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OR Spectrum (2002) 24: 59–78
c _Springer-Verlag 2002
Coordinating supply chain decisions:
an optimization model
Christoph Haehling von Lanzenauer and Karsten Pilz-Glombik
Institut fЁur Produktion, Wirtschaftsinformatik und Operations Research,
Freie UniversitЁat Berlin, Garystrasse 21, 14195 Berlin, Germany
(e-mail: [email protected])
Received: September 19, 2000 / Accepted: July 16, 2001
Abstract. Coordinating information and material .ows are key to effective supply
chain management. The complexity of interactions in and the uncertainties surrounding
supply chains make such coordination dif.cult. However, coordination
can be realized by optimizing the .ows in supply chains with analytical approaches.
Amixed integer programming model is presented to support the tactical decisions of
ordering, producing and transporting under various conditions of information availability
at the loci of decision making. The model is applied to a modi.ed version
of MIT’s well known Beer Distribution Game. The performance of the modeling
approach is contrasted with the results of human decision making under identical
conditions and underlines the enormous potential for performance improvement
analytical decision support can provide. Several methodological aspects for coping
with the dif.culties of solving rather large mixed integer models are presented
and it is shown that they can contribute signi.cantly in dealing with the inherent
computational problems.
Key words: Supply chain management – Optimization – Mixed integer programming
– Uncertainty – Beer distribution game
Introduction
Systems facilitating the movement of materials from sources to sinks are termed
supply chains. Although intra-.rm multi-stage production/inventory systems
(Graves et al., 1993) belong also to this domain, inter-.rm supply chains are nowadays
more at the center of managerial attention (Petrovic and Petrovic, 1998; Slats
et al., 1995; Thomas and Grif.n, 1996). Supply chain management is concerned
with handling effectively and ef.ciently all .ows of materials and information and,
Correspondence to: C. Haehling von Lanzenauer
60 C. Haehling von Lanzenauer and K. Pilz-Glombik
at times, funds within and across the chain. The managerial issues cover problems
with strategic and operational dimensions and include the design and location
of facilities, the speci.cation of supply contracts, the choice of product variety,
the management of inventories, and the selection of transportation modalities, just
to name a few. Since large amounts of funds can be tied up in various forms of
inventories, in different stages of manufacturing as well as in the transportation
pipelines, efforts to continuously improve the process of planning and controlling
all .ows are of paramount importance, in particular under the increasing pressures
for productivity gains and improvements in service in a global environment.
Managing supply chains represents a major challenge due to the complexity
of interactions among the system components, incomplete information available
for decision making, the presence of multiple decision makers pursuing con.icting
objectives and the dynamics of non-stationary conditions. Effectiveness in dealing
with these issues, however, carries also the potential for signi.cant productivity
gains. This potential for improvements is underlined in the results of the “Integrated
Supply Chain Performance Benchmarking Study” indicating that the best in class
has overall logistic cost which are only about half the median cost of all respondents
in the sample (newscope 1997). Similar conclusions are obtained from experiments
using a supply chain management game: The performance of human decision makers
in a simpli.ed and therefore relatively easy context is dismal (Sterman, 1989;
Haehling von Lanzenauer and Pilz-Glombik, 2000). Such evidence underlines the
need for developing approaches which support more effective management of supply
chains.
The purpose of this paper is to contribute to the improvement of supply chain
decision making. In particular, an optimization model re.ecting a general supply
chain structure and integrating a variety of supply chain decisions is formulated.
Transportation aspects will represent a central issue in the modeling process with
particular emphasis being placed on the possibility of batching. The model is subsequently
applied to the speci.c supply chain con.guration of the Beer Distribution
Game and solved for several problems settings. Computational aspects of solving
the model are presented not only under conditions of uncertainty but also for benchmarking
purposes under certainty. The results obtained from the model are then
contrasted with those derived by human decision makers underlining the enormous
potential of optimization approaches in managing supply chains.
Modeling supply chain interdependencies
The diagram shown in Figure 1 is frequently used to illustrate the interdependent
nature of the stages in the supply chain: It is conceptual and highly aggregate. Nevertheless,
it implies activities in three overlapping sections (sourcing, processing
and distributing) in order to create and deliver goods and services to customers and
suggests the need for resources. Decisions about the acquisitions of resources and
their subsequent use are generally made at three levels: At the strategic level decisions
such as the degree of product variety and the markets to be served, the location,
number and size of facilities as well as the extent of vertical and/or horizontal integration
must be made. Tactical planning will involve supplier selection, (aggregate)
Coordinating supply chain decisions: an optimization model 61
Fig. 1. The supply chain
levels of production, modes of transportation, and amounts being shipped between
plants, distribution centers and customers. Issues such as detailed scheduling, lot
sizing and sequencing, joint order decisions and/or vehicle loading and routing belong
to the operational level. Obviously, planning activities and decision making at
these levels are highly interrelated. The extent to which interdependencies between
the stages of the supply chain and across the planning hierarchies are accounted
for depends both on the information available and the decision making style used
at each locus of decision making.
