# GAME THEORETIC APPROACHES TO PARALLEL MACHINE SCHEDULING DIANA GINETH RAMÍREZ RIOS CLAUDIA MARCELA RODRÍGUEZ PINTO

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1 GAME THEORETIC APPROACHES TO PARALLEL MACHINE SCHEDULING DIANA GINETH RAMÍREZ RIOS CLAUDIA MARCELA RODRÍGUEZ PINTO Undergraduate Project as a requste for graduaton DIRECTOR: Ing. Carlos Paternna Ph.D UNIVERSIDAD DEL NORTE School of Engneerng Industral Engneerng Department Barranqulla, Colomba, February, 2007

2 TITLE OF THE INVESTIGATION: GAME THEORETIC APPROACHES TO PARALLEL MACHINE SCHEDULING ABSTRACT: A problem of schedulng jobs on two dentcal parallel machnes s consdered, that pursues mnmzng two crtera n partcular, makespan and total flow tme. A mechansm was proposed as an approach to solve ths type of problem wth a settng of a 2-player noncooperatve game, under the framework of a 2x2 non-sum zero matrx; each player lookng after one of the crtera suggested n the schedulng problem. The scenaro mpled each job behavng selfshly and attemptng to move to a prevous poston n the machne, whch generated a cost for the job agent, who s attemptng to mnmze the total flow tme. At the same tme, a controllng agent allows movements of jobs between any two machnes, expectng to balance the load on the machnes and mnmzng maxmum completon tme. As a result of the dynamc trade-offs between the agents n repeated games, a Pareto Front set of ponts was obtaned. Key Words: dentcal parallel machnes, makespan, partal makespan, total flow tme, partal flowtme, non-cooperatve game, dynamc trade-offs, repeated games, Pareto front. TÍTULO DEL TRABAJO DE GRADO: LA TEORÍA DE JUEGOS APLICADO A LA PROGRAMACIÓN DE MÁQUINAS EN PARALELO RESUMEN: En un problema de programacón de máqunas déntcas en paralelo que persgue mnmzar dos crteros en partcular, lapso y tempo de termnacón total, un mecansmo basado en la teoría de juegos es propuesto para soluconarlo. Se consdera un juego bpersonal nocooperatvo de 2x2 en el que cada jugador busca mnmzar alguno de estos crteros que propone el problema de produccón. Cada escenaro mplca que los jugadores jueguen de manera smultanea y busquen mnmzar los costos que están relaconados con los crteros a optmzar. El jugador que representa al trabajo tene la opcón de dejar al trabajo en su poscón actual o moverlo a una poscón preva, buscando mnmzar su tempo de termnacón; mentras que el otro jugador, un agente controlador, toma la decsón de dejar al trabajo en la máquna actual o moverlo a otra, esperando balancear la carga de la máquna y mnmzar el lapso. Como resultado de una sere de juegos repetdos entre estos agentes, el Frente de Pareto es construdo, mostrando un conjunto de solucones efcentes al problema. Palabras Clave: máqunas déntcas en paralelo, lapso, tempo de termnacón total, juego bpersonal no-cooperatvo, agentes, juegos repetdos, Frente de Pareto. DIRECTOR: Ing. Carlos Paternna Arboleda, Ph.D AUTORES: Ramírez Ríos, Dana Gneth (1982) Rodríguez Pnto, Clauda Marcela (1982)

3 A nuestros padres y hermanos(as), con mucho carño. Una especal dedcacón a nuestros abuelos, porque el ejemplo que segumos de ustedes nos hace ser mejores personas.

4 v AGRADECIMIENTOS Los autores expresan sus agradecmentos a: Carlos Paternna Arboleda, Ph.D, Ingenero Industral y Drector del Proyecto de Grado por sus valosas orentacones. Jar de La Cruz, MSc. Ingenero Industral, por su constante apoyo en la programacón para obtencón de los resultados. Agradecmentos adconales a aquellas personas que nos brndaron asesoría durante parte de este proceso: Lus E. Ramírez (Ingenero Industral) e Ivan Saavedra (Ingenero de Sstemas).

5 v TABLE OF CONTENTS pg INTRODUCTION 1 1. PROBLEM DESCRIPTION BACKGROUND INFORMATION TERMINOLOGY AND NOTATION Schedulng n producton plannng Game theory IDENTIFICATION OF THE PROBLEM JUSTIFICATION LITERATURE REVIEW PARALLEL MACHINE SCHEDULING Parallel machne schedulng problems Heurstc technques found n lterature MULTICRITERIA OPTIMIZATION THEORY Multcrtera decson makng (MCDM) Multcrtera optmzaton problems Defnton of optmalty Determnng Pareto Optmalty Multcrtera lnear programmng Multcrtera mxed nteger programmng MULTICRITERIA SCHEDULING PROBLEMS GAME THEORY Two person zero-sum game and Nash Equlbrum Resoluton methods for non-domnated strateges Mechansm Desgn Game Theory and Computer Scence APPROACHES OF GAME THEORY IN SCHEDULING OBJECTIVES 77

6 v 3.1. GENERAL OBJECTIVE SPECIFIC OBJECTIVES SCOPE AND LIMITATIONS SCOPE LIMITATIONS HYPOTHESIS SET OF HYPOTHESES CONCEPT VARIABLES DEFINITION OPERATIONAL VARIABLES DEFINITION METHODOLOGY METHODOLOGY APPROACH SOLUTION METHODOLOGY Elements and Assumptons Defnton of the Game Numercal Example Results Obtaned Comparng Results to other Heurstcs An extenson to the results RESEARCH ASSOCIATED COSTS ACTIVITIES SCHEDULE CONCLUSIONS AND FURTHER RESEARCH CONCLUSIONS FURTHER RESEARCH 154 BIBLIOGRAPHY APPENDIX A APPENDIX B

7 v LIST OF TABLES Table 1. Resources and Tasks (PINEDO and CHAO. p 15) Table 2. Typology of Schedulng Problems (T'KINDT and BILLAUT, p17) Table 3. Intal Allocaton of jobs n 2 machnes by Load Balancng. Table 4. Job agent selected for the game (A6) and the possble movements t can make. Table 5. Payoff Matrx for job Agent 5 n the frst teraton. Table 6. Proposed schedule for frst teraton. Table 7. Payoff Matrx for job Agent 9 for teraton 2. Table 8. Proposed schedule for teraton 2. Table 9. Payoff Matrx for job Agent 9 for teraton 3. Table 10. Proposed schedule for teraton 3. Table 11. Payoff Matrx for job Agent 10 for teraton 4. Table 12. Proposed schedule for teraton 4. Table 13. Payoff Matrx for job Agent 2 for teraton 5 Table 14. Proposed schedule for teraton 5. Table 15. Payoff Matrx for job Agent 7 for teraton 6. Table 16. Proposed schedule for teraton 6. Table 17. Payoff Matrx for job Agent 3 for teraton 7. Table 18. Proposed schedule for teraton 7.

8 v Table 19. Payoff Matrx for job Agent 4 for teraton 8. Table 20. Proposed schedule for teraton 8. Table 21. Payoff Matrx for job Agent 2 for teraton 9. Table 22. Proposed schedule for teraton 9. Table 23. Payoff Matrx for job Agent 6 for teraton 10. Table 24. Proposed schedule for teraton 10. Table 25. Payoff Matrx for job Agent 3 for teraton 11. Table 26. Proposed schedule for teraton 11. Table 27. Payoff Matrx for job Agent 8 for teraton 12. Table 28. Proposed schedule for teraton 12. Table 29. Payoff Matrx for job Agent 7 for teraton 13 Table 30. Proposed schedule for teraton 13. Table 31. Payoff Matrx for job Agent 2 for teraton 14 Table 32. Proposed schedule for teraton 14. Table 33. Payoff Matrx for job Agent 2 for teraton 15. Table 34. Proposed schedule for teraton 15. Table 35. Payoff Matrx for job Agent 7 for teraton 16. Table 36. Proposed schedule for teraton 16. Table 37. Payoff Matrx for job Agent 10 for teraton 17. Table 38. Proposed schedule for teraton 17. Table 39. Payoff Matrx for job Agent 8 for teraton 18. Table 40. Proposed schedule for teraton 18.

9 x Table 41. Payoff Matrx for job Agent 10 for teraton 19. Table 42. Proposed schedule for teraton 19 Table 43. Payoff Matrx for job Agent 4 for teraton 20. Table 44. Proposed schedule for teraton 20. Table 45. Payoff Matrx for job Agent 2 for teraton 21. Table 46. Proposed schedule for teraton 21. Table 47. Payoff Matrx for job Agent 2 for teraton 22. Table 48. Proposed schedule for teraton 22. Table 49. Payoff Matrx for job Agent 10 for teraton 23. Table 50. Proposed schedule for teraton 23. Table 51. Payoff Matrx for job Agent 8 for teraton 24. Table 52. Proposed schedule for teraton 24. Table 53. Values of Pareto Solutons of the Schedulng Game for ths example. Table 54. Schedules generated that belong to the weak Pareto Front soluton set. Table 55. Results from other algorthms/heurstcs Table 56. Comparson of the results for each replcaton n the nstance m=2, n=10 Table 57. Comparson of the results for each replcaton n the nstance m=2, n=20 Table 58. Comparson of the results for each replcaton n the nstance m=2, n=30

10 x Table 59. Comparson of the results for each replcaton n the nstance m=3, n=10 Table 60. Comparson of the results for each replcaton n the nstance m=3, n=20 Table 61. Comparson of the results for each replcaton n the nstance m=3, n=30 Table 62. Comparson of the results for each replcaton n the nstance m=4, n=10 Table 63. Comparson of the results for each replcaton n the nstance m=4, n=20 Table 64. Comparson of the results for each replcaton n the nstance m=4, n=30 Table 65. Improvement n the results and robustness of the soluton presented by the proposed model. LIST OF FIGURES Fgure 1. Conflctng functons on Fgure 2. Supportve functons on (CARLSSON and FÚLLER, p.4) (CARLSSON and FÚLLER, p.4) Fgure 3. Weak and Strct Pareto Optma where Z defnes a polyhedron (T'KINDT and BILLAUT. Fgure 3.3. p.48)

11 x Fgure 4. Geometrc Interpretaton of a problem (Pa) (T'KINDT and BILLAUT. Fgure 3.9. p.47) Fgure 5. Geometrc Interpretaton of a problem (P(g,b)) (T'KINDT and BILLAUT. Fgure p.61) Fgure 6. Geometrc Interpretaton of a problem (Pek) (T'KINDT and BILLAUT. Fgure p.66) Fgure 7. Geometrc Interpretaton of a problem (Pq) (T'KINDT and BILLAUT. Fgure p.69) Fgure 8. Geometrc Interpretaton of a problem (P(zref, w)) (T'KINDT and BILLAUT. Fgure p.79) Fgure 9. Supported and non supported Pareto optma (T'KINDT and BILLAUT. Fgure p.85) Fgure 10. Man varables defnton. Fgure 11. Job agent s decson Fgure12. Analyzng β for the agent as an opportunty cost of affectng overall flow tme Fgure 13. Agent 0 decdes to swtch the job to the other machne affectng C j of second machne. Fgure 14. Agent 0 decdes not to swtch job, from the machne concerned, t wll stay that way untl another condton s reached. Fgure 15. Decson Tree for Job Agent Fgure 16. Decson Tree for Agent 0

12 x Fgure 17. Pareto Front for ntal results of the example. Fgure 18. Pareto Front obtaned from the Schedulng Game for ths example. Fgure 19. Pareto Front contrasted wth results from other MCDM* tools. Fgure 20. Comparson of the Pareto Fronts generated n the nstance m=2, n=10. Fgure 21. Comparson of the Pareto Fronts generated n the nstance m=2, n=20. Fgure 22. Comparson of the Pareto Fronts generated n the nstance m=2, n=30. Fgure 23. Comparson of the Pareto Fronts generated n the nstance m=3, n=10. Fgure 24. Comparson of the Pareto Fronts generated n the nstance m=3, n=20. Fgure 25. Comparson of the Pareto Fronts generated n the nstance m=3, n=30. Fgure 26. Comparson of the Pareto Fronts generated n the nstance m=4, n=10. Fgure 27. Comparson of the Pareto Fronts generated n the nstance m=4, n=20. Fgure 28. Comparson of the Pareto Fronts generated n the nstance m=4, n=30.

