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

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

3-3 Early linear programming used lengthy manual mathematical solution procedure called the Simplex Method. 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. Computer Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-4 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 - 2x 2 - 2x 3  0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-5 Beaver Creek Pottery Example QM for Windows (1 of 5) Exhibit 3.7 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-6 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

3-7 Beaver Creek Pottery Example QM for Windows: Data Table (3 of 5) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 3.9

3-8 Beaver Creek Pottery Example QM for Windows: Model Solution (4 of 5) Exhibit 3.10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-9 Beaver Creek Pottery Example QM for Windows: Graphical Display (5 of 5) Exhibit 3.11 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-10 Sensitivity analysis determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations. Changes may be reactions to anticipated uncertainties in the parameters or to new or changed information concerning the model. Beaver Creek Pottery Example Sensitivity Analysis (1 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-11 Maximize Z = $40x 1 + $50x 2 subject to: x 1 + 2x 2  40 4x 1 + 3x 2  120 x 1, x 2  0 Figure 3.1 Optimal Solution Point Beaver Creek Pottery Example Sensitivity Analysis (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-12 Maximize Z = $100x 1 + $50x 2 subject to: x 1 + 2x 2  40 4x 1 + 3x 2  120 x 1, x 2  0 Figure 3.2 Changing the x 1 Objective Function Coefficient Beaver Creek Pottery Example Change x 1 Objective Function Coefficient (3 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-13 Maximize Z = $40x 1 + $100x 2 subject to: x 1 + 2x 2  40 4x 1 + 3x 2  120 x 1, x 2  0 Figure 3.3 Changing the x 2 Objective Function Coefficient Beaver Creek Pottery Example Change x 2 Objective Function Coefficient (4 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-14 The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal. The sensitivity range for the x i coefficient is designated as c i. Objective Function Coefficient Sensitivity Range (1 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-15 objective function Z = $40x 1 + $50x 2 sensitivity range for: x 1 : 25  c 1  x 2 : 30  c 2  80 Figure 3.4 Determining the Sensitivity Range for c 1 Objective Function Coefficient Sensitivity Range for c 1 and c 2 (2 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-16 Minimize Z = $6x 1 + $3x 2 subject to: 2x 1 + 4x 2  16 4x 1 + 3x 2  24 x 1, x 2  0 sensitivity ranges: 4  c 1   0  c 2  4.5 Objective Function Coefficient Fertilizer Cost Minimization Example (3 of 3) Figure 3.5 Fertilizer Cost Minimization Example Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-17 Exhibit 3.12 Objective Function Coefficient Ranges Excel “Solver” Results Screen (1 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-18 Exhibit 3.13 Objective Function Coefficient Ranges Beaver Creek Example Sensitivity Report (2 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-19 Exhibit 3.14 Objective Function Coefficient Ranges QM for Windows Sensitivity Range Screen (3 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Sensitivity ranges for objective function coefficients

3-20 Changes in Constraint Quantity Values Sensitivity Range (1 of 4) 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

3-21 Changes in Constraint Quantity Values Increasing the Labor Constraint (2 of 4) Maximize Z = $40x 1 + $50x 2 subject to: x 1 + 2x 2 + s 1 = 40 4x 1 + 3x 2 + s 2 = 120 x 1, x 2  0 Figure 3.6 Increasing the Labor Constraint Quantity Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-22 Changes in Constraint Quantity Values Sensitivity Range for Labor Constraint (3 of 4) Figure 3.7 Determining the Sensitivity Range for Labor Quantity Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-23 Changes in Constraint Quantity Values Sensitivity Range for Clay Constraint (4 of 4) Figure 3.8 Determining the Sensitivity Range for Clay Quantity Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-24 Exhibit 3.15 Constraint Quantity Value Ranges by Computer Excel Sensitivity Range for Constraints (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-25 Exhibit 3.16 Constraint Quantity Value Ranges by Computer QM for Windows Sensitivity Range (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-26 Changing individual constraint parameters Adding new constraints Adding new variables Other Forms of Sensitivity Analysis Topics (1 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-27 Other Forms of Sensitivity Analysis Changing a Constraint Parameter (2 of 4) Maximize Z = $40x 1 + $50x 2 subject to: x 1 + 2x 2  40 4x 1 + 3x 2  120 x 1, x 2  0 Figure 3.9 Changing the x 1 Coefficient in the Labor Constraint Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-28 Adding a new constraint to Beaver Creek Model: 0.20x x 2  5 hours for packaging Original solution: 24 bowls, 8 mugs, $1,360 profit Exhibit 3.17 Other Forms of Sensitivity Analysis Adding a New Constraint (3 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-29 Adding a new variable to the Beaver Creek model, x 3, for a third product, cups Maximize Z = $40x x x 3 subject to: x 1 + 2x x 3  40 hr of labor 4x 1 + 3x 2 + 2x 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). Other Forms of Sensitivity Analysis Adding a New Variable (4 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-30 Defined as the marginal value of one additional unit of resource. The sensitivity range for a constraint quantity value is also the range over which the shadow price is valid. Shadow Prices (Dual Variable Values) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-31 Maximize Z = $40x 1 + $50x 2 subject to: x 1 + 2x 2  40 hr of labor 4x 1 + 3x 2  120 lb of clay x 1, x 2  0 Exhibit 3.18 Excel Sensitivity Report for Beaver Creek Pottery Shadow Prices Example (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-32 Excel Sensitivity Report for Beaver Creek Pottery Solution Screen (2 of 2) Exhibit 3.19 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-33 Sensitivity Analysis Reduced Costs, the change in the value of Z caused by increasing by one unit a decision variable that has a value of zero in the optimal solution. This will create a non-optimal solution, but provides information on the impact of bringing a new variable into the solution. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-34 Sensitivity Analysis The Reduced Cost is zero for a decision variable that is included (has a non-zero value) in the optimal solution. A positive reduced cost indicates that the value of Z will increase if the decision variable is brought into the solution with a value of one unit. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-35 Sensitivity Analysis Allowable Increase/Decrease, the amount an objective function coefficient can be increased/decreased before the location of the optimal solution is changed. A change in location means that the new optimal solution is bounded by different constraints. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-36 Sensitivity Analysis Shadow (Dual)Prices, the change in the optimal value of Z caused by a one unit increase in the RHS value in a constraint. The shadow price of a constraint is zero if the constraint has slack, i.e., the left hand side (LHS) sum is not equal to the RHS value. A positive shadow price indicates that the optimal value of Z will increase if the constraint RHS value increases. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-37 Sensitivity Analysis Allowable Increase/Decrease, the amount the RHS value of a constraint can be increased/decreased before the location of the optimal solution is changed. A change in location means that the new optimal solution is bounded by different constraints. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-38 Two airplane parts: no.1 and no. 2. 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) Model: Maximize Z = $650x x 2 subject to: 4x x 2  105 (stamping,hr) 6.2x x 2  90 (drilling, hr) 9.1x x 2  110 (finishing, hr) x 1, x 2  0 Example Problem Problem Statement (1 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

3-39 Maximize Z = $650x 1 + $910x 2 subject to: 4x x 2  x x 2  x x 2  110 x 1, x 2  0 s1 = 0, s2 = 0, s3 = hr  c 1  1,  q 1  Example Problem Graphical Solution (2 of 3) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall