1. Notes 4IE 3121 Why Sensitivity Analysis So far: find an optimium solution given certain constant parameters (costs, demand, etc) How well do we know these parameters? Usually not very accurately - rough estimates Do our results remain valid? If the parameters change...... how much does the objective function change?... how much do the optimal values of the decision variables change?

2. Notes 4IE 3122 General Optimization Problem Minimize some cost or maximize benefit Constraints: £ Restrictions on supply of some resource ³ Restriction on satisfying demand for some resource = Both supply restriction and demand requirement Variable-type constraints Decision variable determine the levels of some activity Coefficients = per unit impact of activities

3. Notes 4IE 3123 Changing Constraints Relaxing constraints: Optimal value same or better Tightening constraints: Optimal value same or worse Original Relaxed Tightened

4. Notes 4IE 3124 Crude Oil Model Satisfy Demand Supply Restriction

5. Notes 4IE 3125 Solution (LINDO) LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1) 112.5000 VARIABLE VALUE REDUCED COST X1 2.000000 0.000000 X2 3.500000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 -37.500000 3) 0.000000 0.000000 4) 0.950000 0.000000 5) 7.000000 0.000000 6) 2.500000 0.000000 7) 0.000000 -18.750000 8) 1.500000 0.000000

6. Notes 4IE 3126 Sensitivity ‘Plenty’ of this crude

7. Notes 4IE 3127 RHS Coefficients

8. Notes 4IE 3128 LHS Coefficients

9. Notes 4IE 3129 New Constraints Adding constraints tightens the feasible set Removing constraints relaxes the feasible set What about unmodeled constraints?

10. Notes 4IE 31210 Rate of Change Optimal Value RHS Optimal Value RHS Optimal Value RHS Optimal Value RHS Maximize Minimize Supply Demand

11. Notes 4IE 31211 Objective Function Changes

12. Notes 4IE 31212 Crude Oil: Changing x 1 Coefficient

13. Notes 4IE 31213 Rate of Change Optimal Value Coefficient. Optimal Value Coefficient. Maximize Minimize

14. Notes 4IE 31214 New Activities Adding activities Optimal value same or better Removing activities Optimal value same or worse

15. Notes 4IE 31215 Quantifying Effects Now know the qualitative effects of Changing RHS coefficients Changing LHS coefficients Changing objective function coefficients Adding/deleting constraints Adding/deleting activities How much does the objective change? Quantitative change

16. Notes 4IE 31216 Back to Crude Oil Example Decreasing RHS will make objective better or no worse, but by how much? How much are we willing to pay to have one more barrel available?

17. Notes 4IE 31217 Answer using the Dual

18. Notes 4IE 31218 LINDO Solution LP OPTIMUM FOUND AT STEP 4 OBJECTIVE FUNCTION VALUE 1) 92.50000 VARIABLE VALUE REDUCED COST V1 20.000000 0.000000 V2 35.000000 0.000000 V3 0.000000 0.950000 V4 0.000000 0.000000 V5 0.000000 0.000000

19. Notes 4IE 31219 Interpretation Our cost will be reduced by $20 or $ 35, respectively, if the demand for gasoline or jet fuel is one unit less. Smaller demand for lubricants has no effect on the objective We are not willing to pay anything for availability of more crude!

20. Notes 4IE 31220 What is the Dual? The primal is the original optimization problem The dual is an LP defined on the same input parameters but characterizing the sensitivities of the primal There is one dual variable for each main constraint

21. Notes 4IE 31221 Interpretation The dual variables provide implicit prices for marginal units of the resource modeled by the constraint Zero unless active How much we are willing to pay for more of a resource (supply constraint) How much we benefit from not having to satisfy a requirement (demand constraint)

22. Notes 4IE 31222 What to Optimize? Implicit marginal value (minimization primal) or price (maximization primal) is Maximize value or minimize price!

23. Notes 4IE 31223 Dual Constraints For each activity x j in a minimization primal there is a main dual constraint For a maximization primal, each x j  0 has a main dual constraint

24. Notes 4IE 31224 Optimal Solution If primal has optimal solution Either the primal optimal makes a main inequality active or the corresponding dual is zero Either a nonnegative primal variable has optimal value x j = 0 or the corresponding dual price v j must make the j-th dual constraint active

25. Notes 4IE 31225 Dual of a Min Primal

26. Notes 4IE 31226 Dual of a Max Primal

27. Notes 4IE 31227 Top Brass

28. Notes 4IE 31228 Graphical Solution 2000 1500 1000 500 500 1000 1500 2000 Optimal Solution

29. Notes 4IE 31229 Dual

30. Notes 4IE 31230 Lindo Solution OBJECTIVE FUNCTION VALUE 1) 17700.00 VARIABLE VALUE REDUCED COST V1 0.000000 350.000000 V2 0.000000 400.000000 V3 6.000000 0.000000 V4 1.500000 0.000000

31. Notes 4IE 31231 Interpretation We are willing to pay up to $6/each for additional brass plaques We are willing to pay up to $1.5/foot for more wood We don ’ t need any more brass footballs or soccer balls Our objective is sensitive to these estimates!

32. Notes 4IE 31232 Formulating Duals

33. Notes 4IE 31233 Formulating Duals