LINGO output is given for the following linear programming problem. MIN 12 X1 + 10 X2 + 9 X3 SUBJECT TO 2)5 X1 + 8 X2 + 5 X3 >= 60 3)8 X1 + 10 X2 + 5 X3 >= 80 END LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1)80.000000 \[\begin{array} { c c c } \text { VARIABLE } & \text { VALUE } & \text { REDUCED COST } \\ \text { X1 } & .000000 & 4.000000 \\ \text { X2 } & 8.000000 & .000000 \\ \text { X3 } & .000000 & 4.000000 \end{array}\] $\begin{array}{l} \begin{array}{c} ROW&\text{SLACK OR SURPLUS}&\text{DUAL PRICE}\\ \hline 2)&4.000000 & .000000 \\ 3)&.000000 & -1.000000 \end{array} \end{array}$ NO.ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { OBJ. COEFFICIENT RANGES }$ $\begin{array}{crrr} \underline{\text { VARIABLE }}&\frac{\text { CURRENT }}{\text { COEFFICIENT }} & \frac{\text { ALLOWABLE }}{\text { INCREASE }} & \frac{\text { ALLOWABLE }}{\text { DECREASE }}\\ X 1 & 12.000000 & \text { INFINITY } & 4.000000 \\ X 2 & 10.000000 & 5.000000 & 10.000000 \\ X 3 & 9.000000 & \text { INFINITY } & 4.000000 \end{array}$ \[\begin{array}{l} \begin{array} { c c c c } &&&\text { RIGHTHAND SIDE RANGES }\\ & \text { CURRENT } & \text { ALLOWABLE } && \text { ALLOWABLE } \\ \hline \text { ROW } & \text { RHS } & \text { INCREASE } && \text { DECREASE } \\ 2 & 60.000000 & 4.000000 && \text { INFINITY } \\ 3 & 80.000000 & \text { INFINITY } && 5.000000 \end{array} \end{array}\] a.What is the solution to the problem? b.Which constraints are binding? c.Interpret the reduced cost for x1. d.Interpret the dual price for constraint 2. e.What would happen if the cost of x1 dropped to 10 and the cost of x2 increased to 12?

a.x 1 = 0,x 2 = 8,x 3 = 0,s 1 = 4,s 2 = 0,obj.func

Consider the following linear program: \[\begin{array} { c r } \text { Min } & 6 x _ { 1 } + 9 x _ { 2 } ( \$ \text { cost } ) \\ \text { s.t. } & x _ { 1 } + 2 x _ { 2 } \leq 8 \\ & 10 x _ { 1 } + 7.5 x _ { 2 } \geq 30 \\ &x _ { 2 } \geq 2 \\ &x _ { 1 } , x _ { 2 } \geq 0 \end{array}\] The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 27.000 \[\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\ \text { X1 } & 1.500 & 0.000 \\ \text { X2 } & 2.000 & 0.000 \end{array}\] \[\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\ 1 & 2.500 & 0.000 \\ 2 & 0.000 & - 0.600 \\ 3 & 0.000 & - 4.500 \end{array}\] OBJECTIVE COEFFICIENT RANGES \[\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\ \text { X1 } & 0.000 & 6.000 & 12.000 \\ \text { X2 } & 4.500 & 9.000 & \text { No Upper Limit } \end{array}\] RIGHT HAND SIDE RANGES \[\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\ 1 & 5.500 & 8.000 & \text { No Upper Limit } \\ 2 & 15.000 & 30.000 & 55.000 \\ 3 & 0.000 & 2.000 & 4.000 \end{array}\] a.What is the optimal solution including the optimal value of the objective function? b.Suppose the unit cost of x1 is decreased to $4. Is the above solution still optimal? What is the value of the objective function when this unit cost is decreased to $4? c.How much can the unit cost of x2 be decreased without concern for the optimal solution changing? d.If simultaneously the cost of x1 was raised to $7.5 and the cost of x2 was reduced to $6, would the current solution still remain optimal? e.If the right-hand side of constraint 3 is increased by 1, what will be the effect on the optimal solution?

a.x 1 = 1.5 and x 2 = 2.0,and the objective

The binding constraints for this problem are the first and second. \[\begin{array} { l r } \operatorname { Min } & x _ { 1 } + 2 x _ { 2 } \\ \text { s.t. } & x _ { 1 } + x _ { 2 } \geq 300 \\ &2 x _ { 1 } + x _ { 2 } \geq 400 \\ &2 x _ { 1 } + 5 x _ { 2 } \leq 750 \\ &x _ { 1 } , x _ { 2 } \geq 0 \end{array}\] a.Keeping c2 fixed at 2, over what range can c1 vary before there is a change in the optimal solution point? b.Keeping c1 fixed at 1, over what range can c2 vary before there is a change in the optimal solution point? c.If the objective function becomes Min 1.5x1 + 2x2, what will be the optimal values of x1, x2, and the objective function? d.If the objective function becomes Min 7x1 + 6x2, what constraints will be binding? e.Find the dual price for each constraint in the original problem.

a..8 F1F1F1S1F1F1F10 c 1 F1F1F1S1F1F1F10

A section of output from The Management Scientist is shown here. \[\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\ \hline 2 & 240 & 300 & 420 \end{array}\] What will happen if the right-hand-side for constraint 2 increases by 200?

A) Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the same. B) The value of the objective function will change, but the values of the decision variables and the dual prices will remain the same. C) The same decision variables will be positive, but their values, the objective function value, and the dual prices will change. D) The problem will need to be resolved to find the new optimal solution and dual price. A) Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the same. B) The value of the objective function will change, but the values of the decision variables and the dual prices will remain the same. C) The same decision variables will be positive, but their values, the objective function value, and the dual prices will change. D) The problem will need to be resolved to find the new optimal solution and dual price.

Exam 8: Linear Programming: Sensitivity Analysis and Interpretation of Solution

Use the spreadsheet and Solver sensitivity report to answer these questions. a.What is the cell formula for B12? b.What is the cell formula for C12? c.What is the cell formula for D12? d.What is the cell formula for B15? e.What is the cell formula for B16?f. What is the cell formula for B17?g. What is the optimal value for x1?h. What is the optimal value for x2?i. Would you pay $.50 each for up to 60 more units of resource 1?j. Is it possible to figure the new objective function value if the profit on product 1 increases by a dollar, or do you have to rerun Solver?

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Question 41

To solve a linear programming problem with thousands of variables and constraints

(Multiple Choice)

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Question 42

For a minimization problem,a positive dual price indicates the value of the objective function will increase.

(True/False)

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Question 43

The 100% Rule compares

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Question 44

The LP problem whose output follows determines how many necklaces,bracelets,rings,and earrings a jewelry store should stock.The objective function measures profit; it is assumed that every piece stocked will be sold.Constraint 1 measures display space in units,constraint 2 measures time to set up the display in minutes.Constraints 3 and 4 are marketing restrictions. LINEAR PROGRAMMING PROBLEM MAX 100X1+120X2+150X3+125X4 S.T. 1)X1+2X2+2X3+2X4<108 2)3X1+5X2+X4<120 3)X1+X3<25 4)X2+X3+X4>50 OPTIMAL SOLUTION Objective Function Value = 7475.000 Variable Value Reduced Cost X1 8.000 0.000 X2 0.000 5.000 X3 17.000 0.000 X4 33.000 0.000 Constraint Slack/Surplus Dual Price 1 0.000 75.000 2 63.000 0.000 3 0.000 25.000 4 0.000 -25.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 87.500 100.000 No Upper Limit X2 No Lower Limit 120.000 125.000 X3 125.000 150.000 162.500 X4 120.000 125.000 150.000 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 100.000 108.000 123.750 2 57.000 120.000 No Upper Limit 3 8.000 25.000 58.000 4 41.500 50.000 54.000 Use the output to answer the questions. a.How many necklaces should be stocked? b.Now many bracelets should be stocked? c.How many rings should be stocked? d.How many earrings should be stocked? e.How much space will be left unused?f. How much time will be used?g. By how much will the second marketing restriction be exceeded?

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Question 45

LINGO output is given for the following linear programming problem. MIN 12 X1 + 10 X2 + 9 X3 SUBJECT TO 2)5 X1 + 8 X2 + 5 X3 >= 60 3)8 X1 + 10 X2 + 5 X3 >= 80 END LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1)80.000000 VARIABLE VALUE REDUCED COST X1 .000000 4.000000 X2 8.000000 .000000 X3 .000000 4.000000 ROW SLACK OR SURPLUS DUAL PRICE 2) 4.000000 .000000 3) .000000 -1.000000 NO.ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { OBJ. COEFFICIENT RANGES }$ X1 12.000000 INFINITY 4.000000 X2 10.000000 5.000000 10.000000 X3 9.000000 INFINITY 4.000000 RIGHTHAND SIDE RANGES CURRENT ALLOWABLE ALLOWABLE ROW RHS INCREASE DECREASE 2 60.000000 4.000000 INFINITY 3 80.000000 INFINITY 5.000000 a.What is the solution to the problem? b.Which constraints are binding? c.Interpret the reduced cost for x₁. d.Interpret the dual price for constraint 2. e.What would happen if the cost of x₁ dropped to 10 and the cost of x₂ increased to 12?

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Question 46

The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint.

(True/False)

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Question 47

How is sensitivity analysis used in linear programming? Given an example of what type of questions that can be answered.

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Question 48

There is a dual price for every decision variable in a model.

