1. Advanced Operations Research Models Instructor: Dr. A. Seifi Teaching Assistant: Golbarg Kazemi Kazemi_Golbarg@yahoo.com Kazemi_Golbarg@aut.ac.ir 1

2. Today’s lecture Course Outline Administrivia A little LP review Economic Interpretation Duality theory Shadow price Assignments 2

3. What you will learn Mathematical Modeling – learn a variety of ways of modeling real- world problems as structured mathematical problems Solution Methods – learn to use powerful optimization tools to solve the problems arising in your mathematical models 3

4. Course Outline Linear Optimization Models – Formulations, Sensitivity analysis Data Envelopment Analysis and Resource Allocation – Applications and Methods Integer Programming – Formulations, Solution methods Dynamic Programming – Formulation and Applications Column Generation technique Quadratic programming – Duality Theorem and QP Solution Methods 4

5. Administrivia Prerequisite: – Operation Research 1 Reading material: Wayne L. Winston “ Operation Research: Application and Algorithms”; highly recommended Softwares: LINGO- LINDO- NEOS on-line optimizer- XPRESS-MP 5

6. Administrivia (contd.) Grading – Assignments: 10% – Projects: 10%-15%2 projects – Midterm: 30%,Final: 50% The assignments would be from the imparted subjects each session will be given to you at the end of the TA session and, you should send them in a week to this address: Kazemi_ Golbarg@yahoo.comGolbarg@yahoo.com Feedback, comments, suggestions, questions are always most welcome. 6

7. LP review: Definitions Linear programming problem: – problem of maximizing or minimizing a linear function of a finite number of variables – subject to a finite number of linear constraints: ,  or = constraints max/min f(x) = c 1 x 1 + c 2 x 2 + … + c n x n subject toa i1 x 1 + a i2 x 2 + … + a in x n b i  i=1,…,m Feasible point: x  R n s.t. x satisfies all constraints Feasible region: set of all feasible points P = {x  R n : x satisfies all constraints} == Called a polyhedron 7

8. LP review: more definitions max/min f(x) = c 1 x 1 + c 2 x 2 + … + c n x n subject toa 11 x 1 + a 12 x 2 + … + a 1n x n  b 1 a 21 x 1 + a 22 x 2 + … + a 2n x n  b 2 a 31 x 1 + a 32 x 2 + … + a 3n x n = b 3 Decision variables: should completely describe all decisions to be made Objective function Constraints Optimal solution: feasible solution with best (max/min) objective-function value Optimal value: objective-f’n. value at an optimal solution 8

9. Economic Interpretation Suppose that we have to produce three types of materials 1, 2 and 3 by means two different resources human being and wood: Product 1 profit :5 Product 2 profit: 2 Product 3 profit: 3 Resources1 2 3Max. amount Human resource1 2 28 wood1 4 37 9

10. Continue…. 10

11. Simplex Tableau…. Basic Var. Row No.ZX1X2X3S1S2R.H.S Z01-5-2-3000 S110122108 S220341017 Basic Var. Row No. ZX1X2X3S1S1 S2R.H.S Z01014/3-4/305/335/3 S11002/35/31-1/317/3 X12014/31/30 7/3 11

12. Optimal Tableau… Basic Var.Row No.ZX1X2X3S1S2R.H.S Z01026/504/57/581/5 X31002/513/5-1/517/5 X12016/50-1/52/56/5 12

13. Economic Interpretation Necessary Source for Product 2 How can we produce it? 4/3 Decrease of product 1 Available resource 2 units of S1 4 units of S2 4/3*1=4/3 4/3*3=4 2/3 - 13

14. Interpretation… Lost Profit: 1. 4/3 decrease of product 1: 4/3*5=20/3 2. 2/3 consumption of resource 1: 2/3*0=0 Total lost: 0 + 20/3=20/3 Gained profit by producing 1 unite of product 2: 2 Result: 20/3 – 2= 14/3 So we lost! Note: What’s shadow price?! 14

15. Duality… 15

16. Hints…. 16

17. Duality theorem…. Lemma 1: Given a primal linear program, the dual problem of the dual linear program becomes the original primal problem. Weak Duality theorem: If x 0 is a primal feasible solution and y 0 is dual feasible, then. 17

