Linear Programming Computer Solution and Sensitivity Analysis Chapter

Linear Programming: Computer Solution and Sensitivity Analysis Chapter 3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -1

Chapter Topics n Computer Solution n Sensitivity Analysis Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -2

Computer Solution n n Early linear programming used lengthy manual mathematical solution procedure called the Simplex Method (See CD-ROM Module A). Steps of the Simplex Method have been programmed in software packages designed for linear programming problems. Many such packages available currently. Used extensively in business and government. Text focuses on Excel Spreadsheets and QM for Windows. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -3

Beaver Creek Pottery Example Excel Spreadsheet – Data Screen (1 of 6) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 1 3 -4

Beaver Creek Pottery Example “Solver” Parameter Screen (2 of 6) Exhibit 3. 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -5

Beaver Creek Pottery Example Adding Model Constraints (3 of 6) Exhibit 3. 3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -6

Beaver Creek Pottery Example “Solver” Settings (4 of 6) Exhibit 3. 4 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -7

Beaver Creek Pottery Example Solution Screen (5 of 6) Exhibit 3. 5 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -8

Beaver Creek Pottery Example Answer Report (6 of 6) Exhibit 3. 6 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -9

Linear Programming Problem: Standard Form n Standard form requires all variables in the constraint equations to appear on the left of the inequality (or equality) and all numeric values to be on the right-hand side. n Examples: § x 3 x 1 + x 2 must be converted to x 3 - x 1 - x 2 0 § x 1/(x 2 + x 3) 2 becomes x 1 2 (x 2 + x 3) and then x 1 - 2 x 2 - 2 x 3 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -10

Beaver Creek Pottery Example QM for Windows (1 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 7 3 -11

Beaver Creek Pottery Example QM for Windows – Data Set Creation (2 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 8 3 -12

Beaver Creek Pottery Example QM for Windows: Data Table (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 9 3 -13

Beaver Creek Pottery Example QM for Windows: Model Solution (4 of 5) Exhibit 3. 10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -14

Beaver Creek Pottery Example QM for Windows: Graphical Display (5 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 11 3 -15

Beaver Creek Pottery Example Sensitivity Analysis (1 of 4) n Sensitivity analysis determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations. n Changes may be reactions to anticipated uncertainties in the parameters or to new or changed information concerning the model. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -16

Beaver Creek Pottery Example Sensitivity Analysis (2 of 4) Maximize Z = $40 x 1 + $50 x 2 subject to: x 1 + 2 x 2 40 4 x 1 + 3 x 2 120 x 1, x 2 0 Figure 3. 1 Optimal Solution Point Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -17

Beaver Creek Pottery Example Change x 1 Objective Function Coefficient (3 of 4) Maximize Z = $100 x 1 + $50 x 2 subject to: x 1 + 2 x 2 40 4 x 1 + 3 x 2 120 x 1, x 2 0 Figure 3. 2 Changing the x 1 Objective Function Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Coefficient Hall 3 -18

Beaver Creek Pottery Example Change x 2 Objective Function Coefficient (4 of 4) Maximize Z = $40 x 1 + $100 x 2 subject to: x 1 + 2 x 2 40 4 x 1 + 3 x 2 120 x 1, x 2 0 Figure 3. 3 Changing the x 2 Objective Function Coefficient Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -19

Objective Function Coefficient Sensitivity Range (1 of 3) n The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal. n The sensitivity range for the xi coefficient is designated as ci. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -20

Objective Function Coefficient Sensitivity Range for c 1 and c 2 (2 of 3) objective function Z = $40 x 1 + $50 x 2 sensitivity range for: x 1: 25 c 1 66. 67 x 2: 30 c 2 80 Figure 3. 4 Determining the Sensitivity Range for c 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -21

Objective Function Coefficient Fertilizer Cost Minimization Example (3 of 3) Minimize Z = $6 x 1 + $3 x 2 subject to: 2 x 1 + 4 x 2 16 4 x 1 + 3 x 2 24 x 1, x 2 0 sensitivity ranges: 4 c 1 0 c 2 4. 5 Figure 3. 5 Fertilizer Cost Minimization Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -22

Objective Function Coefficient Ranges Excel “Solver” Results Screen (1 of 3) Exhibit 3. 12 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -23

Objective Function Coefficient Ranges Beaver Creek Example Sensitivity Report (2 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 13 3 -24

