a selection of potential suppliers
to replenish the necessary inventory of items for Stage 1. The inventory is then
moved throughout the serial system. An important consideration is that items
increase in value as they are moved from one stage to another. The reason is that
replenishment and processing costs are incurred every time the item is transferred.
A mixed integer nonlinear programming model is proposed to determine the
optimal inventory policy that coordinates the different stages of the serial system,
while properly allocating orders to selected suppliers in Stage 1, with a minimum
total cost per time unit.
Since Problem (P4.3) from Chapter 4 is integrated to perform the supplier selection and order quantity allocation at Stage 1, the proposed model in this chapter
also considers three different criteria into the analysis, namely price, capacity, and
quality. Since it becomes more difficult to solve the model as the number of stages
involved in the system increases, a lower bound on the optimal total cost per time
unit is obtained and a 98% effective algorithm to obtain a power-of-two inventory
policy is proposed. This policy is derived from the solution that is optimal for the
relaxed problem. The advantage of this policy is that it is easy to compute and
yield near optimal solutions.
A numerical example shows the application of both the proposed mathematical
model and the proposed algorithm. A cost comparison shows that the power-oftwo inventory policy provided by the proposed algorithm is only 0.023% above the
143
total cost given by the lower bound. The relevance of this result is that if the
optimal solution was unknown, the total cost provided by the power-of-two policy
would be known to be within 0.023% of the total cost of the optimal solution.
Chapter
7
Conclusions and Future Research
7.1
Conclusions
This research addresses the strategic importance of supplier selection and order
quantity allocation, emphasizing the impact of such decisions on the different stages
comprising a supply chain.
The methodology proposed in the first part of this research integrates all steps
in the supplier selection process. This methodology consists of three phases. Phase
1 offers an easy way to pre-screen a large of potential suppliers to a manageable
number using the ideal solution approach. Phase 2 further analyzes the remaining suppliers from the pre-screening process by means of AHP. The advantage is
that quantitative and qualitative criteria can be incorporated in the analysis. In
Phase 3, managers can evaluate the impact of changing business conditions and
obtain the proper allocation of demand to each selected supplier by means of goal
programming. In this final phase, multiple scenarios can be analyzed by setting
different priorities to different criteria.
The proposed order allocation model used in the third phase of the proposed
methodology is characterized as one-time decision. This is equivalent to solving a
single-period problem where no inventory management over time is considered. For
145
this reason, the second part of this research considers the importance of inventory
management in determining the optimal order quantity from selected suppliers.
Three mixed integer nonlinear programming models are proposed to obtain optimal inventory policies that simultaneously determine how much, how often, and
from which suppliers to order. The mathematical models minimize the setup,
holding, and purchasing costs per time unit under suppliers’ capacity and quality
constraints. The first model allows independent order quantities for each supplier
while the second and third models restrict all order quantities to be of equal size.
A closed-form solution is derived for the third model to determine the optimal
inventory policy for the case when two potential suppliers are considered. In the
proposed mathematical models, sometimes the optimal value of orders allowed
within an order cycle (M ) that minimizes the total cost per time unit may result
in a large cycle time. A practical solution to this problem is that a company can
restrict M to a reasonable small value. In doing so the models are simplified (easier