13 1 INTRODUCTION Wthout a doubt, the world today s the result of a huge course n evoluton, and certanly manknd has not only wtnessed, but also sculpted all surroundng aspects n socety. For nstance, from the begnnngs of the ndustral revoluton untl today, ndustry has completely changed n terms of labor and resource usage. In early stages of the revoluton, factores focused entrely on mass producton and effcency. Under those gven condtons they were consdered compettve and those condtons alone enabled organzatons to be productve, guaranteeng a long lastng envronment and proftable outcomes. Nowadays, durng the knowledge and nformaton era, thngs have changed qute a lot; now success does not only depend on ncreasng producton, snce ths can merely make a dfference. Markets are every day more demandng and dynamc, so producton must be forecast-based, accordng to the expected demand. Products and servces may also guarantee great qualty, effcent resource usage, processng on tme, mantanng the rght level of nventory, or gettng the product to the customer on tme. Now, runnng and controllng so many varables at the same tme can be very rsky, and certanly complex, for any company; thus, takng under consderaton several prortes at the same tme s needed; beng dynamc

14 2 whle makng decsons of ths knd. As Bernard Roy sad takng account of several crtera enable us to propose to the decson maker a more realstc soluton 1. It s not easy to make decsons these days, even when all busnesses seek for compettveness to acheve stablty under very dynamc condtons, they reach for the best machnery and the latest technology, but that s not enough. They encounter problems almost daly, for example, the machne was not ready because the materals needed dd not arrve on tme, the operators dd not prepare the machne properly, a machne has just broke down, so t dd not start as expected, and so many other nterferences that may take place any tme wthn a productve system. Strategc decson makng s crucal for these crcumstances because t requres the most convenent decsons out of all the possble choces. Schedulng theory came along n a tme where producton plannng has become rather necessary for organzatons. It has turned so mportant these days that ts effectveness can determne the permanence and fdelty of the clents n a busness. In order to reach ths, t s necessary that productve systems evolve over tme. Recently, t has been qute obvous how technology has taken over and how the need for quck answers has become vtal for enterprses permanence n all sorts of ndustres. Intellgent Busness (IB) and ndustral engneerng have been 1 T KINDT, Vncent and BILLAUT, Jean-Charles. Multcrtera Schedulng: Theory, Models and Algorthms. Germany: Sprnger-Verlag, p.1

15 3 ntroduced to management and along wth them, a whole new concept of decson makng, n whch decsons have to come from optmal solutons. The decson makng process takes place every day, especally n busnesses. There s no sngle area n a frm that does not requre a person to make decsons. More mportantly, decsons have become so crucal, that there s no tme to thnk about the best one, t s a matter of takng rsks all the tme for the sake of the organzaton. Yet, decson makers have to make the best decsons among the branch of so many possbltes. Strategc thnkng has done an mportant role n decson makng; hence, t has been appled to such dverse scopes wthn manknd graspng problems nowadays. Game theory has ntroduced ths new way of thnkng and ts applcatons have become wdely known these days. In addton to ths, many nformaton systems have the capablty to decde strategcally and so the process s aded for the decson maker pontng and settng out a better llustrated map for the decson maker. Although game theory has been appled to a varety of felds, ths paper wll focus on producton programmng, specfcally n dealng wth schedulng problems on dentcal parallel machnes. For ths type of envronment t can be assumed that ntellgent agents control t. Through ths perspectve game theory can be ntroduced as a mean to solve dfferent crtera, each proposed by two ntellgent agents, where each agent defends ts crtera by assumng certan strateges that

16 4 take nto account each other s decsons. That s, one agent wll choose the best strategy knowng that the other agent s also choosng ts best strategy.

17 5 1. PROBLEM DESCRIPTION 1.1. BACKGROUND INFORMATION All sort of problems can become crtcal n a productve system and, for these stuatons, producton plannng s present to be ahead of them. Moreover, at a schedulng phase, some of the crtera that need to be taken under consderaton for decson makng are: Obtan a hgh utlzaton of machnes and personnel. Mnmze the number of extra hours of labor. Mnmze nventory mantenance costs. Delays n the producton that can be convenent for the customer. Mnmze work-n-process costs. Mnmze the manufacturng costs due to tme spent settng up machnes or dle tme of the machnes. Even though the seral producton system ntroduced by Henry Ford at the begnnngs of the ndustral age consttuted a breakthrough for the economy n ts tme, t s no longer effectve; snce too much costng was brought upon n wasted materal and extreme resource usage. As the producton system became very

18 6 mportant n the ndustry there was more than a need of producng n large quanttes. But not just producng and watng for everythng to be sold, markets became more demandng, so factores faced a new challenge, clent satsfacton. Therefore, qualty started to evolve and t was not only a matter of detectng a defect n the fnal product, and controllng qualty through nspectons, but a whole new concept of qualty had to emerge, that was; qualty management and preventon of defects, qualty s not controlled, qualty s thus created. For that reason, a constant supervson durng the process of transformaton of the product was ndeed needed. Nevertheless, qualty grew much more and now busnesses practce more, commonly, the so called total qualty process, whch nvolves the long awated and hope for customer satsfacton. Respondng on tme to the customer, dspatchng products on tme and respondng effectvely to ther demands, are just some of the actons that manufacturng busnesses need to take as part of ther total qualty process. Producton schedulng arses as a foundaton for operatonal success n manufacturng processes. The tools used to measure t and the methods mplemented have revolutonzed today s productve systems. For nstance, Pnedo and Chao (1999) state ther perspectve unto the schedulng approach by the followng quote, the schedulng functon n a company uses mathematcal technques or heurstc methods to allocate resources to the

19 7 processng of tasks. 2 They classfy the most mportant elements n schedulng as shown n ths chart: RESOURCES TASKS Machnes at a workshop Takeoffs Operatons and at Landng a workshop n an Runways at an arport arport Crews Processng at a constructon unts n a ste Stages Computer n a constructon programs to project be computng envronment executed Table 1. Resources and Tasks (PINEDO and CHAO. p 15) The schedulng theory frst appeared n the md 1950 s as a result of the need for organzng the producton. For Carler and Chrétenne (1988) 3, schedulng s to forecast the processng of a work by assgnng resources to tasks and fxng ther start tmes. ( ) The dfferent components of a schedulng problem are the tasks, the potental constrants, the resources, and the objectve functon ( ) The tasks must be programmed to optmze a specfc objectve ( ) of course, often t can be more realstc n the practce to consder several crtera. The last phrase shows the mportance to the author quoted below, from consderng several crtera, but that s because the real stuaton wll not gve rse to problems one at a tme, and those knds of problems cannot be sad to be completely determnstc and known (nether does one crtera problem). As shown above, the crtera that are usually taken n consderaton n a productve system are related to tme or to costs. In practce, t s more lkely to fnd more than just one factor to consder and to establsh correspondng results because systems cannot 2 PINEDO, Mchael and CHAO, Xul. Op. Ct. p T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p. 5

20 8 be taken as solated, snce a lot of factors may devate the soluton forecasted by one sngle crtera model. Such models are too dealstc and wll never correspond to realty. Multple Crtera Decson Makng and Multple Crtera Optmzaton started wth Pareto at the end of the 19 th century. Snce then, ths dscplne has grown and developed, especally these last thrty years. To ths day, many decson support systems have mplemented methods to manage conflctng crtera, by usng mathematcal theory of optmzaton under multple objectves. On the other hand, game theory formally starts wth Zermelo, whose studes show that games such as chess are n fact resolvable. Borel (1921) and Von Neuman(1951) are doubtlessly the best known to be the poneers n mnmax equlbrum, specfcally n sum-zero games *. Nevertheless the mportant breakthrough came not untl the early fortes, when the book Theory of Games and Economc Behavor, wrtten by John Von Neumann and Oscar Morgenstern, s fnally publshed. Ths book really came along to formalze the wrtngs n an extended way and ntroduced the concept of strategy n extensve games and proposed some applcatons. Yet n the 50 s, there was a great development of ths theory, varous publcatons were made n Prnceton lke, an ntroductory book by Luce and Raffa (1950); Kuhn (1953), who defned the concept of nformaton n * Sub-zero games are games n whch, whle one player gans some proft, the other loses the same amount.

21 9 games; Shapley (1953), who establshed a way to attack cooperatve games ** ; and fnally, John Nash (1950), who defned the Nash equlbrum n zero-sum games. These last nvestgatons were fnanced by the Unted States Department of Defense, snce sum-zero games could be appled to mltary strateges. Moreover, Harsany (1967) extended the theory of games to games wth ncomplete nformaton *** and then Selten (1975) defned the concept of perfect equlbrum n a sub game for games wth ncomplete nformaton and a generalzaton to the case of games wth mperfect nformaton. In 1994, the Real Academy of Scences n Sweden awarded wth a Nobel Prce n Economy to the mathematcan, John Nash and the economsts, John Harsany and Renhard Selten, for ther poneer analyss of the equlbrum n noncooperatve games, whch proved to be very useful for modern economc applcatons. Today game theory has proved to be an mportant tool and Nash s contrbuton was fundamental. ** Cooperatve games are games n whch players can agree wth each other on the decsons they take. *** Games wth ncomplete nformaton are due to the uncertanty that the players have wth all the characterstcs of the game.

22 TERMINOLOGY AND NOTATION Schedulng n producton plannng Manufacturng systems are descrbed by varous factors, lke for example, the number of resources, the confguraton of resources and ts automatzaton. All of these dfferent characterstcs can be represented n many dfferent schedulng models. To understand schedulng models, there are mportant terms and notatons that must be consdered n order to understand algorthms and heurstcs used n multple crtera schedulng theory. It s mportant to clearly dstngush between the varables that defne the problem, ( ) and those varables that descrbe the soluton produced by the schedulng process. To emphasze ths dstncton we have adopted the conventon that lower-case letters denote the gven varables and captal letters denote those that are determned by schedulng. The symbols h, x, y, z and Q, X, Y, Z, wll be used for those whch apply to an ndvdual secton. 4 The number of jobs s denoted by n and the number of machnes by m. To name a specfc job, J represents job number. To name a specfc machne, M k represents machne number k. 4 CONWAY, Rchard, et.al. Theory of Schedulng. USA: Dover Publcatons, p.9

23 11 Vncent T Kndt and Jean-Charles Bllaut, n ther book of Multcrtera Schedulng, descrbed the dfferent classfcaton of dfferent schedulng problems and the confguratons of the resources used 5. Types of schedulng problems wthout assgnment: Sngle machne: Any job s processed n one machne. Ths s one of the most mportant types of problems because n practce, solutons to more complex problems are often found by analyzng t as a sngle machne. Sngle-machne models are also mportant n decomposton approaches, where schedulng problems n complcated envronments are broken down nto smaller, sngle-machne schedulng problems. 6 Flow Shop (F): Jobs have the same route and are processed n a seres of machnes n the same order. Whenever a job completes ts processng on one machne t jons the queue on the next. A subset from ths group s the Flexble Flow Shop whch contans a number of stages n seres wth a number of machnes n parallel at each stage. Job Shop (J): Each job has a route of ts own but the machnes are n the same order. The smplest ones assume that a job may be processed at most once. In others a job may vst a machne several tmes on ts route 5 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p PINEDO, Mchael and CHAO, Xul. Operatons Schedulng: Wth Applcatons n Manufacturng and Servces. USA: McGraw-Hll, p.15

24 12 system. These confguratons arse n many ndustres as the alumnum fols ndustry or the semconductors ndustry. Open Shop (O): Jobs do not have a defnte route to follow and they can be processed n the machnes wth any order. Mxed Shop (X): Some jobs have a certan route, others do not. Confguratons of Machnes: Identcal machnes (P): machnes that have the same processng tme. Independent machnes (R): Processng tme of operaton O, j on machne M k s P, j, k. Tradtonal schedulng and assgnment problems: Parallel Machnes (P/Q/R): Problems that have only one stage and jobs have only one operaton Hybrd Flow shop (HF): Problems where jobs have the same route and varous stages n the same order. General Job Shop (GJ): Problems where each job has ts own route. General Open Shop (GO): Problems where jobs do not have a fxed routng. When dealng wth schedulng problems, there are always one or varous constrants that need to be measured or taken n consderaton for the soluton. Some of the most explct constrants are the processng tmes (p ), due dates (d ),

25 13 release dates (r ) and weghts (w ). When solvng schedulng problems, some optmalty crtera s needed n order to evaluate schedules accordng to the prortes n the producton. It s mportant to notce that the dfference between a crtera and a constrant only depends on the decson maker: For example, statng that no job should be late regardng ts due date leaves no margn n the schedule calculaton. We may even fnd a stuaton where no feasble schedule exsts. On the other hand, mnmsng the number of late jobs allows us to guarantee that there wll always be a soluton even though to acheve ths certan operatons mght be late. ( ) the dfference between a crteron and a constrant s only apparent to the decson maker ( ). 7 Explct constrants n a problem are of varous types; some of the ones more used n theory and practce are shown: Release dates (r j ): Sequence dependent setup s jk means setup tme between job j and k. Ths s used when set up tme depend on the job that s placed on each machne. Preemptons (prmp): a feld that has ths constrant mples that t s not necessary to keep a job on a machne, once started untl completon. It s allowed to nterrupt the processng of a job and put a dfferent one. It s assumed that the amount of processng a preempted job already has receved s not lost. When (prmp) s omtted then preemptons are not allowed. 7 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p.12

26 14 Breakdowns: Imply that machnes are not contnuously avalable. The tme s assumed to be fxed, and for parallel machnes t s avalable at any pont n tme, that s breakdowns can be put as functons of tme. Machne elgblty restrctons (M j ): When ths feld s present n parallel machnes envronment, ths (M j ) denotes the set of machnes that can process the job j. Crtera are classfed n mnmax crtera and mnsum crtera; the frst one refers to mnmze the maxmum value of a set of functons and the second one refers to mnmze the sum of the functons. Some of the most common crtera used n lterature 8 are: C max " makespan" max C Wth C, beng the completon tme of the job J, 1,2,...,n F max " flowtme" max F C r F, for r beng the release date and Fmax the maxmum tme spent on a job. I max max I k I max, whch I k s the sum of dle tmes of resource M k L max max, lateness max L wth L C d T max max : tardness max T ; T max 0;C d E max max : earlness max E ; E max 0;d C 8 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p.13

27 15 C 1 C (average completon tme) or C (total completon tme) n w C 1 n w C (average weghted completon tme) or w C (total completon tme) F 1 F (average flow tme) or F (total flow tme) n w F 1 n w F (average weghted flow tme) U U ; whch s the number of late jobs wth U = 1 f job J s late and 0 f ts not. U w (weghted number of late jobs) E (average number of early jobs) w E (weghted number of early jobs) There are two basc approaches when schedulng problems are analyzed: Backward Schedulng: by ths approach schedulng problems are analyzed takng the due date as a set pont and determnng the date on whch each operaton must start by usng nter-operatonal acceptance tme laps (1989) 9. The rsk taken by usng ths approach can be that the startng date can be past date from present. 9 COMPANY, Ramon, Planeacón y Programacón de la produccón España: Barcelona, 1989.p

28 16 Forward Schedulng: Problems are analyzed from the release dates and on untl due dates. The rsks taken can nclude not attanng to fnsh on the proclamed due dates. Independently from the approach mpled, Gantt charts are used n both; programmng settng results and data transmssons. From the types of problems stated above there s a specal typology that dentfes one problem from the other. Table 1.1 descrbes ths typology n a graphcal way. Nevertheless, there are other typologes that need to be consdered n problems: Determnstc or stochastc: problems mght have all ts characterstcs well known, whle other problems can have ts characterstcs descrbed by random varables. Untary or repettve: Operatons n a problem can correspond to a unque product or can appear to be cyclcal. Statc or dynamc: all data of the problem can be known at the same tme or can be calculated and processed durng the arrval of new operatons.