Various thrusts have been proposed to support the decision making process
in supply chains. Leading edge information and communication technologies are
clearly the central element in linking all stages of the supply chain into a network
(GЁunther et al., 1998). Rapid information .ows combined with the elimination of
much paperwork and the assurance of error-free and timely data promise cost reductions
and sustained competitive advantages. Other efforts suggest initiatives for
ef.cient consumer response, vendor managed inventories or a move from push- towards
pull-systems. While these thrusts have all led to signi.cant productivity gains
(see various contributions in: Hadjiconstantinou, 1999; Poirier and Reiter, 1996),
they are typically applied to sections of the entire supply chain. Further opportunities
exist in combining such efforts and attempting an integrated optimization of the
overall supply chain (Shapiro, 1999; Cohen and Huchzermeier, 1999). Although
supply chain optimization appears to be a priority among senior managers (KPMG
Benchmarking Study 1999), the term optimization has only a vague connotation for
most practitioners. While managers seem to understand under optimization ideas
such as “building an optimized network”, “supply chain optimization being a worthy
quest and a feasible concept”, “business partnering with optimization meaning
that the parties to the relationship have found the maximum value” or “optimization
having to become a strategy in itself” (Poirier and Reiter, 1996, pp. XIV, XV,
3, 83, 270), they are unaware of the rigor of optimization models in general and
mathematical programming methods in particular. Several approaches to support
integrated supply chain optimization have been proposed or are under development
(Shepherd and Lapide, 2000). Models for integrated optimization however remain
dif.cult to formulate and to solve not only in light of the complexities characterizing
supply chains but also due to the different levels in the planning hierarchy, the
possibility of con.icting criteria pursued at stages and the lack of certainty about
relevant data and information.
In the context of this paper coordinated decision making is attempted for a
situation where the supply chain network con.guration is already in place. Thus,
the emphasis will be tactical and the concern will be to capture the interactions
of order, production and transportation decisions between facilities. Of particular
interest in the analysis will be the transportation decisions which have to be made
62 C. Haehling von Lanzenauer and K. Pilz-Glombik
Fig. 2. General network structure
Fig. 3. De.nition of supply chain variables
for a given con.guration of transportation lanes. Figure 2 describes the general
structure for such a supply chain and the information and material .ows between
plants m and stages k with k = 0 and k = K + 1 representing the source and
sink respectively. The interdependencies in the supply chain are portrayed from the
perspective of plant k,m which is therefore termed the focal company. The distinct
path each product is taking through the sourcing, processing and distributing section
is the result of a series of purchasing, production, inventory and transportation
decisions. Furthermore, the nature of the end product structure has implications for
the number and volume of parts and/or products that move between sections and
adjacent stages. In particular, the logistics in the sourcing section will differ for end
products having a converging or a diverging product structure. The many different
parts and unique products which .ow simultaneously through the entire supply
chain will often share some or all of the chain’s facilities (production facilities,
warehouses, transportation modes, shelf space). Thus, the distinct paths of the
parts/products through the supply chain are highly interconnected.
Regardless of the particular approach taken to reach supply chain decisions,
these decisions will manifest themselves in several state variables describing the
supply chain at given points in time which, in turn, can be related to performance
measures. The relevant decision and state variables for the focal companymat stage
k are detailed in Figure 3. The information and material .ows between the focal
company and its adjacent stages are restricted to a single upstream and downstream
plant l and n respectively.
Coordinating supply chain decisions: an optimization model 63
The state variable I1 represents raw materials and parts inventory and is the
result of incoming shipments as well as internal production decisions. The variables
P de.ne production/processing started in a given period. At the end of each period
there may exist an inventory of completed products, identi.ed as I2, and/or a lost
sales or backorder position de.ned as L and B respectively. The variables S and O
represent incoming and outgoing shipments and orders from/to plant l and from/to
plant n. The subscript t speci.es the period when a .owleaves a particular location.
The arrival of a .ow is indicated by an adjustment in the time index t which re.ects
delays in sending orders, in production and in shipments (od., pd. and sd.). Since
the superscripts on the delay parameters indicating origin and destination of a
.ow are the same as on the variables themselves, they are replaced for ease of
presentation by an *. Using these de.nitions I1 and I2 are determined by (1) and
(2).
I1k,m
t = I1k,m
t-1 +
M
_l=1
S
(k-1,l)(k,m)
t-sd. t
Pk,m
ЃН k = 1. . . K,m = 1. . . M, t = 1. . . T (1)
I2k,m
t = I2k,m
t-1 + Pk,m
t-pd. M
_n=1
S
(k,m)(k+1,n)
t
ЃН k = 1. . . K,m = 1. . . M, t = 1. . . T (2)
Expressions (1) and (2) imply that shipments arriving at plant m were made sd
periods earlier while shipments from plant m are made during period t. Similarly,
any production/processing started in t will be completed pd periods later. If stage
k represents the special case of a distribution rather than a production facility, (1)
and (2) can be combined by eliminating the production variables.