to solve) and the length of the order cycle can be shortened. It is shown that by
doing so, the increase in total cost per time unit can be justified by the advantages of having a shorter cycle time. Having a short cycle time offers the following
advantages: facilitates the interaction with suppliers, simplifies the inventory control process, allows companies to evaluate suppliers’ performance periodically, and
allows companies to incorporate product life considerations of the procured items.
Although the transportation cost has seldom been considered in the supplier
selection literature, it has been shown that incorporating inventory and transportation costs simultaneously is essential in achieving absolute optimal solutions to the
problem of supplier selection and order quantity allocation. Under the assumption
that shipments from selected suppliers are less-than-truckload (LTL), approximate
and optimal policies are derived. To derive approximate policies, LTL transportation freight rates are modeled using continuous functions. Optimal policies are
derived by modeling actual LTL transportation costs as a continuous piecewise
146
linear function using binary variables. In this way, the effectiveness of continuous functions to estimate LTL transportation freight rates is studied. The use of
continuous functions is recommended when the number of potential suppliers is
extremely large or when no optimization software is available to solve the mathematical formulation that provides the optimal solution. A solution procedure is
proposed to study the case involving multiple TL or a combination of TL and LTL.
This procedure first finds the smallest number of trucks where all items would take
advantage of the largest discount in transportation, η, (e.g., the discount of shipping η full TL’s). Then, it finds the optimal order quantity that is between η and
η + 1 trucks by solving a mathematical model.
Finally, a serial supply chain system is studied. A model is developed to determine an optimal inventory policy that coordinates the different stages of the
system while properly allocating orders to selected suppliers in Stage 1. A lower
bound on the optimal total cost per time unit is found and a 98% effective powerof-two inventory policy is derived for the system under consideration. This policy
is advantageous since it is simple to compute and yields near optimal solutions. A
numerical example shows that this policy has a total cost that is at most 0.023%
above the optimal inventory policy.
An advantage of the models proposed in this research is that they can easily be
solved using commercial optimization software, such as LINGO [129] and GAMS [130].
Since the models are computationally efficient, a decision maker (e.g. purchasing
manager) can easily implement these models to compute the optimal order quantity
allocation for selected suppliers. The proposed methodology in Chapter 3 and the
optimization models in Chapters 4–6 can be directly applied to industries, such as
automotive, for which a single-sourcing strategy is not often suitable because of
the large number of parts and products purchased.
The following section provides future directions resulting from this research.
147
7.2
Future Research
In this research, deterministic demand and leadtimes have been assumed. In addition to a wide variety of deterministic extensions to this research, future research
would include extending the proposed models to the case of stochastic demand and
leadtimes. This would give a more accurate approach to real world environments
in which uncertainty is always present.
A direct extension to the methodology proposed in Chapter 3 would be to consider multiple items. Additionally, since the methodology has been proposed as
a single-period decision model, it can be easily implemented in an e-procurement
environment. An e-procurement system would include the following general steps:
at the beginning of each procurement period, the buyer identifies the decision criteria, specifies the required levels for these criteria, and solicits bids from suppliers.
Then, the buyer evaluates and selects winning bids using the proposed methodology.
The models proposed in Chapter 4 consider a deterministic demand over time