18. Corollaries…… Corollary 1 If x 0 is primal feasible, y 0 is a dual feasible, and, then x 0 and y 0 are optimal solutions to the respective problem Corollary 2 If the primal problem is unbounded above, then the dual problem is infeasible. 18

19. (Contd.) Corollary 3 If the dual problem is unbounded below, then the primal problem is infeasible. Note ： The converse of the above two is not true. 19

20. Strong Duality Theorem…. If either the primal or the dual problem has a finite optimal solution, then so does the other and they achieve the same optimal objective value. If either problem has an unbounded objective value, then the other has no feasible solution. If either problem has no feasible solution, then the other has either no feasible solution or unbounded. 20

21. Complementary slackness Primal Problem Dual Problem Max c T x Min b T y s.t. s.t. For the primal problem, define ： primal slackness vector ( ) Dual problem, define ： dual slackness vector ( ) 21

22. Continue…… For any primal feasible solution x and dual feasible solution y, we know i.e. is the duality gap between the primal feasible solution x and dual feasible solution y. This duality gap vanishes, if and only if and. 22

23. Complementary slackness Let x be a primal feasible solution and y be a dual feasible solution to a symmetric pair of linear problem. Then x and y become an optimal solution pair if and only if the complementary slackness conditions: either or x j = 0 either or y i = 0 are satisfied. 23

24. Karush-Kuhn-Tucker (K-K-T) Optimality Conditions Theorem Given a linear programming problem in its standard form, vector x is an optimal solution to the problem if and only if then exist vector y and r, s such that 1. Ax+s=b, (primal feasibility) 2. (dual feasibility) 3. (complementary slackness) ( ) for canonical form 24

25. Revised simplex What?! This method is a modified version of the Primal Simplex Method. Why?! It is designed to exploit the fact that in many practical applications the coefficient matrix {a ij } is very sparse, namely most of its elements are equal to zero. 25

26. Continue…. Bottom line: Don’t update all the columns of the simplex tableau: update only those columns that you need! 26

27. Example… BASIC VAR. ROW NO. ZX1X2X3S1S2R.H.S Z01-4-3-6000 S1103131030 S2202230140 27 Basic Var. Row NO. ZX1X2X3S1S2R.H.S Z011001170 X3104/3012/3-1/320/3 X22010 110

28. Continue…. 28

29. Continue…. 29

30. Economic Interpretation of the dual shadow price, marginal price, equilibrium price. Given the standard L.P. model, the primal problem can be viewed as a process of providing different services ( ) to meet a set of customer demands (Ax=b) in a maximum profit manner (max c T x). 30

31. Continue…….. For a non-degenerate optimal solution x* obtain from revised simplex method, we have with an optimal value since for a small enough in demand, we know B -1 (b+  b) ＞ 0 and is an optimal B.F.S. to the following problem: Max c T x s.t. Ax=b+  b 31

32. Continue….. The optimal value of above problem is where y is the dual variables. The above equation says the increment cost of satisfying an incremental demand  b is equal to y* T  b. y i * can be though of the marginal cost, of the providing one unit of the i-th demand at optimum. 32

33. Continue….. i.e. it indicates the maximum unit profit one has received for satisfying additional demands when an optimal is achieved. Called shadow price or marginal cost. 33

34. Assignments! 1. Use revised simplex method to solve the following LP: 34

35. Continue… 2. Interpret the Simplex tableaus (first and second and the last table are given) below as a production problem (Write a scenario for it, and a model then interpret it) 35 Basic Var. Row No. ZX1X2S1S2S3R.H.S Z01-3-50000 S110101004 S2200201012 S3300200118

36. 36 Basic Var. Row No. ZX1X2S1S2S3R.H.S Z01-3005/2030 S110101004 X2200101/206 S33030016 Basic Var. Row No. ZX1X2S1S2S3R.H.S Z010003/2136 S1100001/3-1/32 X2200011/206 X130110-1/31/32

37. Continue…. 3. In the previous question determine the shadow prices, and if the sources vary from 4,12 and 18 to 5,14,18 what happens to the profit?! 37

38. I will be waiting for your assignments until next Sunday 12:00 PM Don’t forget …. Thanks a lot 38