Shortages will either results in lost sales or backorders. The levels of lost sales
L or backorders B in period t at stage k and plant m are given in
Lk,m
t=
M
_n=1
O
(k+1,n)(k,m)
t-od. M
_n=1
S
(k,m)(k+1,n)
t
ЃН k = 1. . . K,m = 1. . . M, t = 1. . . T (3)
Bk,m
t = Bk,m
t-1 +
M
_n=1
O
(k+1,n)(k,m)
t-od. M
_n=1
S
(k,m)(k+1,n)
t
ЃН k = 1. . . K,m = 1. . . M, t = 1. . . T (4)
with the variable O representing the orders received at stage k and plant m which
were placed od periods earlier from plant n at stage k + 1. The possibility of
external orders for components or parts received by plant m is omitted for sake
of simplicity but without loss of generality. The formulations of lost sales and
backorders presented in (3) and (4) are in aggregate form and imply that shortages
could be allocated in very uneven pattern. If shortages should be allocated on a
64 C. Haehling von Lanzenauer and K. Pilz-Glombik
discretionary or pro-rata basis, the expressions for lost sales or backorders must be
de.ned separately for each pair of plants at adjacent stages. Furthermore, a shipment
made in t cannot exceed current orders plus any backordered demand. It should
however be noted that shipments are decision variables which implies that shortages
either in form of lost sales or backorders could be incurred simultaneously since it
may not be economical to deliver although suf.cient stock is available. Naturally,
either lost sales or backorders will occur implying that either (3) or (4) will be used
but not both.
The total amount of production/processing carried out at stage k and plant m
during period t must be restricted by the capacity available Kap at that plant. Since
production/processing may take longer than one period the capacity restriction is
in the form of
Pk,m
t+
и
_ф=1
Pk,m
t-ф ЎЬ Kapk,m
ЃН k = 1. . . K,m = 1. . . M, t = 1. . . T, и = pdk,m - 1. (5)
In the expressions (1) to (4) a shipment leaving plant m at stage k in period t is
designated for plant n at stage k +1. Such shipment can only be made if suf.cient
transportation capacity is available. The transportation capacity is determined by
the .eet of vehicles available at plant m and stage k which is the result of strategic
decision making.
Let the index j(j = 1, 2, . . . , J) represent individual vehicles available for
deliveries at stage k and plant m. Each vehicle has a capacity of cj,t units and can
be used in period t regardless of any assignments in prior periods. The selection of
a particular vehicle for shipping purposes at plant m is made via a binary variable
Xj,t Ѓё {0, 1}. Constraint (6) ensures that a shipment can be made only if suf.cient
transportation capacity is available and (7) restricts the use of vehicle j to a single
destination in period t.
S
(k,m)(k+1,n)
t ЎЬ
J
_j=1
X
(k,m)(k+1,n)
j,t ·
j
ck,m
ЃН k = 1. . . K,m = 1. . . M, t = 1. . . T, n = 1. . . M (6)
M
_n=1
X
(k,m)(k+1,n)
j,t ЎЬ
1
ЃН k = 1. . . K,m = 1. . . M, t = 1. . . T, j = 1. . . J (7)
Expressions (6) and (7) imply further that vehicle j at plant m and stage k is only
used for shipping to plant n at stage k + 1. Thus, no split shipments are possible.
This form of modeling transportation is meaningful if all transportation activities
are outsourced, and it is also consistent with the tactical focus of this paper.
If transportation capacity is owned or leased the availability of vehicle j in
period t depends also on the vehicle’s use in prior periods since deliveries may
take longer than a single period. Thus, an additional constraint would have to be
Coordinating supply chain decisions: an optimization model 65
included to re.ect that a vehicle j may still be in use. Furthermore, to permit joint
shipments from one or more plants at stage k to several plants at the adjacent stage
downstream one would not only require a modi.cation of the above restrictions
but also the simultaneous solution of the associated operational problem of vehicle
routing. In addition, it may be possible that a vehicle used for shipping from stage
k to stage k + 1 will not be returned to its origin but is used for deliveries to
plants located at other stages. Such arrangements would avoid costly empty return
trips. These aspects concerning other transportation possibilities, however, will not
be pursued in this paper. For a recent analysis of related transportation issues see
ZЁapfel and Wasner (2000).
Because stages k = 0 (source) and k = K + 1 (sink) differ from the regular
stages, the set of restrictions presented so far holds only for k = 1, 2, . . . , K. In
particular, k = 0 has no predecessor and, therefore, it does not place orders nor
will it receive shipments; correspondingly, k = K + 1has no successor and will
not receive orders nor will it make shipments. In the above restrictions which are
de.ned for k = 1, 2, . . . , K the decision variable O remains unspeci.ed for the
.rst stage in the network. (8) insures that a shipment leaving k = 0 can only be
made if an order has been placed by k = 1.
S
(0,m)(1,n)
t= O
(1,n)(0,m)
t-od. ЃН
m = 1. . . M, n = 1. . . M, t = 1. . . T (8)
Finally, customer demand occurring at the sink must be integrated into the model.
This is accomplished in (9) by setting the orders received at stage K equal to the
externally given customer requirements R.