for an infinite planning horizon. A direct extension would be to analyze the case
of known variable demand in N periods, as in the dynamic inventory problem.
This problem would be characterized by a sequence of supplier selection and order
quantity allocation decisions at every period over the N -period horizon.
The multiple item case is also important for inventory models with transportation costs, like the ones presented in Chapter 5. Companies often consolidate
multiple items to save on costs due to transportation. A more complex model is
necessary to represent such scenario.
Another extension would be to combine a discount structure for the purchasing
price similar to that of the transportation cost in Chapter 5. By optimizing these
costs simultaneously, a better policy for the overall system may be achieved.
Additionally, in Chapter 5 a model was presented considering actual freight
rates in order to provide optimal solutions. In this model, the transportation costs
148
are represented as a continuous piecewise linear function (of the weight shipped)
using binary variables. This model can be easily extended to consider piecewise
linear functions that are not continuous. This is the case when the capacity of
a TL is exceeded and an extra truck might be needed. In such situations, there
exists a sudden increase in total transportation cost, which may make the function
discontinuous. To model this case, each segment of the non-continuous function
can represented by a binary variable with its corresponding cost function.
A direct extension to the serial system analyzed in Chapter 6 is to incorporate
transportation costs for the items being moved throughout the system. Since supply chain management is concerned with optimizing over the entire system, an ideal
model would combine supplier selection, inventory control, and transportation decisions. Additionally, it has been assumed that decisions for the serial system are
centralized. That is, there is a single decision maker who makes the inventory decisions for all stages to coordinate the inventory and minimize the cost of the entire
system. An immediate extension to this research is the comparison of centralized
and decentralized systems. Intuitively a centralized system should outperform the
decentralized system because the centralized case decisions are made to optimize
the entire system, while in a decentralized system each stage is going to make its
own independent decisions. These independent decisions may not be the best for
the entire system.
In addition to the serial system studied in Chapter 6, results obtained in Chapters 4 and 5 can be extended to more general supply chain networks. For example, an extension of interest would be to link the use of multiple-suppliers to
the traditional one-warehouse, multi-retailer system. Since transportation is a key
determinant in distribution effectiveness, transportation costs reflecting economies
of scale, as the ones studied in this research, need to be incorporated into the
inventory policy considerations.
Over the last years, supply chain disruptions have had a significant impact on
149
companies’ short and long-term performance because they have failed to consider
risk factors in their strategies. Some of the risks in the supply process are: disruptions during the transfer of products due to uncontrollable events, uncertain
supply yields, uncertain supply lead times, etc. Incorporating these factors is fundamental for companies to be able to develop alternative supply strategies in case
of disruptions.
Finally, due to the conflicting criteria considered in the supplier selection problem, the single-objective models for order quantity allocation developed in this
research could be extended as multi-criteria inventory models where the tradeoffs
associated with these criteria can be quantified.
Appendix
A
Transportation-Inclusive Models
with Different-Size Order Quantities
This appendix will extend the models in Chapter 5 to study the case where order
quantities allocated to selected suppliers may be of different sizes, Qi .
A.1
Model Considering Continuous Functions
