1 1 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by JOHN LOUCKS St. Edward’s University INTRODUCTION TO MANAGEMENT SCIENCE, 13e Anderson Sweeney Williams Martin

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 3 Linear Programming: Sensitivity Analysis and Interpretation of Solution n Introduction to Sensitivity Analysis n Graphical Sensitivity Analysis n Sensitivity Analysis: Computer Solution n Simultaneous Changes

3 3 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction to Sensitivity Analysis n Sensitivity analysis (or post-optimality analysis) is used to determine how the optimal solution is affected by changes, within specified ranges, in: the objective function coefficients the objective function coefficients the right-hand side (RHS) values the right-hand side (RHS) values n Sensitivity analysis is important to a manager who must operate in a dynamic environment with imprecise estimates of the coefficients. n Sensitivity analysis allows a manager to ask certain what-if questions about the problem.

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 n LP Formulation

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 n Graphical Solution (objective function coefficient)

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 n Graphical Solution (objective function coefficient) ―3/2 <= slope of objective function <= ―7/10

7 7 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Objective Function Coefficients n The range of optimality for each coefficient provides the range of values over which the current solution will remain optimal. n Objective function coefficient’s range (range of optimality) is just for one variable given that all others are not changed n What if the coefficients are 13 and 8 for S and D respectively. but which is out of range of but which is out of range of ―3/2 <= slope of objective function <= ―7/10

8 8 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Right-Hand Sides n Let us consider how a change in the right-hand side for a constraint might affect the feasible region and perhaps cause a change in the optimal solution. n The improvement in the value of the optimal solution per unit increase in the right-hand side is called the dual price. n The range of feasibility is the range over which the dual price is applicable. n As the RHS increases, other constraints will become binding and limit the change in the value of the objective function.

9 9 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 n Graphical Solution (Right Hand Side)

10 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 n Graphical Solution (Right Hand Side of Constraint 1) Intersection of constraints (3) & (4) : ( , ) (1) 7/10* * = (1) 7/10* * = Intersection of S-axis & (3) : (708, 0) (1) 7/10* *0 = (1) 7/10* *0 = 495.6

11 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Computer Solutions n Management Scientist

12 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 n Graphical Solution (Right Hand Side of Constraint 2) No upper limit Intersection of constraints (1) & (3) : (540, 252) (1) 1/2* /6*252 = 480 (1) 1/2* /6*252 = 480

13 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Dual Price n The improvement in the value of the optimal solution per unit increase in the right-hand side is called the dual price. n The dual price for a nonbinding constraint is 0. For >= constraints, dual price of 0 surplus is ― For = constraints, dual price of 0 surplus is ― For <= constraints, dual price of 0 slack is + n A negative dual price indicates that the objective function will not improve if the RHS is increased. n The range of feasibility (range of RHS) is the range over which the dual price is applicable (not changed).

14 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution n Simultaneous Changes Until now, the sensitivity analysis information is based on the assumption that only one coefficient changes Until now, the sensitivity analysis information is based on the assumption that only one coefficient changes n 100% rule More than 2 objective coefficients or more than 2 RHS More than 2 objective coefficients or more than 2 RHS Optimal solution basis (positive valued decision variables) are not changed if sum of all the (changes / allowable changes) ratios is less than 1. Optimal solution basis (positive valued decision variables) are not changed if sum of all the (changes / allowable changes) ratios is less than 1.

15 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution n Objective function coefficients Ex1 : Ex1 : Ex2 : Ex2 : Ex3 : Ex3 : (not simultaneously binding) (not simultaneously binding)

16 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution n RHS values Ex1 : Ex1 : Ex2 : Ex2 : Ex3 : Ex3 : (not simultaneously binding) (not simultaneously binding)

17 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution n RHS values Global optimal solution found. Global optimal solution found. Objective value: Objective value: Infeasibilities: Infeasibilities: Total solver iterations: 3 Total solver iterations: 3 Variable Value Reduced Cost Variable Value Reduced Cost S S D D Row Slack or Surplus Dual Price Row Slack or Surplus Dual Price

18 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution n RHS values Global optimal solution found. Global optimal solution found. Objective value: Objective value: Infeasibilities: Infeasibilities: Total solver iterations: 2 Total solver iterations: 2 Variable Value Reduced Cost Variable Value Reduced Cost S S D D Row Slack or Surplus Dual Price Row Slack or Surplus Dual Price

19 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Computer Solution n RHS values Global optimal solution found. Global optimal solution found. Objective value: Objective value: Infeasibilities: Infeasibilities: Total solver iterations: 2 Total solver iterations: 2 Variable Value Reduced Cost Variable Value Reduced Cost S S D D Row Slack or Surplus Dual Price Row Slack or Surplus Dual Price

