Slides by JOHN LOUCKS St. Edward’s University

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© 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.© 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 byJOHNLOUCKSSt. Edward’sUniversity

INTRODUCTION TO MANAGEMENT SCIENCE, 13e

AndersonSweeneyWilliams

Martin

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Chapter 3 Linear Programming: Sensitivity Analysis

and Interpretation of Solution Introduction to Sensitivity Analysis Graphical Sensitivity Analysis Sensitivity Analysis: Computer Solution Simultaneous Changes

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Introduction to Sensitivity Analysis 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 right-hand side (RHS) values

Sensitivity analysis is important to a manager who must operate in a dynamic environment with imprecise estimates of the coefficients.

Sensitivity analysis allows a manager to ask certain what-if questions about the problem.

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Example 1 LP Formulation

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Example 1 Graphical Solution (objective function

coefficient)

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Example 1 Graphical Solution (objective function

coefficient)

―3/2 <= slope of objective function <= ―7/10

3 72 9 10

6.3 13.5

S

S

C

C

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Objective Function Coefficients The range of optimality for each coefficient

provides the range of values over which the current solution will remain optimal.

Objective function coefficient’s range (range of optimality) is just for one variable given that all others are not changed

What if the coefficients are 13 and 8 for S and D respectively.

but which is out of range of

―3/2 <= slope of objective function <= ―7/10

6.3 13.56.667 14.286

S

D

CC

13 1.6258

S

D

CC

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Right-Hand Sides 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.

The improvement in the value of the optimal solution per unit increase in the right-hand side is called the dual price.

The range of feasibility is the range over which the dual price is applicable.

As the RHS increases, other constraints will become binding and limit the change in the value of the objective function.

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Example 1 Graphical Solution (Right Hand Side)

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Example 1 Graphical Solution (Right Hand Side of

Constraint 1)

Intersection of constraints (3) & (4) : (474.545, 350.182) (1) 7/10*474.545 + 1*350.182 = 682.364

Intersection of S-axis & (3) : (708, 0) (1) 7/10*708 + 1*0 = 495.6

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© 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 Management Scientist

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Example 1 Graphical Solution (Right Hand Side of

Constraint 2)

No upper limit

Intersection of constraints (1) & (3) : (540, 252) (1) 1/2*540 + 5/6*252 = 480

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Dual Price The improvement in the value of the optimal

solution per unit increase in the right-hand side is called the dual price.

The dual price for a nonbinding constraint is 0.

For >= constraints, dual price of 0 surplus is ― For <= constraints, dual price of 0 slack is +

A negative dual price indicates that the objective function will not improve if the RHS is increased.

The range of feasibility (range of RHS) is the range over which the dual price is applicable (not changed).

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Sensitivity Analysis: Computer Solution Simultaneous Changes

• Until now, the sensitivity analysis information is based on the assumption that only one coefficient changes

100% rule• More than 2 objective coefficients

or more than 2 RHS• Optimal solution basis (positive valued

decision variables) are not changedif sum of all the (changes / allowable changes) ratios is less than 1.

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Sensitivity Analysis: Computer Solution Objective function coefficients

• Ex1 :

• Ex2 :

• Ex3 : (not simultaneously binding)

13, 8S DC C 13 31.6258 2

3 1100 100 128.6 1003.5 2.3333

11.5, 8.25S DC C 3 11.5 71.3942 8.25 10

1.5 0.75100 100 75 1003.5 2.3333

13, 14S DC C 3 5100 100 180.3 100

3.5 5.28571

3 13 70.92862 14 10

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Sensitivity Analysis: Computer Solution RHS values

• Ex1 :

• Ex2 :

• Ex3 : (not simultaneously binding)

1 3670, 650RHS RHS 40 58100 100 121.7 100

52.36316 128

1 3650, 650RHS RHS

1 3670, 808RHS RHS

20 58100 100 83.5 10052.36316 128

40 100100 100 128.5 10052.36316 192

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Sensitivity Analysis: Computer Solution RHS values Global optimal solution found. Objective value: 7390.898 Infeasibilities: 0.000000 Total solver iterations: 3

Variable Value Reduced Cost S 395.4528 0.000000 D 381.8189 0.000000

Row Slack or Surplus Dual Price 1 7390.898 1.000000 2 11.36416 0.000000 3 84.09247 0.000000 4 0.000000 8.727289 5 0.000000 12.72711

1 3670, 650RHS RHS

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1 3650, 650RHS RHS

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1 3670, 808RHS RHS

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Sensitivity Analysis: Second Example (p.110)

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Dual price• The improvement of the objective function

value per 1 unit increase of the RHS.• Total production requirement and Processing

time are binding• Dual price of processing time is 1• Dual price of total minimum (350) is -4

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.