29 17 SCHEDULING Sngle Machne Flow Shop Job Shop Open Shop SCHEDULING AND ASSIGNMENT WITH STAGES non duplcated machnes Parallel Machnes Hybrd Flow Shop General Job Shop General Open Shop GENERAL SCHEDULING AND ASSIGNMENT Parallel Machnes wth General assgnment Shop problems wth General assgnment Open Shop wth General assgnment common set of machnes Table 2. Typology of Schedulng Problems (T'KINDT and BILLAUT. Fgure 1.1. p.15) One of the frst to propose a notaton for schedulng problems was Conway (1967) 10, yet the most frequently used n lterature was ntroduced by Graham (1979) 11, whch s dvded nto: α І β І γ 12. α contans the typology shown n the Table 1.1., descrbes the structure of the problem. α = α 1 α 2. The sub-felds α 1 and α 2 refer to the type of schedulng problem and the number of machnes avalable, respectvely. β contans the explct constrants of the problem. γ contans the crteron or crtera to be optmzed n the problem. These are some of the basc rules for solvng certan types of problems 13 : 10 T KINDT, Vncent and BILLAUT, Jean-Charles. Op.Ct. p Ibd. p Ibd. p Ibd. p.21

30 18 SPT (Shortest Processng Tme): Orders the jobs takng frst the one that has the shortest processng tme. LPT * (Longest Processng Tme) s the converse rule. SRPT (Shortest Remanng Processng Tme) s the preemptve verson of the SPT rule and LRPT (Longest Remanng Processng Tme) s the converse rule. WSPT (Weghted Shortest Processng Tme frst): Sequences the jobs n ncreasng order of ther rato p /w. EDD (Earlest Due Date): Orders the jobs by takng frst the earlest due date. EST (Earlest Startng Tme): Sometmes jobs requre of a release tme, so f t s the case, jobs can be ordered by takng the one that has the earlest startng tme. FAM (Frst Avalable Machne): In parallel machnes, t can be necessary to use ths rule by placng the job n the next avalable machne. SPT-FAM (Shortest Processng Tme - Frst Avalable Machne): When assgnng jobs to the machnes, sometmes t s mportant to consder the shortest processng tme frst. EDD-FAM (Earlest Due Date - Frst Avalable Machne): When assgnng jobs to the machnes, t can also be mportant to consder the earlest due date frst. p d Lmax * LPT s commonly used when the objectve s mnmzng the makespan (C max )

31 19 FM (Fastest Machne Frst): When machnes are not dentcal, t mght be needed to process n the fastest machne frst. SPT and EDD can also complement ths rule. WSPT-FAM ( Weghted Shortest Processng Tme - Frst Avalable Machne): p w C In order to solve all types of schedulng problems, specal procedures are used where the rules mentoned above are taken n consderaton. It s mportant to note that when a schedulng problem belongs to a class P, there s an exact polynomal algorthm to solve t. Otherwse, f the problem belongs to the class NP-hard, there are two possble solutons, ether a heurstc * s proposed to calculate a problem n polynomal tme, or an algorthm s used to calculate the optmal soluton, but ts maxmum complexty ** s exponental Game Theory Game theory s the study of the strategc nteracton between two or more ndvduals (also known as players) who take decsons that wll affect n some way, dependng on what one expects from the other or others. There are two ways to descrbe games, n the strategc form and n the extensve form. To understand the * A heurstc s an approxmated algorthm. ** The complexty of an algorthm s measured both n tme and n memory space. Yet, n ths nvestgaton the complexty s gven by the tme, calculatng the number of teratons the algorthm takes to be processed.

32 20 concepts used n game theory, Fudenberg and Trole, n ther book Game Theory, refer to the notaton used for games n strategc form Games In Strategc Form It s composed of three basc elements: A set of players Є I, assumed to be n a fnte set {1, 2,,I } The pure-strategy space S for each player The payoff functons u, whch gves each player what von Neumann and Morgenstern called ther utlty u (s) for each set of strateges s=(s 1,, s I ). Snce players are denoted wth, the player s opponents wll be referred to as -. Ths termnology s used to emphasze that a player s objectve s to mnmze hs own payoff functon and ths wll affect, postvely or negatvely, the other player or players. Furthermore, there are mportant terms that need to be taken n consderaton: Equalzng strategy: Ths strategy results n the same average payoff for all players no matter what each player does. Value of the game: The average value of the payoff gven that both players have played n a proper way. Mnmax strategy: The optmal strategy that results n the value of the game. 14 FUDENBERG, Drew and TIROLE, Jean. Game Theory. Massachusetts: MIT Press, p. 4

33 21 Pure Strategy (s ): The optmal strategy that comes from choosng only one of the strateges that a player has. Mxed strategy ( ): The optmal strategy s the result of takng varous proportons of the pure strateges, n whch the randomness of the dstrbuton for each player s statstcally ndependent from that of the opponents and the payoffs to each player are the expected values of the payoffs for those pure strateges. Utlty theory: It states that the payoff must be evaluated by ts utlty to the player rather than the numercal monetary value. Common knowledge: Structure of a game that assumes that all players know the structure of the strategc form, and know that ther opponents know t, and know that ther opponents know that they know, and so on ad nfntum. 15 : Mathematcal expectaton of probablty dstrbuton of a player functon, gven that player s usng strategy, ths notaton represents the mathematcal expectaton of the payoff functon, gven that player s usng strategy, 1, 2, n, 1 1, 2,... n,, 2...,..., n 1, 2, n, Games n an Extensve Form 15 FUDENBERG, Drew and TIROLE, Jean. Op.Ct. p.4 16 OWEN, Gullermo. Game Theory. San Dego: Academc Press p5.

34 22 By an n-person game n extensve form s meant. a topologcal tree wth a dstngushed vertex A called the startng pont of ; a functon, called the payoff functon, whch assgns an n-vector to each Termnal vertex of ; a partton of the non termnal vertces of nto n+1 sets S 0, S1,..., S n, called the player sets; a probablty dstrbuton, defned at each vertex of S 0, among the mmedate followers of ths vertex. for each =1,2,,n, a subpartton of S nto subsets j S 0, called nformaton subsets, such that two vertces n the same nformaton set have the same number of mmedate followers and no vertex can follow another vertex n the same nformaton set. the set for each nformaton set j I j S, an ndex set onto the set of mmedate followers of each vertex of j I together wth a 1-1 mappng of j S. Condton states that there s a startng pont, gves a payoff functon, dvdes the moves nto chance moves S 0 and personal moves whch correspond to the n players S,..., 0, S1 Sn, ; defned a randomzaton scheme at each chance move; dvdes a player s moves nto nformaton sets : he knows whch nformaton set to be, but not whch vertex of the nformaton set OWEN, Gullermo. Game Theory. San Dego: Academc Press.1995 p2

35 IDENTIFICATION OF THE PROBLEM Today s productve systems confront a realty that requres them to be each day more compettve. Industres all around the world are usng ther best technology to make better decsons when plannng producton. Although technology s advancng day after day, the man mportance reles on the ways decsons are made. Yet, modelng productve systems n order to plan the producton can turn out to be a farly complex job. These complex problems cannot be solved wth basc schedulng algorthms and that s why today s researchers have studed these problems n many dfferent confguratons. Tanaev (1994), Pnedo (1995), Blazewcz(1996) and Brucker(1998) 18 are just some of the authors that have proposed algorthms and heurstcs to solve schedulng problems followng dfferent objectves n each one of them. Yet, the most mportant breakthrough n schedulng has been wth problems nvolvng multple objectves. These are the knd of problems that model real lfe stuatons n ndustres nowadays and t s necessary to understand the dfferent approaches that have been done to solve them. More specfcally, ths nvestgaton wll analyze a partcular type of problem confguraton, parallel machne schedulng problems. Usually, the schedule for ths type of confguraton results from the arrangement of each one of the n jobs, 18 T KINDT, Vncent and BILLAUT, Jean-Charles. Op.Ct. p.21

36 24 assgned to m number of machnes and ths depends on the avalablty of the machne (Frst Avalable Machne), or, n the case of non-dentcal machnes, t can depend on the velocty of the machne (Fastest Machne Frst). To mnmze flow tme, an algorthm that constructs a lst n order of non-decreasng processng tme (SPT) s wdely used. On the other hand, n order to mnmze makespan (C max ), a lst n order of decreasng processng tme s constructed (LPT). Assumng that for ths type of problem, a schedule that mnmzes makespan s found. Does ths mean that nventory mantenance costs are also mnmzed? Or would t mean that work n process s reduced? The hghest chance s that none of ths may happen. Consderng the alternatves n a specfc schedulng envronment, the decson maker must gve prortes accordng to these, and construct an approprate sequence to meet the specfc requrements. Even so, some of the objectves consdered mght be conflctng * and thus, focusng n reducng one of the prortes may lead to an excessve ncrease n the other crtera values, resultng n a net loss. The decson maker must take nto account these consderatons, approprately choosng the combnaton that could result mnmzng the maxmum loss. ** * Crtera are consdered to be conflctng f the reducton of one leads to ncrease the other. ** Mnmax or Maxmn Strategy : a term commonly used n game theory stated n a two-person zero sum game.

37 JUSTIFICATION In today s ndustres, a producton planner faces not only the problem of attanng a good schedule sequence to provde results to meet clent s needs, but must also consder mnmzng total costs, or nventory costs; for nstance. Ths s why; focusng on sngle crteron may just hnder fndng a more ntegral soluton. Fndng the proper combnaton of crtera not only provdes a more robust soluton, but may also approach to real systems envronment n a more drect way. Amplfyng the exstng approaches to multcrtera schedulng problems usng mathematcal models such as the ones n game theory, broadens ths feld for further research n these topcs, as well as havng more alternate routes to fnd Pareto Fronts. However, ths s stll a new topc and there are a lot of gaps and unknowns yet to dscover, so the results gathered may dffer slghtly from the ones found through Pareto analyss. The need to open new ways to tackle problems just opens a gate that n the future may turn to more knowledge. Ths way, schedulng theory wll eventually evolve and therefore strengthen actual possble applcatons. The nteracton of both schedulng and game theory s therefore, postve for both research areas. On one sde, game theoretc applcatons can nvolve more topcs than the usual ones treated n economc decsons and bology. On the other sde, schedulng, beng so mportant

38 26 throughout the 20 th century, especally for the last 50 years, has obtaned good solutons for short term decsons and wll contnue brngng results to meet today s clent s changng and demandng needs.

39 27 2. LITERATURE REVIEW 2.1. PARALLEL MACHINE SCHEDULING Parallel machne schedulng problems Whle analyzng the confguraton of these type of problems, the followng assumptons wll be taken n consderaton: Jobs are ndependent They arrve smultaneously The setup s sequence ndependent There s one or more machnes to perform the processng All machnes are dentcal wthn the system When analyzng an envronment of parallel machnes the next matrx s therefore useful to analyze what s gong on n the shop: Machnes m 1 p 11 p 12 p 1m 2 p 21 Jobs... n p n1 p nm

40 28 Here p j s the tme to perform the sngle operaton of job on the machne j. Ths of course assumes that the machne j performs the whole operaton. And the smplest case nvolves dentcal machnes, whch mples that all elements on a gven row are equal. If there are dfferent numbers wthn the same row, then the jobs have dfferent processng tme that depends on the machnes, ths s common where the resource s people and therefore there mght be specaltes for each workng performances. If the machnes alone have dfferent performance rates or speeds then the whole column for each machne has a number that corresponds to that acquanted speed. Otherwse the subscrpts can be omtted and p alone can be used. It s assumed for ths research that all machnes are dentcal. A key queston n the stuaton concernng parallel machnes arses: Is t better to dvde a sngle job so that the process tme of ths job s mnmzed? As stated by CONWAY (1967) 19 : If some dvson of the job s allowed, then better schedules are possble, but the determnaton of the schedule s more dffcult. Ths practce s reasonably common n some types of ndustres, especally the ones that may have operatons that are repettons of some smaller elements n work on these peces. 19 CONWAY, et. al., Op. Ct. p.75

41 29 For nstance, let m be the number of dentcal machnes and n jobs to be performed, each wth a processng tme p. There s a total of mp work to be done, and f ths s dvded equally among the machnes, they wll fnsh smultaneously after p tme unts, Regardless of how the jobs are assgned to ndvdual machnes. If the job s dvded as sad, then the frst job wll fnsh at p, the second tme at m 2 p, the m thrd at 3 p, and so on. Therefore, the average flow tme can be obtaned by m addng al these terms and dvdng them by m. The followng sequence wll be obtaned. F p m 1 ( m 2 m 3 m... m ) m p m 2 m 1 2 m 1 m p Takng these to the lmts t s easy to see that when there are too many machnes, that s, when the lmt tends to ths procedure can mprove flow tme up to 50%, and n the extreme when there are only two machnes the ncreasng percentage s 25. One could see m machnes workng smultaneously on a sngle job, as a sngle machne wth m tmes the power of the basc machne. So, from a schedulng pont of vew, t s better to provde requred capacty to a sngle machne than to an equvalent number of separate machnes. However consderatons of relablty work operate n opposte drectons.

