Linear Programming
• Introduction
George B Dantzig developed LP in 1947. It is a problem solving
approach designed to help managers/decision makers in
planning relative trade-off in resource allocations.
The word “programming” in LP should not be confused with
computer programming it simply means ‘choosing a course of
action’.
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Common Terms in LP
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Optimization
Objective Function
Constraints – Inequality/Equality
RHS values
Non-negativity
Binding Constraints
Redundant Constraints
Sensitivity Analysis
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Solution Techniques
– Algebraic
– Graphic
– Simplex Algorithm
– Spreadsheet approach [Solver]
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Four Common Elements in LP
1.
We always Maximize or minimize a problem.
2. All resources have some constraints/limits.
3. There are always choices that can be made.
4. All relationships are assumed to be linear.
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Steps in Problem Formulation
1. Understand the Problem
2. Identify the Variables
3. Define the objective Function
4. Formulate/write out the Constraints
5. Solve and check.
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Formulating LP Problems
The product-mix problem at Shader Electronics
 Two products
1. Shader X-pod, a portable music player
2. Shader BlueBerry, an internetconnected color telephone
 Determine the mix of products that will
produce the maximum profit
Formulating LP Problems
Hours Required
to Produce 1 Unit
Department
Electronic
Assembly
Profit per unit
X-pods
(X1)
BlueBerrys
(X2)
Available Hours
This Week
4
2
$7
3
1
$5
240
100
Decision Variables:
X1 = number of X-pods to be produced
X2 = number of BlueBerrys to be produced
Formulating LP Problems
Objective:
Maximize:
ST:
Constraints:
7 X1 + 5 X2
4 X1 + 3 X2  240 (Electronic)
2 X1 + 1 X2  100 (Assembly )
X1 , X2 <= 0
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Number of BlueBerrys
Formulating LP Problems
120
100
Assembly
80
60
40
Electronic
20
0
20
40
60
80
Number of X-Pods
100
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LP Applications
1. Scheduling school buses to minimize total
distance traveled
2. Allocating police patrol units to high crime
areas in order to minimize response time
to 911 calls
3. Scheduling tellers at banks so that needs
are met during each hour of the day while
minimizing the total cost of labor
LP Applications
4. Selecting the product mix in a factory to
make best use of machine- and laborhours available while maximizing the
firm’s profit
5. Picking blends of raw materials in feed
mills to produce finished feed
combinations at minimum costs
6. Determining the distribution system that
will minimize total shipping cost
LP Applications
7. Developing a production schedule that will
satisfy future demands for a firm’s product
and at the same time minimize total
production and inventory costs
8. Allocating space for a tenant mix in a new
shopping mall
so as to maximize
revenues to the
leasing company