(True/False)

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Question 49

Consider the following linear program: Min 6+9(\ cost ) s.t. +2\leq8 10+7.5\geq30 \geq2 ,\geq0 The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 27.000 Variable Value Reduced Cost X1 1.500 0.000 X2 2.000 0.000 Constraint Slack/Surplus Dual Price 1 2.500 0.000 2 0.000 -0.600 3 0.000 -4.500 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 0.000 6.000 12.000 X2 4.500 9.000 No Upper Limit RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 5.500 8.000 No Upper Limit 2 15.000 30.000 55.000 3 0.000 2.000 4.000 a.What is the optimal solution including the optimal value of the objective function? b.Suppose the unit cost of x₁ is decreased to $4. Is the above solution still optimal? What is the value of the objective function when this unit cost is decreased to $4? c.How much can the unit cost of x₂ be decreased without concern for the optimal solution changing? d.If simultaneously the cost of x₁ was raised to $7.5 and the cost of x₂ was reduced to $6, would the current solution still remain optimal? e.If the right-hand side of constraint 3 is increased by 1, what will be the effect on the optimal solution?

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Question 50

Sensitivity analysis information in computer output is based on the assumption of

(Multiple Choice)

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Question 51

An objective function reflects the relevant cost of labor hours used in production rather than treating them as a sunk cost.The correct interpretation of the dual price associated with the labor hours constraint is

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Question 52

The dual price for a percentage constraint provides a direct answer to questions about the effect of increases or decreases in that percentage.

(True/False)

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Question 53

When the right-hand sides of two constraints are each increased by one unit,the objective function value will be adjusted by the sum of the constraints' dual prices.

(True/False)

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Question 54

The binding constraints for this problem are the first and second. Min +2 s.t. +\geq300 2+\geq400 2+5\leq750 ,\geq0 a.Keeping c₂ fixed at 2, over what range can c₁ vary before there is a change in the optimal solution point? b.Keeping c₁ fixed at 1, over what range can c₂ vary before there is a change in the optimal solution point? c.If the objective function becomes Min 1.5x₁ + 2x₂, what will be the optimal values of x₁, x₂, and the objective function? d.If the objective function becomes Min 7x₁ + 6x₂, what constraints will be binding? e.Find the dual price for each constraint in the original problem.

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Question 55

For any constraint,either its slack/surplus value must be zero or its dual price must be zero.

(True/False)

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Question 56

Explain the two interpretations of dual prices based on the accounting assumptions made in calculating the objective function coefficients.

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Question 57

A section of output from The Management Scientist is shown here. Constraint Lower Limit Current Value Upper Limit 2 240 300 420 What will happen if the right-hand-side for constraint 2 increases by 200?

(Multiple Choice)

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Question 58

The 100% Rule does not imply that the optimal solution will necessarily change if the percentage exceeds 100%.

(True/False)

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Question 59

Given the following linear program: Max 5+7 s.t. \leq6 2+3\leq19 +\leq8 ,\geq0 The graphical solution to the problem is shown below.From the graph we see that the optimal solution occurs at x₁ = 5,x₂ = 3,and obj.func.= 46. $Given the following linear program: \begin{array} { l r } \operatorname { Max } & 5 x _ { 1 } + 7 x _ { 2 } \\ \text { s.t. }& x _ { 1 } \leq 6 \\ & 2 x _ { 1 } + 3 x _ { 2 } \leq 19 \\ & x _ { 1 } + x _ { 2 } \leq 8 \\ & x _ { 1 } , x _ { 2 } \geq 0 \end{array} The graphical solution to the problem is shown below.From the graph we see that the optimal solution occurs at x<sub>1</sub> = 5,x<sub>2</sub> = 3,and obj.func.= 46. a.Calculate the range of optimality for each objective function coefficient. b.Calculate the dual price for each resource.$ a.Calculate the range of optimality for each objective function coefficient. b.Calculate the dual price for each resource.

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Question 60

Showing 41 - 60 of 64

To solve a linear programming problem with thousands of variables and constraints

For a minimization problem,a positive dual price indicates the value of the objective function will increase.

The 100% Rule compares

The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint.

How is sensitivity analysis used in linear programming? Given an example of what type of questions that can be answered.

There is a dual price for every decision variable in a model.

Sensitivity analysis information in computer output is based on the assumption of

An objective function reflects the relevant cost of labor hours used in production rather than treating them as a sunk cost.The correct interpretation of the dual price associated with the labor hours constraint is

The dual price for a percentage constraint provides a direct answer to questions about the effect of increases or decreases in that percentage.

When the right-hand sides of two constraints are each increased by one unit,the objective function value will be adjusted by the sum of the constraints' dual prices.

For any constraint,either its slack/surplus value must be zero or its dual price must be zero.

Explain the two interpretations of dual prices based on the accounting assumptions made in calculating the objective function coefficients.

A section of output from The Management Scientist is shown here. Constraint Lower Limit Current Value Upper Limit 2 240 300 420 What will happen if the right-hand-side for constraint 2 increases by 200?

The 100% Rule does not imply that the optimal solution will necessarily change if the percentage exceeds 100%.

Introduction

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Utility and Game Theory

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Integer Linear Programming

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