42 30 If machnes are dentcal a pseudo-processng tme s defned for each job as p p' If machnes are unform then ths p s gven by p m m j p j However, there are many cases n whch ths s not possble, n such cases; the schedulng procedure conssts on assgnng a job to both a partcular machne and to a poston n sequence on that machne. Let j k be the job whch s n the kth poston n sequence on the j th machne and on the j th machne: n j be the number of jobs processed The mean flow tme s gven by: F m nj j 1 k 1 ( n j n k 1) p j k One can eventually nterchange jobs n equvalent postons n sequence wthout affectng n mean flow-tme, yet SPT rule does not guarantee to mnmze maxmum flow tme. An example taken from CONWAY shows how sometmes ths cannot be acheved: Let there be four jobs wth processng tme 1,2,3,10 to be processed, the two shortest processng tme schedules A and B, have maxmum flow-tmes of 12 and 11, respectvely. Schedule C s not a shortest processng tme schedule; t has a maxmum flow tme of 10 but a greater mean flow-tme. Schedule A B C Machne Machne

43 31 For the m dentcal machne case, n whch each job must be assgned to an ndvdual machne, no optmal procedure has been offered, but stll there are bounds such as the followng, called, weghted lower bound for weghted mean flow tme: F m m m( n n 1) u F u It s known that no greater bound s possble by exhbtng a set of jobs that actually attan ths bound. Even when t s not possble to fnd solutons to gven problems such as C C max max ( LPT ) ( OPT ) because most problems for parallel machnes are NP-hard; other mportant prortes n parallel machne envronments arse, such as makespan. By mnmzng makespan the sequence obtaned s gong to be the shortest one, and many tmes, dependng on the constrants these can be acheved. As stated before, n parallel machnes preemptons play a more mportant role than wth sngle machnes. For these models there are optmal schedules. In order to show the advantages that preemptons allow for parallel machne envronments t s useful to see how effcent both models can be. THE MAKESPAN WITHOUT PREEMPTIONS: Frst, t has been demonstrated that the problem Pm Cmax s NP-hard. Durng the last couple of decades, many heurstcs have been proposed; a very

44 32 common one s the (LPT) rule *, n whch the largest jobs are assgned to the m machnes. After that, whenever a machne s freed, the longest job among those not yet processed s put on the machne. So ths heurstc tres to place the shorter jobs toward the end of the schedule, ths way t balances the load. In order to gve an ndcaton of the effcency of ths algorthm Pnedo(2002) 20 determnes a lower bound gven by the followng expresson: C C max max ( LPT ) ( OPT ) For these types of problems the gven nequalty s always true: C C max max ( LPT ) ( OPT ) m C max (LPT) denotes the makespan of the LPT schedule and C max (OPT) denotes the makespan of the (possble unknown) schedule. COMPLETION TIME WITHOUT PREEMPTIONS: When the objectve s completon tme, that s Pm C j then the SPT rule gves an optmal soluton, and thus mnmzes ths gven objectve, but for nstance the problem Pm w j C j s NP hard. So ths result cannot be generalzed to parallel machnes. It has been shown that the WSPT heurstc s a good heurstc, the worst case on ths heurstc leads to the lower bound: * See Termnology and Notatons, algorthms and heurstcs. 20 PINEDO, Mchael. Schedulng: Theory, Algorthms, and systems. New Jersey. Upper Saddle Rver.2002.p 94.

45 33 w C ( WSPT ) j w C ( OPT ) THE MAKESPAN WITH PREEMPTIONS: j j j 1 (1 2 2) Sometmes allowng preemptons smplfy the analyss, for example the problem Pm prmp C max, sometmes even lnear programmng LP formulaton can be used to obtan nformaton about the optmal soluton, take PINEDO (2002) 21 lnear programmng formulaton of the problem (LP): Mnmze C subject to max m 1 m 1 n 1 x j x x x j j j 0 P C C j max max j j 1,2,..., n 1,2,..., n 1,2,..., m 1,2,..., m j 1,2,..., n Where x j represents the total tme job j spends on machne. The frst set of constrants makes sure the jobs receve the requred amount of processng. The second enforces that the total amount of processng each job receves s less or equal to the makespan. The thrd set makes sure that the total amount of processng on each machne s less than the makespan. The soluton of course does not prescrbe an actual schedule, t just specfes the amount of tme job j 21 Ibíd. p.105

46 34 should spend on machne, and from ths pont a schedule can be constructed. 22 : Takng nto account the fact that C n max j max p1, m j 1 C * max p Ths bound allows constructng a smple algorthm that fnds an optmal soluton: Algorthm (mnmzng Makespan wth Preemptons) Step1. Take the n jobs and process them one after another on a sngle machne n any sequence. The makespan s then equal to the sum of the n processng tmes and s less than or equal to mc* max. Step2. Take a sngle machne schedule and cut t nto m parts. The frst part consttutes the nterval 0, C *, the second part the nterval max C *,2C, the thrd part of the nterval 2C * max,3c * max max * max and so on. Step3. Take as the schedule for machne 1 n the bank of parallel machnes the processng sequence of the frst nterval; take as the schedule for the machne 2 the processng sequence of the frst nterval; and so on. Another way to obtan an optmal soluton s through one of the most used strateges, s the LRPT rule *. Ths schedule s structurally appealng, n the theoretcal pont of vew, but n the practcal pont of vew, t has drawbacks 22 PINEDO, Mchael. Op. Ct. p.106 * See Termnology and Notatons, Secton

47 35 because most of the tmes the number of preemptons n the determnstc approach s nfnte Heurstc Technques found n Lterature 23 Kurz and Askn (2001) presented three: Slcng (SL), Mult-Ft (MMF) and Multple Inserton (MI). SL solved the problem frst as f t was a sngle machne and then the sequence was dvded n the m machnes. MMF starts by assgnng the jobs to the machnes and then solves the problem of assgnng to each machne usng the technques of soluton gven by the Travelng Salesman Problem (TSP). MI orders the jobs frst by SPT and then assgns the jobs n the machnes by placng the job frst on the machne that has the least partal makespan. Franca, 24 et.al. (1994) developed a heurstc technque that mnmzes the makespan and t conssts on three steps. Franca, et.al. (1994) developed a heurstc that does not consder set-up tmes and ts objectve s to mnmze makespan n three steps: jobs are classfed n each machne n order to mantan the machnes wth approxmately the same load, then they are balanced by 23 GUTIERREZ, Elécer and MEJÍA, Gonzalo. Evaluacón de los algortmos Genétcos para el problema de Máqunas en Paralelo con Tempos de Alstamento Dependentes de la Secuenca y Restrccones en las Fechas de Entrega.. Unversdad de los Andes: Enero 25, (Pdf document form the World Wde Web) p.6 +el+problema+de+maqunas+en+paralelo.pdf 24 Franca, et.al.

48 36 passng jobs from the machne wth more load to the one wth less and, fnally, the machnes are balanced once more by swtchng jobs between them MULTICRITERIA OPTIMIZATION THEORY Multcrtera Decson Makng (MCDM) In the process of decson makng there are a set of tools that permt a correct approach to an optmal soluton of a problem. Many authors have presented sgnfcant contrbutons and, n general, the MCDM approach s more of a descrpton where possble solutons are defned, ncludng the attrbutes and evaluaton of the crtera, but most mportantly, there s a utlty functon where the crtera s ncorporated. Ths utlty functon has to be maxmzed durng ths process and that s how optmal solutons are reached. There has been a growng nterest and actvty n the area of multple crtera decson makng (MCDM), especally n the last 20 years. Modelng and optmzaton methods have been developed n both crsp and fuzzy envronments. 25 There are several axoms presented by Boysseu (1984) and Roy (1985) 26 that are fundamental to MCDM: 25 CARLSSON, Chrster and FÚLLER, Robert. Multple Crtera Decson Makng:The Case of Interdependence. p.1 ultple.pdf 26 T KINDT, Vncent and BILLAUT, Jean-Charles. Op.Ct. p.43

49 37 The decson maker always maxmzes, mplctly or explctly, a utlty functon. An optmal soluton exsts for every stuaton. No comparable soluton exsts, t wll always need to have to choose or sort between a par of decsons. Decson maker s preferences can depend upon two bnary relatons: preference (P) and ndfference (I). Apart from the fundamentals, T Kndt descrbes n MCDM two dfferent approaches 27 : Multple Attrbute Utlty Theory (MAUT): Proposed by Von Newman and Morgensten n 1954 s more of a stochastc approach that s done when decsons are subject to uncertanty at a crtera level. Analytcal Herarchy Process (AHP): Proposed by Saaty n 1986, n whch crtera s classfed n groups usng a herarchcal analyss n form of a tree and each crteron has been weghted n the utlty functon. Yet there are also some lmtatons to MCDM because problems are sad to be unrealstc and ths makes the theory less useful than what t should be. Accordng to Zeleny (1992) 28, MCDM s not useful when there s tme pressure, when the problem s more completely defned, when usng a strct herarchcal decson 27 T KINDT, Vncent and BILLAUT, Jean-Charles. Op.Ct. p CARLSSON, Chrster and FÚLLER, Robert. Op.Ct. p. 2

50 38 system, when there s changng envronment, when there s lmted or partal knowledge of the problem and when there s collectve decson makng n busnesses; all ths because t reduces the number of crtera beng consdered, leavng behnd other possble alternatves. Some authors, lke Carlsson and Fuller, agree that the tradtonal assumpton used n MCDM, n whch the crtera are taken as ndependent, s very lmted and deal to be appled to today s busness decson makng. Reeves and Franz ntroduced a multcrtera lnear programmng problem, where they presume the decson maker has to determne hs preferences n terms of the objectves but he must have more than an ntutve understandng of the trade-offs he s probably dong wth the objectves. For ths reason, an assumpton s made and that s, that a decson maker s taken to be a ratonal thnker and wth a complete understandng of the whole stuaton n whch hs preferences have some bass wth the use of a utlty functon. It has been unversally recognzed that there s no such thng as an optmal soluton vald for any multobjectve problem. In lterature, much has been found n terms of dfferent approaches to solvng MCDM problems. Delgado, et. al. (1990) used, for example, fuzzy sets and possblty theory not only to nvolve MCDM but also, multobjectve programmng. Also, Felx (1992) worked wth fuzzy relatons between crtera by presentng a novel theory for multple attrbute decson makng.

51 39 Carlsson, on the other hand, used fuzzy Pareto optmal set of non-domnated alternatves to fnd the best compromse soluton to MCDM problems wth nterdependent crtera. 29 In order to understand more about the nterdependences between crtera, t s mportant to notce the problem defned by Carlsson n terms of multple objectves: max x x X where f ( x),..., f n 1 f 1 : n k ( x) are objrectve functons s a decson var able and x s a subset of n. Defnton 30 : Let f and f j be the two objectve functons of the problem defned above.. f supports f j on X ( denoted f f j ) f f ( x') f ( x) entals f ( x') f ( x), for all x', x X; j j j. f s n conflct wth f j on X (denoted f f j ) f f ( x' ) f ( x) entals f ( x' ) f ( x), for all x', x X; j. Otherwse, f and f are ndependent on X. j j 29 CARLSSON, Chrster and FÚLLER, Robert. Op. Ct. p.4 30 Ibíd. Def p.4

52 40 Fgure 1. Conflctng functons on (CARLSSON Fgure and 1. Conflctng FULLER. functons Fgure 1. on p.4) (CARLSSON and FÚLLER. Fgure 1. p.4) Fgure 2. Supportve functons on (CARLSSON Fgure 2. and Supportve FULLER. functons Fgure 2. on p.5) (CARLSSON and FÚLLER. Fgure 2. p.5) In tradtonal MCDM t has been found that the crtera should be ndependent, yet there are some methods that deal wth conflctve objectves but do not recognze other nterdependences that can be present, whch makes the problem more unrealstc. Zeleny (1992) 31 recognzed that there are objectves that mght support each other when he shows the fallacy wth usng weghts ndependent from crteron performance Multcrtera Optmzaton Problems When schedulng problems have more than one objectve, they are sad to be multcrtera-based. It s mportant to understand the theory that they have consdered to solve these types of problems. The multcrtera optmzaton theory takes bascally a set of prortes establshed by the decson maker and provdes the best soluton under ther preferences. T Kndt shows a mathematcal defnton of the multcrtera optmzaton problems expressng them as a specal case of 31 CARLSSON, Chrster and FÚLLER, Robert. Op. Ct. p.12

53 41 vector optmzaton problems where the soluton space s S and the crtera space, Z(S), are vectoral eucldan spaces of fnte dmenson, Q and K respectvely 32 : Mn Z( x) wth Z( x) Subject to x S. e. S S x / g Q 1 ( x);...; g and Z( S) Z ( x);...; Z M 1 ( x) K T 0 K wth 1 ( x) T Q, K Defnton of Optmalty 33 Let Q S be a set of solutons and Z S) K ( the mage n the crtera space of S by K crtera Z. K x, y : x x y y x x y, y, 1,..., K 1,..., K Ths s vald for K > 2, because for sngle crteron problems (K=1), there s no way to compare between two solutons, for whch the optmal soluton s gven rght away. In the case of multple objectves, ths s no longer the case because there wll be varous solutons that mnmze several crtera and they need to be compared. To approach t, Pareto Optma, a general defnton of optmalty, s used. There are three types of Pareto optma that have been defned by several authors: weak, strct and proper Pareto optma. 32 T KINDT, Vncent and BILLAUT, Jean-Charles. Op.Ct. p Ibd. p. 47