Objective Function Coefficient Ranges QM for Windows Sensitivity Range Screen (3 of 3) Sensitivity ranges for objective function coefficients Exhibit 3. 14 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -25

Changes in Constraint Quantity Values Sensitivity Range (1 of 4) n The sensitivity range for a right-hand-side value is the range of values over which the quantity’s value can change without changing the solution variable mix, including the slack variables. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -26

Changes in Constraint Quantity Values Increasing the Labor Constraint (2 of 4) Maximize Z = $40 x 1 + $50 x 2 subject to: x 1 + 2 x 2 + s 1 = 40 4 x 1 + 3 x 2 + s 2 = 120 x 1, x 2 0 Figure 3. 6 Increasing the Labor Constraint Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -27

Changes in Constraint Quantity Values Sensitivity Range for Labor Constraint (3 of 4) Figure 3. 7 Determining the Sensitivity Range for Labor Quant Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -28

Changes in Constraint Quantity Values Sensitivity Range for Clay Constraint (4 of 4) gure 3. 8 Determining the Sensitivity Range for Clay Quantity Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -29

Constraint Quantity Value Ranges by Computer Excel Sensitivity Range for Constraints (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 15 3 -30

Constraint Quantity Value Ranges by Computer QM for Windows Sensitivity Range (2 of 2) Exhibit 3. 16 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -31

Other Forms of Sensitivity Analysis Topics (1 of 4) n Changing individual constraint parameters n Adding new constraints n Adding new variables Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -32

Other Forms of Sensitivity Analysis Changing a Constraint Parameter (2 of 4) Maximize Z = $40 x 1 + $50 x 2 subject to: x 1 + 2 x 2 40 4 x 1 + 3 x 2 120 x 1, x 2 0 Figure 3. 9 Changing the x 1 Coefficient in the Labor Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Constraint Hall 3 -33

Other Forms of Sensitivity Analysis Adding a New Constraint (3 of 4) Adding a new constraint to Beaver Creek Model: 0. 20 x 1+ 0. 10 x 2 5 hours for packaging Original solution: 24 bowls, 8 mugs, $1, 360 profit Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 17 3 -34

Other Forms of Sensitivity Analysis Adding a New Variable (4 of 4) Adding a new variable to the Beaver Creek model, x 3, for a third product, cups Maximize Z = $40 x 1 + 50 x 2 + 30 x 3 subject to: x 1 + 2 x 2 + 1. 2 x 3 40 hr of labor 4 x 1 + 3 x 2 + 2 x 3 120 lb of clay x 1, x 2, x 3 0 Solving model shows that change has no effect on the original solution (i. e. , the model is not sensitive to this change). Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -35

Shadow Prices (Dual Variable Values) n Defined as the marginal value of one additional unit of resource. n The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -36

Excel Sensitivity Report for Beaver Creek Pottery Maximize. Prices Z = $40 x Shadow Example (1 of 2) 1 + $50 x 2 subject to: x 1 + 2 x 2 40 hr of labor 4 x 1 + 3 x 2 120 lb of clay x 1, x 2 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 18 3 -37

Excel Sensitivity Report for Beaver Creek Pottery Solution Screen (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3. 19 3 -38

Example Problem Statement (1 of 3) n Two airplane parts: no. 1 and no. 2. n Three manufacturing stages: stamping, drilling, finishing. Decision variables: x 1 (number of part no. 1 to produce) x 2 (number of part no. 2 to produce) n n Model: Maximize Z = $650 x 1 + 910 x 2 subject to: 4 x 1 + 7. 5 x 2 105 (stamping, hr) 6. 2 x 1 + 4. 9 x 2 90 (drilling, hr) 9. 1 x 1 + 4. 1 x 2 110 (finishing, hr) x 1, x 2 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -39

Example Problem Graphical Solution (2 of 3) Maximize Z = $650 x 1 + $910 x 2 subject to: 4 x 1 + 7. 5 x 2 105 6. 2 x 1 + 4. 9 x 2 90 9. 1 x 1 + 4. 1 x 2 110 x 1, x 2 0 s 1 = 0, s 2 = 0, s 3 = 11. 35 hr 485. 33 c 1 1, 151. 43 89. 10 q 1 137. 76 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -40

Example Problem Excel Solution (3 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -41

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 3 -42