Chapter 7 Duality and Sensitivity in Linear Programming

7.1 Generic Activities versus Resources Perspective Objective Functions as Costs and Benefits Optimization models objective functions usually can be interpreted as minimizing some measure of cost or maximizing some measure of benefit. [7.1] Choosing a Direction for Inequality Constraints The most natural expression of a constraint is usually the one making the right-hand-side constant non- negative. [7.2] 0.3 x x 2  2.0 (gasoline) x x 2  -2.0

7.1 Generic Activities versus Resources Perspective Inequalities as Resource Supplies and Demands Optimization model constraints of the  form usually can be interpreted as restricting the supply of some commodity or resource. [7.3] x 1  9 (Saudi) Optimization model constraints of the  form usually can be interpreted as requiring satisfaction of a demand for some commodity or resource. [7.4] 0.4 x x 2  1.5 (jet fuel)

7.1 Generic Activities versus Resources Perspective Equality Constraints as Both Supplies and Demands Optimization model equality constraints usually can be interpreted as imposing both a supply restriction and a demand requirement on some commodity or resource. [7.5] Variable-Type Constraints Non-negativity and other sign restriction constraints are usually best interpreted as declarations of variable type rather than supply or demand limits on resources. [7.6]

7.1 Generic Activities versus Resources Perspective Variables as Activities Decision variables in optimization models can usually be interpreted as choosing the level of some activity. [7.7] LHS Coefficients as Activity Inputs and Outputs Non-zero objective function and constraint coefficients on LP decision variables display the impacts per unit of the variable’s activity on resources or commodities associated with the objective and constraints. [7.8]

Inputs and Outputs for Activities Two Crude Example 1000 barrels of Saudi petroleum processed (x1) Availability 1000 barrels Cost $ unit gasoline.4 unit jet fuel.2 unit lubricants

7.2 Qualitative Sensitivity to Changes in Model Coefficients Relaxing versus Tightening Constraints Relaxing the constraints of an optimization model either leaves the optimal value unchanged or makes it better (higher for a maximize, lower for a minimize). Tightening the constraints either leaves the optimal value unchanged or makes it worse. [7.9]

Relaxing Constraints x2x2 x1x1 7 8 x1  9x1  9 x2  6x2  x2x2 x1x1 7 8 x1  9x1  9 x2  6x2  6

Tightening Constraints x2x2 x1x1 7 8 x1  9x1  9 x2  6x2  x2x2 x1x1 7 8 x1  9x1  9 x2  6x2  6

Swedish Steel Blending Example min 16 x x 2 +8 x 3 +9 x x x x 7 s.t.x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 = x x x x 4  x x x x 4  x x x 5  x x x 5  x x x 6  x x x 6  x x 7  x x 7  13.0 x 1  75 x 2  250 x 1 …x 7  0 (7.1)

Effect of Changes in RHS Optimal Value RHS x 1  75 Current Slope -  Slope Slope Slope 0.00

Effect of Changes in RHS Optimal Value RHS x x x 6  10.0 Current Slope +  Slope 8.57 Slope Slope 0.00 Slope

Effect of Changes in RHS and LHS Changes in LP model RHS coefficients affect the feasible space as follows: [7.10] Changes in LP model LHS constraint coefficients on non-negative decision variables affect the feasible space as follows: [7.11] Constraint TypeRHS IncreaseRHS Decrease Supply (  ) RelaxTighten Demand (  ) TightenRelax Constraint TypeCoefficient Increase Coefficient Decrease Supply (  ) TightenRelax Demand (  ) RelaxTighten

Effect of Adding or Dropping Constraints Adding constraints to an optimization model tightens its feasible set, and dropping constraints relaxes its feasible set. [7.12] Explicitly including previous un-modeled constraints in an optimization model must leave the optimal value either unchanged or worsened. [7.13]

Effect of Changing Rates of Constraint Coefficient Impact Coefficient changes that help the optimal value in LP by relaxing constraints help less and less as the change becomes large. Changes that hurt the optimal value by tightening constraints hurt more and more. [7.14]

Effect of Changing Rates of Constraint Coefficient Impact Optimal Value RHS Maximize objective Optimal Value RHS Supply (  ) Demand (  )

Effect of Changing Rates of Constraint Coefficient Impact Optimal Value RHS Minimize objective Optimal Value RHS Supply (  ) Demand (  )

Effects of Objective Function Coefficient Changes Changing the objective function coefficient of a non- negative variable in an optimization model affects the optimal value as follows: [7.15] Model Form (Primal) Coefficient Increase Coefficient Decrease Maximize objectiveSame or betterSame or worse Minimize objectiveSame or worseSame or better

Changing Rates of Objective Function Coefficient Impact Maximize objective Optimal Value coef Objective function coefficient changes that help the optimal value in LP help more and more as the change becomes large. Changes that hurt the optimal value less and less. [7.16] Optimal Value coef Minimize objective

Effect of Adding or Dropping Variables Adding optimization model activities (variables) must leave the optimal value unchanged or improved. Dropping activities will leave the value unchanged or degraded. [7.17]

7.3 Quantitative Sensitivity to Changes in LP Model Coefficients Primals and Duals Defined The primal is the given optimization model, the one formulating the application of primary interest. [7.18] The dual is a subsidiary optimization model, defined over the same input parameters as the primal but characterizing the sensitivity of primal results to changes in inputs. [7.19]

Dual Variables There is one dual variable for each main primal constraint. Each reflects the rate of change in primal value per unit increase from the given RHS value of the corresponding constraint. [7.20] The LP dual variable on constraint i has type as follows: [7.21] Primal i is  i is  i is = Minimize objective i  0 i  0 Unrestricted (URS) Maximize objective i  0 i  0 Unrestricted (URS)

Two Crude Example min 20 x x 2 s.t. 0.3 x x 2  2.0 : 1 (gasoline) 0.4 x x 2  1.5 : 2 (jet fuel) 0.2 x x 2  0.5 : 3 (lubricants) x 1  9 : 4 (Saudi) x 2  6 : 5 (Venezuelan) x 1, x 2  0 1  0, 2  0, 3  0, 4  0, 5  0 (7.4) (7.5)

Dual Variables as Implicit Marginal Prices Dual variables provide implicit prices for the marginal unit of the resource modeled by each constraint as its RHS limit is encountered. [7.22] Variable, 1, $1000s/1000 barrels, is the implicit price of gasoline at the margin when demand for gasoline is at 2000 barrels. Variable, 4, reflects the marginal impact of the Saudi availability constraint at its current level of 9000 barrels.

Implicit Activity Pricing in Terms of Resources Produced and Consumed

Optimal Value Equality between Primal and Dual (7.6)

Main Dual Constraints to Enforce Activity Pricing

Primal Complementary Slackness between Primal Constraints and Dual Variable Values 3 = 0, 4 = 0, 5 = 0

Dual Complementary Slackness between Dual Constraints and Primal Variable Values

7.4 Formulating LP Duals Form of the Dual for Non-negative Primal Variables

Two Crude Example (7.7) Primal: min 20 x x 2 s.t. 0.3 x x 2  x x 2  x x 2  0.5 x 1  9 x 2  6 x 1, x 2  0

Duals of LP Models with Non-positive and Unrestricted Variables Max Form Primal ElementCorresponding Dual Element Obj. Constraint

Duals of LP Models with Non-positive and Unrestricted Variables Min Form Primal ElementCorresponding Dual Element Obj. Constraint

Dual of the Dual Is the Primal The dual of the dual of any linear program is the LP itself. [7.30]

7.5 Primal-to-Dual Relationships Weak Duality between Objective Values (7.8)

Weak Duality between Objective Values The primal objective function evaluated at any feasible solution to a minimize primal is greater than or equal to (  ) the objective function value of the corresponding dual evaluated at any dual feasible solution. For a maximize primal it is (  ). [7.31]

Strong Duality between Objective Values If either a primal LP or its dual has an optimal solution, both do, and their optimal objective function values are equal. [7.32]

Dual Optimum as a By-product (7.10) (7.11)

Unbounded and Infeasible Cases If either a primal LP model or its dual is unbounded, the other is infeasible. [7.34] The following shows which outcome pars are possible for a primal LP and its dual: [7.35] PrimalDual OptimalInfeasibleUnbounded OptimalPossibleNever InfeasibleNeverPossible UnboundedNeverPossibleNever

7.6 Computer Outputs and What If Changes of Single Parameters RHS ranges in LP sensitivity outputs show the interval within which the corresponding dual variable value provides the exact rate of change in optimal value per unit change in RHS (all other data held constant) [7.36]

7.6 Computer Outputs and What If Changes of Single Parameters Dropping a constraint can change the optimal solution only if the constraint is active at optimality. [7.39] Adding a constraint can change the optimal solution only if that optimum violates the constraint. [7.40] An LP variable can be dropped without changing the optimal solution only if its optimal value is zero. [7.41] A new LP variable can change the current primal optimal solution only if its dual constraint is violated by the current dual optimum. [7.42]

7.7 Bigger Model Changes, Re- optimization, and Parametric Programing Ambiguity at Limits of the RHS and Objective Coefficient Ranges At the limits of the RHS and objective function sensitivity ranges, rates of optimal value change are ambiguous, with one value applying below the limit and another above. Computer outputs may show either value. [7.43]

Two Crude Example x2x2 x1x1 7 8 Gasoline Jet Fuel Lubricants Saudi Venezuelan B 1 =2.0 Optimal Dual Lower Range Upper Range

Two Crude Example x2x2 x1x1 7 8 Gasoline Jet Fuel Lubricants Saudi Venezuelan B 1 =2.625 Optimal Dual Lower Range Upper Range Optimal Dual Lower Range Upper Range

Two Crude Example x2x2 x1x1 7 8 Gasoline Jet Fuel Lubricants Saudi Venezuelan B 1 =3.25 Optimal Dual Lower Range Upper Range

Connection between Rate Changes and Degeneracy Rates of variation in optimal value with model constants change when the collection of active primal or dual constraints changes. [7.44] Degeneracy, which is extremely common in large-scale LP models, limits the usefulness of sensitivity by- products from primal optimization because it leads to narrow RHS and objective coefficient ranges and ambiguity at the range limits. [7.45]

Re-Optimization to Make Sensitivity Exact If the number of “what-if” variations does not grow too big, re-optimization using different values of model input parameters often provides the most practical avenue to good sensitivity analysis. [7.46]

Parametric Variation of One Coefficient Parametric studies track the optimal value as a function of model inputs. If the number of “what-if” variations does not grow too big, re-optimization using different values of model input parameters often provides the most practical avenue to good sensitivity analysis. [7.46]

Parametric Variation of One Coefficient Parametric studies track the optimal value as a function of model inputs. Parametric studies of optimal value as a function of a single-model RHS or objective function coefficient can be constructed by repeated optimization using new coefficient values just outside the previous applicable sensitivity range. [7.47]

Parametric Variation of One Coefficient: Two Crude Example Optimal Value RHS Slope +  (infeasible) Slope Slope 0.00 Slope CaseRHSDualLower Rang Upper Range Base Model Variant  Variant  ++ + Variant  - 1.125

Assessing Effects of Multiple Parameter Changes Elementary LP sensitivity rates of change and ranges hold only for a single coefficient change, with all other data held constant. [7.48] If demand increase in jet fuel (b 2 ) is twice as the increase for gasoline (b 1 ). b 1 new = (1+  ) b 1 base b 2 new = (1+2  ) b 2 base b i new = b i base +  b i  b 1 =2.0  b 2 =3.0 (7.14)

Assessing Effects of Multiple Parameter Changes The effect of a multiple change in RHS with step  can be analyzed parametrically by treating  as a new decision variable with constraint coefficient -  b i that detail the rates of change in RHS’s bi and a value fixed by a new equality constrain. [7.49] min 20 x x 2 s.t.0.3 x x  x x  x x 2  0.5 x 1  9 x 2  6  = b 6 x 1, x 2  0,  URS

Parametric Variation of One Coefficient: Two Crude Example Optimal Value  Slope +  (infeasible) Slope Slope

Parametric Change of Multiple Objective Function Coefficients The effect of multiple change in objective function with step  can be analyzed parametrically by treating objective rates of change -  c i as coefficients in a new equality constraint having RHS zero and a new unrestricted variable with objective coefficient . [7.50]