0perationsManagement 1 Contoh LP Urutan Simpleks

Transparency Masters to accompany Heizer/Render – Principles of Operations Management, 5e, and Operations Management, 7e

© 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458B-1

Operations Operations ManagementManagement

Linear ProgrammingLinear ProgrammingModule BModule B



OutlineOutline REQUIREMENTS OF A LINEAR PROGRAMMING

PROBLEM FORMULATING LINEAR PROGRAMMING

PROBLEMS Shader Electronics example

GRAPHICAL SOLUTION TO A LINEAR PROGRAMMING PROBLEM Graphical representation of Constraints Iso-Profit Line Solution Method Corner-Point Solution Method



Outline - ContinuedOutline - Continued

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

Values Changes in the Objective Function Coefficient

SOLVING MINIMIZATION PROBLEMS LINEAR PROGRAMMING APPLICATIONS

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

THE SIMPLEX METHOD OF LP



When you complete this chapter, you should be able to :

Identify or Define: Objective function Constraints Feasible region Iso-profit/iso-cost methods Corner-point solution Shadow price

Learning ObjectivesLearning Objectives



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

Learning Objectives - ContinuedLearning Objectives - Continued



Mathematical technique Not computer programming

Allocates scarce resources to achieve an objective

Pioneered by George Dantzig in World War II Developed workable solution called Simplex

Method in 1947

What is Linear Programming?What is Linear Programming?



Scheduling school busses to minimize total distance traveled when carrying students

Allocating police patrol units to high crime areas in order to minimize response time to 911 calls

Scheduling tellers at banks to that needs are met during each hour of the day while minimizing the total cost of labor

Examples of Successful LP Examples of Successful LP ApplicationsApplications



Examples of Successful LP Examples of Successful LP Applications - ContinuedApplications - Continued

Picking blends of raw materials in feed mills to produce finished feed combinations at minimum costs

Selecting the product mix in a factory to make best use of machine- and labor-hours available while maximizing the firm’s profit

Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the leasing company



Requirements of a Linear Requirements of a Linear Programming ProblemProgramming Problem

1 Must seek to maximize or minimize some quantity (the objective function)

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

3 Must be alternative courses of action from which to choose

4 Objectives and constraints must be expressible as linear equations or inequalities



Formulating Linear Programming Formulating Linear Programming ProblemsProblems

Assume: You wish to produce two products (1) Walkman

AM/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 and 100

hours of assembly time available Profit on a Walkman is $7; profit on a Watch-TV $5



Formulating Linear Programming Formulating Linear Programming Problems - continuedProblems - continued

Let: X1 = number of Walkmans X2 = number of Watch-TVs

Then: 4X1 + 3X2 240 electronics constraint 2 X1 + 1X2 100 assembly

constraint 7X1 + 5X2 = profit maximize profit



Draw graph with vertical & horizontal axes (1st quadrant only)

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

Find feasible region Find optimal solution

Corner point method Iso-profit line method

Graphical Solution MethodGraphical Solution Method



Shader Electronic Shader Electronic CompanyCompany Problem Problem

Hours Required toProduce 1 Unit

Department X1Walkmans

X2Watch-TV’s

Available HoursThis Week

Electronic 4 3 240

Assembly 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



Shader Electronic Company Shader Electronic Company ConstraintsConstraints

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80

Number of Walkmans (X1)

Num

ber o

f Wat

ch-T

Vs (X

2)

Electronics(Constraint A)Assembly(Constraint B)



Shader Electronic Company Shader Electronic Company Feasible RegionFeasible Region

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80


Num

ber o

f Wat

ch-T

Vs (X

2)

FeasibleRegion




Shader Electronic CompanyShader Electronic CompanyIso-Profit LinesIso-Profit Lines

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80


Num

ber o

f Wat

ch-T

Vs (X

2)

7*X1 + 5*X

2 = 210

7*X1 + 5*X2 = 420


Iso-profit line



Shader Electronic Company Shader Electronic Company Corner Point SolutionsCorner Point Solutions

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80


Num

ber o

f Wat

ch-T

Vs (X

2)

Iso-profit line


Possible Corner Point Solution



Shader Electronic Company Shader Electronic Company Optimal SolutionOptimal Solution

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80


Num

ber o

f Wat

ch-T

Vs (X

2)

Optimal solution

Iso-profit line





Shader Electronic Company Shader Electronic Company Optimal SolutionOptimal Solution

0

20

40

60

80

100

120

0 10 20 30 40 50 70 80


Num

ber o

f Wat

ch-T

Vs (X

2)

Optimal solution

Iso-profit line



X1 = 30

X2 = 40

60



Decision variables X1 = tons of BW chemical produced X2 = tons of color chemical produced

Objective Minimize Z = 2500X1 + 3000X2

Constraints X1 30 (BW); X2 20 (Color) X1 + X2 60 (Total tonnage) X1 0; X2 0 (Non-negativity)

Formulation of SolutionFormulation of Solution



Simplex Steps for MaximizationSimplex Steps for Maximization1. Choose the variable with the greatest positive 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 ratio3. 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 this tableau.

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



Simplex Steps for MinimizationSimplex 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 this tableau. If there

are any Cj-Zj numbers less than zero, return to step 1.



Sensitivity AnalysisSensitivity 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



You’re an analyst for a division of Kodak, which makes BW & color chemicals. At least 30 tons of BW and at least 20 tons of color must be made each month. The total chemicals made must be at least 60 tons. How many tons of each chemical should be made to minimize costs?

BW: $2,500 BW: $2,500 manufacturing cost manufacturing cost per monthper month

Color: $ 3,000 manufacturing cost per month

© 1995 Corel Corp.

Minimization ExampleMinimization Example



Graphical SolutionGraphical Solution

X1

Feasible Region

0

20

40

60

80

0

Tons

, Col

or C

hem

ical (

XTo

ns, C

olor

Che

mica

l (X 22))

20 40 60 80Tons, BW Chemical (X1)

BW

Color

Total

Find values for X1 + X2 60.

X1 30, X2 20.

X1

X2



Optimal Solution: Optimal Solution: Corner Point MethodCorner Point Method

Feasible Region

0

20

40

60

80

0

Tons

, Col

or C

hem

ical

Tons

, Col

or C

hem

ical

20 40 60 80Tons, BW Chemical

BW

Color

Total

A

B

Find corner points



Assembly Constraint RHS Assembly Constraint RHS Increased by 10Increased by 10

X10

20

40

60

80

100

0 20 40 60

Original assembly constraint

Assembly constraint increased by 10

Sol’n

Sol’n

X2

Original Solution

Electronics Constraint

New Solution

Feasible Region



Assembly Constraint RHS Assembly Constraint RHS Decreased by 10Decreased by 10

X10

20

40

60

80

100

0 20 40 60

Original assembly constraint

Sol’n

Sol’n

X2

Assembly constraint

decreased by 10

Original Solution

New Solution



0

10

20

30

40

50

60

0 10 20 30 40 50 60

A Minimization ProblemA Minimization Problem

Feasible region

X1 = 30 X2 = 20

x1 + x2 = 60

a

b

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0perationsManagement 1 Contoh LP Urutan Simpleks