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Sensitivity Analysis: Note and Comments Degeneracy

• Consider the available Sewing time is 480 which is calculated with 1/2*540 + 5/6*252

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Global optimal solution found. Objective value: 7668.000 Infeasibilities: 0.000000 Total solver iterations: 2



Objective Coefficient Ranges:

Current Allowable Allowable Variable Coefficient Increase Decrease S 10.00000 3.500000 3.700000 D 9.000000 5.285714 2.333333

Righthand Side Ranges:

Current Allowable Allowable Row RHS Increase Decrease 2 630.0000 0.000000 134.4000 3 480.0000 INFINITY 0.000000 4 708.0000 192.0000 0.000000 5 135.0000 INFINITY 18.00000

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• 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.• 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. 100% rule works only when sum of ratios are

less than 100.

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Example 3 (more than 2 variables) Consider the following linear

program:Max 10S + 9D + 12.85Ls.t. 0.7S + 1D + 0.8L <

630 0.5S + 5/6D + 1L <

600 1S + 2/3D + 1L <

708 0.1S + 0.25D + 0.25L

< 135 x1, x2 > 0

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Example 3

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Example 3 (more than 2 variables) Interpretation

• Deluxe model is not produced• Finishing (Constraint 3) and Inspection and

Packaging (Constraint 4) are binding• Range of objective function for Deluxe is

― infinity < current 9 < 10.15• 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*8.1 + 0.25*19 – 9(sum of dual prices consumed to produce 1 unit of Deluxe)

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Example 3 (more than 2 variables) Primal problem vs. Dual problem

Max 10S + 9D + 12.85L 0.7S + 1D + 0.8L <

630 0.5S + 5/6D + 1L <

600 1S + 2/3D + 1L <

708 0.1S + 0.25D + 0.25L <

135 S, D, L > 0

Min 630C + 600W + 708F + 135I 0.7C + 0.5W + 1F + 0.1I –R1 =

10 1C + 5/6W + 2/3F + 0.25I –R2 =

9 0.8C + 1W + 1F + 0.25I –R3

= 12.85 C, W, F, I, R1, R2, R3 > 0

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Example 3 (more than 2 variables) Primal problem vs. Dual problem

• Primal problem maximize the total profit contribution with the constraints of limited available resources

• Dual problemminimize the total cost allocation to resources with the constraints of guaranteeing the minimum profitability.

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Example 3 (more than 2 variables) Alternative optimal solution (p.116 Fig. 3.7)

• Profit contribution of Deluxe is 10.15• Slack of Constraint 1 is 0, but the dual price is

also 0.• 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.

Extra constraint (p.117 Fig. 3.8)• 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.

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Example 4 (Bluegrass Farms Problem, p.118)

Decision variablesS = pounds of standard horse feed product to feedE = pounds of vitamin-enriched oat product to feedA = pounds of new vitamin and mineral feed additive

Min 0.25S + 0.5E + 3A 0.8S + 0.2E + 0.0A > 3 1S + 1.5E + 3.0A > 6 0.1S + 0.6E + 2.0A > 4 1S + 1E + 1A < 6

S, E, A > 0

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Interpretation• What’s the optimal decision?• What’s the optimal cost?• Which constraint has slack/surplus?• What the dual prices for binding constraints?

0.919 of maximum weight meansif 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 functionWhat will happen if the standard horse feed product is free

• Explain with the ranges of RHS

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Example 5 (Electronic Communication Problem,

p.123) Maximize or minimize What are the constraints

How many? Decision variables

M = number of unit to produce for the marine equipment distribution channelB = number of units to produce for the business equipment distribution channelR = number of units to produce for the national retail chain distribution channelD = number of units to produce for the direct mail distribution channel

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p.123) Model Formulation

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p.123)

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p.123)

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p.123) Interpretation

• What’s the optimal decision?• What’s the optimal cost?• What should be the profit for the direct mail

channel in order to produce some for the direct model?

• Which constraint has slack/surplus?• What the dual prices for binding constraints?• Explain with the ranges of objective function• 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?

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Ch.3 Homework Q29 on p.149

• Formulate the model• Solve with Excel

In Excel, you choose all options of 보고서 after 해찾기 to get the output of sensitivity analysis• Solve with LINGO• Answer all questions on p.149 Q29.• Put all output answers in one file except Excel

file and upload through mis3nt.gnu.ac.kr

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End of Chapter 3

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