This model uses the total cost per time unit from Problem (P4.1) where order
quantities allocated to selected suppliers are different (Qi ). However, a constant
holding cost (h) is used instead of a holding cost dependent on purchasing price
(a). This was done in order to make comparisons between the models presented
in Chapter 5 and the ones presented here. The total cost per time unit follows:
r
P
ZF Qi = d ·
i=1
r
P
Ji ki
Ji Qi
r
P
h
+ · i=1
r
2 P
i=1
i=1
r
P
Ji Q2i
+d·
Ji Qi
Ji Qi pi
i=1
r
P
i=1
Ji Qi
r
P
dw
+
·
100
Ji Qi Fy i
i=1
r
P
Ji Qi
i=1
r
P
Ji Qi li
dh i=1
+
· P
,
r
Y
Ji Qi
i=1
(A.1)
151
where the first term represents the setup cost, the second term denotes the holding cost, the third term is the purchasing cost, the fourth term accounts for
the transportation cost, and the fifth term denotes the cost corresponding to
in-transit inventory. The corresponding capacity constraints are,
dJj Qj ≤ cj
r
X
Ji Qi , for j = 1, . . . , r,
i=1
and the quality constraint is,
r
X
Ji Qi qi ≥ qa
i=1
r
X
Ji Qi .
i=1
In Chapter 4, Ri = Ji Qi was defined to linearize constraints because the term
Ji Qi makes them nonlinear. After substituting Ri and rearranging terms, the
complete formulation is the following:
(PA.1)
minimize
subject to
ZF Qi
" r
r
r
X
h X Ri2 X
d
Ji ki +
·
+
Ri pi
= P
r
2d
J
i
i=1
i=1
Ri i=1
i=1
#
r
r
h X
w X
·
Ri Fy i + ·
Ri li ,
+
100 i=1
Y i=1
R j d ≤ cj
r
X
Ri , j = 1, . . . , r,
i=1
r
X
i=1
Ri qi ≥ qa
r
X
Ri ,
i=1
Ri = Qi Ji , i = 1, . . . , r,
Ji ≥ 0, integer, i = 1, . . . , r,
Ri , Qi ≥ 0, i = 1, . . . , r.
The following equations, listed respectively, replace Fyi in the objective function
152
when Langley’s function or the Power function are used to estimate freight rates,
A.2
Fy i = Ai + αi Qi w,
(A.2)
Fy i = ai (Qi w)bi .
(A.3)
Model Considering Actual Freight Rates
In this section, Problem (P5.2) is extended to consider order quantities of different
sizes from different suppliers, Qi . The model is as follows:
(PA.2)
minimize
subject to
ZAQi
" r
r
r
X
X
h X Ri2
d
·
Ji ki +
+d·
Ri pi
= P
r
2d
J
i
i=1
i=1
Ri i=1
i=1
#
r
r
X
h X
Ri li ,
+
Ji · T Ci (Qw) + ·
Y i=1
i=1
Rj d ≤ cj
r
X
Ri , j = 1, . . . , r,
i=1
r
X
Ri qi ≥ qa
i=1
r
X
Ri ,
i=1
Ri = Qi Ji , i = 1, . . . , r,
Qi · w =
u
i +1
X
bi,k · λi,k , i = 1, . . . , r,
k=1
T Ci (Qw) =
u
i +1
X
gi,k · λi,k , i = 1, . . . , r,
k=1
λi,k ≤ Zi,k , i = 1, . . . , r; k = 1,
λi,k ≤ Zi,k−1 + zi,k , i = 1, . . . , r; k = 2, . . . , ui ,
λi,k ≤ Zi,k−1 , i = 1, . . . , r; k = ui + 1,
u
i +1
X
k=1
λi,k = 1, i = 1, . . . , r,
153
u
i +1
X
Zi,k = 1, i = 1, . . . , r,
k=1
Zi,k ∈ {0, 1}, i = 1, . . . , r; k = 1, . . . , ui ,
Qi ≥ 0, integer, i = 1, . . . , r,
Ji ≥ 0, integer, i = 1, . . . , r.
A.3
Illustrative Example
In this section, a numerical example is presented to analyze the impact of transportation costs on supplier selection and order quantity allocation decisions for the
models where orders of different size are considered, Qi . The same data and parameters as in Section 5.5.6.1 (Chapter 5) are used. The functions generated from
fitting Eqs. (A.2) and (A.3) to the effective rates of each supplier are summarized
in Table A.1.
Table A.1. Summary of Freight Rate Continuous Estimates
Supplier
Langley’s Fn ($/CWT)
R2 value
Power’s Fn ($/CWT)
R2 value
1
2
3
Fy 1 = 61.7-0.00127 (Q1 w)
Fy 2 = 80.3-0.00129 (Q2 w)
Fy 3 = 48.2-0.00109 (Q3 w)
0.763
0.746
0.758
Fy 1 = 1586.21 (Q1 w)−0.4028
Fy 2 = 789.97 (Q2 w)−0.2831
Fy 3 = 2247.57 (Q3 w)−0.4757
0.947
0.935
0.938
A.3.1
Analysis of Results
The same results as in Section 5.5.6.2 are analyzed here: WTA, LF, LFA, PF,
PFA, and AO. These are obtained as follows:
1. WTA (Problem (P4.1) + actual transportation cost): results are obtained
in two steps. First, Problem (P4.1), which neither considers transportation
nor in-transit inventory costs, is solved to obtain the total cost per time unit
154
(ZS∗ ), the order allocation, and quantity allocated to selected suppliers (Q∗ ).
Second, the transportation and in-transit inventory costs are computed. The
freight rates for the three suppliers are obtained from Tables 5.5–5.7 considering the shipping weight (Q∗ · w). The transportation cost per time unit is
given by
r
P
dw
·
100
Ji FiA
i=1
r
P
,
(A.4)
Ji
i=1