54 z 4 z 5 z 6 z 7 z 8 42 Defnton of weak Pareto optma: x S s a weak Pareto optmum f and only f y S such that = 1,,K, Z (y) < Z (x). Ths set of weak Pareto optma, WE, defnes the trade-off curve n the crtera space, whch s called the effcency curve. Ths s the more general class of Pareto optma, the other two types are subsets of ths one. See Fgure 3 for an example of the set of ponts that represent the weak and the strct Pareto optma. Z 2 z 0 z 1 z 2 z 3 z 0, z 1, z 2, z 3,z 4, z 5, z 6, z 7,z 8 : weak Pareto optma z 2, z 3,z 4, z 5, z 6 : strct Pareto optma z 0, z 1, z 7,z 8 : non strct Pareto optma z 0, z 2, z 3,z 4, z 5, z 6, z 8 : extreme weak Pareto optma z 0, z 1, z 2, z 3,z 4, z 5, z 6, z 7 z 8 : extreme strct Pareto optma Z 1 Fgure 3. Weak and Strct Pareto Optma where Z defnes a polyhedron (T KINDT and Fgure 3. Weak and Strct Pareto BILLAUT. Optma where Fgure Z 3.3. defnes p.48) a polyhedron (T'KINDT and BILLAUT. Fgure 3.3. p.48) Defnton of strct Pareto optma: x S s a strct Pareto optmum f and only f y S such that = 1,,K, Z (y) < Z (x) wth at least one strct nequalty. E s the set of strct Pareto optma of S and E WE. Defnton of proper Pareto optma [Geoffron, 1968]: Let x, y S, y x and I y = { [1;K] / Z (y) < Z (x)}. x S s a proper Pareto optmum f and only f x s a strct Pareto optmum and M > 0 such that

55 43 y S, y x, I y = Z ( x) Z ( y) I y, ( j, 1 < j < K wth Z j (x) < Z j (x) ) such that M Z ( y) Z ( x) j j PRE s the set of proper Pareto optma of S and PRE E. Notce that ths defnton s only vald f each crteron Z can reach s mnmum value Determnng Pareto Optmalty When reachng for Pareto optma, the decson maker has to look for the best trade-off solutons between conflctng crtera, and t s assumed to be done by optmzng a utlty functon. When searchng for the soluton, the decson maker must choose for an algorthm or heurstc that can determne the whole Pareto optma set. The decson maker provdes weghts to the dfferent crtera beng analyzed n order to determne the prortes. In lterature many ways have been used to determne Pareto optma, t s just a matter of choosng the correct one dependng on the qualty of the calculable solutons and the ease of the applcaton T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p.54

56 44 Determnaton by Convex Combnaton of Crtera [Geoffron, 1968] 35 Let S be the convex set of solutons and K crtera Z convex on S. x 0 s a proper Pareto optmum f and only f K, wth 0 ;1 and K 1 1 such that x 0 s an optmal soluton of the problem ( P ): Mn g( Z( x)) wth g( Z( x)) Subject to x S K 1 Z ( x) The above theorem, Geoffron s Theorem, the parameters cannot be equal to zero because, otherwse, not all the results found wll correspond to proper Pareto optma. So another condton s needed to determne a weak Pareto optma: Let S be the convex set of solutons and K crtera Z convex on S. x 0 s a set of weak Pareto optmum f and only f K, wth 0 ;1 and K 1 1 such that x 0 s an optmal soluton of the problem ( P ). 36 T Kndt 37 ntroduces how graphcal representatons of the dfferent optmzaton problems can be done by usng level curves. For mnmzng the convex 35 T KINDT, Vncent and BILLAUT, Jean-Charles.. p Ibd. Op.Ct Lemma 2. p Ibd. p.58

57 45 combnaton of crtera, problem (P ) can be represented by defnng frst the set of level curves n the decson space, usng the condtons for ths specfc approach: Let X _( a) x S / K 1 Z ( x) a wth 0;1 and K 1 1 By wrtng L ( a) Z( X _( a)) n order to construct the curves n the graphs, the curve of mnmal value g* s found, where the lne L (g*) s tangent to Z n the crtera space. See fgure 4 for a geometrc representaton of the problem descrbed above. Z 2 a 1 < a 2 < g*< a 3 < a 4 < a 5 Z L = (a 5 ) L L = (a 1 ) = (a 4 ) L= (a 2 ) L L= (g* = (a ) 3 ) Z 1 Fgure 4. Geometrc Interpretaton of a of a problem (P ) (P ) (T KINDT (T'KINDT and BILLAUT. Fgure p.59) p.59) Determnaton by Parametrc Analyss [Soland, 1979] Before statng the theorem that condtons ths new method, t s necessary to defne what a strctly ncreasng functon s. A functon K f ; s strctly ncreasng f and only f K x, y, x y, x y f ( x) f ( y).

58 46 Theorem 38 Let G Y be the set of strctly ncreasng functons from K to whch are lower bounded on Z, and g GY. x 0 S that x 0 s an optmal soluton of problem P (g,b) : Mn g( Z( x)) Subject to x Z( x) S b s a strct Pareto optmum f and only f such Accordng to T Kndt, the problem (P (g,b) ) can be nterpreted geometrcally by the use of level curves. Let S' x S / Z( x) b, X ( a) x S' / g( Z( x)) a and L ( a) Z( X ( a)). Ths tme, the optmal soluton s found by searchng for the level curve that has ts mnmal value g* such that L (g*) s tangental to Z n crtera space. So the ntercepton between both spaces L (g*) and Z defnes the decson space of the strct Pareto optma. See fgure T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. Theorem 4 p.60

59 47 Z 2 a 1 < a 2 < g* b 2 Z b 1 L = (g* ) L = (a 1 ) L = (a 2 ) Z 1 Fgure 5. Geometrc Interpretaton of a problem (P (g,b) ) Fgure 5. Geometrc Interpretaton of a problem (P (g,b) ) (T KINDT (T'KINDT and and BILLAUT. BILLAUT. Fgure Fgure p.61) p.61) Determnaton by Means of the -constrant Approach 39 Ths method s used to mnmze a crteron assumng that the others (K-1) are upper bounded and t enables the decson maker to fnd a strct Pareto Optma. The followng theorem was proposed n [Yu,1947] ( Theorem 5. p.62) and s today frequently used by many authors: x 0 S s a strct Pareto optmum f and only f k k k k k K 1 k 1; K ( 1 ;...; K 1; k 1;...; K ) such that Z(x 0 ) s a unque crtera vector correspondng to the optmal soluton of the followng problem ( P k ): 39 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p.62

60 48 Mn Z k ( x) Subject to x S Z ( x) k, 1; K, k The last theorem s harder to apply because of the constrant of unqueness consdered. However, there s another theorem developed by [Mettnen, 1994] (Theorem 6. p.62-63) that does not take ths nto account, nstead, t develops weak Pareto optma rather than a strct one: Let x 0 S. If k k k k k K 1 k 1; K, and f ( 1 ;...; K 1; k 1;...; K ) such that x 0 s an optmal soluton of the followng problem ( P k ): Mn Z k ( x) Subject to x S Z ( x) k, 1; K, k Accordng to T Kndt, ths problem can also be nterpreted by means of level curves. Let k 1; K and k ( k 1 ;...; k K 1 ; k k 1 ;...; k K ) K 1. Let us defne S k x S / Z ( x) k 1; K, k, X ( a) k x S k / Z ( x) k a and L ( a*) k Z( X ( a) k ).

61 49 k To solve t, the mnmum value a* must be determned such that Z ( S ) s tangental to L *) k ( a n the crtera space. Yet, a value x* s a strct Pareto optmum k k f k, S such that L ( a*) Z( x*). See fgure 6. Z 2 a 1 < a 2 < g* Z(S k ) Z 1 L = (g* ) L = (a 1 ) L = (a 2 ) Fgure Fgure 6. Geometrc 6. Interpretaton of of a a problem (P k) ( P k ) (T'KINDT and BILLAUT. Fgure p.66) (T KINDT and BILLAUT. Fgure p.66) Use of the Tchebycheff Metrc 40 Ths metrc was proposed by [Bowman, 1976] and s practcally used to look for the closest soluton to a reference crtera vector or reference pont *. Before descrbng ths metrc, t s necessary to consder some defntons:. d d d T d 41 z z1 ;...; zk s the deal pont f and only f z mn( Z ( x)), 1,..., K x S T 1 K K d. Let K vectors z z ;...; z verfyng z z, 1,...,. So a gans matrx 42 j, j K s defned by G z, 1,..., K, j 1,...,. 40 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p Ibíd. Def. 25 p Ibíd. Def. 26, p.67

62 50. Let G be the gans matrx. The nadr s defned as a crtera vector, noted z defned by z max ( G j, ), 1,..., K j 1,..., K. na na 43 v. ut z s a utopan pont, f and only f ut z domnates d z wth at least one strct nequalty A reference pont s known as every vector, noted z ref, whch s consdered to be an objectve to reach. The objetve s to fnd the closest possble soluton to ths pont n order to optmze the functon. Ponts mentoned above n the last defntons are also consdered reference ponts. Bowman s defnton of the Tchebycheff metrc 45. Let z 1 and z 2 K. The Tchebycheff metrc s a measure of the dstance n the K between z 1 and z 2 defned by: z 1 z 2 T max ( z 1,..., K 1 z 2 ). The Tchebycheff pont, noted (z * ), s a specal reference pont, such that, z 1 0 * mn( Z ( x)) wth S x S / Z ( x) mn ( Z ( x')) and S S. 0, x S x' S,..., K ) K Bowman s Theorem T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. Def. 27 p Ibíd. Def. 28, p Ibíd. Def. 67. p.67

63 51 If x 0 K S s a strct Pareto optmum then * 0, *,..., * ) 2 K such that 0 x s an optmal soluton to the problem ( P ): Mn Z( x) subject to x S ( z * *) T The geometrc nterpretaton of the problem P can be done by the use of level curves. Let x * and be fxed, X ( a) x S / Z ( x) ( z * *) a and L ( a) Z( L ( a)). So by determnng the T mnmal value a *, such that L (a*), thus, the solutons for X (a*) are found. See fgure 7. Z 2 a 1 < a 2 < g* Z z*- L = (g* ) L = (a 2 ) Z 1 L = (a 1 ) Fgure Fgure 7. Geometrc 7. Geometrc Interpretaton of of a a problem (P ) (P (T KINDT ) (T'KINDT and and BILLAUT. Fgure Fgure p.69) 46 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. Theorem 7. p.67

64 52 Determnaton by Goal-Attanment Approach 47 Ths approach s smlar to the last one mentoned, but the dfference les on how the soluton s searched. Ths requres for the decson maker to defne a goal for the crtera and so t looks for the soluton that gets closer to ths goal. [Gembck, 1979] and [Werzbck,1990] proposes the followng theorem 48 : x 0 S s a weak Pareto optmum f and only f ref K z a reference pont and w K a weghts vector such that 0 x s an optmal soluton of the problem ( P ) (, w) z ref stated below. Max g( Z( x)) wth g( Z( x)) subject to x S 1 mn ( 1,..., K w ( z ref Z ( x))) A geometrc nterpretaton of ths problem s done by T Kndt by projectng the pont onto the trade-off curve n a drecton specfed by the weghts value w. 49 See fgure 8 that descrbes the case where a soluton s found and where no feasble soluton s found. Z 2 Z 2 Z Z z ref z ref Z 1 Z 1 Fgure 8. Geometrc Interpretaton of a problem (P (z ref, w) ) (T'KINDT and BILLAUT. Fgure p.79) Fgure 8. Geometrc Interpretaton of a problem ( P ) (T KINDT and BILLAUT. Fgure p.79) ( z ref, w) 47 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p Ibd. Theorem 12. p T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct.. p.79

65 53 Determnaton by the Use of Lexcographcal Order 50 Ths method s used when no trade-off s allowed n the problem, so t s defned accordng to a lexcographcal order, Z Z... Z 1 2 K, and noted mn Lex ( Z). In order to obtan a soluton to ths problem, two condtons must be satsfed: Z s lower bounded on each subset soluton, 0 x, the soluton for x 0 1 S and that S. So, to determne the optmal K S must be found wth: S S S 1 2 K x x 0 x 0 0 S / Z S S 1 1 / Z K ( x 2 / Z 0 ) ( x K 0 ) ( x 0 mn( Z ) x S 1 mn( Z 1 x S ( x)), 2 mn( Z x S K ( x)),..., K ( x)) Multcrtera Lnear Programmng Even though, the approaches shown n the prevous secton are used for many dfferent hypothess made, there s a smpler way to solve them through Multcrtera Lnear Programmng. The model s presented by T Kndt 51 : 50 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p Ibd. p.83

66 54 Mn Z Mn Z x 1 K subject to ax wth Z wth Z b Q 1 K Q 1 c j j 1 Q j 1 c x j K j x j c 1 x c K x wth A representng the coeffcents matrx ( M x Q ) and b representng the constants vector of dmenson M. The crteron convex polyhedron defned by the set of solutons S. Z s a lnear functon and so Z s a Some of the applcatons of Multcrtera Lnear Programmng nclude the determnaton of strct Pareto optma by convex combnaton of crtera and by - constrant approach Multcrtera Mxed Integer Programmng Some of the approaches studed above have a lack of convexty hypotheses on Z, whch determnes that some non supported solutons appear. Gven ths cases, Multcrtera Mxed Integer Programmng models the supported and the non

67 55 supported Pareto optma. T Kndt shows an example to explan the dfference between these two terms: Set Z s the set of ponts represented and co(z) s the convex hull defned by Z. We have co K K K ( Z) z / 0;1, 1 and z Z, z. Snce x 0 does not 1 1 belong to the border of co(z), t s consdered to be non supported strct Pareto optma. Also, pont x 4 represents a weak Pareto optma. See fgure 9 to observe the graphcal nterpretaton of the problem. Z 2 co (Z) z 1, z 2, z 3, z 6 : supported strct Pareto optma z 5 z 1 z 4 z 0 : non supported strct Pareto optma z 5 : supported weak Pareto optma z 4 : non supported weak Pareto optma z 2 z 0 z 3 z 6 Z 1 Fgure Fgure 8. Supported 9. and and non non supported Pareto optma (T'KINDT (T KINDT and BILLAUT. Fgure p.85) The resoluton methods lke the parametrc analyss, the Tchebycheff metrcs and the goal-attanment approach, does not present any problem n determnng the non supported and supported Pareto optma.

