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

If the range of feasibility for b₁ is between 16 and 37,then if b₁ = 22 the optimal solution will not change from the original optimal solution.

The following linear programming problem has been solved by The Management Scientist.Use the output to answer the questions. LINEAR PROGRAMMING PROBLEM MAX 25X1+30X2+15X3 S.T. 1)4X1+5X2+8X3<1200 2)9X1+15X2+3X3<1500 OPTIMAL SOLUTION Objective Function Value = 4700.000 Variable Value Reduced Cost X1 140.000 0.000 X2 0.000 10.000 X3 80.000 0.000 Constraint Slack/Surplus Dual Price 1 0.000 1.000 2 0.000 2.333 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 19.286 25.000 45.000 X2 No Lower Limit 30.000 40.000 X3 8.333 15.000 50.000 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 666.667 1200.000 4000.000 2 450.000 1500.000 2700.000 a.Give the complete optimal solution. b.Which constraints are binding? c.What is the dual price for the second constraint? What interpretation does this have? d.Over what range can the objective function coefficient of x₂ vary before a new solution point becomes optimal? e.By how much can the amount of resource 2 decrease before the dual price will change?f. What would happen if the first constraint's right-hand side increased by 700 and the second's decreased by 350?

The decision variables represent the amounts of ingredients 1,2,and 3 to put into a blend.The objective function represents profit.The first three constraints measure the usage and availability of resources A,B,and C.The fourth constraint is a minimum requirement for ingredient 3.Use the output to answer these questions. a.How much of ingredient 1 will be put into the blend? b.How much of ingredient 2 will be put into the blend? c.How much of ingredient 3 will be put into the blend? d.How much resource A is used? e.How much resource B will be left unused?f. What will the profit be?g. What will happen to the solution if the profit from ingredient 2 drops to 4?h. What will happen to the solution if the profit from ingredient 3 increases by 1?i. What will happen to the solution if the amount of resource C increases by 2?j. What will happen to the solution if the minimum requirement for ingredient 3 increases to 15?

The range of feasibility measures

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. $\text { Input Section }$ $\text { Objective Function Coefficients }$ x y 5 4 Constraints Req'd \#1 4 3 60 \#2 2 5 50 \#3 9 8 144 $\text { Output Section }$ Variables 9.6 7.2 Profit 48 28.8 76.8 Constraint Usage Slack \#1 60 1.35-11 \#2 55.2 -5.2 \#3 144 -2.62-11 a.Give the original linear programming problem. b.Give the complete optimal solution.

Sensitivity analysis is concerned with how certain changes affect

Relevant costs should be reflected in the objective function,but sunk costs should not.

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.

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

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

Consider the following linear program: Max 3+4 (\ Profit) s.t. +3\leq12 2+\leq8 \leq3 ,\geq0 Objective Function Value = 20.000 Variable Value Reduced Cost X1 2.400 0.000 X2 3.200 0.000 Constraint Slack/Surplus Dual Price 1 0.000 1.000 2 0.000 1.000 3 0.600 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 1.333 3.000 8.000 X2 1.500 4.000 9.000 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 9.000 12.000 24.000 2 4.000 8.000 9.000 3 2.400 3.000 No Upper Limit a.What is the optimal solution including the optimal value of the objective function? b.Suppose the profit on x₁ is increased to $7. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $7? c.If the unit profit on x₂ was $10 instead of $4, would the optimal solution change? d.If simultaneously the profit on x₁ was raised to $5.5 and the profit on x₂ was reduced to $3, would the current solution still remain optimal?

Decreasing the objective function coefficient of a variable to its lower limit will create a revised problem that is unbounded.

Output from a computer package is precise and answers should never be rounded.

Sensitivity analysis is often referred to as

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

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 Variable Value Reduced Cost X1 - 0.000 X2 24.000 - X3 - 3.500 Constraint Slack/Surplus Dual Price 1 0.000 15.000 2 - 0.000 3 - 0.000 4 0.000 - OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 5.400 12.000 No Upper Limit X2 2.000 9.000 20.000 X3 No Lower Limit 7.000 10.500 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 145.000 150.000 156.667 2 - - 64.000 3 - - 80.000 4 110.286 116.000 120.000

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.

If the dual price for the right-hand side of a  constraint is zero,there is no upper limit on its range of feasibility.

The 100 percent rule can be applied to changes in both objective function coefficients and right-hand sides at the same time.

The amount that the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution is the