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

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

The cost that varies depending on the values of the decision variables is a

Eight of the entries have been deleted from the LINGO 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 VARIABLE VALUE REDUCED COST X1 - 1.312500 X2 - - X3 27.500000 - ROW SLACK OR SURPLUS DUAL PRICE 2) - -.125000 3) - -.781250 NO.ITERATIONS= 2 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES VARIABLE X1 6.000000 - - X2 7.500000 1.500000 2.500000 X3 10.000000 5.000000 3.571429 RIGHTHAND SIDE RANGES CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 2400.000000 1100.000000 900.000000 3 400.000000 240.000000 125.714300

The LP model and LINGO output represent a problem whose solution will tell a specialty retailer how many of four different styles of umbrellas to stock in order to maximize profit.It is assumed that every one stocked will be sold.The variables measure the number of women's,golf,men's,and folding umbrellas,respectively.The constraints measure storage space in units,special display racks,demand,and a marketing restriction,respectively. MAX 4 X1 + 6 X2 + 5 X3 + 3.5 X4 SUBJECT TO 2)2 X1 + 3 X2 + 3 X3 + X4 <= 120 3)1.5 X1 + 2 X2 <= 54 4)2 X2 + X3 + X4 <= 72 5)X2 + X3 >= 12 END OBJECTIVE FUNCTION VALUE 1)318.00000 VARIABLE VALUE REDUCED COST X1 12.000000 .000000 X2 .000000 .500000 X3 12.000000 .000000 X4 60.000000 .000000 ROW SLACK OR SURPLUS DUAL PRICE 2) .000000 2.000000 3) 36.000000 .000000 4) .000000 1.500000 5) .000000 -2.500000 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES VARIABLE X1 X2 X3 X4 4.000000 1.000000 2.500000 6.000000 .500000 INFINITY 5.000000 2.500000 .500000 3.500000 INFINITY .500000 RIGHTHAND SIDE RANGES CURRENT ALLOWABLE ALLOWABLE ROW RHS INCREASE DECREASE 2 120.000000 48.000000 24.000000 3 54.000000 INFINITY 36.000000 4 72.000000 24.000000 48.000000 5 12.000000 12.000000 12.000000 Use the output to answer the questions. a.How many women's umbrellas should be stocked? b.How many golf umbrellas should be stocked? c.How many men's umbrellas should be stocked? d.How many folding umbrellas should be stocked? e.How much space is left unused?f. How many racks are used?g. By how much is the marketing restriction exceeded?h. What is the total profit?i. By how much can the profit on women's umbrellas increase before the solution would change?j. To what value can the profit on golf umbrellas increase before the solution would change?k. By how much can the amount of space increase before there is a change in the dual price?l. You are offered an advertisement that should increase the demand constraint from 72 to 86 for a total cost of $20. Would you say yes or no?

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

The amount by which an objective function coefficient can change before a different set of values for the decision variables becomes optimal is the

The dual price for a < constraint

The amount of a sunk cost will vary depending on the values of the decision variables.

The reduced cost for a positive decision variable is 0.

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 Variable Value Reduced Cost X1 0.000 1.500 X2 80.000 0.000 X3 0.000 1.000 Constraint Slack/Surplus Dual Price 1 5.000 0.000 2 40.000 0.000 3 0.000 -1.250 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 2.500 4.000 No Upper Limit X2 0.000 5.000 6.000 X3 5.000 6.000 No Upper Limit RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 80.000 85.000 No Upper Limit 2 No Lower Limit 280.000 320.000 3 280.000 320.000 340.000 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.

Which of the following is not a question answered by standard sensitivity analysis information?

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 Variable Value Reduced Cost X1 13.333 0.000 X2 10.000 0.000 X3 0.000 10.889 Constraint Slack/Surplus Dual Price 1 0.000 -0.778 2 0.000 5.556 3 23.333 0.000 OBJECTIVE COEFFICIENT 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 RIGHT HAND SIDE RANGES 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?

A constraint with a positive slack value

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

If a decision variable is not positive in the optimal solution,its reduced cost is

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

The dual price measures,per unit increase in the right hand side,

A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective.

Decision variables must be clearly defined before constraints can be written.

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. $\text { Input Section }$ $\text { Objective Function Coefficients }$ x y 4 6 Constraints Avail. \#1 3 5 60 \#2 3 2 48 \#3 1 1 20 $\text { Output Section }$ Variables 13.333333 4 Profit 53.333333 24 77.333333 Constraint Usage Slack \#1 60 1.789-11 \#2 48 -2.69-11 \#3 17.333333 2.6666667 a.Give the original linear programming problem. b.Give the complete optimal solution.