20 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Second Example (p.110)

21 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Second Example (p.110)

22 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Second Example (p.110) n Dual price The improvement of the objective function value per 1 unit increase of the RHS. The improvement of the objective function value per 1 unit increase of the RHS. Total production requirement and Processing time are binding Total production requirement and Processing time are binding Dual price of processing time is 1 Dual price of processing time is 1 Dual price of total minimum (350) is -4 Dual price of total minimum (350) is -4 n Notes Dual price is an extra cost. If the profit contribution is calculated considering the purchasing cost of the resource, the price we are willing to pay for that resource is purchasing cost + dual price for 1 unit. Dual price is an extra cost. If the profit contribution is calculated considering the purchasing cost of the resource, the price we are willing to pay for that resource is purchasing cost + dual price for 1 unit.

23 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Note and Comments n Degeneracy Consider the available Sewing time is 480 which is calculated with 1/2* /6*252 Consider the available Sewing time is 480 which is calculated with 1/2* /6*252

24 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Note and Comments n Degeneracy Consider the available Sewing time is 480 which is calculated with 1/2* /6*252 Consider the available Sewing time is 480 which is calculated with 1/2* /6*252 Global optimal solution found. Global optimal solution found. Objective value: Objective value: Infeasibilities: Infeasibilities: Total solver iterations: 2 Total solver iterations: 2 Variable Value Reduced Cost Variable Value Reduced Cost S S D D Row Slack or Surplus Dual Price Row Slack or Surplus Dual Price Objective Coefficient Ranges: Objective Coefficient Ranges: Current Allowable Allowable Current Allowable Allowable Variable Coefficient Increase Decrease Variable Coefficient Increase Decrease S S D D Righthand Side Ranges: Righthand Side Ranges: Current Allowable Allowable Current Allowable Allowable Row RHS Increase Decrease Row RHS Increase Decrease INFINITY INFINITY INFINITY INFINITY

25 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis: Note and Comments n Degeneracy Consider the available Sewing time is 480 which is calculated with 1/2* /6*252 Consider the available Sewing time is 480 which is calculated with 1/2* /6*252 In the standard form number of variables (2+3=5), number of constraints 3. Thus, basic solution has (set 2 variables to 0, and solve simultaneous equations ( 연립방정식 ). Now, at the optimal solution 3 variables are 0. In the standard form number of variables (2+3=5), number of constraints 3. Thus, basic solution has (set 2 variables to 0, and solve simultaneous equations ( 연립방정식 ). Now, at the optimal solution 3 variables are 0. Dual price of binding constraints is 0. Dual price of binding constraints is 0. Constraint 2 (Sewing) has 0 slack, but dual price is 0 Constraint 2 (Sewing) has 0 slack, but dual price is 0 Range of Feasibility (range of RHS) for constraints 2, 3 and 4 are only one direction. Range of Feasibility (range of RHS) for constraints 2, 3 and 4 are only one direction. n 100% rule works only when sum of ratios are less than 100.

26 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 (more than 2 variables) n Consider the following linear program: Max 10S + 9D L s.t. 0.7S + 1D + 0.8L < S + 5/6 D + 1L < S + 5/6 D + 1L < 600 1S + 2/3 D + 1L < 708 1S + 2/3 D + 1L < S D L < S D L < 135 x 1, x 2 > 0 x 1, x 2 > 0

27 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3

28 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 (more than 2 variables) n Interpretation Deluxe model is not produced Deluxe model is not produced Finishing (Constraint 3) and Inspection and Packaging (Constraint 4) are binding Finishing (Constraint 3) and Inspection and Packaging (Constraint 4) are binding Range of objective function for Deluxe is ― infinity < current 9 < Range of objective function for Deluxe is ― infinity < current 9 < Reduced cost : the amount that an objective function coefficient would have to improve in order for the corresponding decision variable becomes positive. Reduced cost : the amount that an objective function coefficient would have to improve in order for the corresponding decision variable becomes positive. Reduced cost of Deluxe is 1.15 = 1 * 0 + 5/6*0 + 2/3* *19 – 9 (sum of dual prices consumed to produce 1 unit of Deluxe)

29 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 (more than 2 variables) n Primal problem vs. Dual problem Max 10S + 9D L 0.7S + 1D + 0.8L < S + 1D + 0.8L < S + 5/6 D + 1L < S + 5/6 D + 1L < 600 1S + 2/3 D + 1L < 708 1S + 2/3 D + 1L < S D L < S D L < 135 S, D, L > 0 S, D, L > 0 Min 630C + 600W + 708F + 135I 0.7C + 0.5W + 1F + 0.1I –R1 = C + 0.5W + 1F + 0.1I –R1 = 10 1C + 5/6 W + 2/3 F I –R2 = 9 1C + 5/6 W + 2/3 F I –R2 = 9 0.8C + 1W + 1F I –R3 = C + 1W + 1F I –R3 = C, W, F, I, R1, R2, R3 > 0 C, W, F, I, R1, R2, R3 > 0

30 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 (more than 2 variables) n Primal problem vs. Dual problem Primal problem maximize the total profit contribution with the constraints of limited available resources Primal problem maximize the total profit contribution with the constraints of limited available resources Dual problem minimize the total cost allocation to resources with the constraints of guaranteeing the minimum profitability. Dual problem minimize the total cost allocation to resources with the constraints of guaranteeing the minimum profitability.