O
(K+1,n)(K,m)
t-od. = R
(K+1,n)(K,m)
t ЃН
m = 1. . . M, t = 1. . . T (9)
While (8) and (9) deal with the physical boundaries of the supply chain, conditions
must also be speci.ed which exist at the beginning and the end of a given planning
horizon. In particular, opening values must be provided at each stage and plant
for the state variables inventory, lost sales or backorders and production already in
process as well as for the .owvariables orders already placed and shipments already
made but not yet received. Such boundary conditions will avoid the dif.culties
of having in the model formulation variables with time indices which are either
negative or extend beyond the end of the planning horizon T. This latter possibility
arises from delays of od. periods in receiving an order and correspondingly from
delays of sd. periods in making shipments. Details for the opening and ending
conditions are speci.ed in Section 4 of this paper. Finally, some of the state and
.ow variables could also be subject to further physical (e.g. inventory capacity) or
logical (e.g. shipments can only be made every other period) restrictions.
The formulation presented so far implies that a single product with a simple
product structure (i.e. Veredelungsfertigung) .ows through the supply chain. Permitting
several end products, each made from a variety of components or parts
requires at least information about the input-output relationships in form of either
a Gozinto graph or a bill of material matrix. Modeling more general product structures
makes it necessary to identify each item (raw material, intermediate and end
product) and expressing the interdependencies among items. This is easily accomplished
by introducing an item index for all variables and by including additional
66 C. Haehling von Lanzenauer and K. Pilz-Glombik
restrictions. The restrictions must describe the input-output relationships between
intermediate and end products and may include weight and/or volume considerations
for transportation purposes. Given the focus of this paper these extensions are
not considered.
To evaluate the economic activities in a supply chain one or several performance
measures are required. In a decentralized decision making environment it is possible
to use different performance measures at each plant and stage. Aggregating such
criteria into a single measurement for the entire supply chain would obviously
create dif.culties. It is therefore suggested to determine the contribution margin
for the entire chain by adding the respective terms at all plants and stages. Let
rvk,m represent the revenue received by plant m at stage k for a unit shipped to
plant n at stage k + 1 and vck,m the variable purchasing cost incurred at plant m
and stage k per unit delivered from stage k-1 and plant l. Correspondingly, ick,m
and bck,m are de.ned as the unit cost (per time period) for inventory and backorder
respectively and tck,m as the .xed cost of sending vehicle j from plantmat stage k
to plant n at k +1 regardless of full or less-than-full truckloads. The present value
of the contribution margin for the supply chain over the entire planning horizon can
then be expressed by (10) with с being an appropriate discounting term.
Z=
T
_t=1
K
_k=1
M
_m=1_ M
_n=1
S
(k,m)(k+1,n)
t-sd. ·
M
rvk,m -
_l=1
S
(k-1,l)(k,m)
t-sd. ·
vck,m_сt
T
_t=1
K
_k=1
M
_m=1 _Ik,m
t · ick,m + Bk,m
t · bck,m_сt
T
_t=1
K
_k=1
M
_m=1.
.
J
_j=1
M
_n=1
X
(k,m)(k+1,n)
t,j · tc
(k,m)(k+1,n)
j.
ясt (10)
It should be noted that the revenue received by plant m at stage k will equal the
variable purchasing cost incurred at plant n at stage k +1; therefore, the .rst term
in the performance function could be simpli.ed by considering revenues only at
stage K and variable purchasing cost only at stage 1.
The expressions (1) to (10) can be considered as a general “accounting” framework
which is used to track and measure supply chain variables and to capture
systems performance. Other than ensuring that certain restrictions are not violated
nothing is implied about how the supply chain decisions at any stage are or should
be made. In fact, any set of supply chain decisions – however derived – can be
“imported”. While the set of restrictions is used to check for consistency, the valuefunction
(10) will evaluate the systems performance. As such the model formulation
has strictly descriptive character.
If, however, the performance function (10) subject to the various restrictions is
to be maximized, the formulation represents a mixed integer programming (MIP)
model as soon as the respective non-negativity conditions
I1, I2, P, L,B,O, S ЎЭ 0 (11)
Coordinating supply chain decisions: an optimization model 67
are in place and the initial as well as ending conditions are implemented. Now
the model takes on normative character with endogenously generated supply chain
decisions. The (optimal) solution will consist of coordinated decisions at all plants
and stages in the supply chain. This latter avenue will be our prime concern.
Supply chain decision making
Supply chain decisions can either be made in a decentralized fashion at each
plant/stage or in a centralized mode for the entire chain by a planning agency.
The particular form of decision making is not only a function of the organizational
structures existing in the supply chain but it is also in.uenced by the location of
information and the level of certainty in the data provided to the decision makers.
With the availability of only local information (i.e. no information sharing),
supply chain decisions at plant m at stage k (i.e. how much to order from stage
k - 1, how much to produce and what to ship to stage k + 1) must be made
in a decentralized mode. These decisions will be based on local information about
current inventory/backorder levels, any upstream deliveries and downstream orders
which have been received as well as on forecasts about these .ows in future periods.
Of course, the decision making process will also use economic data and can be
supported by a variety of methods as outlined by Inderfurth (1999).
