0perationsmanagement-1

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Operations

Management

Linear Programming

Module B

Transparency Masters toaccompany Heizer/RenderPrinciples of OperationsManagement, 5e, and Operations

2004 by Prentice Hall, Inc., Upper SaddleRiver, N.J. 07458

B-1


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Outline

REQUIREMENTS OF A LINEARPROGRAMMING PROBLEM

FORMULATING LINEAR PROGRAMMING

PROBLEMS Shader Electronics example

GRAPHICAL SOLUTION TO A LINEARPROGRAMMING PROBLEM

Graphical representation of Constraints

Iso-Profit Line Solution Method

Corner-Point Solution Method



B-2


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Outline - Continued

SENSITIVITY ANALYSIS Sensitivity Report Change in the Resources of the Right-Hand-

Side Values

Changes in the Objective FunctionCoefficient

SOLVING MINIMIZATION PROBLEMS

LINEAR PROGRAMMING APPLICATIONS

Production Mix Example Diet Problem Example Production Scheduling Example Labor Scheduling Example

THE SIMPLEX METHOD OF LPTransparency Masters toaccompany Heizer/RenderPrinciples of OperationsManagement, 5e, and Operations


B-3


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Learning Objectives - Continued

When you complete this chapter, you should

be able to :

Describe or Explain:

How to formulate linear models

Graphical method of linear programming

How to interpret sensitivity analysis



B-5


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What is Linear Programming?

Mathematical technique

Not computer programming

Allocates scarce resources to achieve anobjective

Pioneered by George Dantzig in World

War II Developed workable solution called

Simplex Method in 1947



B-6


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Examples of Successful LPApplications

Scheduling school busses to minimize total

distance traveled when carrying students

Allocating police patrol units to high crimeareas in order to minimize response time

to 911 calls

Scheduling tellers at banks to that needsare met during each hour of the day while

minimizing the total cost of labor



B-7


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Examples of Successful LPApplications - Continued

Picking blends of raw materials in feedmills to produce finished feedcombinations at minimum costs

Selecting the product mix in a factory tomake best use of machine- and labor-hoursavailable while maximizingthe firms profit

Allocating space for a tenant mix in a newshopping mall so as to maximize revenuesto the leasing company



B-8


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Requirements of a LinearProgramming Problem

1 Must seek to maximize or minimizesomequantity (the objective function)

2 Presence of restrictions or constraints -limits ability to achieve objective

3 Must be alternative courses of actionfromwhich to choose

4 Objectives and constraints must beexpressible as linearequations orinequalities



B-9


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Formulating Linear Programming

ProblemsAssume:

You wish to produce two products (1) WalkmanAM/FM/Cassette and (2) Watch-TV

Walkman takes 4 hours of electronic work and 2

hours assembly Watch-TV takes 3 hours electronic work and 1

hour assembly

There are 240 hours of electronic work time and100 hours of assembly time available

Profit on a Walkman is $7; profit on a Watch-TV$5



B-10


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Formulating Linear Programming

Problems - continued

Let:

X1 = number of Walkmans

X2

= number of Watch-TVs

Then:

4X1 + 3X2 240 electronics constraint

2 X1 + 1X2 100 assemblyconstraint

7X1 + 5X2 = profit maximize profit



B-11


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Graphical Solution Method

Draw graph with vertical & horizontal

axes (1st quadrant only)

Plot constraints as lines, then as planes Use (X1,0), (0,X2) for line

Findfeasible region

Find optimalsolution Corner point method

Iso-profit line method



B-12


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Shader Electronic Company Problem

Transparency Masters to accompanyHeizer/Render Principles of OperationsManagement, 5e, and Operations Management,7e


B-13

Hours Required toProduce 1 Unit

Department X1Walkmans

X2Watch-TVs

Available HoursThis Week

Electronic 4 3 240Assembly 2 1 100

Profit/unit $7 $5

Constraints: 4x1 + 3x2 240 (Hours of Electronic Time)2x1 + 1x2 100 (Hours of Assembly Time)

Objective: Maximize: 7x1 + 5x2


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Shader Electronic CompanyConstraints



B-14

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80

Number of Walkmans (X1)

NumberofWatch-T

Vs(X2

)

Electronics(Constraint A)

Assembly(Constraint B)


