Question 1

Use the following Management Scientist output to answer the questions. LINEAR PROGRAMMING PROBLEM MAX 31X1+35X2+32X3 S.T. 1) 3X1+5X2+2X3>90 2) 6X1+7X2+8X3<150 3) 5X1+3X2+3X3<120 OPTIMAL SOLUTION Objective Function Value = 763.333 OBJECTIVE COEFFICIENT RANGES RIGHT HAND SIDE RANGES a.Give the solution to the problem. b.Which constraints are binding? c.What would happen if the coefficient of x₁ increased by 3? d.What would happen if the right-hand side of constraint 1 increased by 10?

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Question 2

Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a minimization objective function and all ≥ constraints.  ​ 
a.Give the original linear programming problem.
b.Give the complete optimal solution.

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Question 3

​How can the interpretation of dual prices help provide an economic justification for new technology?

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Question 4

Eight of the entries have been deleted from the LINDO output that follows. Use what you know about linear programming to find values for the blanks.
MIN 6 X1 + 7.5 X2 + 10 X3
SUBJECT TO
2) 25 X1 + 35 X2 + 30 X3 >= 2400
3) 2 X1 + 4 X2 + 8 X3 >= 400
END
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE
1) 612.50000   NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED:

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Question 5

In a linear programming problem, the binding constraints for the optimal solution are
5X + 3Y ≤ 30
2X + 5Y ≤ 20
a.Fill in the blanks in the following sentence:As long as the slope of the objective function stays between _______ and _______, the current optimal solution point will remain optimal.
b.Which of these objective functions will lead to the same optimal solution?1) 2X + 1Y 2) 7X + 8Y 3) 80X + 60Y 4) 25X + 35Y

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Question 6

​Describe each of the sections of output that come from The Management Scientist and how you would use each.

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Question 7

LINDO 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 NO. ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: 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 8

Portions of a Management Scientist output are shown below. Use what you know about the solution of linear programs to fill in the ten blanks.
LINEAR PROGRAMMING PROBLEM
MAX 12X1+9X2+7X3
S.T.
1) 3X1+5X2+4X3<150
2) 2X1+1X2+1X3<64
3) 1X1+2X2+1X3<80
4) 2X1+4X2+3X3>116
OPTIMAL SOLUTION
Objective Function Value = 336.000   OBJECTIVE COEFFICIENT RANGES  RIGHT HAND SIDE RANGES

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Question 9

​How would sensitivity analysis of a linear program be undertaken if one wishes to consider simultaneous changes for
both the right-hand-side values and objective function.

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Question 10

Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all ≤ constraints.  ​ 
a.Give the original linear programming problem.
b.Give the complete optimal solution.

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Question 11

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

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Question 12

The optimal solution of the linear programming problem is at the intersection of constraints 1 and 2. Max 2x₁ + x₂ s.t. 4x₁ + 1x₂ ≤ 400 4x₁ + 3x₂ ≤ 600 1x₁ + 2x₂ ≤ 300 x₁ , x₂ ≥ 0 a.Over what range can the coefficient of x₁ vary before the current solution is no longer optimal? b.Over what range can the coefficient of x₂ vary before the current solution is no longer optimal? c.Compute the dual prices for the three constraints.

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Question 13

The binding constraints for this problem are the first and second. Min x₁ + 2x₂ s.t. x₁ + x₂ ≥ 300 2x₁ + x₂ ≥ 400 2x₁ + 5x₂ ≤ 750 x₁ , x₂ ≥ 0 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 14

​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 15

Use the following Management Scientist output to answer the questions.
MIN 4X1+5X2+6X3
S.T.
1) X1+X2+X3<85
2) 3X1+4X2+2X3>280
3) 2X1+4X2+4X3>320
Objective Function Value = 400.000   OBJECTIVE COEFFICIENT RANGES  RIGHT HAND SIDE RANGES  
a.What is the optimal solution, and what is the value of the profit contribution?
b.Which constraints are binding?
c.What are the dual prices for each resource? Interpret.
d.Compute and interpret the ranges of optimality.
e.Compute and interpret the ranges of feasibility.

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Question 16

​Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of
feasibility.

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Quiz 3: Linear Programming: Sensitivity Analysis and Interpretation of Solution

Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a minimization objective function and all ≥ constraints. a.Give the original linear programming problem. b.Give the complete optimal solution.

How can the interpretation of dual prices help provide an economic justification for new technology?

Describe each of the sections of output that come from The Management Scientist and how you would use each.

How would sensitivity analysis of a linear program be undertaken if one wishes to consider simultaneous changes for both the right-hand-side values and objective function.

Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all ≤ constraints. a.Give the original linear programming problem. b.Give the complete optimal solution.

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

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

Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of feasibility.

Quiz 3: Linear Programming: Sensitivity Analysis and Interpretation of Solution

Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a minimization objective function and all ≥ constraints. ​ a.Give the original linear programming problem. b.Give the complete optimal solution.

​How can the interpretation of dual prices help provide an economic justification for new technology?

​Describe each of the sections of output that come from The Management Scientist and how you would use each.

​How would sensitivity analysis of a linear program be undertaken if one wishes to consider simultaneous changes for both the right-hand-side values and objective function.

Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all ≤ constraints. ​ a.Give the original linear programming problem. b.Give the complete optimal solution.

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

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

​Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of feasibility.