Optimización Combinatoria: Una Introducción

1. Optimización Combinatoria: Una Introducción Dr. Juan Frausto Solís Tecnológico de Monterrey Campus Cuernavaca

2. CONTENTS • Course Data • Subjects • The rules of the game • Introduction to LP: Linear Programming to Optimization Theory Modeling Problems Some Complexity Remarks

3. Course´s Data • Dr. Juan Frausto Solís • juan.frausto@itesm.mxa • n.frausto@itesm.mx • http://campus.cva.itesm.mx/jfrausto/ • Address: • Reforma 182-A Col Lomas de Cuernavaca • Temixco Morelos, 62589 Mexico

4. Course´s Data • Dairigido a: • Estudiantes de doctorado • Estudiantes de Maestrìa con posibilidad de hacer investigaciòn independiente • Propósito del curso: • Una introducción a los métodos de Optimización Combinatoria. Habilidades para investigación original en el área.

5. Temas • 1. Introducción • 2. Linear Programming (LP). Método simplex General • 3. Variantes del Método Simplex. RSM; Dual • 4. Degeneración • 5. Interior-Point Methods (IPM) • 6. Evolutive Programming. Simplex Genetic, Simplex Annealing • 7. Simplex Cosine • 8. Optimización Combinatoria en Algoritmos de Redes: SVM; NN • 9. Problemas duros Clásicos: SAT, Graph Coloring,.. • 10. Problemas duros no clásicos: GRASPING, Futbol Robótico, Inversión Bursatil,..

6. Bibliography • G.B. Dantzig, Linear Programming, Springer • David G: Luemberger, Programación Lineal y No Lineal, Addison Wesley Iberoamerica • Jorge Nocedal, Operations Research, Springer • A. D. Belegundu, Optimization Concepts and Applications in Engineering, Prentice Hall • Hamdy Taha, Operations Research, An Introduction. Prentice Hall • Autores: Golberg, Gary Johnson, Papadimitrou,..

7. Rules of the Game

8. Optimización Combinatoria

9. 1947: G.W. Dantzig propose LP to set • general objectives and arrive at a detailed • schedule to meet these goals.  Many Extensions collectively known As Mathematical Programming: Non LP, Int Prog, Stocastic Prog, Comb Opt, Network Flow Maximization

10. optimize z = f(x) Subject to x One variable Multi-variable Linear No Linear x z y  Type of Problem Tipo de Problema  x Constraint Unconstraint Continuos Integer z Multiobjective One-objective OPTIMIZATION AREA

11. Math Prog = Optimization Theory What Mathematical Programming is?

12. Math Prog: Branch of Math dealing with techniques for Maximizing or Minimizing an Objective function, subject to linear, no linear, and integer constraints. LP: Spetial case of MP concerned with Maximizing or Minimizing a linear Objective function in many variables and subject to Linear equality and inequality constraints Math Prog Vs LP

13. Why the Programming word is into LP? For many applications, the solution Can be interpreted as a program, Namely, a statement of the time And quantity of actions to Performed by the system so that it May move from its given status Towards some defined objective

14. Small Problems: constraints 1000 Medium Problems: 1000  constraints 2000 Large Problems: 2000  constraints10000 Very Large Problems: constraints10,000 Size of LP Problems

15. Some Simple Examples

16. A product Mix Problem

17. DES-k manufactures four models of desks. Each one is first constructed in the carpentry shop and is next sent to the finishing Shop, where it is varnished, waxed, and polished. The number of man-hours of labor required in each shop and the number of hours available in each shop are known. The raw materials and supplies are available in adequate supply and all desks produced can be sold. The DES-k wants to know the optimal product mix, that is, the quantities to make of each type of desk that will maximize profit. The case of DES-K, a founiture company

18. Many activities compete for resources, such as machine capacity at a plant. The available quantities of some resources may be insufficient to accomodate all the demands placed on them. Moreover, some activities may consume several resources in producing desired outputs. LP models allow resources to be allocated across the entire sytem being analized to determine how scarce resources can be optimal used. Resource Allocation Model in a Manufacturing Company

19. Ajax sells three types of computers with the following • Net Profit for each sold computer: • Personal Computer Alpha: $350 • NoteBook Computer Beta: $470 • Workstation Gamma : $610 • Net profit equals the sales price of each computer minus • the direct costs of´purchasing components, producing • Computer cases, and assembling and testing the • Computer. We assume that all production during the week will be sold immediatly. Ajax Computer Co. Has a Resource Allocation Problem

20. This week, 120 hours are available on The A-Line Test equipment Where assembled ´s and ´s are tested, and 48 hours are available On the C-Line test equipment where ´s assembled Are tested. The testing of each computer takes 1 hour. Ajax test capacity

21. Production is constrained on the availability of 2000 labor hours for product assembly. Each  requires 10 labor hours, each  requires 15 labor hours and Each  requires 20 labor hours. Other activities are involved, but are not very important. We want to allocate these resources to maximize net profits for the week. Ajax Production constraint for this week: Labor availability

22. Building an LP Model

23. Formulating Linear Programs • Before you can put a problem into a computer and efficiently find a solution, you must first abstract it, which means you have to build a mathematical model.

