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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

• 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.

Formulating LP Problems

The product-mix problem at Shader Electronics

Two products

1. Shader X-pod, a portable music player

2. Shader BlueBerry, an internet-connected color telephone

Determine the mix of products that will produce the maximum profit

Formulating LP Problems

X-pods BlueBerrys Available HoursDepartment (X1) (X2) This Week

Hours Required to Produce 1 Unit

Electronic 4 3 240

Assembly 2 1 100

Profit per unit $7 $5

Decision Variables:X1 = number of X-pods to be producedX2 = number of BlueBerrys to be produced

Formulating LP Problems

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Objective:

Maximize:

ST:Constraints:

) (Assembly 10012 £+

57 X2+

)(Electronic 24034 £+X1

<= 0X2X1 ,X1

X1

X2X2

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Formulating LP Problems

120

100

80

60

40

20

0

Nu

mb

er o

f B

lueB

erry

s

20 40 60 80 100Number of X-Pods

Assembly

Electronic

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 labor-hours 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