If global information about system variables is available (i.e. global information
sharing) supply chain decisions can be made in a in decentralized or in a centralized
mode. At any point in time all inventory/back order positions, all order, production
and transportation decisions are shared with all loci of decision making. Although
supply chain decisions derived in a decentralized mode with global information
should contribute to the quality of decisions, the complex interdependencies among
systems variables, the inability to interpret data and the dif.culties of dealing with
time delays appear to block signi.cant improvements in systems performance by
human decision makers. Optimal supply chain decisions producing coordinated
information and material .ows can be derived under centralized decision making
with the simultaneous consideration of all interdependencies. Such coordinated
decisions are possible with the above model by optimizing the performance function
de.ned over all stages of the supply chain. The success with which such model can
be solved and the efforts required depend on the size of the model and to some extend
on the nature of the constraints, particularly those involving integer variables.
An important element in managing the supply chain are the external customer
requirements.With no prior information about customer requirements supply chain
decisions in a given period are based on the actual order received and on forecasts
about future requirements. After implementing current decisions the actual customer
requirements are used to update the state variables as well as for forecasting
purposes. Based on these forecasts the supply chain decisions for the remaining
part of the planning horizon will be redetermined. This process of partial implementation,
learning from newer information and re-solving the updated model is
repeated.
With perfect information the external customer requirements in each period
are known ahead of times. Depending on the degree of information sharing these
68 C. Haehling von Lanzenauer and K. Pilz-Glombik
customer requirements are known either only to the last stage (local information)
or they are available at all locations including a central planning agency (global
information). Supply chain decisions can then be derived either in a decentralized
or centralized mode. Although the assumption of perfect information represents an
unlikely scenario, it permits the derivation of upper bound solutions and can thus
be used for benchmarking purposes.
Finally, supply chain decisions could be derived using some information (e.g.
expected value, variance, the full distribution) about external customer requirements.
The supply chain model presented can either be solved as described above
for the scenario of no prior information or one could attempt to use approaches
based on stochastic programming (Birge and Louveaux, 1997; Lucas et al., 1999).
The latter avenue is not pursued in light of the dif.culties of solving these problems.
Implementing the model and results
Problem setting
The model formulated for the general supply chain structure has been implemented
in the context of a modi.ed version of the Beer Distribution Game. The Beer Distribution
Game was developed at MIT’s Sloan School of Management to introduce
students to the concepts of computer simulation in general and Industrial Dynamics
in particular (Sterman, 1989). Today the game is used worldwide as an introduction
to the problems of and as a training ground in supply chain decision making.
Although the Beer Distribution Game constitutes a distribution rather than a production
setting and its supply chain con.guration is a special case of the general
supply chain structure, it has been selected for evaluation purposes since it permits
the possibility of contrasting model-based and human decision making.
The game represents primarily a distribution system consisting of four sequentially
arranged stages, each consisting of only one facility where a single commodity
is produced, distributed through several stages and .nally consumed. Figure 4 illustrates
the structure of the supply chain with k = 0 representing production at
the factory (source) and k = 5 the customers (sink) being served by the retailer.
Since production is not a central issue in the game, the details presented in
the modeling section above are of little relevance. For example, the distinction
between inventory I1 and I2 is not necessary. For sake of simplicity no value-adding
takes place in the supply chain which is also free of any physical restrictions. The
order placing, the production and shipping processes are characterized by delays
of one and two periods respectively. Each stage interacts with its adjacent stages by
ordering and receiving from an upstream and by shipping to a downstream stage
Fig. 4. Supply chain of the beer distribution game
Coordinating supply chain decisions: an optimization model 69
Table 1. Vehicle information
Type of vehicles A B C D E
Capacity 12 6 5 3 2
Cost [$] 30 18 15 12 10
Table 2. Transportation arrangements
Transportation arrangement Fleet of vehicles
Wholesaler Distributor Factory
TA-I: Homogeneous / in.exibleA A A
TA-II: Heterogeneous C, D, E, E B, B A
TA-III: Homogeneous / .exible C, D, E, E C, D, E, E C, D, E, E
the single commodity. In the original version of the game any stage in the chain
must satisfy incoming orders to the extent possible. Shortages at any stage will
result in backorders which have to be satis.ed with priority. Further details of and
procedures for playing the game are available in Sterman (1989) and Haehling von
Lanzenauer and Pilz-Glombik (2000).
For the purpose of this paper the game is somewhat modi.ed: The performance
measure will – in addition to – holding and backorder cost also include .xed transportation
cost, variable purchasing cost and revenues. The transportation costs are
incurred by the shipping stage and depend on the transportation mode selected.
The .xed transportation costs must be paid regardless of full or less-than-full shipments;
thus, in this modi.ed version of the game it can make sense not to ship a
small order although suf.cient stock is available. Since no transportation is necessary
to the factory and since end customers obtain the commodity at the retailer,
shipment decisions are required only at stages k = 1to 3. Transportation is carried
out by .eets of vehicles. Information about the vehicles available and a description
of three different transportation arrangements – each consistent with restrictions
(6) and (7) – are given in Tables 1 and 2 respectively. Although the composition
of .eets can differ from stage to stage, the .eet of vehicles in each transportation
arrangements has the same total capacity. A transportation arrangement is called
homogeneous if the same .eet exists at all stages. A .eet with differently sized
vehicles is termed .exible. Consistent with reality the cost per unit capacity rises
with smaller vehicles. As in the original version of the game holding and backorder
cost at all four stages are $1and $ 2 per unit and time period. The value of revenues
received by the retailer minus the variable purchasing cost at the factory is set at
$20 per unit which insures that with reasonably simple transportation decisions an
overall positive contribution for the entire supply chain is possible.
