Introduction to Management Science 9 th Edition by
Introduction to Management Science 9 th Edition by Bernard W. Taylor III Chapter 3 Linear Programming: Computer Solution and Sensitivity Analysis © 2007 Pearson Education Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 1
Chapter Topics n Computer Solution n Sensitivity Analysis Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 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. Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 3
Beaver Creek Pottery Example Excel Spreadsheet – Data Screen (1 of 6) Exhibit 3. 1 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 4
Beaver Creek Pottery Example “Solver” Parameter Screen (2 of 6) Exhibit 3. 2 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 5
Beaver Creek Pottery Example Adding Model Constraints (3 of 6) Exhibit 3. 3 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 6
Beaver Creek Pottery Example “Solver” Settings (4 of 6) Exhibit 3. 4 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 7
Beaver Creek Pottery Example Solution Screen (5 of 6) Exhibit 3. 5 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 8
Beaver Creek Pottery Example Answer Report (6 of 6) Exhibit 3. 6 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 9
Linear Programming Problem: Standard Form 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. 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 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 10
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. Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 11
Beaver Creek Pottery Example Sensitivity Analysis (2 of 4) Maximize Z = $40 x 1 + $50 x 2 subject to: 1 x 1 + 2 x 2 40 4 x 1 + 3 x 2 120 x 1, x 2 0 Figure 3. 1 Optimal Solution Point Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 12
Beaver Creek Pottery Example Change x 1 Objective Function Coefficient (3 of 4) Maximize Z = $100 x 1 + $50 x 2 subject to: 1 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 Coefficient Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 13
Beaver Creek Pottery Example Change x 2 Objective Function Coefficient (4 of 4) Maximize Z = $40 x 1 + $100 x 2 subject to: 1 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 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 14
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 ci. Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis as 15
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 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 16
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 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 17
Objective Function Coefficient Ranges Excel “Solver” Results Screen (1 of 3) Exhibit 3. 12 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 18
Objective Function Coefficient Ranges Beaver Creek Example Sensitivity Report (2 of 3) Exhibit 3. 13 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 19
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. Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 20
Changes in Constraint Quantity Values Increasing the Labor Constraint (2 of 4) Maximize Z = $40 x 1 + $50 x 2 subject to: 1 x 1 + 2 x 2 40 4 x 2 + 3 x 2 120 x 1, x 2 0 Figure 3. 6 Increasing the Labor Constraint Quantity Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 21
Changes in Constraint Quantity Values Sensitivity Range for Labor Constraint (3 of 4) Sensitivity range for: 30 q 1 80 hr Figure 3. 7 Determining the Sensitivity Range for Labor Quantity Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 22
Changes in Constraint Quantity Values Sensitivity Range for Clay Constraint (4 of 4) Sensitivity range for: 60 q 2 160 lb Figure 3. 8 Determining the Sensitivity Range for Clay Quantity Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 23
Constraint Quantity Value Ranges by Computer Excel Sensitivity Range for Constraints (1 of 2) Exhibit 3. 15 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 24
Other Forms of Sensitivity Analysis Topics (1 of 4) n Changing individual constraint parameters n Adding new constraints n Adding new variables Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 25
Other Forms of Sensitivity Analysis Changing a Constraint Parameter (2 of 4) Maximize Z = $40 x 1 + $50 x 2 subject to: 1 x 1 + 2 x 2 40 4 x 2 + 3 x 2 120 x 1, x 2 0 Figure 3. 9 Changing the x 1 Coefficient in the Labor Constraint Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 26
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 Exhibit 3. 17 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 27
Other Forms of Sensitivity Analysis Adding a New Variable (4 of 4) Adding a new variable to the Beaver Creek model, x 3, 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). Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 28
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. Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 29
Excel Sensitivity Report for Beaver Creek Pottery Shadow Prices Example (1 of 2) Maximize Z = $40 x 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 Exhibit 3. 18 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 30
Excel Sensitivity Report for Beaver Creek Pottery Solution Screen (2 of 2) Exhibit 3. 19 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 31
Example Problem Statement (1 of 3) n n Two airplane parts: no. 1 and no. 2. Three manufacturing stages: stamping, drilling, milling. Decision variables: x 1 (number of part no. 1 to produce) x 2 (number of part no. 2 to produce) 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 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 32
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. 43 137. 76 q 1 89. 10 Figure 3. 10 Graphical Solution Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 33
Example Problem Excel Solution (3 of 3) Exhibit 3. 20 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 34
End of Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis 35
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