where FiA indicates the actual freight rates obtained ($/CWT). The in-transit
transportation cost per time unit is calculated as follows:
r
P
Ji li
dh i=1
.
r
Y P
Ji
(A.5)
i=1
The resulting costs from Eqs. (A.4) and (A.5) are added to the cost found in
step one, ZS∗ .
2. LF (Langley’s function): results are determined by solving Problem (PA.1)
using the Langley’s functions (Fy 1 , Fy 2 , and Fy 3 ) provided in the second
column of Table A.1. These results consider estimated transportation and
inventory costs simultaneously.
3. LFA (LF with actual transportation costs): these results are calculated in
two steps. First, using the order quantity obtained in LF, the corresponding
actual freight rates for the selected suppliers are determined from Tables 5.5–
5.7. Second, the total transportation cost is recalculated using these actual
freight rates in Eq. (A.4).
4. PF (Power function): results were found by solving Problem (PA.1) using
the power functions (Fy 1 , Fy 2 , and Fy 3 ) provided in the fourth column of
155
Table A.1. These results consider estimated transportation and inventory
costs simultaneously.
5. PFA (PF with actual transportation costs): these results are found in two
steps. First, using the order quantity from PF, the corresponding actual
freight rates for the selected suppliers are determined from Tables 5.5–5.7.
Second, the total transportation cost is recalculated using these actual freight
rates in Eq. (A.4).
6. AO (Absolute Optimal solution): results are obtained by solving Problem
(PA.2), which considers actual transportation costs.
Table A.2 shows the solutions obtained.
Table A.2. Solutions to Illustrative Example (Different Qi ’s)
WTA
LFA
PFA
AO
Order Allocation
J1
J2
J3
Ordering Quantities
Q1
Q2
Q3
Total Cost
($/month)
% Deviation
(from AO)
1
6
11
3
168
303
570
625
38,502.86
34,378.91
34,268.34
33,679.95
14.32
2.08
1.75
–
8
0
0
0
1
5
8
4
168
0
0
0
168
242
523
313
Conclusions from Table A.2 are discussed next:
• By incorporating transportation costs and inventory costs simultaneously,
as in LA, PA, and AO, the manufacturer can take advantage of economies
of scale in shipping. The order quantity for WTA is considerably smalles
than those for LFA, PFA, and AO. Since WTA is determined based on a
model that only optimizes inventory costs its resulting order quantity does
not take advantage of economies of scale in transportation. After adding
transportation and in-transit inventory costs. WTA results in the worst
solution in terms of total cost per time unit (14.32% greater than that of
AO).
156
• In LA, PA, and AO, supplier 2 has been eliminated from the set of selected
suppliers. Supplier 2 offers the highest average freight rate of of the three
suppliers.
• In contrast to the case where Q’s are the same, order allocation from LA, PA,
and AO are different. Additionally, the order quantities allocated to supplier
1 are larger than those allocated to supplier 3. It is evident that benefits of
economies of scale in transportation are greater for supplier 1.
• Costs comparisons between Table 5.9 (Chapter 5) and Table A.2 indicate
that the difference in total costs among the different results is marginal.
For example, the difference in the optimal total cost per time unit (AO)
between the case where Q’s are the same and the case where the Qi ’s are
different is only $139.05/month. This represents a 0.41% improvement from
the case of same Q’s to the case of different Qi ’s. This difference is smaller
than expected. By allowing the order quantities to be different sizes (Qi ),
economies of scale were expected to be greater resulting in a larger difference
between the total costs. That is, one would expect the order quantity from
a given supplier offering the lowest average freight rate to be larger than the
suppliers with more expensive average freight rates, but this sometimes is
impossible due to the capacity and quality restrictions of the problem.
Table A.3 shows the transportation costs generated for WTA, LFA, PFA, and
AO. These results were obtained for fixed values of M , from 2 to 25.
Not considering inventory and transportation costs simultaneously results and
then adding the actual transportation costs to the solution results in an average