24. Formulating Linear Problem • The process of building a math model is so important than solving it, because this process provides insight about how the system works. • The model also helps organize essential information about it.

25. It is easy to formulate a Problem? • To formulate a problem of the real word could be most difficult than to solve it!! • This is because of the richness, variety, and ambiguity than exists in the real world or • Because of our ambiguos understanding of the real world.

26. How to get a Model: The row (material balance) Approach

27. Understanding the Problem Step 0

28. Step 1: Define the Decision Variables • X= Number of ´s to be assembled tested and sold during the week • X = Number of  ´s to be assembled tested and sold during the week • X= Number of  ´s to be assembled tested and sold during the week

29. Objective: To Maximize the Net Profit Net Profit= 350 X+ 470X + 610X Max Z = 350 X+ 470X + 610X Step 2.What is the Objective  Defining the Objective Function

30. Step 3: Constraints A Line Test Capacity: X+ X  120 C line Test Capacity X  48 Labor availability 10 X+ 15X + 20 X  2000

31. Step 4: No Negativity variables • X 0 • X  0 • X  0

32. Max Z = 350 X+ 470X + 610X • Subjecto to: • X+ X  120 A-Line test Capacity • X  48 C-Line Test Capacity • 10 X+ 15X + 20 X  2000 Labor availability • X 0; X  0 ; X  0 Final Step: Joining Everything

33. Max Z = 350 X+ 470X + 610X • Subjecto to: • X+ X  120 • X  48 • 10 X+ 15X + 20 X  2000 • X 0; X  0 ; X  0 Model Structure Right Hand Sides Non Negative Variables

34. Max Z = 350 X+ 470X + 610X • Subjecto to: • X+ X  120 • X  48 • 10 X+ 15X + 20 X  2000 • X 0; X  0 ; X  0 1 0 10 X  Structure Model: Activities Asociated with each Decision variable Is an activity Describing the rate At which the Decision variable Consumes Resources. Activity

35. How to get a Model? • Row approach  Material Balance Approach • Column Approach  Recipe/Activity Approach

36. DES-K With Data • DES-K manufactures four models of desks. Each desk is first constructed in the carpentry shop and then is sent to the finiship shop, where it is varnished, washed and polished. The number of man hours of labor requerid on each shop is:

37. DES-K With Data • Because of limitations in capacity of the plant,no more than 6,000 man hours can be expected in the carpentry shop and 4000 in the finishing shop in the next six months. The profit (revenue minus costs) from the sale of each item is:

38. DES-K With Data • Determine the optimal production mix assuming that the raw material and supplies are available in adequate supply amd all desks produced will be sold. • That is, determine the quantities to make each type of product which maximize profit

39. Column (Recipe/Activity)Approach • A System: It is decomposable into a number of elementary functions: the activities.

40. Activity Final or Intermediate products men,material, equipment items Column (Recipe/Activity)Approach • Activity: Like a recipe in a cookbook. • Activity level: Quantity of each activity

41. Column (recipe/Activity)Approach

42. STEPs on Column (recipe/Activity)Approach

43. Step 1: Define the activity Set • Decompose the entire system into all of its elementary functions, the activities or processes and • Choose a unit for each type of activity or process in terms of which its quantity or level can be measured.

44. STEP 1: Define The activity Set.Exemple: DES-K • Activity: • Manufacturing a desk • Activity Level: Number of desks manufactured = Decision Variable. • Manufacturing Desk1: X1 • Manufacturing Desk2: X2 • Manufacturing Desk3: X3 • Manufacturing Desk4: X4

45. Step 2: Define the set itemsExemple: DES-K • 1. Object Classes: Desks • 2. Items: • Capacity carpentry shop (in man hours) • Capacity finishing shop (in man hours) • Costs (measured in dollars)

46. Ex: aij= amount of time in shop i required to manufacture one desk j Step 3: Define The Input-Output Coefficients aij • Determine the quantity of each item i consumed or produced by the operation of each activity j at its unit level.

47. 4 hours of carpentry capacity $12 Manufacturing 1 unit of Desk 1 1 hour of finishing capacity STEP 3: Input-Output Coefficient

48. STEP 3: Input-Output Coefficients

49. Specify: • Exogenous amounts of each item being supplied (required) to the system Step 4. Specify the Exogenous Flows bi • Capacity in the carpentry and finishing are inputs to each activities: • 6000 maximun (carpentry) • 4000 maximun(finishing shop)

50. Step 5. Balance Equations • Assign unknown activity levels x1,x2,x3,..., • For each item write the material balance 4x1+9x2+7x3+10x4  6000 x1+ x2 +3x3+40x4  4000 f.O: Z= 12x1+20x2+18x3+40x4