Exam 8: LP Sensitivity Analysis

LINDO output is given for the following linear programming problem. MIN 12 X1 + 10 X2 + 9 X3 SUBJECT TO END LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1)80.000000 2) $5 \mathrm { X } 1 + 8 \mathrm { X } 2 + 5 \mathrm { X } 3 > = 60$ 3) $8 X 1 + 10 X 2 + 5 X 3 > = 80$ VARIABLE VALUE REDUCED COST 1 .000000 4.000000 2 8.000000 .000000 3 .000000 4.000000 NO.ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED: ROW SLACK OR SURPLUS DUAL PRICE 2) 4.000000 .000000 3) 000000 -1.000000 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?

Use the following Management Scientist output to answer the questions. LINEAR PROGRAMMING PROBLEM MAX 31X1+35X2+32X3 S.T. OPTIMAL SOLUTION Objective Function Value = 763.333 1) $3 X 1 + 5 X 2 + 2 X 3 > 90$ 2) $6 \mathrm { X } 1 + 7 \mathrm { X } 2 + 8 \mathrm { X } 3 < 150$ 3) $5 \mathrm { X } 1 + 3 \mathrm { X } 2 + 3 \mathrm { X } 3 < 120$ Variable Value Reduced Cost 1 13.333 0.000 2 10.000 0.000 3 0.000 10.889 OBJECTIVE COEFFICIENT RANGES Constraint Slack/Surplus Dual Price 1 0.000 -0.778 2 0.000 5.556 3 23.333 0.000 RIGHT HAND SIDE RANGES Variable Lower Limit Current Value Upper Limit X1 30.000 31.000 No Upper Limit X2 No Lower Limit 35.000 36.167 X3 No Lower Limit 32.000 42.889 Constraint Lower Limit Current Value Upper Limit 1 77.647 90.000 107.143 2 126.000 150.000 163.125 3 96.667 120.000 No Upper Limit 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?

Any change to the objective function coefficient of a variable that is positive in the optimal solution will change the optimal solution.

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 END LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 2) $25 X 1 + 35 X 2 + 30 X 3 > = 2400$ 3) $2 X 1 + 4 X 2 + 8 X 3 > = 400$ 1)612.50000 NO.ITERATIONS= 2 RANGES IN WHICH THE BASIS IS UNCHANGED: ROW SLACK OR SURPLUS DUAL PRICE 2) -.125000 3) -.781250

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

If the range of feasibility indicates that the original amount of a resource,which was 20,can increase by 5,then the amount of the resource can increase to 25.

If the optimal value of a decision variable is zero and its reduced cost is zero,this indicates that alternative optimal solutions exist.

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.

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

A negative dual price for a constraint in a minimization problem means

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. OPTIMAL SOLUTION Objective Function Value = 336.000 1) $3 \mathrm { X } 1 + 5 \mathrm { X } 2 + 4 \mathrm { X } 3 < 150$ 2) $2 \mathrm { X } 1 + 1 \mathrm { X } 2 + 1 \mathrm { X } 3 < 64$ 3) $1 \mathrm { X } 1 + 2 \mathrm { X } 2 + 1 \mathrm { X } 3 < 80$ 4) $2 X 1 + 4 X 2 + 3 X 3 > 116$ OBJECTIVE COEFFICIENT RANGES RIGHT HAND SIDE RANGES Variable Lower Limit Current Value Upper Limit 1 5.400 12.000 No Upper Limit 2 2.000 9.000 20.000 3 No Lower Limit 7.000 10.500

​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.

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

​The dual price for a < constraint

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

​Sensitivity analysis is concerned with how certain changes affect

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

The optimal solution of the linear programming problem is at the intersection of constraints 1 and 2. ​ Max 2+ s.t. 4+1\leq400 4+3\leq600 1+2\leq300 ,\geq0 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.

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

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