Decision Modeling

1. Decision Modeling Sensitivity Analysis inLinear Programming

2. Sensitivity Analysis • Sensitivity analysis is the study of the effect on a solution of a small change in a model coefficient • E.g., what happens if our pricing forecast is wrong? • In constrained optimization, this is a process that takes place after the optimal solution has been found • Solver will generate a sensitivity analysis report upon request.

3. Oak Products LP Formulation max 56C + 40M subject to 8C + 4M ≤ 1280 (long dowels) 4C + 12M ≤ 1600 (short dowels) 4C + 4M ≤ 760 (legs) C ≤ 140 (heavy seats) M ≤ 120 (light seats) C + M ≥ 100 (union agreement) C, M ≥ 0 (non-negativity)

4. Graphical Analysis of LP • On-screen demonstration

5. Important Notes • The following applies to all LP solutions: • The optimal solution of an LP will never occur at an interior point of the feasible region • Geometrically, a binding constraint is one that passes through the optimal solution • Geometrically, a non-binding constraints is one that does not pass through the optimal solution • If a non-binding constraint is removed, the optimal solution does not change

6. Important Notes • The following applies to all LP solutions: • Adding a constraint will either impair the objective function value or leave it unchanged • Adding a decision variable will either improve the objective function value or leave it unchanged • Think of the decision variable as present in the original formulation, but constrained to the value of zero

7. Unbounded and Infeasible Models • Unbounded models typically occur when one or more important constraints have been left out of the model • In this case, the objective function can be improved infinitely by changing the value of the decision variables • Infeasible (or inconsistent) models arise when there is no feasible area for the decision variables • In this case, it is impossible to satisfy all constraints at the same time

8. Sensitivity Analysis • How sensitive is the optimal solution to inexact data? • The answer to this question will help determine the credibility of the model’s recommendations • If the objective function changes very little with large changes in the value of a particular coefficient, you would not be concerned with uncertainty in that coefficient’s value • If the objective function varies widely with small changes in that coefficient, you cannot tolerate uncertainty in its value

9. Sensitivity Analysis • Sensitivity analysis is based on changing the value of one coefficient while holding all other coefficients constant • Change in objective function coefficients will leave the feasible region unchanged, but will perhaps result in a different corner solution • Change in constraint coefficients generally changes the feasible region (the set of values that the decision variables may take) • Larger feasible area will improve the objective function value or leave it unchanged

10. Sensitivity Report in Solver • When Solver has found an optimal solution, you can select a sensitivity report • Report has one section on objective function coefficients • By how much must a coefficient change before the optimal solution changes? • Report has one section on constraint coefficients • What is the value of an additional unit of a (scarce) resource?

11. Sensitivity Report in Solver • On-screen demonstration

12. Objective Function Coefficients • If a coefficient is changed by less than the allowable amount, the current solution remains optimal • In a MAX model, increasing profitability of an activity cannot reduce the level of that activity • In a MIN model, decreasing cost of an activity cannot reduce the level of that activity • “Reduced cost” refers to the latter case and tells you by how much cost must decrease for an activity currently not in the solution before it enters the optimal solution (i.e., its decision variable becomes different from zero)

13. Constraint Coefficients • Solver report analyzes RHS coefficients • Shadow price: the value of an additional unit of a resource (i.e., the RHS) • Zero shadow price indicates that the constraint is not binding (there are unused resources of that type) • Non-zero shadow price valid only within a range of changes • As RHS changes, eventually some constraint is no longer binding in the new optimum • Shadow price is reduced due to different trade-offs • Eventually, the constraint is no longer binding

14. Constraint RHS • Tightening a constraint means that you make it more difficult achieve • Feasible region is reduced due to RHS change • Loosening a constraint means that you make it easier to achieve • Feasible region is enlarged do to RHS change • “Allowable increase/decrease” indicates the range of RHS changes within which the shadow price is valid

15. Degeneracy • If an optimal solution has a larger number of binding constraints than the number of decision variables with non-zero value, the solution is called degenerate • Sensitivity report might result in some anomalies • E.g., allowable increase/decrease for a shadow price might be zero • You must be careful when interpreting the sensitivity report of a degenerate solution

16. Shortcomings of Solver Sensitivity Report • Sensitivity report provides information only about small changes in a single coefficient at a time • The report displays sensitivity only for effects on objective function value • Report does not analyze changes in LHS constraint coefficients • Don’t like it? • Re-solve model (systematically) with other coefficient values

17. Exercise 1 Eastern Steel is mining iron ore from four different mines to be used in a new steel alloy product. Product requirements are so that elements A, B, and C must be present in certain minimum quantities (pounds per ton) in the blended ore. The different ores contain different amounts of these elements. Formulate the minimum cost blending problem.

18. Exercise 2 Buster Sod operates an 800-acre farm. Sod’s principal activities are raising wheat, alfalfa and beef. Sod has just been given his water allotments for next year (1,000 acre-feet) and is busy preparing his production plan for next year. He estimates that beef prices next year will be $500 per ton and that wheat will sell at $2 per bushel. Also, he will be able to sell alfalfa for $22 per ton, but if he has to buy alfalfa to feed his cattle, he has to pay $28 per ton. The wheat yield is 70 bushels per acre and the alfalfa yield is 4 tons per acre.