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Shader Electronic CompanyFeasible Region



B-15

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80


Nu

mberofWatch-TVs(X

2)

FeasibleRegion




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Shader Electronic CompanyIso-Profit Lines



B-16

0

20

40

60

80

100120

0 10 20 30 40 50 60 70 80


NumberofWatch-T

Vs(X

2)



Iso-profit line


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Shader Electronic Company

Corner Point Solutions



B-17

020

40

60

80

100

120

0 10 20 30 40 50 60 70 80


Num

berofWatch-T

Vs(X

2)

Iso-profit line



Possible CornerPoint Solution


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Optimal Solution



B-18

020

40

60

80

100

120

0 10 20 30 40 50 60 70 80


Num

berofWatch-T

Vs(X

2)

Optimal solution

Iso-profit line



Possible Corner

Point Solution
http://localhost/var/www/apps/conversion/tmp/scratch_9/module_B.PPT


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Optimal Solution



B-19

020

40

60

80

100

120

0 10 20 30 40 50 70 80


Num

berofWatch-TV

s(X

2)

Optimal solution

Iso-profit line



Possible CornerPoint Solution

X1 = 30

X2 = 40

60
http://localhost/var/www/apps/conversion/tmp/scratch_9/module_B.PPT


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Formulation of Solution

Decision variables X1 = tons of BW chemical produced

X2 = tons of color chemical produced

Objective MinimizeZ= 2500X1 + 3000X2 Constraints

X130 (BW); X220 (Color)

X1 +X260 (Total tonnage)

X1 0;X2 0 (Non-negativity)



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Simplex Steps for Maximization

1. Choose the variable with the greatest positive Cj- Zj toenter the solution

2. Determine the row to be replaced by selecting thatone with the smallest (non-negative) quantity-to-pivotcolumn ratio

3. Calculate the new values for the pivot row

4. Calculate the new values for the other row(s)

5. Calculate the Cj and Cj-Zj values for this tableau.

If there are any Cj-Zj numbers greater than zero, returnto step 1.



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Simplex Steps for Minimization1 Choose the variable with the greatest negative

Cj- Zj to enter the solution2 Determine the row to be replaced by selecting

that one with the smallest (non-negative)quantity-to-pivot column ratio

3 Calculate the new values for the pivot row4 Calculate the new values for the other row(s)

5 Calculate the Cj and Cj-Zj values for thistableau. If there are any Cj-Zj numbers less

than zero, return to step 1.



B-22


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Sensitivity Analysis

Projects how much a solution might change if

there were changes in variables or input data.

Shadow price (dual) - value of one additional

unit of a resource



B-23


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Minimization Example

Youre an analyst for a division of

Kodak, which makes BW & color

chemicals. At least 30 tons of BW

and at least 20 tons of colormust

be made each month. The total

chemicals made must be at least 60

tons. How many tons of each

chemical should be made tominimize costs?



B-24

BW: $2,500manufacturing costper month

Color: $ 3,000 manufacturing costper month

1995 Corel Corp.


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Graphical Solution



B-25

X1

Feasible

Region

0

20

40

60

80

0

Tons,

ColorChe

mical(X

2)

20 40 60 80Tons, BW Chemical (X1)

BW

Color

Total

Find values forX1 +X2 60.

X130, X220.

X1

X2

Optimal Solution:


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Optimal Solution:Corner Point Method



B-26

FeasibleRegion

0

20

40

60

80

0

Tons,

ColorChe

mical

20 40 60 80Tons, BW Chemical

BW

Color

Total

A

B

Find corner points

A bl C t i t RHS


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Assembly Constraint RHSIncreased by 10



B-27

X10

20

40

60

80

100

0 20 40 60

Originalassemblyconstraint

Assembly constraintincreased by 10

Soln

Soln

X2

Original Solution

Electronics

Constraint

NewSolution

FeasibleRegion

A bl C t i t RHS


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Assembly Constraint RHSDecreased by 10



B-28

X10

20

40

60

80

100

0 20 40 60

Originalassemblyconstraint

Soln

Soln

X2

Assemblyconstraint

decreased by 10

OriginalSolution

NewSolution


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A Minimization Problem



B-29

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Feasible

region

X1 = 30 X2 = 20

x1 + x2 = 60

a

b

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