Exam 8: LP Sensitivity Analysis

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

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

Consider the following linear program: The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 27.000 MIN $6 x _ { 1 } + 9 x _ { 2 } ( \$$ cost $)$ s.t. +2 \leq8 10+7.5 \geq30 \geq2 , \geq0 ​ Variable Value Reduced Cost 1 1.500 0.000 2 2.000 0.000 OBJECTIVE COEFFICIENT RANGES Constraint Slack/Surplus Dual Price 1 2.500 0.000 2 0.000 -0.600 3 0.000 -4.500 RIGHT HAND SIDE RANGES Variable Lower Limit Current Value Upper Limit 1 0.000 6.000 12.000 2 4.500 9.000 No Upper Limit ​ 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?

The binding constraints for this problem are the first and second. Min $\mathrm { x } _ { 1 } + 2 \mathrm { x } 2$ s.t. $x _ { 1 } + x _ { 2 } \geq 300$ $2 \mathrm { x } _ { 1 } + \mathrm { x } _ { 2 } \geq 400$ $2 \mathrm { x } 1 + 5 \mathrm { x } _ { 2 } \leq 750$ $\mathrm { x } _ { 1 } , \mathrm { x } _ { 2 } \geq 0$ 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.

A constraint with a positive slack value

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

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.

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

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

Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a variable in a constraint.

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.

Consider the following linear program: The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 20.000 MIAX $3 x _ { 1 } + 4 x 2 ( \$$ Profit $)$ s.t. $x _ { 1 } + 3 x _ { 2 } \leq 12$ $2 x _ { 1 } + x _ { 2 } \leq 8$ $x 1 \leq 3$ $x _ { 1 } , x _ { 2 } \geq 0$ Variable Value Reduced Cost 1 2.400 0.000 2 3.200 0.000 OBJECTIVE COEFFICIENT RANGES Constraint Slack/Surplus Dual Price 1 0.000 1.000 2 0.000 1.000 3 0.600 0.000 RIGHT HAND SIDE RANGES Variable Lower Limit Current Value Upper Limit X1 1.333 3.000 8.000 2 1.500 4.000 9.000 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?

When the cost of a resource is sunk,then the dual price can be interpreted as the

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

​Sensitivity analysis is often referred to as

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

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

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

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