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Sensitivity Analysis in Linear Programming: Maximizing Profit

Explore sensitivity analysis in LP through a LEGO production scenario, examining profit changes with varying input values and resource availability. Learn how to generate a Sensitivity Report in Excel Solver.

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Sensitivity Analysis in Linear Programming: Maximizing Profit

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  1. What-If Analysis for Linear Programming

  2. Sensitivity Analysis—“What  if?” Weekly supply of raw materials: 8 Small Bricks 6 large Bricks Products: Chair Profit = $15/Chair Table Profit = $20/Table

  3. Example: The Lego Production Problem (Revisited) Had 8 small bricks and 6 large bricks to produce tables and chairs. The optimal solution is to produce 2 tables and 2 chairs. What if the profit per table really is $25? Does the optimal solution change? What if the profit per table really is $35? Does the optimal solution change? Thus, there is some range of values for this profit over which the optimal solution does not change, but beyond which the optimal solution does change. What if one additional large brick becomes available? Does this enable increasing the total profit? The amount of any increase is called the shadow price for large bricks. What if a second additional large brick becomes available? Does this enable further increasing the total profit by the amount of the shadow price? Thus, there is a certain range of values (called the range of feasibility) for the number of available large bricks over which the shadow price is applicable.

  4. Formulation

  5. Generating the Sensitivity Report Solve the problem using the Solver:

  6. Then, choose “Sensitivity” under Reports.

  7. Questions Answered by Excel • What is the optimal solution? • What is the profit for the optimal solution? • If the net profit per table changes, will the solution change? • If the net profit per chair changes, will the solution change? • If more (or less) large bricks are available, how will this affect profit? • If more (or less) small bricks are available, how will this affect profit?

  8. The Sensitivity Report

  9. Net Profit from Tables = $35

  10. Seven Large Bricks

  11. Nine Large Bricks

  12. Summary of Output from Computer Solution • Changing Cells: • Final Value The value of the variable in the optimal solution • Reduced Cost Increase in the objective function value per unit increase in the value of a zero-valued variable (for small increases). • Allowable Defines the range of the cost coefficients in the • Increase/ objective function for which the current solution • Decrease (value of the variables in the optimal solution) will not change.

  13. Constraints: • Final Value The usage of the resource in the optimal solution. • Shadow price The change in the value of the objective function per unit increase in the right hand side of the constraint: • ∆Z = (Shadow Price)(∆RHS) • (Note: only valid if change is within the allowable range for RHS values—see below.) • Constraint The current value of the right hand side of the • R.H. Side constraint (the amount of the resource available). • Allowable Defines the range of values of the RHS for which • Increase/ the shadow price is valid and hence for which the • Decrease new objective function value can be calculated. (NOT the range for which the current solution will not change.)

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