computational methods for management and economics carla gomes l.
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  1. Computational Methods forManagement and EconomicsCarla Gomes Module 7b Duality and Sensitivity Analysis Economic Interpretation of Duality (slides adapted from: M. Hillier’s, J. Orlin’s, and H. Sarper’s)

  2. Post-optimality Analysis • Post-optimality – very important phase of modeling. • Duality plays and important role in post-optimality analysis • Simplex provides several tools to perform post-optimality analysis

  3. Post-optimality analysis for LP

  4. Economic Interpretation of Duality • LP problems – quite often can be interpreted as allocating resources to activities. • Let’s consider the standard form: xi >= 0 , (i =1,2,…,n)

  5. What if we change our resources – can we improve our optimal solution? • Resources – m (plants) • Activities – n (2 products) • Wyndor Glass problemoptimal product mix --- allocation of resourcesto activities i.e., choose the levels of the activities that achieve best overall measure of performance

  6. Sensitivity Analysis How would changes in the problem’s objective function coefficients or right-hand side values change the optimal solution?

  7. Dual Variables (Shadow Prices) • y1*= 0  dual variable (shadow price) for resource 1 • y2*= 1.5  dual variable (shadow price) for resource 2 • y3*= 1  dual variable (shadow price) for resource 3 How much does Z increase if we increase resource 2 by 1 unit (i.e., b2 = 12  b2=13)?

  8. Graphical Analysis of Dual variables – Variation in RHS Increasing level of resource 2 (b2) (5/3,13/2) 2w=13  Z=3(5/3)+5(13/2)=37.5 ∆ Z=1.5 = y2* Z=3(2)+5(6)=36 (2,6) Why is y1*=0?

  9. Economic Interpretation of Dual Variables The dual variable associated with resource i (also called shadow price), denoted by yi*, measures the marginal value of this resource, i.e., the rate at which Z could be increased by (slightly) increasing the amount of this resource (bi), assuming everything else stays the same. The dual variable yi* is identified by the simplex method as the coefficient of the ith slack variable in row 0 of the final simplex tableau.

  10. Dual Variables: binding and non-binding constraints • The shadow prices (dual variables) associated with non-binding constraints are necessarily 0 (complementary optimal slackness)  there is a surplus of non-binding resource and therefore increasing it will not increase the optimal solution. Economist refer to such resources as free resources (shadow price =0) • Binding constraints on the other hand correspond to scarce resources – there is no surplus. In general they have a positive shadow price.

  11. Does Z always increase at the same rate if we keep increasing the amount of resource 2? (0,9) b2=18 (5/3,13/2) 2w=13  Z=3(5/3)+5(13/2)=37.5 ∆ Z=1.5 = y2* Z=3(2)+5(6)=36 (2,6) What if b2 > 18 (i.e., 2W>18)?  the optimal solution will stay at (0,9) for b2>=18

  12. 2w=13  Z=3(5/3)+5(13/2)=37.5 ∆ Z=1.5 = y2* Z=3(2)+5(6)=36 b2=6 Does Z always decrease at the same rate if we decrease resource 2? (5/3,13/2) (2,6) If b2 < 6 the solution will no longer vary proportionally. The optimal solution varies proportionally to the variation in b2 only if 6 <= b2 <=18. In other words, the current basis remains optimal for 6 ≤ b2 ≤ 18, but the decision variable values and z-value will change.

  13. A dual variable yi* gives us the rate at which Z could be increased by increasing the amount of resource i slightly. • However this is only true for a small increase in the amount of the resource. I.e., this definition applies only if the change in the RHS of constraint i leaves the current basis optimal. It also assumes everything else stays the same. • Another interpretation of yi* is: if a premium price must be paid for the resource i in the market place, yi* is the maximum premium (excess over the regular price) that would be worth paying.

  14. Optimal Basis in the Wyndor Glass Problem • How can we characterize (verbally) the optimal basis of the Wyndor Glass problem? • Plant 1 – unutilized capacity (non-binding constraint) • Plant 2 – fully utilized capacity (binding constraint) • Plant 3 - fully utilized capacity (binding constraint)

  15. How do we interpret the intervals? • If we change one coefficient in the RHS, say capacity of plant 2, by D the “basis” remains optimal, that is, the same equations remain binding. • So long as the basis remains optimal, the shadow prices are unchanged. • The basic feasible solution varies linearly with D. If D is big enough or small enough the basis will change.

  16. The dual price or shadow price for the i th constraint of an LP is the amount by which the optimal z-value is improved (increased in a max problem or decreased in a min problem) if the rhs of the i th constraint is increased by one. This definition applies only if the change in the rhs of constraint i leaves the current basis optimal. The dual variables or shadow prices are valid in a given interval.

  17. Sensitivity analysis for c1 How much can we vary c1 without changing the current basic optimal solution?

  18. Sensitivity analysis for c1 Our objective function is: Z= c1 D+5W=k slope of iso-profit line is: isoprofit line How much can c1 vary until the slope of the iso-profit line equals the slope of constraint 2 and constraint 3?

  19. How much can c1 vary until the slope of the iso-profit line equals the slope of constraint 2 and constraint 3? • Slope of constraint 2 0 • Slope of constraint 3  -3/2

  20. Importance of Sensitivity Analysis Sensitivity analysis is important for several reasons: • Values of LP parameters might change. If a parameter changes, sensitivity analysis shows it is unnecessary to solve the problem again. For example in the Wyndor problem, if the profit contribution of product 1 changes to 5, sensitivity analysis shows the current solution remains optimal. • Uncertainty about LP parameters. In the Wyndor problem for example, if the capacity of plant 1 decreases to 2, the optimal solution remains a weekly rate of 2 doors and 6 windows. Thus, even if availability of capacity of plant 1 uncertain, the company can be fairly confident that it is still optimal to produce a weekly rate of 2 doors and 6 windows.

  21. Does the shadow price always have an economic interpretation? • Not necessarily • For example,there is no economic interpretation for dual variables associated with ratio constraints

  22. Glass Example • x1 = # of cases of 6-oz juice glasses (in 100s) • x2 = # of cases of 10-oz cocktail glasses (in 100s) • x3 = # of cases of champagne glasses (in 100s) max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs) 10 x1 + 20 x2 + 10 x3  150 (wareh. cap. in ft2) x1  8 (6-0z. glass dem.) x1  0, x2  0, x3  0 (from AMP and slides from James Orlin)

  23. Z* = 51.4286 Decision Variables • x1 = 6.4286 (# of cases of 6-oz juice glasses (in 100s)) • x2 = 4.2857 (# of cases of 10-oz cocktail glasses (in 100s)) • x3 = 0 (# of cases of champagne glasses (in 100s)) Slack Variables • s1* = 0 • s2* = 0 • s3* = 1.5714 Dual Variables • y1* = 0.7857 • y2* = 0.0286 • y3* = 0 Complementary optimal slackness conditions

  24. Consider constraint 1. 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs) • Let’s look at the objective function if we change the production time from 60 and keep all other values the same. The dual /shadow Price is 11/14.

  25. More changes in the RHS The shadow Price is 11/14 until production = 65.5

  26. What is the intuition for the shadow price staying constant, and then changing? • Recall from the simplex method that the simplex method produces a “basic feasible solution.” The basis can often be described easily in terms of a brief verbal description.