31 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 (more than 2 variables) n Alternative optimal solution (p.116 Fig. 3.7) Profit contribution of Deluxe is Profit contribution of Deluxe is Slack of Constraint 1 is 0, but the dual price is also 0. Slack of Constraint 1 is 0, but the dual price is also 0. Range of optimality (range of objective function coefficient) has one direction Range of optimality (range of objective function coefficient) has one direction If the primal problem has an alternative optima, the dual is degenerate and vice versa. If the primal problem has an alternative optima, the dual is degenerate and vice versa. n Extra constraint (p.117 Fig. 3.8) Deluxe should be produces at least 30% of standard bag. D > 0.3S  –0.3S + D > 0 Deluxe should be produces at least 30% of standard bag. D > 0.3S  –0.3S + D > 0 Dual price –1.38 means that the total profit will decrease if Deluxe is produce 1 more than 30% of standard bag. Dual price –1.38 means that the total profit will decrease if Deluxe is produce 1 more than 30% of standard bag.

32 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 4 (Bluegrass Farms Problem, p.118) n Decision variables S = pounds of standard horse feed product to feed E = pounds of vitamin-enriched oat product to feed A = pounds of new vitamin and mineral feed additive Min 0.25S + 0.5E + 3A 0.8S + 0.2E + 0.0A > 3 0.8S + 0.2E + 0.0A > 3 1S + 1.5E + 3.0A > 6 1S + 1.5E + 3.0A > 6 0.1S + 0.6E + 2.0A > 4 0.1S + 0.6E + 2.0A > 4 1S + 1E + 1A < 6 1S + 1E + 1A < 6 S, E, A > 0 S, E, A > 0

33 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 4 (Bluegrass Farms Problem, p.118)

34 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 4 (Bluegrass Farms Problem, p.118) n Interpretation What’s the optimal decision? What’s the optimal decision? What’s the optimal cost? What’s the optimal cost? Which constraint has slack/surplus? Which constraint has slack/surplus? What the dual prices for binding constraints? of maximum weight means if maximum weight requirement is increased, some cheaper product will be feed to meet the requirements of ingredients by allowing more weights What the dual prices for binding constraints? of maximum weight means if maximum weight requirement is increased, some cheaper product will be feed to meet the requirements of ingredients by allowing more weights Explain with the ranges of objective function What will happen if the standard horse feed product is free Explain with the ranges of objective function What will happen if the standard horse feed product is free Explain with the ranges of RHS Explain with the ranges of RHS

35 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5 (Electronic Communication Problem, p.123) n Maximize or minimize n What are the constraints How many? n Decision variables M = number of unit to produce for the marine equipment distribution channel B = number of units to produce for the business equipment distribution channel R = number of units to produce for the national retail chain distribution channel D = number of units to produce for the direct mail distribution channel

36 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5 (Electronic Communication Problem, p.123) n Model Formulation

37 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5 (Electronic Communication Problem, p.123)

38 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5 (Electronic Communication Problem, p.123)

39 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 5 (Electronic Communication Problem, p.123) n Interpretation What’s the optimal decision? What’s the optimal decision? What’s the optimal cost? What’s the optimal cost? What should be the profit for the direct mail channel in order to produce some for the direct model? What should be the profit for the direct mail channel in order to produce some for the direct model? Which constraint has slack/surplus? Which constraint has slack/surplus? What the dual prices for binding constraints? What the dual prices for binding constraints? Explain with the ranges of objective function Explain with the ranges of objective function Explain with the ranges of RHS Explain with the ranges of RHS What if the production requirement of 600 is changed to 601? How much of the advertising budget is allocated to business distributors?

40 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Ch.3 Homework Ch.3 Homework n Q29 on p.149 Formulate the model Formulate the model Solve with Excel Solve with Excel In Excel, you choose all options of 보고서 after 해찾기 to get the output of sensitivity analysis Solve with LINGO Solve with LINGO Answer all questions on p.149 Q29. Answer all questions on p.149 Q29. Put all output answers in one file except Excel file and upload through mis3nt.gnu.ac.kr Put all output answers in one file except Excel file and upload through mis3nt.gnu.ac.kr

41 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 3