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## Why Sensitivity Analysis

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**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? IE 312**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 IE 312**Changing Constraints**• Relaxing constraints: • Optimal value same or better • Tightening constraints: • Optimal value same or worse Original Relaxed Tightened IE 312**Crude Oil Model**Satisfy Demand Supply Restriction IE 312**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 IE 312**Sensitivity**‘Plenty’ of this crude IE 312**RHS Coefficients**IE 312**LHS Coefficients**IE 312**New Constraints**• Adding constraints tightens the feasible set • Removing constraints relaxes the feasible set • What about unmodeled constraints? IE 312**Optimal**Value Optimal Value Optimal Value Optimal Value RHS RHS RHS RHS Rate of Change Supply Demand Maximize Minimize IE 312**Objective Function Changes**IE 312**Rate of Change**Maximize Minimize Optimal Value Optimal Value Coefficient. Coefficient. IE 312**New Activities**• Adding activities • Optimal value same or better • Removing activities • Optimal value same or worse IE 312**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 IE 312**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? IE 312**Answer using the Dual**IE 312**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 IE 312**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! IE 312**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 IE 312**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) IE 312**What to Optimize?**• Implicit marginal value (minimization primal) or price (maximization primal) is • Maximize value or minimize price! IE 312**Dual Constraints**• For each activity xj in a minimization primal there is a main dual constraint • For a maximization primal, each xj 0 has a main dual constraint IE 312**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 xj= 0 or the corresponding dual price vj must make the j-th dual constraint active IE 312**Dual of a Min Primal**IE 312**Dual of a Max Primal**IE 312**Top Brass**IE 312**2000**1500 1000 500 500 1000 1500 2000 Graphical Solution Optimal Solution IE 312**Dual**IE 312**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 IE 312**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! IE 312**Formulating Duals**IE 312**Formulating Duals**IE 312