Table of Contents Chapter 5 (What-If Analysis for Linear Programming) Continuing the Wyndor Case Study (Section 5.2)5.2 Changes in One Objective Function Coefficient (Section 5.3)5.3–5.9 Simultaneous Changes in Objective Function Coefficients (Section 5.4)5.10–5.17 Single Changes in a Constraint (Section 5.5)5.18–5.23 Simultaneous Changes in the Constraints (Section 5.6)5.24–5.26 Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

Wyndor (Before What-If Analysis) 5-2

Using the Spreadsheet to do Sensitivity Analysis The profit per door has been revised from $300 to $200. No change occurs in the optimal solution. 5-3

Using the Spreadsheet to do Sensitivity Analysis The profit per door has been revised from $300 to $500. No change occurs in the optimal solution. 5-4

Using the Spreadsheet to do Sensitivity Analysis The profit per door has been revised from $300 to $1,000. The optimal solution changes. 5-5

Using Solver Table to do Sensitivity Analysis 5-6

Using Solver Table to do Sensitivity Analysis 5-7

Using the Sensitivity Report to Find the Allowable Range 5-8

Graphical Insight into the Allowable Range The two dashed lines that pass through the solid constraint boundary lines are the objective function lines when P D (the unit profit for doors) is at an endpoint of its allowable range, 0 ≤ P D ≤

Using the Spreadsheet to do Sensitivity Analysis The profit per door has been revised from $300 to $450. The profit per window has been revised from $500 to $400. No change occurs in the optimal solution. 5-10

Using the Spreadsheet to do Sensitivity Analysis The profit per door has been revised from $300 to $600. The profit per window has been revised from $500 to $300. The optimal solution changes. 5-11

Using Solver Table to do Sensitivity Analysis 5-12

Using Solver Table to do Sensitivity Analysis 5-13

Using Solver Table to do Sensitivity Analysis 5-14

The 100 Percent Rule The 100 Percent Rule for Simultaneous Changes in Objective Function Coefficients: If simultaneous changes are made in the coefficients of the objective function, calculate for each change the percentage of the allowable change (increase or decrease) for that coefficient to remain within its allowable range. If the sum of the percentage changes does not exceed 100 percent, the original optimal solution definitely will still be optimal. (If the sum does exceed 100 percent, then we cannot be sure.) 5-15

Graphical Insight into 100 Percent Rule The estimates of the unit profits for doors and windows change to P D = $525 and P W = $350, which lies at the edge of what is allowed by the 100 percent rule. 5-16

Graphical Insight into 100 Percent Rule When the estimates of the unit profits for doors and windows change to P D = $150 and P W = $250 (half their original values), the graphical method shows that the optimal solution still is (D, W) = (2, 6) even though the 100 percent rule says that the optimal solution might change. 5-17

Using the Spreadsheet to do Sensitivity Analysis The hours available in plant 2 have been increased from 12 to 13. The total profit increases by $150 per week. 5-18

Using the Spreadsheet to do Sensitivity Analysis The hours available in plant 2 have been further increased from 13 to 18. The total profit increases by $750 per week ($150 per hour added in plant 2). 5-19

Using the Spreadsheet to do Sensitivity Analysis The hours available in plant 2 have been further increased from 18 to 20. The total profit does not increase any further. 5-20

Using Solver Table to do Sensitivity Analysis 5-21

Using the Sensitivity Report 5-22

Graphical Interpretation of the Allowable Range 5-23

Using the Spreadsheet to do Sensitivity Analysis One available hour in plant 3 has been shifted to plant 2. The total profit increases by $50 per week. 5-24

Using Solver Table to do Sensitivity Analysis 5-25

The 100 Percent Rule The 100 Percent Rule for Simultaneous Changes in Right-Hand Sides: The shadow prices remain valid for predicting the effect of simultaneously changing the right-hand sides of some of the functional constraints as long as the changes are not too large. To check whether the changes are small enough, calculate for each change the percentage of the allowable change (decrease or increase) for that right-hand side to remain within its allowable range. If the sum of the percentage changes does not exceed 100 percent, the shadow prices definitely will still be valid. (If the sum does exceed 100 percent, then we cannot be sure.) 5-26