  27. The verbal description for the optimum basis for the glass problem: • Produce Juice Glasses and cocktail glasses only • Fully utilize production and warehouse capacity z = 5 x1 + 4.5 x2 6 x1 + 5 x2 = 60 10 x1 + 20 x2 = 150 x1 = 6 3/7 (6.4286) x2 = 4 2/7 (4.2857) z = 51 3/7 (51.4286)

  28. The verbal description for the optimum basis for the glass problem: • Produce Juice Glasses and cocktail glasses only • Fully utilize production and warehouse capacity z = 5 x1 + 4.5x2 6 x1 + 5 x2 = 60 + D 10 x1 + 20 x2 = 150 For D = 5.5, x1 = 8, and the constraint x1 8 becomes binding. x1 = 6 3/7 + 2D/7 x2 = 4 2/7 – D/7 z = 51 3/7 + 11/14 D

  29. How do we interpret the intervals? • If we change one coefficient in the RHS, say production capacity, by D the “basis” remains optimal, that is, the same equations remain binding. • So long as the basis remains optimal, the shadow prices are unchanged. • The basic feasible solution varies linearly with D. If D is big enough or small enough the basis will change.

  30. Illustration with the glass example: max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs) 10 x1 + 20 x2 + 10 x3  150 (wareh. cap. in ft2) x1  8 (6-0z. glass dem.) x1  0, x2  0, x3  0 The shadow price is the “increase” in the optimal value per unit increase in the RHS. If an increase in RHS coefficient leads to an increase in optimal objective value, then the shadow price is positive. If an increase in RHS coefficient leads to a decrease in optimal objective value, then the shadow price is negative.

  31. Illustration with the glass example: max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs) 10 x1 + 20 x2 + 10 x3  150 (wareh. cap. in ft2) x1  8 (6-0z. glass dem.) x1  0, x2  0, x3  0 Claim: the shadow price of the production capacity constraint cannot be negative. Reason: any feasible solution for this problem remains feasible after the production capacity increases. So, the increase in production capacity cannot cause the optimum objective value to go down.

  32. Illustration with the glass example: max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs) 10 x1 + 20 x2 + 10 x3  150 (wareh. cap. in ft2) x1  8 (6-0z. glass dem.) x1  0, x2  0, x3  0 Claim: the shadow price of the “x1  0” constraint cannot be positive. Reason: Let x* be the solution if we replace the constraint “x1  0” with the constraint “x1  1”. Then x* is feasible for the original problem, and thus the original problem has at least as high an objective value.

  33. Signs of Shadow Prices for maximization problems • “  constraint” . The shadow price is non-negative. • “  constraint” . The shadow price is non-positive. • “ = constraint”. The shadow price could be zero or positive or negative.

  34. Signs of Shadow Prices for minimization problems • The shadow price for a minimization problem is the “increase” in the objective function per unit increase in the RHS. • “  constraint” . The shadow price is non-positive. • “  constraint” . The shadow price is non-negative • “ = constraint”. The shadow price could be zero or positive or negative. • Please answer with your partner.

  35. The shadow price of a non-binding constraint is 0. “Complementary Slackness.” max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs) 10 x1 + 20 x2 + 10 x3  150 (wareh. cap. in ft2) x1  8 (6-0z. glass dem.) x1  0, x2  0, x3  0 In the optimal solution, x1 = 6 3/7. Claim: The shadow price for the constraint “x1 8” is zero. Intuitive Reason: If your optimum solution has x1 < 8, one does not get a better solution by permitting x1 > 8.

  36. Is the shadow price the change in the optimal objective value if the RHS increases by 1 unit. • That is an excellent rule of thumb! It is true so long as the shadow price is valid in an interval that includes an increase of 1 unit.

  37. The shadow price is valid if only one right hand side changes. What if multiple right hand side coefficients change? • The shadow prices are valid if multiple RHS coefficients change, but the ranges are no longer valid.

  38. Reduced Costs

  39. Do the non-negativity constraints also have shadow prices? • Yes. They are very special and are called reduced costs? • Look at the reduced costs for • Juice glasses reduced cost = 0 • Cocktail glasses reduced cost = 0 • Champagne glasses red. cost = -4/7

  40. What is the managerial interpretation of a reduced cost? • There are two interpretations. Here is one of them. • We are currently not producing champagne glasses. How much would the profit of champagne glasses need to go up for us to produce champagne glasses in an optimal solution? • The reduced cost for champagne classes is –4/7. If we increase the revenue for these glasses by 4/7 (from 6 to 6 4/7), then there will be an alternative optimum in which champagne glasses are produced.

