Download x - UTC.edu

Chapter 2
An Introduction to Linear Programming :
Graphical and Computer Methods
Professor Ahmadi
Slide 1
Learning Objectives








Understand basic assumptions and properties of
linear programming (LP)
General LP notation
LP formulation of the model
A Maximization Problem
Graphical solution
A Minimization Problem
Special Cases
Formulating a Spreadsheet Model
Slide 2
Introduction

Management decisions in many organizations involve
trying to make most effective use of resources.
• Include machinery, labor, money, time, warehouse
space, and raw materials.
• May be used to produce products - such as
computers, automobiles, or clothing or
• Provide services - such as package delivery, health
services, or investment decisions.
Slide 3
Mathematical Programming

Mathematical programming is used for resource allocation
problems.

Linear programming (LP) is the most common type of
mathematical programming.

One assumes that all the relevant input data and
parameters are known with certainty in models
(deterministic models).

Computers play an important role in the advancement
and use of LP.
Slide 4
Development of a LP Model

LP has been applied extensively to various problems areas • medical, transportation, operations, petroleum
• financial, marketing, accounting,
• human resources, agriculture, and others

Development of all LP models can be examined in a three
step process:
• (1) formulation
• (2) solution and
• (3) interpretation.
Slide 5
Properties of a LP Model
1.
All problems seek to maximize or minimize some
quantity, usually profit or cost (called the objective
function).
2.
LP models usually include restrictions, or constraints,
that limit the degree to which one can pursue the
objective.
3.
There must be alternative courses of action from which
one can choose.
4.
The objective and constraints in LP problems must be
expressed in terms of linear equations or inequalities.
Slide 6
Three Steps of Developing LP Problem
Formulation.
• Process of translating problem scenario into a simple LP model
framework with a set of mathematical relationships.
Solution.
• Mathematical relationships resulting from the formulation
process are solved to identify an optimal solution.
Interpretation and What-if Analysis.
• Problem solver or analyst works with manager to

Interpret results and implications of problem solution.

Investigate changes in input parameters and model
variables and impact on problem solution results.
Slide 7
Basic Assumptions of a LP Model
1.
Conditions of certainty exist.
2.
Proportionality in the objective function and constraints
(1 unit – 3 hours, 3 units 9 hours).
3.
Additivity (the total of all activities equals the sum of
the individual activities).
4.
Divisibility assumption that solutions need not
necessarily be in whole numbers (integers).
Slide 8
Linear Equations and Inequalities

This is a linear equation:
2X1 + 15X2 = 10

This equation is not linear:
5X+4X2 + 15X3 = 100

LP uses, in many cases, inequalities like:
X1 + X2  C
or
X1 + X2  C
Slide 9
LP Solutions




The maximization or minimization of some quantity
is the objective in all linear programming problems.
A feasible solution satisfies all the problem's
constraints.
Changes to the objective function coefficients do not
affect the feasibility of the problem.
An optimal solution is a feasible solution that results
in the largest possible objective function value, z,
when maximizing or smallest z when minimizing.
Slide 10
LP Solutions



A feasible region may be unbounded and yet there
may be optimal solutions. This is common in
minimization problems and is possible in
maximization problems.
The feasible region for a two-variable linear
programming problem can be a single point, a line, a
polygon, or an unbounded area.
Any linear program falls in one of three categories:
• is infeasible
• has a unique optimal solution or alternate optimal
solutions
• has an objective function that can be increased
without bound (Unbounded)
Slide 11
LP Solutions





A graphical solution method can be used to solve a
linear program with two variables.
If a linear program possesses an optimal solution,
then an extreme point will be optimal.
If a constraint can be removed without affecting the
shape of the feasible region, the constraint is said to
be redundant.
A non-binding constraint is one in which there is
positive slack or surplus when evaluated at the
optimal solution.
A linear program which is over-constrained so that
no point satisfies all the constraints is said to be
infeasible.
Slide 12
Guidelines for Model Formulation





Understand the problem thoroughly.
Write a verbal statement of the objective function and
each constraint.
Define the decision variables.
Write the objective function in terms of the decision
variables.
Write the constraints in terms of the decision variables.
Slide 13
A Simple Maximization Problem
Olympic Bike is introducing two new lightweight
bicycle frames, the Deluxe and the Professional, to be
made from special aluminum and steel alloys. The
anticipated unit profits are $10 for the Deluxe and $15
for the Professional.
The number of pounds of each alloy needed per
frame is summarized below. A supplier delivers 100
pounds of the aluminum alloy and 80 pounds of the
steel alloy weekly. How many Deluxe and Professional
frames should Olympic produce each week?
Aluminum Alloy Steel Alloy
Deluxe
2
3
Professional
4
2
Slide 14
Max. Example: Olympic Bike Co.

