1. Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Operations Management - 5 th Edition Chapter 13 Supplement Roberta Russell & Bernard W. Taylor, III Linear Programming

2. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-2 Lecture Outline   Model Formulation   Graphical Solution Method   Linear Programming Model   Solution   Solving Linear Programming Problems with Excel   Sensitivity Analysis

3. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-3 A model consisting of linear relationships representing a firm’s objective and resource constraints Linear Programming (LP) LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints

4. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-4 Types of LP

5. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-5 Types of LP (cont.)

6. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-6 Types of LP (cont.)

7. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-7 LP Model Formulation   Decision variables mathematical symbols representing levels of activity of an operation   Objective function a linear relationship reflecting the objective of an operation most frequent objective of business firms is to maximize profit most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost   Constraint a linear relationship representing a restriction on decision making

8. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-8 LP Model Formulation (cont.) Max/min z = c 1 x 1 + c 2 x 2 +... + c n x n subject to: a 11 x 1 + a 12 x 2 +... + a 1n x n (≤, =, ≥) b 1 a 21 x 1 + a 22 x 2 +... + a 2n x n (≤, =, ≥) b 2 : a m1 x1 + a m2 x 2 +... + a mn x n (≤, =, ≥) b m a m1 x1 + a m2 x 2 +... + a mn x n (≤, =, ≥) b m x j = decision variables b i = constraint levels c j = objective function coefficients a ij = constraint coefficients

9. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-9 LP Model: Example LaborClayRevenue PRODUCT(hr/unit)(lb/unit)($/unit) Bowl1440 Mug2350 There are 40 hours of labor and 120 pounds of clay available each day Decision variables x 1 = number of bowls to produce x 2 = number of mugs to produce RESOURCE REQUIREMENTS

10. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-10 LP Formulation: Example Maximize Z = $40 x 1 + 50 x 2 Subject to x 1 +2x 2  40 hr(labor constraint) 4x 1 +3x 2  120 lb(clay constraint) x 1, x 2  0 Solution is x 1 = 24 bowls x 2 = 8 mugs Revenue = $1,360

11. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-11 Graphical Solution Method 1.Plot model constraint on a set of coordinates in a plane 2.Identify the feasible solution space on the graph where all constraints are satisfied simultaneously 3.Plot objective function to find the point on boundary of this space that maximizes (or minimizes) value of objective function

12. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-12 Graphical Solution: Example 4 x 1 + 3 x 2  120 lb x 1 + 2 x 2  40 hr Area common to both constraints 50 50 – 40 40 – 30 30 – 20 20 – 10 10 – 0 0 – |10 60 50 20 30 40 x1x1x1x1 x2x2x2x2

13. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-13 Computing Optimal Values x 1 +2x 2 =40 4x 1 +3x 2 =120 4x 1 +8x 2 =160 -4x 1 -3x 2 =-120 5x 2 =40 x 2 =8 x 1 +2(8)=40 x 1 =24 4 x 1 + 3 x 2  120 lb x 1 + 2 x 2  40 hr 40 40 – 30 30 – 20 20 – 10 10 – 0 0 – |10 20 30 40 x1x1x1x1 x2x2x2x2 Z = $50(24) + $50(8) = $1,360 24 8

14. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-14 Extreme Corner Points x 1 = 224 bowls x 2 =  8 mugs Z = $1,360 x 1 = 30 bowls x 2 =  0 mugs Z = $1,200 x 1 = 0 bowls x 2 =  20 mugs Z = $1,000 A B C |20 30 40 10 x1x1x1x1 x2x2x2x2 40 40 – 30 30 – 20 20 – 10 10 – 0 0 –

15. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-15 4x 1 + 3x 2  120 lb x 1 + 2x 2  40 hr 40 40 – 30 30 – 20 20 – 10 10 – 0 0 – B |10 20 30 40 x1x1x1x1 x2x2x2x2 C A Z = 70x 1 + 20x 2 Optimal point: x 1 = 30 bowls x 2 =  0 mugs Z = $2,100 Objective Function

16. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-16 Minimization Problem CHEMICAL CONTRIBUTION BrandNitrogen (lb/bag)Phosphate (lb/bag) Gro-plus24 Crop-fast43 Minimize Z = $6x 1 + $3x 2 subject to 2x 1 +4x 2  16 lb of nitrogen 4x 1 +3x 2  24 lb of phosphate x 1, x 2  0

17. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-17 14 14 – 12 12 – 10 10 – 8 8 – 6 6 – 4 4 – 2 2 – 0 0 – |22|222 |44|444 |66|666 |88|888 |10 12 14 x1x1x1x1 x2x2x2x2 A B C Graphical Solution x 1 = 0 bags of Gro-plus x 2 = 8 bags of Crop-fast Z = $24 Z = 6x 1 + 3x 2

18. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-18 Simplex Method  A mathematical procedure for solving linear programming problems according to a set of steps  Slack variables added to ≤ constraints to represent unused resources x 1 + 2x 2 + s 1 =  40 hours of labor x 1 + 2x 2 + s 1 =  40 hours of labor 4x 1 + 3x 2 + s 2 =  120 lb of clay 4x 1 + 3x 2 + s 2 =  120 lb of clay  variables subtracted from ≥ constraints to represent excess above resource requirement. For example  Surplus variables subtracted from ≥ constraints to represent excess above resource requirement. For example 2x 1 + 4x 2 ≥  16 is transformed into 2x 1 + 4x 2 ≥  16 is transformed into 2x 1 + 4x 2 - s 1 =  16 2x 1 + 4x 2 - s 1 =  16  Slack/surplus variables have a 0 coefficient in the objective function Z = $40x 1 + $50x 2 + 0s 1 + 0s 2 Z = $40x 1 + $50x 2 + 0s 1 + 0s 2

19. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-19 Solution Points with Slack Variables

20. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-20 Solution Points with Surplus Variables

21. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-21 Solving LP Problems with Excel Click on “Tools” to invoke “Solver.” Objective function Decision variables – bowls (x 1 )=B10; mugs (x 2 )=B11 =C6*B10+D6*B11 =C7*B10+D7*B11 =E6-F6 =E7-F7

22. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-22 Solving LP Problems with Excel (cont.) After all parameters and constraints have been input, click on “Solve.” Objective functionDecision variables C6*B10+D6*B11≤40 C7*B10+D7*B11≤120 Click on “Add” to insert constraints

23. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-23 Solving LP Problems with Excel (cont.)

24. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-24 Sensitivity Analysis

25. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-25 Sensitivity Range for Labor Hours

26. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-26 Sensitivity Range for Bowls

27. Copyright 2006 John Wiley & Sons, Inc.Supplement 13-27 Copyright 2006 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without express permission of the copyright owner is unlawful. Request for further information should be addressed to the Permission Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information herein.