  41. Why are they called the reduced costs? Nothing appears to be “reduced” • The reduced costs can be obtained by treating the shadow prices are real costs. This operation is called “pricing out.”

  42. Pricing Out shadow price ……11/14 ……1/35 …….0 max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 10 x1 + 20 x2 + 10 x3  150 1 x1  8 x1  0, x2  0, x3  0 Pricing out treats shadow prices as though they are real prices. The result is the “reduced costs.”

  43. Pricing Out of x1 shadow price ……11/14 ……1/35 …….0 max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 10 x1 + 20 x2 + 10 x3  150 1x1  8 x1  0, x2  0, x3  0 5 - 6 x 11/14 - 10 x 1/35 - 1 x 0 = 5 – 33/7 – 2/7 = 0 Reduced cost of x1 =

  44. Pricing Out of x2 shadow price ……11/14 ……1/35 …….0 max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 10 x1 + 20x2 + 10 x3  150 1 x1  8 x1  0, x2  0, x3  0 4.5 - 5 x 11/14 - 20 x 1/35 - 0 x 0 = 4.5 – 55/14 – 4/7 = 0 Reduced cost of x2 =

  45. Pricing Out of x3 shadow price ……11/14 ……1/35 …….0 max 5 x1 + 4.5 x2 + 6 x3 ($100s) s.t 6 x1 + 5 x2 + 8 x3 60 10 x1 + 20 x2 + 10x3  150 1 x1  8 x1  0, x2  0, x3  0 6 - 8 x 11/14 - 10 x 1/35 - 0 x 0 = 6 – 44/7 – 2/7 = -4/7 Reduced cost of x3 =

  46. Can we use pricing out to figure out whether a new type of glass should be produced? shadow price ……11/14 ……1/35 …….0 max 5 x1 + 4.5 x2 + 7 x4 ($100s) s.t 6 x1 + 5 x2 + 8 x4 60 10 x1 + 20 x2 + 20x4  150 1 x1  8 x1  0, x2  0, x4  0 7 - 8 x 11/14 - 20 x 1/35 - 0 x 0 = 7 – 44/7 – 4/7 = 1/7 Reduced cost of x4 =

  47. Pricing Out of xj shadow price ……y1 ……y2 ……… ……ym max 5 x1 + 4.5 x2 + cj xj ($100s) s.t 6 x1 + 5 x2 + a1j xj 60 10 x1 + 20 x2 + a2jxj  150 ……….. ………. + amjxj = bm x1  0, x2  0, x3  0 Reduced cost of xj = ?

  48. Brief summary on reduced costs • The reduced cost of a non-basic variable xj is the “increase” in the objective value of requiring that xj >= 1. • The reduced cost of a basic variable is 0. • The reduced cost can be computed by treating shadow prices as real prices. This operation is known as “pricing out.” • Pricing out can determine if a new variable would be of value (and would enter the basis).

  49. Summary • The shadow price is the unit change in the optimal objective value per unit change in the RHS. • The shadow price for a “ 0” constraint is called the reduced cost. • Shadow prices usually but not always have economic interpretations that are managerially useful. • Non-binding constraints have a shadow price of 0. • The sign of a shadow price can often be determined by using the economic interpretation • Shadow prices are valid in an interval. • Reduced costs can be determined by pricing out

  50. Reduced Costs • The reduced cost of a variable x is the shadow price of the “x  0” constraint. It is also the negative of cost coefficient for x in the final tableau. • Suppose in the previous example that we required that x3 1? What is the impact on the optimal objective value? What is the resulting solution? By the previous slide, the impact is -4/7.