Duality

1. Duality 虞台文 大同大學資工所 智慧型多媒體研究室

2. Content • The Dual of an LP in General Form • Complementary Slackness • The Shortest-Path Problem and Its Dual • Dual Information in the Tableau • The Dual Simplex Algorithm

3. Duality The Dual of an LP in General Form 大同大學資工所 智慧型多媒體研究室

4. > < An LP in General Form

5. Standard Form

6. Standard Form

7. Standard Form

8. In fact, the simplex algorithm is to find a basisB from A such that the above criterion can be satisfied. ^ ^ Optimality Criterion Determined from the basis we choose.

9. Optimality Criterion ’Rm

10. Inequalites RestrictedVariables  ’Rm

11. UnrestrictedVariables ’Rm

12. Equalites UnrestrictedVariables  ’Rm

13. Equalites ’Rm

14. > < Equalites  UnrestrictedVariables ’Rm

15. > < Inequalites ’Rm

16. > < Inequalites  RestrictedVariables ’Rm

17. > < Constraints(Primal)  Constraints(Dual) ’Rm

18. > < > < Primal  Dual ? ? ?

19. > < > < Primal  Dual

20. > < > < Theorem 1 If an LP has an optimal solution, so does its dual, at the optimality their costs are equal.

21. Clearly, is a feasible solution of the dual. The costlower-bound of the primal. The constraint of the dual. Theorem 1 Has an optimum Pf) Let x and  be feasible solutions to the primal and the dual, respectively. the cost ot primal the profit of dual  Upper bounded Lower bounded optimal solution of primal The profitupper-bound of the dual.

22. Theorem 2 The dual’s dual is the primal.

23. Primal Dual Dual > < > < > < > < The dual’s dual is the primal. Theorem 2 Pf) Dual’s dual

24. Primal Dual Dual > < > < > < > < The dual’s dual is the primal. Theorem 2 Pf) Dual’s dual

25. Unit Price Required Units Food Nutrient c1 b1 c2 b2 c3 . . . . . . . . . . . . cn bm aij: number of units of nutrient i in one unit food j. xj: number of units of food j in the diet. Example: The Diet Problem Minimize Subject to

26. Unit Price Required Units Food Nutrient c1 b1 c2 b2 c3 . . . . . . . . . . . . cn bm aij: number of units of nutrient i in one unit food j. xj: number of units of food j in the diet. Example: The Diet Problem Minimize Subject to Maximize Subject to

27. Unit Price Required Units Food Nutrient c1 b1 c2 b2 c3 . . . . . . . . . . . . cn bm Minimize the total cost we need to pay so as to get the required nutrients. aij: number of units of nutrient i in one unit food j. xj: number of units of food j in the diet. Example: The Diet Problem Minimize Subject to The amount of the jth food needed? ? Maximize Subject to ?

28. Unit Price Required Units Food Nutrient c1 b1 c2 b2 c3 . . . . . . . . . . . . cn bm aij: number of units of nutrient i in one unit food j. Suppose that there is a pillmaker who can produce all the nutrients needed. xj: number of units of food j in the diet. Example: The Diet Problem Minimize Subject to Maximize the total price that a consumer needs to pay so as to get the nutrients needed, i.e., to maximize the profit. Maximize Subject to The selling price for one unit of ith nutrient.

29. Unit Price Required Units Food Nutrient c1 b1 c2 b2 c3 . . . . . . . . . . . . cn bm aij: number of units of nutrient i in one unit food j. One must get enough nutrients. xj: number of units of food j in the diet. Example: The Diet Problem Minimize Subject to The unit price for each nutrient must be reasonable, i.e., low enough. Maximize Subject to

30. Unit Price Required Units Food Nutrient c1 b1 c2 b2 c3 . . . . . . . . . . . . cn bm aij: number of units of nutrient i in one unit food j. xj: number of units of food j in the diet. Example: The Diet Problem Minimize Subject to Maximize Subject to

31. Unit Price Required Units Food Nutrient c1 b1 c2 b2 c3 . . . . . . . . . . . . cn bm aij: number of units of nutrient i in one unit food j. xj: number of units of food j in the diet. Example: The Diet Problem Minimize Subject to Consumer’s View. Maximize Subject to Producer’s View.

32. Unit Market Price Unit Value Raw Material Stock Product 1 b1 1 2 b2 2 3 b3 . . . . . . . . . . . . . . . m bm n Managing a Production Facility

33. Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 2 c2 3 b3 . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. Managing a Production Facility xj: the number of units of the jth product produced.

34. Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 2 c2 3 b3 . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced. Maximize Subject to

35. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Minimize 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced.

36. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Minimize 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced.

37. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 2 c2 3 b3 . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced. Minimize Subject to Maximize Subject to

38. Minimize Maximize Subject to Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Maximize Minimize Subject to 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced.

39. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Minimize 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. Maximize the total profit. The Resource Allocation Problem xj: the number of units of the jth product produced. The amount of the jth item produced? ? ?

40. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Minimize 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced. Minimize the loss due to inventory, i.e., the lost opportunity cost. The estimated value increment for one unit of the ithmaterial

41. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Minimize 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. One can use up each material at most. The Resource Allocation Problem xj: the number of units of the jth product produced. Don’t underestimate the value increment for each material.

42. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Minimize 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced.

43. Maximize Subject to Unit Market Price Unit Value Raw Material Unit Profit Stock Product 1 b1 1 c1 2 b2 Minimize 2 c2 3 b3 Subject to . . . . . . . . . . . . . . . . . . m bm n cn aij: number of units of the ith raw material needed to produce one unit of the jth product. The Resource Allocation Problem xj: the number of units of the jth product produced. Production Manager’s View Comptroller’s View

44. Duality Complementary Slackness 大同大學資工所 智慧型多媒體研究室

45. > < > < Complementary Slackness

46. the optimum of the primal the optmum of the dual Theorem 3Complementary Slackness x  Pf) Facts:

47. the optimum of the primal the optmum of the dual Theorem 3Complementary Slackness x  Pf) Facts: u + v =0 Define x and  are optima u + v =0

48. It is easily to verify that x is feasible. Example Does x=(0, 0.5, 0, 2.5, 1.5)’ solve the LP?

49. > < Example Does x=(0, 0.5, 0, 2.5, 1.5)’ solve the LP? Primal Dual

50. x2 x4 x5 > < feasible Example Does x=(0, 0.5, 0, 2.5, 1.5)’ solve the LP? Primal Dual    0 0.5 0 2.5 1.5 x is an optimum  = (2.5, 1, 1)’