While customer requirements in the original version of the game are characterized
by a ramp which is well suited for the purposes of simulation and Industrial
Dynamics, a seasonal pattern is used in this paper which is more appropriate for the
supply chain setting. Customer requirements in period t are generated by a random
70 C. Haehling von Lanzenauer and K. Pilz-Glombik
Fig. 5. Demand pattern
draw from a normal distribution N(5; 0, 5); this value is then multiplied by a given
seasonal factor and .nally rounded to the nearest nonnegative integer. This process
is repeated for every period of the planning horizon. A string of 36 such generated
values is called a random realization of customer requirements. Figure 5 presents
not only a single realization but also the mean, the maximum and the minimum
obtained by 100 such random realizations. The single as well as the 100 realizations
are used in the subsequent evaluations of human and model based decision making.
Starting with an initial inventory of twelve units at all stages, ten (twice .ve)
units in all transportation pipelines and also .ve units in each order entry, supply
chain decisions must be made for 36 periods. In order to avoid a behavior which
“drives the system into the ground”, ending conditions of twelve units in each
inventory must/should be met both by human decision makers as well as in the
modeling context. If the ending conditions differ from this target a penalty of $ 10
per unit short, up to a maximum of $ 120, is applied. Alternatively, the game could
also be played by setting the planning horizon to, say, 50 periods but the game and
the model is already terminated after 36 periods.
Human and model-based decision making with no prior information
The results presented below assume that supply chain decisions are made after the
current period requirements have been received but without any knowledge about
future requirements.
Human decision makers (i.e. teams of 8 to 12 students, 2 or 3 at each station)
are introduced to the game during a short introductory session in which they gain
understanding of the rules by playing the game for a few practice periods. The
supply chain decisions made by these human decision makers are based on global
Coordinating supply chain decisions: an optimization model 71
information sharing; the decisions are obtained period by period using any imaginable
decision making style with whatever decision support and forecasting system
a team wishes to choose. The performance of all teams in a sample of 16 is dismal.
Not a single team was able to generate results which come even close to the
model-based performance: While only one of the 16 teams was able to generate a
small positive contribution margin for the entire supply chain, most teams .nished
with a signi.cant loss position. Their order and shipment decisions re.ect a lack of
coordination and lead generally to very substantial inventory and backlog positions
at all stages. The pattern of inventory and backlog over time typically display the
well known bullwhip phenomenon. These observations are consistent with other
experiments reported in Haehling von Lanzenauer and Pilz-Glombik (2000).
Detailed results for two of the better performing teams are presented below
for the single randomly generated realization of customer requirements of Figure 5
and transportation arrangementsTA-I andTA-II. These transportation arrangements
have been selected for two reasons: TA-I is well suited to demonstrate coordinated
decision making and TA-II is consistent with real life where a more .exible .eet
is used the closer one is to the end customer. The pattern of order/production and
shipment decisions as well as the corresponding inventory and backlog positions
at all stages including the value of the performance function and its breakdown
into the various cost components under human decision making are shown in the
left half of Figure 6. It should be noted that for purposes of compact presentation
backlogs are shown in Figure 6 as negative inventory levels, although they have
been de.ned as non-negative variables.
Due to the nonlinear transportation cost it may be economical to experience
inventory and backlogs simultaneously, a fact evidenced in Figure 6 for the wholesaler,
distributor and the factory. Careful inspection of the charts will however
indicate that this phenomenon does not occur at the retailer, since no transportation
cost exist.
The rather low levels of backlogs shown in Figure 6 are in part the result of the
fact that these human supply chain decisions are obtained in a second round. While
the decisions in the .rst round are based on local information only, global information
sharing is permitted in the second round. Thus, a certain amount of “learning”
has materialized after the .rst round although the actual customer requirements
differ in both rounds.
The model-based results can be thought of being determined by a central planning
agency using the above supply chain model supported by a .rst order exponential
smoothing forecasting system. Given the complete lack of information
about the customer requirements at the beginning of the planning horizon responsiveness
is an attribute of the forecasting system that must be of key importance.
Thus, the smoothing constant is set at 0,9. With the accumulation of actual customer
requirements (i.e. learning), efforts to recognize a pattern in the underlying
demand structure such as seasonality or a trend and then integrating the identi.ed
pattern into the forecasting scheme are not part of this paper but are currently under
investigation. Although the model is solved with a forecast of future customer
requirements for the entire planning horizon, only the current period supply chain
decisions are implemented. As soon as available the actual customer requirements
72 C. Haehling von Lanzenauer and K. Pilz-Glombik
Fig. 6. Supply chain decisions with no prior information
are used for revising the forecasts and for updating the state variables. The model
is then re-solved for the remaining part of the planning horizon. This process is
repeated for every period and is therefore directly comparable to human decision
making. Although the sequence of model based solutions constitutes good and certainly
improved decision making, it does not necessarily represent optimal supply
chain decisions under conditions of uncertainty. Aspects describing the computational
procedures for solving the rather large MIP models as well as technical
information regarding optimality gaps, solution times etc. are discussed below.
