1. 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. 07458 B-1 Operations Management Linear Programming Module B

2. 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. 07458 B-2 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

3. 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. 07458 B-3 Outline - 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

4. 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. 07458 B-4 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

5. 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. 07458 B-5 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 - Continued

6. 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. 07458 B-6  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?

7. 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. 07458 B-7  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

8. 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. 07458 B-8 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

9. 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. 07458 B-9 Requirements of a Linear Programming Problem 1Must seek to maximize or minimize some quantity (the objective function) 2Presence of restrictions or constraints - limits ability to achieve objective 3Must be alternative courses of action from which to choose 4Objectives and constraints must be expressible as linear equations or inequalities

10. 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. 07458 B-10 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  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

11. 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. 07458 B-11 Formulating Linear Programming Problems - continued Let:  X 1 = number of Walkmans  X 2 = number of Watch-TVs Then:  4X 1 + 3X 2  240electronics constraint  2X 1 + 1X 2  100assembly constraint  7X 1 + 5X 2 = profitmaximize profit

12. 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. 07458 B-12  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 Graphical Solution Method

13. 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. 07458 B-13 Shader Electronic Company Problem Hours Required to Produce 1 Unit DepartmentX1X1 Walkmans X2X2 Watch-TV’s Available Hours This Week Electronic43240 Assembly21100 Profit/unit$7$5 Constraints: 4x 1 + 3x 2  240 (Hours of Electronic Time) 2x 1 + 1x 2  100 (Hours of Assembly Time) Objective:Maximize: 7x 1 + 5x 2

14. 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. 07458 B-14 Shader Electronic Company Constraints 0 20 40 60 80 100 120 01020304050607080 Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Electronics (Constraint A) Assembly (Constraint B)

15. 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. 07458 B-15 Shader Electronic Company Feasible Region 0 20 40 60 80 100 120 01020304050607080 Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Feasible Region Electronics (Constraint A) Assembly (Constraint B)

16. 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. 07458 B-16 Shader Electronic Company Iso-Profit Lines 0 20 40 60 80 100 120 01020304050607080 Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) 7*X 1 + 5*X 2 = 210 7*X 1 + 5*X 2 = 420 Electronics (Constraint A) Assembly (Constraint B) Iso-profit line

17. 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. 07458 B-17 Shader Electronic Company Corner Point Solutions 0 20 40 60 80 100 120 01020304050607080 Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Iso-profit line Electronics (Constraint A) Assembly (Constraint B) Possible Corner Point Solution

18. 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. 07458 B-18 Shader Electronic Company Optimal Solution 0 20 40 60 80 100 120 01020304050607080 Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Optimal solution Iso-profit line Electronics (Constraint A) Assembly (Constraint B) Possible Corner Point Solution

19. 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. 07458 B-19 Shader Electronic Company Optimal Solution 0 20 40 60 80 100 120 010203040507080 Number of Walkmans (X 1 ) Number of Watch-TVs (X 2 ) Optimal solution Iso-profit line Electronics (Constraint A) Assembly (Constraint B) Possible Corner Point Solution X 1 = 30 X 2 = 40 60

20. 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. 07458 B-20  Decision variables  X 1 = tons of BW chemical produced  X 2 = tons of color chemical produced  Objective  Minimize Z = 2500 X 1 + 3000 X 2  Constraints  X 1  30 (BW); X 2  20 (Color)  X 1 + X 2  60 (Total tonnage)  X 1  0; X 2  0 (Non-negativity) Formulation of Solution

21. 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. 07458 B-21 Simplex Steps for Maximization 1.Choose the variable with the greatest positive C j - Z j to enter the solution 2.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 row 4.Calculate the new values for the other row(s) 5.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.

22. 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. 07458 B-22 Simplex Steps for Minimization 1Choose the variable with the greatest negative C j - Z j to enter the solution 2Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity- to-pivot column ratio 3Calculate the new values for the pivot row 4Calculate the new values for the other row(s) 5Calculate the C j and C j -Z j values for this tableau. If there are any C j -Z j numbers less than zero, return to step 1.

23. 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. 07458 B-23 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

24. 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. 07458 B-24 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 manufacturing cost per month Color: $ 3,000 manufacturing cost per month © 1995 Corel Corp. Minimization Example

25. 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. 07458 B-25 Graphical Solution X1X1 Feasible Region 0 20 40 60 80 0 Tons, Color Chemical (X 2 ) 20406080 Tons, BW Chemical (X 1 ) BW Color Total Find values for X 1 + X 2  60. X 1  30, X 2  20. X1X1 X2X2

26. 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. 07458 B-26 Optimal Solution: Corner Point Method Feasible Region 0 20 40 60 80 0 Tons, Color Chemical 20406080 Tons, BW Chemical BW Color Total A B Find corner points

27. 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. 07458 B-27 Assembly Constraint RHS Increased by 10 X1X1 0 20 40 60 80 100 0204060 Original assembly constraint Assembly constraint increased by 10 Sol’n X2X2 Original Solution Electronics Constraint New Solution Feasible Region

28. 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. 07458 B-28 Assembly Constraint RHS Decreased by 10 X1X1 0 20 40 60 80 100 0204060 Original assembly constraint Sol’n X2X2 Assembly constraint decreased by 10 Original Solution New Solution

29. 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. 07458 B-29 A Minimization Problem Feasible region X 1 = 30 X 2 = 20 x 1 + x 2 = 60 a b