68 MULTICRITERIA SCHEDULING PROBLEMS The prncpal objectve of schedulng s to optmze the objectve functon started by the problem by defnng a schedule that best fts t. The resultng soluton corresponds to the Pareto optmum for the multcrtera schedulng problem. Accordng to the notaton presented before **, the schedulng problems are referred to n a general way by usng the three-feld notaton. The last feld,, denotes a lst of crtera that need to be consdered to solve the problems. When there s more than one crteron, ths corresponds to a multple objectve problem: Z 1, Z 2,..., Z K, where Z s the crteron to be mnmzed. It s just Z f t s a sngle crteron problem. For better understandng of ths type of problems, a new feld s ntroduced and t corresponds to the resoluton methods, studed prevously and used to solve these types of problems. Snce these resoluton methods wll be taken n consderaton later n the nvestgaton, t s necessary to get acquanted wth the notaton used 52 : F l ( Z 1,..., Z K crtera. ), f the objectve s to mnmze a lnear convex combnaton of Z 1,..., Z u 1, Z u,..., Z K ), f the objectve s to mnmze only the crteron Z u, ( 1 by usng the constrant approach. ** See secton 1.2. Termnology and Notaton. 52 T KINDT, Vncent and BILLAUT, Jean-Charles. Op. Ct. p.110

69 57 P Z,..., Z ) f the objectve s to mnmze by usng parametrc analyss. ( 1 K F T ( Z 1,..., Z K ) f the objectve functon s a dstance known as an deal soluton and calculated by Tchebycheff metrc. F Z,..., Z ) f the objectve s to mnmze by usng the goal-attanment s ( 1 K approach. Lex Z,..., Z ) ndcates that no trade-off s authorzed so they must order ( 1 K the crtera begnnng wth the most mportant one. Kumar, Marathe, Parthasarathy and Srnvasan have mplemented approxmaton algorthms for schedulng on multple machnes n order to solve a b-crtera problem based on mnmzng makespan and weghted completon tme. They proposed a sngle randomzed roundng algorthm that combnes the power of LST and randomzaton n order to obtan a smultaneous optmzaton of multple objectves. Wth ths, they obtan a (2, 3/2) b-crtera approxmate algorthm for makespan and weghted completon tme GAME THEORY In game theoretc lterature much has been sad wth regards to many types of games and each one of them has approached an applcaton to the dfferent areas of study. All the theory that surrounds t makes t a very dynamc and extensve

70 58 feld. Yet, t s necessary to know the fundamental aspects of game theory and for ths there are varous authors that had contrbuted to the understandng of t. The man objectve n game theory s to develop ratonal crtera n order to decde over two or more strateges. There are two basc assumptons gven for ths: players are ratonal thnkers and players choose ther strategy to maxmze ther own beneft Two person zero-sum game and Nash Equlbrum One of the smplest forms of a game s the one that nvolves two players and whose sum of the utltes s equal to zero, sometmes referred to as strctly compettve games 53, where =1,2 u (s) = 0 for all s. The non-zero sum games may be more practcal n many applcatons; yet, for purposes of the analyss shown, t s mportant to understand ths type of a game before gong any further. The benefts for each player are shown n a matrx, n the form of payoffs. Usually the payoffs are postve to show earnngs and negatve to show losses. It s necessary to note that the games consdered are fnte games, whch means that each player s set of strateges s fnte. There are several ways to approach these types of problems dependng on the stuaton of each player n terms of 53 OWEN. Op.Ct. p.11

71 59 strateges and respectve payoff functons. To llustrate ths n a better way, an example s descrbed n Fudenberg and Trole s book, for whch players 1 and 2 have three pure strateges each. Player 1 has strategy U,M,D (upper, mddle and down) and player 2 has L, M, R (left, mddle and rght). The chart above shows the resultng matrx. L M R U 4,3 5,1 6,2 M 2,1 8,4 3,6 D 3,0 9,6 2,8 Each player has one strategy to choose, yet, sometmes the player can choose more than one strategy. When ths possblty s contemplated, then the payoff for the players can be estmated rather than fxed. The payoff of player I to a mxed strategy profle s gven by the followng expresson: s S I j 1 j ( s j ) u ( s) In the example above, the vector representng the mxed strategy of player 1 s ( D ( U ), ( M ), ( )). The profles for each player are: j 1, , 3 and j 1 0,, The payoff for ths gven profles, can be calculated as shown below: u ( 1 1) 1 ( ) 1 ( ) 1 ( ) 11 2 and u ( 1 2 ) 27. 6

72 60 But, before gong any further on mxed strateges, there are varous ways of obtanng optmal strateges. Startng wth the smplest form, ths s by detectng those domnated strateges n the matrx for each player. Gven the last example, note that for player 2, R gves a hgher payoff than M does, no matter what player 1 chooses: L R U 4,3 6,2 M 2,1 3,6 L 3,0 2,8 Lkewse, for player 1, U wll gve a hgher payoff for both M and L: L R U 4,3 6,2 At ths pont of the game, player two s able to choose the strategy that best satsfes hs needs, the strategy that gves hm/her the greatest utlty; n ths case, t s L. So the par of strateges chosen by both players are: U for player 1 and L for player 2, representng for them a payoff of 4 and 3 respectvely. In ths case, they have pure and strctly domnated strateges, where solutons are ndependent and n equlbrum, whch means that the soluton wll be always (4,3) no matter what each player does ndependently. It s mportant to note the defnton for Nash Equlbrum, whch s ntroduced by the famous Noble prce wnner, John Nash. Fudenberg and Trole 54 present t as defnton 1.2 A mxed-strategy profle * s a Nash equlbrum f, for all players, 54 FUDENBERG, Drew and TIROLE, Jean. Op.Ct p.11

73 61 u ( *, *) u ( s, *) for all s S For pure strateges, t satsfes the same condtons as the mxed strateges, only that the probabltes can only take values of 0 or 1. Harsany (1973b) 55 ntroduces a strct Nash Equlbrum, where each player has a unque best response to hs rvals strateges. So the pure strategy s * s strct for all and all s s*, u ( *, *) u ( s*, *). Notce that ths strct equlbrum happens only wth pure strateges. It seems that the strct equlbra are more compellng than the equlbra where players are ndfferent to ther equlbrum strategy and even to a non-equlbrum response. It s also sad that due to varous small changes n the nature of the game, strct equlbra are robust. In general, Nash Equlbra gve reasonable predctons to how a game s played and t s the only one that has the property of common knowledge between players. There s no ncentve to play dfferently when a game has Nash Equlbra because players can detect t. On the other hand, n a non-nash profle, players can make decson mstakes durng the optmzaton of ther own payoff functon or n the predcton of the other s possble moves. Ths type of mstakes s the reason why most economc applcatons of game theory restrct Nash equlbra Resoluton methods for non-domnated strateges 55 FUDENBERG, Drew and TIROLE, Jean. Op.Ct. p.11-12

74 62 Not many applcatons have domnated strateges, and there are stll other solutons that can stll be Nash Equlbra. There are two ways to solve a 2 by 2 game when there s no terated domnance. The frst one s to search for saddle ponts, whch are what we already know as Nash Equlbra. The second way s to fnd the mxed strateges, whch have to be found f there s no saddle pont found. To fnd the saddle ponts, the Mnmax Theorem s ntroduced. The Mnmax Theorem s proven by Von Neumman and Morgenstern and s the most mportant n game theory. Owen presents the defnton of ths theory, but before defnng the theory t s necessary to know some addtonal terms needed to understand the defnton. Gven a matrx game A, A a a 11 1 a 1 j a j, where a j represents s the payoff to each player for choosng strategy s whle the opponent chooses the strategy s j, there s v 1 that represents the gan-floor of player 1 and v 2 represents the lost-celng of player 2. These values are defned by Owen 56 as: v max mn a v mn max a 1 j and 2 j j j 56 OWEN. Op.Ct. p.14

75 63 So to fnd the saddle ponts player 1 should not wn less than v 1 and player 2 should not lose more than v 2, satsfyng ths condton: v 1 v 2.. Mnmax Theorem For any functon F(x,y) defned on any Cartesan product X X Y, max mn F( x, y) x X y Y mn max F( x, y). y Y x X Hence we have, v 1 v 2. Ths theorem s used to fnd other possble solutons that cannot be treated wth terated domnance. The followng example of a game matrx s gven to show ths: L R mn(1) U 1,6 6,5 1 L 5,2 2,4 2 mn(2) 2 4 In ths example, by gettng the maxmum value of the mnmum of each column and of each row, t s possble to reach for a saddle pont. In ths example, one saddle pont s found, yet there are other possbltes. L R mn(1) U 1,1 6,2 1 L 5,3 1,1 1 mn(2) 1 1 L R mn(1) U 3,2 6,2 3 L 4,3 5,1 4 mn(2) 2 1 The frst game matrx s shown below, descrbes a stuaton where 2 saddle ponts are found. The second game matrx descrbes a game wth no saddle pont; n ths case, t s necessary to fnd the mxed strateges.

76 Mechansm desgn Mechansm desgn s a subfeld of mcroeconomcs and game theory, whch s used to obtan an optmal system-wde soluton to a decentralzed optmzaton problem wth multple self nterested agents, each wth prvate nformaton about ther preferences. In a mechansm desgn problem, an agent s asked to nput ther confdental nformaton to the system and ths one, n response, provdes an acton and an outcome, accompaned by an ncentve to promote truth-revelaton n ther partcpaton, n order to reach for an optmal soluton. In order to understand how the mechansm works, t s mportant to recall some notatons used. The way an agent recognzes ts preferences, a type must be declared. Let denote the type of an agent, from a set of possble types. Let u o, ) denote the utlty of agent for the outcome o, gven type. For ( the agent to choose for a course of acton, t must have a set of strateges to choose from. Let s ( ) denote the strateges of agent gven type, where s the set of all possble strateges avalable to the agent. Let u s,...,, ) ( 1 s I denote the utlty of agent at the outcome of the game, gven preferences and strateges profle s s,..., s ) selected by each agent. ( 1 I

77 65 There are three soluton concepts used for solvng these partcular type of problems; two of them have already been ntroduced earler n ths chapter, Nash equlbrum and domnated strategy equlbrum, a thrd one, Bayesan-Nash equlbrum s also beng used. Accordng to Nash equlbrum, every agent maxmzes ts utlty wth strategy s, gven ts preferences and the strategy of the other agents. To play Nash equlbrum n a one-slot game, every agent must have perfect nformaton about the preferences of the other agent, agent ratonalty must also be common knowledge 57. A robust soluton concept s the domnated strategy equlbrum, where each agent has the same utlty-maxmzng strategy, for all strateges of the rest of the agents. It does not make any assumptons about the nformaton handled by the agents and does not requre an agent to beleve the other agents behave ratonally to choose ts own strategy. In mechansm desgn, domnant strategy mplementatons of socal choce functons are much more desrable than Nash mplementatons. The last concept, the Bayesan-Nash equlbrum, n comparson to Nash equlbrum, agent s strategy s ) must be a best response to the dstrbuton over strateges of other agents, gven the nformaton of ther preferences n a dstrbuted functon. Ths type of soluton makes more reasonable assumptons about agent nformaton than Nash equlbrum, but a weaker soluton concept compared to the domnant strategy equlbrum. These three soluton concepts are applcable both for statc and dynamc games; n statc games, every agent chooses ts strategy smultaneously, ( 57 PARKES, Davd. Chapter 2. Classc Mechansm Desgn. Harvard Unversty. p (Pdf document va www) <http://www.eecs.harvard.edu/~parkes/pubs/ch2.pdf>

78 66 and n dynamc games, actons are based on observaton and learnng from other agents preferences throughout the course of the game. The system-wde goal then s defned wth a socal choce functon, f : 1 x... x, I whch selects the optmal soluton, f ( ), gven the agent types, ( 1,..., I ). At frst, the mechansm, M (,...,, g( )) 1 I, defnes a set of strateges avalable and the method used to select the fnal outcome based on agent s strateges. Accordng to game theory, the mechansm mplements socal choce functon f ( ) f the outcome computed wth equlbrum agent strateges s a soluton to the socal choce functon for all possble agent preferences. Ths equlbrum concept may be Nash, Bayesan-Nash, domnant or any other. The socal functon has many propertes, for example, t s Pareto optmal f no agent can ever be happer wthout makng at least one other agent less happy, t s effcent f t maxmzes the total value over all agents, t can also be budget balanced so no net transfers out or nto the system. Both allocatve effcency and budget-balance mply Pareto optmalty. The type of mechansm may vary dependng on ts propertes, for example, the agent s preferences may be descrbed by a quaslnear functon f ther utlty s decomposed nto a valuaton functon that depends on a choce rule and a payment functon whch s assgned based on the strategy profle. The mechansm s not just subject of the agent s preferences, also on the equlbrum concepts and

79 67 partcpaton condtons (ndvdual-ratonalty s appled those players outsde the mechansm). Partcularly, the drect revelaton mechansm s characterzed by an ncentve compatblty property; ths means that agents report the truthful nformaton about ther preferences n equlbrum, out of ts own self nterest. Ther strategy s to report a type ˆ s ( ), based on ts actual preferences. The outcome specfcaton s gven by a postve real valued objectve functon g (o, ). The requred output s the outcome o F that mnmzes g. The drect revelaton mechansm characterzes n the mplementaton of domnant strateges, whch means that t s strategy proof. Yet, the case can be such that the soluton s obtaned from Bayesan-Nash equlbrum, but ths only happen f s *, where every agent s expected utlty maxmzng strategy n equlbrum wth every other agent s to report ts true proft. VCG (Vckrey-Clark-Groves) mechansm s appled to mechansm desgn optmzaton problems where the objectve functon s smply the sum of all agents valuatons and mplements domnated strategy equlbrum solutons. When ntroducng transfers, t depends on the characterstcs of the mechansm. If quaslnear preferences are assumed, then the transfer functon, t, takes part of the utlty functon, u ( o, ) v ( k, ) t. In order to mplement an effcent outcome, k * max v ( k, ˆ ), and compute transfers, k j ˆ) t ( v j j ( k, j ) v j j ( k*, j ) ˆ ˆ