deviation of 89% from the optimal solution. Modeling freight rates using Langley’s function results in transportation costs that are 28% higher than the optimal
transportation costs. Similarly, modeling freight rates using the Power function
leads to transportation costs that are 7% higher than the optimal transportation
157
Table A.3. Analysis of Transportation Costs
M
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
WTA
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
10,987.45
10,799.30
11,977.36
12,177.12
12,286.09
12,137.07
11,990.26
11,990.24
11,990.24
11,990.23
11,990.25
11,990.24
12,053.94
11,996.91
11,990.24
11,990.24
11,990.24
11,990.24
11,990.24
11,990.24
11,990.24
11,990.24
11,990.24
11,990.24
Average
% Deviation
% Dev
from AO
71.9%
92.4%
87.3%
103.3%
118.9%
89.8%
87.5%
87.5%
100.1%
87.5%
87.5%
87.5%
88.5%
87.6%
87.5%
87.5%
87.5%
87.5%
87.5%
87.5%
87.5%
87.5%
87.5%
87.5%
H
89%
LFA
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
7,854.47
8,875.89
7,854.47
8,436.71
7,854.47
8,194.73
7,854.47
8,066.93
7,854.47
7,987.75
8,293.56
7,933.84
8,194.73
7,894.74
8,122.31
7,865.09
8,066.93
7,841.82
8,023.19
7,840.29
7,987.75
8,144.21
7,958.46
8,102.28
% Dev
from AO
22.9%
58.1%
22.9%
40.8%
39.9%
28.2%
22.9%
26.2%
31.1%
24.9%
29.7%
24.1%
28.2%
23.5%
27.0%
23.0%
26.2%
22.7%
25.5%
22.6%
24.9%
27.4%
24.5%
26.7%
H
28%
PFA
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
6,393.28
6,970.62
6,393.28
6,791.88
6,393.28
6,775.73
6,834.25
6,780.84
6,791.88
6,789.18
6,778.43
6,507.04
6,775.73
6,791.88
6,777.39
6,781.20
6,780.84
6,776.78
6,791.88
6,775.73
6,783.12
6,776.56
6,778.43
6,778.42
% Dev
from AO
0.0%
24.2%
0.0%
13.4%
13.9%
6.0%
6.9%
6.1%
13.4%
6.2%
6.0%
1.8%
6.0%
6.2%
6.0%
6.1%
6.1%
6.0%
6.2%
6.0%
6.1%
6.0%
6.0%
6.0%
AO
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
6,393.25
5,613.12
6,393.25
5,990.72
5,613.12
6,393.28
6,393.22
6,393.30
5,990.72
6,393.28
6,393.25
6,393.28
6,393.28
6,393.29
6,394.01
6,393.26
6,393.30
6,393.27
6,393.26
6,393.28
6,393.28
6,393.22
6,393.30
6,393.28
H
7%
costs. Therefore, the Power function is a better estimate of the actual freight rates
than Langley’s function.
Appendix
B
A Serial Inventory System with
Supplier Selection at All Stages
In Chapter 6, a serial N -stage system with supplier selection at Stage 1 was studied.
In this appendix, a more general serial supply chain system is considered where
the supplier selection process takes place at all the stages of the serial system (see
Figure B.1).
Supplier
1
Supplier
2
…
…
Stage 2
Stage 1
Supplier
r1
Supplier
1
Supplier
2
…
Supplier
r2
Stage N
Supplier
S
li
1
Supplier
S
li
2
…
Supplier
S
pplier
rN
Figure B.1. Supplier Selection Decisions at All Stages
This system could be an assembly operation of a product that needs to be
assembled at different facilities. The objective is to coordinate the inventory levels
of the parts being produced at every stage while procuring the required materials
for assembly from a set of potential suppliers.
159
B.1
Model Development
The mathematical model that solves the problem described above is developed in
this section. The same assumptions as in Chapter 6 are followed. The number of
suppliers available at each Stage j is rj , for j = 1, . . . , N . Some of the notation
from Chapter 6 has been redefined as follows:
Data
rj – number of available suppliers at Stage j
hj – inventory holding cost per unit and time unit at Stage j
ej – echelon unit holding cost at installation j
kij – setup cost for placing an order to ith supplier at Stage j
pij – unit price of ith supplier at Stage j
cij – capacity of ith supplier at Stage j
qij – perfect rate of ith supplier at Stage j
qaj – minimum acceptable perfect rate of parts at Stage j
Variables
Jij – number of orders of ith supplier per order cycle at Stage j
Qj – order quantity at Stage j
An optimal policy that minimizes the total cost per time unit of the system
should follow Qj = nj Qj+1 , where nj is a fixed positive integer. In this section, Qj
is replaced by sj QN , where sj = nj nj+1 · · · nN −1 . Therefore, the general holding
cost for the system is,
N
QN X
·
ej sj .
2 j=1
(B.1)
The setup cost for all stages is as follows:

d
QN

rj
P

N
X
 i=1 Jij kij 


 .
·
rj
 P

j=1
sj
Jij
i=1
(B.2)
160
The total purchasing cost per time unit is,

rj
P

N
X
 i=1 Jij pij 
.

d·
rj

 P
j=1
Jij
(B.3)
i=1
The capacity constraints at each installation stage are,
dJij ≤ cij
rj
X
Jij , i = 1, . . . , rj ; j = 1, . . . , N,
(B.4)
i=1
and the quality constraints are the following:
rj
X
Jij qij ≥ qaj , i = 1, . . . , rj .
(B.5)
i=1
Notice that the total variable cost per time unit for the system is composed of
the holding and setup costs,

N
d
QN X
·
e j sj +
2 j=1
QN

rj
P

N
 i=1 Jij kij 
X
 ,


·
rj

 P
j=1
Jij
sj
(B.6)
i=1
Since the constraints are independent of QN , the optimal order quantity at
installation N is found by taking the first derivative of Eq. (B.6) with respect to
QN , setting it to zero, and solving for QN . This yields,
v  

u
rj
P
u
u  N  Jij kij 
u  P  i=1

u 2d   rj

u
P
j=1
u
s
J
j
ij
u
i=1
Q∗N = u
.
u
N
P
t
e j sj
j=1
(B.7)
161
By substituting Q∗N into Eq. (B.6), the following total variable cost is obtained,
v


 rj
u
P
u
N
N
u X
X
 i=1 Jij kij 
u

,

u2d
e j sj 
rj

 P
t
j=1
j=1
sj
Jij
(B.8)
i=1
and the final mixed integer nonlinear programming model including Eq. (B.8) and
the total purchasing cost per time unit in the objective function and the capacity
and quality constraints is,
(PB.1)
minimize
v

 rj

u
P
u
u X
N
N
X
 i=1 Jij kij 
u


ej sj 
ZN = u

 sj Mj 
t2d
j=1
j=1

rj
P

J
p
N
X
 i=1 ij ij 


+d·
 Mj  ,
j=1
subject to
dJij ≤ cij Mj , i = 1, . . . , rj ; j = 1, . . . , N,
(B.9)
rj
X
Jij qij ≥ qaj Mj , j = 1, . . . , N,
(B.10)
Jij = Mj , j = 1, . . . , N,
(B.11)
i=1
rj
X
i=1
sj ≥ aj sj+1 , j = 1, . . . , N − 1,
(B.12)
Mj ≥ 1, integer, j = 1, . . . , N,
(B.13)
Jij ≥ 0, integer, i = 1, . . . , rj , j = 1, . . . , N.
(B.14)
where Mj indicates the total number of orders allocated to all selected suppliers in
one order cycle at Stage j. Constraint (B.12) ensures integrality for order quantities
between stages, Qj = sj QN , j = 1, ..., N − 1. Problem (PB.1) can be solved using
commercial optimization software like LINGO [129] and GAMS [130]. However, the
162
problem will become extremely difficult to solve as the number of stages and the
number of suppliers (at each stage) increase. For this reason, it is recommended
that a power-of-two policy, similar to the one presented in Chapter 6, be derived
in future research.
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Vita
Abraham Mendoza
Abraham Mendoza studied at the Universidad Panamericana (Guadalajara, Mexico), where he received a B.S. degree in Industrial Engineering in May 2000. In
2002, he was awarded a scholarship from the National Council of Science and
Technology to pursue graduate studies in the Department of Industrial and Manufacturing Engineering at the Pennsylvania State University, where he received his
M.S. and Ph. D. degrees in Industrial Engineering and Operations Research in
2004 and 2007, respectively. While at Penn State, he was the recipient of the 2006
Material Handling Education Foundation Honors Scholarship, became a member of
the Alpha Pi Mu Honor Society, and was a teaching assistant for the masters program of Quality and Manufacturing Management for five semesters. His research
interests include supply chain logistics, operations management, transportation,
and facility layout/material handling. He is a member of the Council of Logistics Management (CSCMP), Institute for Operations Research and Management
Sciences (INFORMS), and Institute of Industrial Engineers (IIE).