Model Formulation
• Verbal Statement of the Objective Function
Maximize total weekly profit.
• Verbal Statement of the Constraints
Total weekly usage of aluminum alloy < 100 pounds.
Total weekly usage of steel alloy < 80 pounds.
• Definition of the Decision Variables
x1 = number of Deluxe frames produced weekly.
x2 = number of Professional frames produced weekly.
Slide 15
Max. Example: Olympic Bike Co.

Model Formulation (Continued)
Max 10x1 + 15x2
s.t.
(Total Weekly Profit)
2x1 + 4x2 < 100 (Aluminum Available)
3x1 + 2x2 < 80 (Steel Available)
x1, x2 > 0
(Non-negativity)
Slide 16
Max. Example: Olympic Bike Co.

Graphical Solution Procedure
x2
3x1 + 2x2 < 80 (Steel)
40
Optimal x1 = 15, x2 = 17 1/2, z = $ 412.50
35
30
2x1 + 4x2 < 100 (aluminum)
25
20
MAX 10x1 + 15x2
15
10
5
5
10
15
20
25
30
35
40
45
50
x1
Slide 17
Slack and Surplus Variables




A linear program in which all the variables are nonnegative and all the constraints are equalities is said
to be in standard form.
Standard form is attained by adding slack variables
to "less than or equal to" constraints, and by
subtracting surplus variables from "greater than or
equal to" constraints.
Slack and surplus variables represent the difference
between the left and right sides of the constraints.
Slack and surplus variables have objective function
coefficients equal to 0.
Slide 18
A Simple Minimization Problem

Solve graphically for the optimal solution:
Min z = 5x1 + 15x2
s.t.
x1 + x2 > 500
x1
< 400
x2 > 200
x1, x2 > 0
Slide 19
Special Cases



Alternative Optimal Solutions
Infeasible Solutions
Unbounded Problem
Slide 20
Alternative Optimal Solutions

In the graphical method, if the objective function line is
parallel to a boundary constraint in the direction of
optimization, there are alternate optimal solutions, with
all points on this line segment being optimal.
Example:
Max z = 8x1 + 6x2
s.t.
4x1 + 3x2 < 12
9x1 + 12 x2 < 36
x1, x2 > 0
Slide 21
Example: Alternative Optimal Solutions

There are numerous optimal points. The objective
function (8x1 + 6x2) is parallel to the first constraint.
x2
4x1 + 3x2 <
12
4
9x1 + 12 x2 < 36
3
Objective
function
3
4
x1
Slide 22
Example: Infeasible Problem

Solve graphically for the optimal solution:
Max
s.t.
z = 2x1 + 6x2
4x1 + 3x2 < 12
2x1 + x2 > 8
x1, x2 > 0
Slide 23
Example: Infeasible Problem

There are no points that satisfy both constraints, hence
this problem has no feasible region, and no optimal
solution.
x2
8
4x1 + 3x2 < 12
2x1 + x2 > 8
4
3 4
x1
Slide 24
Example: Unbounded Problem

Solve graphically for the optimal solution:
Max
s.t.
z = 3x1 + 4x2
x1 + x2 > 5
3x1 + x2 > 8
x1, x2 > 0
Slide 25
Example: Unbounded Problem

The feasible region is unbounded and the objective
function line can be moved parallel to itself without
bound so that z can be increased infinitely.
x2
3x1 + x2 > 8
8
5
x1 + x2 > 5
Max 3x1 + 4x2
2.67
5
x1
Slide 26
Using Excel for solving LP problems

Use “Solver” in Excel and solve the LP problems given previously.
• Tools/Solver
End of chapter 2
Slide 27