80 68 Where max ˆ. k k v j j ( k, j ) Ths transfer functon must guarantee both an optmal strategy and a balanced budget. But t has been shown that for ths mechansm t s mpossble to mplement a soluton n domnant strateges and satsfy balanced budget constrant for every possble message profle. A smple way to solve ths budget balancng problem n domnant strateges s to ntroduce an extra agent to the mechansm, agent 0, whose preferences are known and has no preferences over the solutons, and whose only nterest reles on the transfers, u ( ) 0 t 0. Agent 0 wll collect all the payments of the agents so f, t ˆ ) 0 *( ) t N ( j, then ths mechansm guarantees both a balanced budget and a selecton of an optmal soluton Game Theory and Computer Scence Game theory has been contnuously used n the branch of computer scence that can be observed as smple nterpretatons of zero-sum games for analyzng problems n onlne computaton to more complex aspects of game theory n artfcal ntellgence. Agent-based smulaton s been advancng n the area of computaton and best descrbes how game theoretc prncples are benefcal to ther models. As observed n mechansm desgn, the fgure of agents s used to model objects wth some degree of decson, whose actons depend on these decsons. Lus

81 69 Mateus Rocha n hs research project 58 as part of the Complex Systems and Applcatons Group n New Mexco, Unted States, referred to agents as an entty that must be able to step out of the dynamcs of an envronment, and make a decson about what acton to take next: Snce choce s a term loaded wth many connotatons from theology, phlosophy, cogntve scence, and so forth, I prefer to dscuss nstead the ablty of some agents to step out of the dynamcs of ts nteracton wth an envronment and explore dfferent behavor alternatves. In physcs we refer to such a process as dynamcal ncoherence [Pattee,1993]. In computer scence, Von Neumann, based on the work of Turng on unversal computng devces refer to these systems as memory-based systems. That s, systems capable of engagng wth ther envronments beyond concurrent statedetermned nteracton by usng memory to store descrptons and representatons of ther envronments. Such agents are dynamcally ncoherent n the sense that ther next state or acton s not solely dependent on the prevous state, but also on some (random-access) stable memory that keeps the same value untl t s accessed and does not change wth the dynamcs of the envronment-agent nteracton. Ths s how agents have been defned as part of computatonal models, yet aspects can become nterestng when analyzng how these models are based on game theoretc strateges, where the model ams to study only the decson strateges and evoluton of the strateges over tme. They also follow a synchronous behavor, whch means that all agents are updated smultaneously and there s an outcome as part of ths behavor. The terated Prsoner s Dlemma s an example of an dealzed model for many real-world phenomena, lke the armraces and evolutonary bology. It conssts of 2 ndvduals whch are arrested together but placed n separated rooms. As they are questoned, no communcaton s allowed between them, but they are offered to testfy aganst each other. If one betrays the other, he gets a suspended sentence whle the other gets the whole sentence. If both testfy aganst each other, the testmony s 58 MATEUS ROCHA, Lus. From Artfcal Lfe to Semotc Agent Models. Los Alamos, NM: Los Alamos Natonal Laboratory. October 1, (Pdf document).

82 70 dscredted and they both get a hgh sentence. Yet, f they decde not to testfy, they both get a smaller sentence. Ths model s defned as a non-cooperatve game of 2 players, each one wth 2 strateges each: to betray or to not betray. The terated Prsoner s Dlemma (IPD) game has been wdely used by economsts and other researchers to dscover the potental emergence of mutually cooperatve behavor among non-altrustc agents 59. Ths game typcally assumes that ndvdual players have no control over who they play wth; nstead, they are modeled by a mechansm, where randomness s mplemented as part of the smulaton. One of the most mportant conclusons reached by these studes has been that the mutual cooperatve behavor can be reached on the long-run, a pretty large or nfnte number of teratons. Ths s gven by the suffcently large frequency of mutual cooperatve matches and the perceved hgh probablty of future nteractons. The researcher Tesfatson remarked n hs paper about the IPD game: In actualty, soco-economc nteractons are often characterzed by the preferental choce and refusal of partners. The queston arses whether the emergng and long-run vablty of cooperatve behavor n the IPD game would be enhanced f players were more realstcally allowed to choose and refuse ther potental game partners. ( ) The tradtonal IPD game s extended to an IPD/CR game n whch players choose and refuse partners on the bass of contnually updated expected payoffs. 60 A more smply type of game, the zero-sum game s used n computer scence to model what s known as demonc nondetermnsm, whch s based on choosng the worst possble outcome when there s no suffcent nformaton about future events. Randomzaton algorthms are used wth ths model n order to analyze 59 TESFATSION, Legh. How Economsts Can Get Alfe: Abbrevated Verson. Iowa. September 6, (Pdf document) 60 Ibd.

83 71 problems of onlne computaton, whch descrbes a stuaton were ndvduals have to nput data at the same tme and wth ths nformaton, decsons are made APPROACHES OF GAME THEORY IN SCHEDULING In lterature, there s very lttle wrtten about the possble game theoretc nteractons made n game theory. Yet some farly recent papers have ntroduced on the topc. Authors lke T.C. La and Y.N. Sots (1999) propose a way to search for a mnmal set of certan schedules 61 through the use of game theory. For ths, they propose a number of schedulng problems that need of the best expected processng tmes, whch are under the control of a decson maker. At each decson pont of the schedulng problem, a two-person zero sum game wth the decson maker beng player 1 and nature beng player 2. Other authors lke Serafn 62 menton game theory as a way to reach for optmal and non-domnated solutons, and consder specfcally the objectve of mnmzng the maxmum tardness of the jobs whose completon tmes can be known n advance. He also mentons that n mathematcal programmng, ths type of approach s named Unordered Lexco Optma. 61 LAI, T.C. and SOTS, Y.N. Sequencng wth uncertan numercal Data for Makespan. USA: The Journal of Operatonal Research Socety, Vol. 50, No.3. pdf document. p.1 62 SERAFINI, Paolo. Schedulng Jobs on several machnes wth the job splttng property. USA: Operatons Research, Vol.44, No.4. pdf document. p. 620

84 72 Yet, one of the most nterestng contrbutons was presented by Kutanoglu and Wu, n ther papers, An Incentve Compatble Mechansm for Dstrbuted Resource Plannng and On Combnatoral Aucton and Lagrangean Relaxaton for Dstrbuted Resource Schedulng. In the frst paper they mplement a mechansm desgn problem, where job agents are consdered to represent the jobs and ther preferences may be motvated by any constrant consdered local, lke delvery requrements. They defne the game as an n-person non-cooperatve game wth ncomplete nformaton, where the n players are consdered to be the job agents, where each one has a strategy to choose and for each decson, a utlty functon s assgned. In the prevous secton, the mechansm desgn procedure was explaned, yet the goal of the mechansm n these type of schedulng problems s to choose a partcular functon usng the outcome functon h() for a partcular realzaton of agents utlty functons n order to choose an optmal or socally effcent schedule y*. Ths procedure s also known as schedule selecton game. Kutanoglu and Wu used a resource allocaton problem to llustrate ths approach. The local constrant used was the job due date and two smplfcatons were made of the problem, t was decomposed n a seres of sngle machne problems and set up tmes could be added. Frst the mechansm created some canddate schedules usng Lagrangeanbased aucton theoretc algorthm. The utlty was consdered as the negatve value of weghted tardness and the agents performance depends only on ts job allocaton n a schedule and ts transfer. The second paper descrbes how local decson makers base ther dea on local utlty whch means that ther problem s to

85 73 maxmze ther expected reward subject to local constrants. The type of coordnaton s known as a bd where the auctoneer s a bd processor that makes resource allocaton n form of an aucton processng usng bddng nformaton. These approaches of game theory to schedulng have been closely related to the area of computer scence and n ths area, two authors have contrbuted n an extraordnary way, Ronen and hs professor Nsan, followng the technques of mechansm desgn and applyng t to task allocaton problems, especally contrbute n the computatonal possbltes of these mechansms. A formal model s ntroduced by them for studyng optmzaton problems, n order to observe how mechansm desgn can be appled to several of these problems. The model s concerned wth computng functons that depend on nputs that are dstrbuted among n dfferent agents. A problem n ths model has, n addton to the specfcaton of the functon to be computed, a specfcaton of the goals of each of the agents. The soluton, termed a mechansm, ncludes, n addton to an algorthm computng the functon, payments to be handed out to the agents. These payments are ntended to motvate the agents to behave correctly. 63 They defned a task allocaton problem wth k tasks that need to be allocated on n agents. Each agent type s, for each task j, the mnmum amount of tme t j the agent s capable of performng ths task n. The goal s to mnmze the makespan. The valuaton of each agent s the negaton of the sum of the tmes t has spent on 63 NISAN, Noam. Algorthms for Selfsh Agents: Mechansm Desgn for Dstrbuted Computaton. Jerusalem: Insttute of Computer Scence. (pdf document)

86 74 the tasks allocated to t. They denoted the drect revelaton mechansm m(x, p), where x=x(t) s the allocaton algorthm and p=p(t) s the payment. They studed ths task schedulng problem and desgned an n-approxmated mechansm, where n s the number of the agents; he proved that a lower bound of 2 to the approxmate rato that can be acheved by any mechansm (for the case of two agents); and fnally desgned a randomzed mechansm * that beats the determned lower bound. Nsan and Ronen have shown that worst case behavor can be mproved usng randomness wthout weakenng the game theoretc requrements of the mechansm. Fnally they came up wth a Second Chance Mechansm, where the agents are allowed, besdes declarng ther types, to declare an appeal functon where the mechansm s able to compute a better possblty for each agent. In order to work out ths mechansm, an algorthm k must be defned by the mechansm for the correspondng optmzaton problem, n order to produce the best result for each agent. After an teraton s done, the agent can modfy ths appeal functon or t can be automatcally done. Ether way, the mportance of ths method les on the fact that each agent s able to get the best from two solutons, dependng on the stuaton. Many authors have contrbuted to the mechansm proposed by Nssan, accordng to specfc applcatons and stuatons. Koutsoupas and Papadmtrou ntated nvestgatons on the coordnaton rato, whch s the relaton between the cost of * A Randomzed mechansm s a probablty dstrbuton over a famly of mechansms, all sharng the same set of strateges and posble outputs.

87 75 the worst possble Nash equlbrum and that of the socal optmum. Specfcally, they showed that for two dentcal machnes, the worst-case coordnaton rato s exactly 3/2. The task allocaton model they proposed s consdered a problem of allocatng schedulng tasks to machnes accordng to game theoretc assumptons. Each task s consdered a sngle entty and cannot be splt to be assgned n a part to dfferent machnes. In ths type of problem, the parameter to be nvestgated s that of the maxmum cost assocated wth any machne. It s smlar to a routng flow model where tasks are to be routed effcently usng a classcal makespan mnmzaton schedulng problem, each task can be assgned to a sngle machne but the decson to whch machne t s assgned to, s determned by the user s strategy. If the task s scheduled determnstcally to a machne n [m], then t s a pure strategy, but f t s allocated by some probablty dstrbuton t follows a mxed strategy. For dentcal tasks and machnes, a balance determnstc allocaton s consdered where each task s allocated to a machne ( mod m) + 1 n order to reach Nash Equlbrum. It has been shown n lterature that t s NP-hard to fnd the best and the worst pure Nash equlbra, but there exsts a polynomal tme algorthm that computed, for any gven task allocaton problem: a Nash equlbrum wth no hgher cost. Moreover, the exstence of a PTAS (polynomal-tme approxmaton scheme) for the problem of computng Nash equlbrum wth mnmum socal cost s demonstrated. Yet, all algorthms cted above have an undesrable property: they are centralzed and off-lne. Recently, there has been some research on non-centralzed algorthms for fndng Nash equlbra.

88 76 The problem s descrbed as a load balancng process, where each task s reallocated accordng to a selfsh rule, by defnng some strateges. It s assumed that the task s reallocated n a sngle step. For the dentcal machne case, Even- Dar 64, et al., found that f one moves the frst the maxmum weght task n a machne wth a mnmum load, then Nash equlbrum can be reached. Ther approach consders a game of many players (jobs) and actons (machnes) and studes ther asymptotc behavor. Durng ths game, jobs are allowed to select a machne to mnmze ther own cost. The cost that a job observes s determned by the load of the machne, whch s the sum of the weghts of the jobs runnng on t. Durng ths process, at least one job s wllng to change to another machne, untl Nash Equlbrum s reached. Only one job s allowed to move n each step and t s the centralzed controller that selects whch job wll move n the current tme step. The strategy used by the controller s the algorthm used to select whch of the computng jobs wll move. Snce all jobs behave selfshly, t s assumed that when a job mgrates, the observed load on the machne s strctly reduced, whch we refer to as a best-reply polcy, otherwse t s an mprovement polcy. In the case of dentcal machnes, they proved that f one moves the mnmum weghted task, the convergence may take place n exponental number of steps, otherwse, f one moves the maxmum weghted task and ths one follows the best reply polcy, Nash 64 EVEN-DAR, Eyal, KESSELMAN, Alex and MANSOUR, Yshay. Covergence Tme to Nash Equlbrum. Tel-Avv Unversty. (PDF document) p.18

89 77 equlbrum s reached n at most n steps. Ths s one of the reasons why t s mportant to choose the rght schedulng strategy. Recent works had been usng the mechansm of load balancng but wthout usng a centralzed control system. Berenbrnk 65, et al., n ther paper, Dstrbuted Selfsh Load Balancng, they dscussed a natural protocol for the agents whch was mplemented n a strongly dstrbuted settng, wthout any centralzed controller and wth good convergence propertes. In each round, the load of each task from the current machne was beng compared to that of a randomly chosen machne and f the observed load of the other machne was less than that of the current machne, then the job automatcally moved. The followng procedure shows the steps already descrbed: For each task b do n parallel Let b be the current resource of task b Choose resource j b unformly at random Let X b (t) be the current load of resource Let X jb (t) be the current load of resource j If X b (t) > X jb (t) + 1 then Move task b from resource b to j b wth probablty 1-X jb (t)/ X b (t) 65 BERENBRINK, Petra, et. al. Dstrbuted Selfsh Load Balancng. Canada, Natural Scences and Engneerng Research Councl of Canada. May 2, (PDF document) p.17

90 78 The advantage of ths protocol s that t s very smple and there s no need of global nformaton, tasks dd not even need to know the total number of tasks beng allocated, they only needed to know ther observed load on each of the resources.

