Chapter 1, Heizer/Render, 5th edition

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Transcript Chapter 1, Heizer/Render, 5th edition

Operations Management

Linear Programming Module B

Outline



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 - continued



Sensitivity Analysis



Solving Minimization Problems



Linear Programming Applications



Production Mix Example



Diet Problem Example



Production Scheduling Example



Labor Scheduling Example



The Simplex Method of LP

Learning Objectives

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 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

What is Linear Programming?



Mathematical technique



Not computer programming



Allocates scarce resources to achieve an objective



Pioneered by George Dantzig in World War II



Developed workable solution in 1947

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Called Simplex Method

Examples of Successful LP Applications



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 Applications - 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 Programming Problem

1 2 3 4

Must seek to

maximize

minimize

quantity (the objective function) some Presence of restrictions or

constraints

limits ability to achieve objective Must be

alternative courses of action

which to choose from Objectives and constraints must be expressible as

linear

equations or inequalities

Formulating Linear Programming Problems



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

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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 Problems - continued



Let:

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X 1 = number of Walkmans



X 2



Then: = number of Watch-TVs

  

4X 1 2X 1 7X 1 + 3X 2 + 1X 2 + 5X 2

 

240 100 = profit electronics constraint assembly constraint maximize profit

Graphical Solution Method

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



Plot constraints as lines, then as planes

 

Use (X 1 ,0), (0,X 2 ) for line Find feasible region



Find



optimal

solution Corner point method



Iso-profit line method

Shader Electronic Company Problem

Department Electronic Assembly Hours Required to Produce 1 Unit X Walkmans 4 2 1 X Watch-TV’s 3 1 2 Available Hours This Week 240 100 Profit/unit $7 $5 Constraints: 4x 1 2x 1 Objective: + 3x + 1x 2 2   240 (Hours of Electronic Time) 100 (Hours of Assembly Time) Maximize: 7x 1 + 5x 2 PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) B-13 © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

120 100 80 60 40 20 0 0

Shader Electronic Company Constraints

Electronics (Constraint A) Assembly (Constraint B) 10 20 30 40 50 Number of Walkmans (X 1 ) 60 70 80

Shader Electronic Company Feasible Region

120 100 80 60 40 20 0 0 Feasible Region 10 Electronics (Constraint A) Assembly (Constraint B) 20 30 40 50 Number of Walkmans (X 1 ) 60 70 80

Shader Electronic Company Iso-Profit Lines

120 100 80 60 40 20 0 0 10

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render)

Electronics (Constraint A) Assembly (Constraint B) 20 30

B-16

40 50 Number of Walkmans (X 1 ) 60 70 80

Shader Electronic Company Solution

120 100 80 60 40 20 0 0 10

PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render)

ISO-Profit Line (X 1 Electronics (Constraint A) Assembly (Constraint B) Solution Point =30, X 2 =40) 20 30 40 50 Number of Walkmans (X 1 ) 60

B-17

70 80

Shader Electronic Company Solution Corner Point Solution

120 100 Electronics (Constraint A) Assembly (Constraint B) 80 60 40 Possible Corner Point Solution Optimal solution 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1 )

Formulation of Solution

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Decision variables

   

2 = tons of BW chemical produced = tons of color chemical produced Objective

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Minimize Z = 2500X 1 Constraints

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1 + 3000X 2



30 (BW); X 2



20 (Color) + X 2

 

0; X 2 60 (Total tonnage)



0 (Non-negativity)

Simplex Steps for Maximization

3 4 5 1 2

Choose the variable with the greatest positive C j - Z j to enter the solution Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-to-pivot column ratio Calculate the new values for the pivot row Calculate the new values for the other row(s) Calculate the C j and C j -Z j values for this tableau.

If there are any C j -Z j numbers greater than zero, return to step 1.

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

Minimization Example

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 BW: $2,500 manufacturing cost per month

tons . How many tons of each chemical should be made to

minimize

costs?

Color: $ 3,000 manufacturing cost per month

Graphical Solution

80 60 Tons, Color Chemical (X 2 ) 40 20 0 0 PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) Total BW 20 B-23 40 60

Find values for

1 + X 2 60.

X 1



30, X 2



20.

Optimal Solution: Corner Point Method

80 BW

Find corner points.

60 Tons, Color Chemical 40 Total Feasible Region B 20 Color A 0 0 PowerPoint presentation to accompany Operations Management, 6E (Heizer & Render) 20 40 60 Tons, BW Chemical B-24 80 © 2001 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

Simplex Steps for Minimization

Choose the variable with the greatest negative C j - Z j to enter the solution

Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-to pivot column ratio

Calculate the new values for the pivot row

Calculate the new values for the other row(s)

Calculate the C j and C j -Z j there are any C j -Z j step 1.

values for this tableau. If numbers less than zero, return to