The model-based supply chain decisions for the same realization of customer requirements
and transportation arrangements are shown in the right half of Figure 6.
Coordinating supply chain decisions: an optimization model 73
These results not only generate substantial positive contribution margins but more
importantly they re.ect coordinated decision making. For transportation arrangement
TA-I this coordination is highly visible not only in the order and shipment
decisions but also in the inventory and backlog positions. Other than at the retailer
inventory and backlog positions are greatly reduced or entirely eliminated after the
initial stock has been depleted. For the heterogeneous transportation arrangement
TA-II coordination in decision making can also be observed. The greater .exibility
in TA II is particularly visible in the smaller shipments made by the wholesaler
who has several options in choosing vehicles. The small inventory positions at the
distributor and the somewhat larger at the wholesaler impact the overall performance
and appear to be also the result of full truckload shipments and the fact that
the solution presented is still far from optimality. As expected an occasional backorder
occurs although suf.cient inventory is available. This constellation simply
indicates that it is not economical to transport a small order, in particular if only
large and expensive vehicles must be used. For both transportation arrangements
the inventory and backlog pattern at the retailer – the only stage facing market
uncertainty – result from the unknown seasonal shape of customer requirements
(see Fig. 5) and the rather simple forecasting procedure. Although it is meaningful
to include in the face of uncertainty a safety stock at the retailer, the issue is not
pursued in this paper since it is discussed in detail in Haehling von Lanzenauer and
Pilz-Glombik (2000).
Although not shown in Figure 6 similar .ndings hold for transportation arrangement
TA-III and other realizations of customer requirements. The results in Figure 6
contrast different approaches to supply chain decision making under uncertainty
and emphasize the enormous potential of realizing signi.cant bene.ts from using
analytical support systems.
Model-based decision making with varying degrees of information availability
The model-based approach will now be evaluated under varying degrees of information
availability about external customer requirements.With perfect information
the actual customer requirements over the entire planning horizon are known beforehand
and the model has therefore to be solved only once. Even for the case
of perfect information it is dif.cult to solve the large MIP model and to prove optimality.
In the case of perfect expectations only the expected values of customer
requirements in every period are given but not their actual values. Once the actual
customer requirements are known, they are used for updating the state variables and
the model will be re-solved. Thus, as in the case of no prior information the model
has to be solved several times. Care has therefore to be exercised in comparing the
different cases since the time used to reach the results differ. The key in this section
is not to quantify the exact performance differences under identical conditions but
to obtain an indication of the consequences of having better information available
about customer requirements. Nevertheless, attempts have been made to use an
“equivalent” amount of time whenever possible.
Table 3 presents the average, the minimum and maximum values of the performance
function (10) resulting from 100 random realizations of customer require74 C. Haehling von Lanzenauer and K. Pilz-Glombik
Table 3. Mean/min/max contribution for different information availability
Transportation Contribution [$]
arrangement No prior information Perfect information Perfect expectation
Mean Min Max Mean Min Max Mean Min Max
TA-I 1321 842 1651 1833 1309 2186 1737 1146 2023
TA-II 1002 627 1302 1606 1196 1902 1477 935 1783
TA-III 869 496 1188 1635 1259 1861 1461 1077 1672
ments using the various combinations of information availability and transportation
arrangements. Although all contribution margins are positive and substantial, the
average performance assuming no prior information is – as expected – well below
the corresponding values for the cases of perfect information and perfect expectation.
From the results given in Table 3 it can be concluded that a method of
forecasting reliably the expected requirements would contribute very signi.cantly
to overall systems performance. Thus, an unbiased forecasting scheme is of key
importance. The results of Table 3 also suggest that heterogeneity and/or .exibility
in the .eet of vehicles will not improve systems performance, at least not
for the parameters used in the three transportation arrangements and the solution
times allowed. The consequences different variances or different distributions for
the customer requirements generating process have on performance are still under
investigation.
The pattern of orders, inventory/backlogs and shipments for the same single
realization of customer requirements used above are presented in Figure 7 for the
cases of perfect information as well as perfect expectation and for transportation
arrangements TA-I and TA-II. Figure 7 illustrates coordinated decision making
which results from the model-based approach. As before this phenomenon is more
visible for transportation arrangement TA-I. It is interesting to note that for TA-I
and perfect expectations the initial stock positions held at upstream stations are
immediately transferred to the retailer. It appears that this result is – at least in part
– due to the fact that holding and backorder cost are the same at all stages. The
small inventory positions held at the wholesaler and distributor under TA-II seem
to be the result of the larger .exibility available in choosing vehicles and possibly
a lack of optimality.
Computational considerations of solving the MIP models
The MIP formulations presented in the optimization approach differ substantially
in size depending on the particular constellation modeled. While all formulations
include nearly 1000 continuous variables, the number of binary variables ranges
from 108 for transportation arrangement TA-I to 432 for TA-III. Regular and upper
bound restrictions amount to over 1000 depending on the transportation arrangement
used. Over and above the number of variables and restrictions is also affected
by the solution mode which at times require additional variables and/or restrictions
Coordinating supply chain decisions: an optimization model 75
Fig. 7. Model based decisions with different information availability
and at others permit the successive elimination of variables. All calculations were
carried out in a Microsoft Windows 98 environment using AMPL and CPLEX 5.0
on a PC with a Pentium III and 450 Mhz. The models were solved in two modes: a
standard process and a windowing process.
The standard process solves the model as formulated using standard parameters
within AMPL and CPLEX for preprocessing as well as for branching and bounding
rules. The standard process has been used for the generation of results which
are presented in Table 3. For each combination of transportation arrangement and
availability of information about customer requirements 100 randomly generated
realizations of customer requirements have been evaluated. The entries in Table 3
76 C. Haehling von Lanzenauer and K. Pilz-Glombik
report the mean as well as the maximum and minimum values of the performance
function (10) for these 100 realizations. Under conditions of perfect information,
when the model has to be solved only once for a given realization, a time limit of
5 minutes (300 seconds) was used for the branch and bound phase. The resulting
.nal gaps range between 5 and 10%, they are effected by the transportation arrangement
and vary signi.cantly from one realization of requirements to the next. No
attempt has been made to prove optimality. As explained above the model has to be
solved repeatedly (looping) under conditions of no prior information and perfect
expectations. A constant time limit of 30 seconds has been allowed for the branch
and bound phase in each loop, although the problem size decreases as the looping
process approaches the end of the planning horizon. Given the uncertainty present
under conditions of no prior information and perfect expectation no statements can
be made regarding the optimality gap after the allocated solution times have been
exhausted.
To improve the quality of the solution which is obtained in a predetermined
amount of time or to reduce the time necessary to reach a speci.ed gap a windowing
process has been developed. For a similar approach see Stadtler (2000). In this
process the original model is transformed into a modi.ed (relaxed) formulation
by reducing the number of binary variables used in the solution process. This is
achieved by de.ning binary variables only for a predetermined number of immediate
time periods (time window) and using continuous variables for the remainder
of the planning horizon. With fewer binary variables it is easier to .nd a good solution
to this modi.ed formulation. After .xing the supply chain decisions in the
periods prior to the current period, the time window is shifted towards the end of
the planning horizon and the process is repeated. Once the time window reaches
the end of the planning horizon a solution to the modi.ed problem represents also
a feasible solution to the original problem.
The windowing concept can also be applied to stages. The original problem
formulation is modi.ed by using binary variables only at some downstream (i.e.
those closer to the customer) stages (stage window) and continuous variables at
the remaining stages. Solving the relaxed problem for the entire planing horizon
will be considerably easier. After .xing the supply chain decisions for the stages
belonging to the stage window, the process is repeated by shifting the stage window
upstream. The process will be terminated as soon as none of the binary variables
remain relaxed. Finally, a combination of time and stage windowing is possible.
Figure 8 illustrates the steps and loops of the combined windowing process when
forecasting is required which includes all cases other than perfect information.
The windowing process has been implemented with a time window of 10 periods
or/and a stage windowof 1 assuming no prior information about future customer
requirements. The average performance results and solutions times required for 100
realizations of customer requirements are presented in Table 4. As can be observed
the average systems contribution can be improved as one moves from no window
(i.e. standard process) via a time and stage window to the combined window.
Although this improvement holds for all transportation arrangements, it is most
pronounced (in %) for transportation arrangement TA-III. It should also be noted
that the time required to obtain better solutions decreases with the time and comCoordinating supply chain decisions: an optimization model 77
Fig. 8. Windowing-process and model relaxation
Table 4. Mean contribution and mean optimization time using different solution methods
with no prior information
Transportation No window Time-window Stage-window Time- and
arrangement (standard) (tw = 10) (sw = 1) stage-window
(tw = 10; sw = 1)
C [$] Minutes C [$] Minutes C [$] Minutes C [$] Minutes
TA-I 1321 5 1338 2 1383 10 1408 2
TA-II 1002 13 1093 7 1112 18 1136 10
TA-III 869 15 884 7 1079 17 1127 7
bined windows but grows for the stage window. This underlines the effectiveness of
the windowing process, in particular the use of time windows. In general, one can
conclude that sizable MIP models supporting supply chain decision making can be
solved with reasonable efforts. The effects of implementing the windowing process
with other parameters con.gurations and scenarios are still under investigation.
Summary and conclusions
Optimizing material and information .ows in supply chains under various conditions
of information availability remains a real management challenge. Analytical
approaches supporting the supply chain decision making process can contribute
signi.cantly to improve systems performance. A MIP model which includes tactical
transportation decisions has been developed and was evaluated under varying
degrees of information availability. The modelwas implemented for a modi.ed version
of MIT’s well known Beer Distribution Game. Its use in this context permits
a contrast between human and model-based decision making. The results obtained
so far indicate a huge potential for white collar productivity gains from analytical
approaches to supply chain management. Finally, aspects of solving the computationally
dif.cult model are discussed.
78 C. Haehling von Lanzenauer and K. Pilz-Glombik
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