91 79 3. OBJECTIVES 3.1. GENERAL OBJECTIVE To use the prncples establshed n Game Theory n order to obtan solutons for parallel machne schedulng problems under multple crtera and test ts effectveness n producton decson makng by comparng them to preexstng heurstcs and algorthms used under multcrtera schedulng. 3.2 SPECIFIC OBJECTIVES To understand the mportance of Game Theory n the context of productve systems programmng by determnng the sequence of jobs that can gve a quck and robust soluton to schedulng problems nvolvng parallel machnes, n order to generate alternate soluton sources for these types of confguratons. To prove whether Game Theory can approach schedulng problems under multple objectves, by establshng schedules through the swtchng of jobs accordng to trade-offs among ntellgent agents wthn the system. To establsh comparsons wth other heurstcs that do not use game theoretc prncples, n order to test the robustness of Game Theory n ths feld, through the contrast of solutons generated by those other heurstcs and the ones attaned by Game Theory.

92 80 4. SCOPE AND LIMITATIONS 4.1. SCOPE Allocatng resources n dynamc envronments s a very wde topc and can be approached from many dfferent perspectves. Ths nvestgaton s focused on parallel machnes related problems. There are many problems that arse from dfferent envronments up to ths, ncludng flexble flow shops. Lterature on flexble flow shops s based mostly on TOC approaches, thus these are almost always treated focusng on the bottlenecks. Wthn these bottlenecks further analyss can be made when realzng that a workng center may be consdered as a group of parallel machnes. Hence, ths research wll not focus on the whole flexble flow shop frame but on same based procedure statons parallel arranged. There are also several problems that arse from parallel machnes. Those nclude dentcal machnes, related machnes, and unrelated machnes. Ths nvestgaton wll not consder related and unrelated machnes focused problems. Its man scope wll be wthn the dentcal machnes consderaton. Another typcal classfcaton for these knds of problems wll be the consderaton of preempton and consderaton of dvdng a sngle job nto parts so they can be thoroughly processed once at a tme by splttng the job among all avalable

93 81 machnes. Lterature ncludes algorthms for preempton consderaton and also when preempton s not allowed. But as our approach s manly on tradeoffs, and job swtchng among machnes, we wll not be able to consder preempton. Ths s clearly a lmtaton snce multcrtera schedulng theory s able to mprove the man varables wthn a parallel machne envronment by allowng preempton on jobs, as seen n secton 2.11 (Lterature Revew on Parallel machne schedulng problems concernng completon tme and makespan for schedules when preempton s allowed) LIMITATIONS Assumptons of ndependent setup tmes, release tmes for all jobs consdered equal, not consderng due dates nor deadlnes, not takng nto account the unrelated machnes problems, lmt our research, however the hghest nvolved constran les n the complexty that schedulng problems may have, especally when they are NP hard, and lterature s not well llustrated by effectve allocaton algorthms. So our comparsons once we generate new schedules have to be lmted to a seres of data gathered from results found n the papers such as the papers by Gupta and Ho 66, and Archer 67, Amr Ronen 68, Noam NISAN GUTPA and Ho. Bcrtera optmzaton of the makespan and mean flow tme on two dentcal parallel machnes 67 ARCHER, Aaron, TARDOS Eva Truthful Mechansms for One-Parameter Agents

94 82 Another lmtaton s the solaton we mght gve to the perspectve of the problems, gven that the consdered or man focused approaches wll be made just for parallel machne confguratons. It s known that most systems have more than one staton, and could be consdered a flexble jobshop, however these mght brng specfc approaches, and thus these consderatons mght make feasble other nvestgatons that consder more general flexble jobshops. The convergence of game theoretc approaches to consder the problem of allocatng resources n a schedulng envronment s almost a new approach, drect contanng lterature mght not be entrely avalable; nevertheless, further papers and artcles have been publshed n the nternet, contanng these consderatons. As mentoned n the lterature revew, problems have been focused usng job agents, machne agents, onlne and offlne mechansms wth a centralzed or dstrbuted system. In many of these papers convergence to Nash equlbrum and comparsons were made wth other type of game theoretc solutons such as domnant equlbrum and Bayesan Nash equlbrum. Solutons n ths nvestgaton wll be lmted to Nash and domnated solutons. 68 RONEN, Amr. Solvng Optmzaton Problems Among Selfsh Agents. Jerusalem: Hebrew Unversty, (PDF document) 92 p. 69 NISAN, Noam. Algorthms for Selfsh Agents: Mechansm Desgn for Dstrbuted Computaton. Jerusalem: Insttute of Computer Scence. (PDF document) 17p.

95 83 A specfc lmtaton that may be present when allocatng jobs les n the fact that our model s a conservatve one, frst of all t s based on Nash Equlbrum, and secondly we allow jobs to move to the prevous poston, ths means t only consders trades of one tmeslot per job, so changes may not be as drastc.

96 84 5. HYPOTHESIS The followng hypotheses were dentfed accordng to the objectves and problem formulaton; of course all of these, wthn the lterature revew context SET OF HYPOTHESES INVESTIGATION HYPOTHESES The schedulng functon nteracton wth game theory approaches can lead to generate alternate solutons that may smplfy the decson makng process by reducng the complex sample space of schedules n parallel machne related confguratons and provde a range of solutons that belong to the Pareto Front, thus, represent an effectve combnaton of crtera for the sequencng of jobs. The Pareto Front set of ponts obtaned from the game theory approach complement those obtaned through the classcal multcrtera technques. NULL HYPOTHESES The schedulng functon nteractons wth game theoretc approaches do not lead to Pareto Front ponts that represent an effectve combnatoral crtera tradeoff that may smplfy the decson makng process, and thus do not generate alternate solutons.

97 85 The Pareto Front set of ponts obtaned from the game theory approach do not complement those obtaned through the classcal multcrtera technques. ALTERNATE HYPOTHESES The prncples of game theory wll lead to a Pareto Front set of ponts, yet they are not always the best ones due to the complex sample space of possble solutons, but can be consdered for future research. The Pareto Front set of ponts obtaned from the game theory approach represent a set that approxmates that obtaned through classcal multcrtera approaches but may dffer due to randomness nvolved n the process CONCEPT VARIABLES DEFINITION The man varables of the problem are obtaned from the set of hypothess, those nclude: SG T (Schedulng and game theory technque rules) schedulng and game theory nteracton technques allocaton rules for approachng multcrtera parallel machne problems.

98 86 MC T (Other multcrtera technque rules) allocaton rules for other dfferent approachng technques concernng multcrtera parallel machne problems. g Game theory tools. s Schedulng for parallel machne elements. P (Set of ponts n a Pareto Front) Set of ponts that belong to a Pareto Front whch represent robust solutons. MC P Set of Pareto ponts that result from multcrtera approachng technques. SG P Set of Pareto ponts that result from game theory approachng technques OPERATIONAL VARIABLES DEFINITION In order to determne how the hypotheses above are affected, t s necessary to establsh the dependence and nteracton among the varables descrbed, that s n terms of correlaton.

99 87 1. P MC P 4. P SG f T SG 2. T SG f g, s 5. P SG P 3. T MC f s 6. P MC f T MC 7. P SG P MC P Fgure 10. Man varables defnton. Reasons for these results: 1. The set of Pareto ponts that result from multcrtera approachng technques are subset of all ponts that represent all possble Pareto Front. 2. The schedulng and game theory technques depend hghly on what can be obtaned from game theory tools and schedulng parallel machne elements, ts nteracton produces the accordng convergence technque that wll eventually determne other varables as the set of Pareto ponts that result from multcrtera approachng technques. 3. The other multcrtera technques do not depend on game theory mplcaton, so t s just functon of the schedulng nteracton of elements. 4. The set of Pareto ponts that result from game theory approachng technques are a functon of schedulng and game theory nteracton technques. 5. The set of Pareto ponts that result from game theory and schedulng approachng technques are subset of all ponts that represent all possble Pareto Front.

100 88 6. The set of Pareto ponts that result from multcrtera approachng technques are a functon of other multcrtera technques approaches. 7. The unfed set composed of these two subsets does not represent the whole set of ponts possble n a Pareto Front, that s, nfntve ponts can not be found by nether of these approaches.

101 89 6. METHODOLOGY 6.1. METHODOLOGY APPROACH Parallel dentcal machne confguraton, when preempton s not allowed, sketches the research outlook and gves an ntegral scene of the systems ths research wll focus nto. Ths research s not consderng due date based jobs, nor deadlnes, and s assumng equal release dates for all jobs. It s also supposed that the system s not setup dependent, although ths project can be adapted to setup dependent envronments n further research. Dverse authors have focused on gvng solutons towards these knds of envronments; currently, consderatons on ths bass have made multcrtera a man focus snce such models respond more effectvely to the way problems arse n real productve systems. However, all these problems are very complex, and requre very specfc algorthms and heurstcs to evaluate trade-off among crtera, and how the ntegral performance of the sequence s determned. Game theory has turned to a focus towards these envronments applcatons. Several papers have been publshed but stll these approaches are mostly new. Lterature has focused on other means to tackle these problems.

102 90 Through out ths research a descrptve study wll be made, headng for the attempted hypotheses, n order to test them and thus corroborate the fact that dfferent perspectves can be used n order to undertake such problems, gvng rse to other means of attanng solutons to provde backup for decson makng. Also, dscoverng the correlaton among two felds; set of ponts resulted from pure multcrtera Schedulng technques and game theory-schedulng nteracton technques may lead to the fact that the last one s not ndependent of the classcal approach. By classfyng and dentfyng all elements of ths problem, and generatng a logcal model where solutons can be attaned accordng to the parameters prevously stated, a set of ponts can be obtaned as the outcome values of the varables that are the crtera that need to be mproved wthn the resultng schedules, an analyss and synthess method can be used n order to establsh whether the ponts obtaned are truly effcent by comparng the values of these varables wth the outcome of varables obtaned through other multcrtera models such as the heurstcs found n the paper Analytcal Evaluaton of Mult-Crtera Heurstcs 70. In ths paper the author states that an algorthm such as the General SB Routne / sumc, where ths routne conssts of adjustng the bottleneck; and SB stands for Shftng Bottleneck. Another paper key for our multcrtera comparson s Mnmzng Flow Tme subject to Optmal Makespan on Two Identcal Parallel 70 DANIELS Rchard L, Analytcal Evaluaton of Mult-Crtera Heurstcs.

103 91 Machnes 71 where they use Lexcographcal Search n order to obtan values for the makespan and flow tme on dentcal parallel machnes confguraton. The values found through ths routne wll also be compared wth the outcomes from our model. In order to corroborate the robustness of our model we wll also desgn an experment n whch we wll run the model under two dfferent setups, and varyng two dfferent factors wthn t: the number of jobs to be processed and the number of machnes avalable. The two dfferent setups wll be: Agents usng ncentves to pursue them to make changes n the actual schedule, and Agents not usng ncentves on the equatons. By ths analytcal experment we wll try to confrm how solutons under ncentves may brng more nterestng solutons and acheve the frst of our specfc objectve stated whch was to prove that our model ndeed generates alternatve schedules and as stated on the set of hypotheses, nterestng solutons for schedulng problems. Secondary sources taken nto account nclude, books n schedulng theory, game theory, multcrtera schedulng theory and also publshed papers concernng the applcatons and nteracton between game theory. These papers were bascally the drver for our proposed model, ncludng the mechansm desgn 72, Aucton 71 GUTPA, Jatnder N.D; and HO, Johnny C. Mnmzng Flow Tme subject to Optmal Makespan on Two Identcal Parallel Machnes 72 KUTANOGLU, Erhan and WU, Davd. An Incentve Compatble Mechansm for Dstrbuted Resource Plannng.

104 92 Theoretc Modelng 73, Worst case Equlbra 74 and KUTANOGLU s ncentve desgn schedulng that wll manly be obtaned from the nternet as well as from specalzed journals n the data base resources. Another document that was specally mportant for prescrbng our model was Even-Dar s 75 paper where he proposed n jobs wth an assocated agent, over m machnes, and jobs were allowed to select a machne to mnmze ther own cost, ths cost was determned by the load on the machne, whch was the sum of the weghts of the jobs runnng on t. It s stated that at least one job s wllng to change to another machne, untl Nash Equlbrum s reached. In ths paper t s also assumed that only one job s allowed to be moved n each step, whch s slghtly dfferent from whch we want to propose wthn our model. It s also stated that there s a general controller of the entre system who s n charge of allowng or not a movement of a job from one machne to other. The platform under whch ths model s based s Vsual Basc macros for Excel, where a model s presented accordng to the assumptons stated, and the steps that needed to be followed were tagged wthn the algorthm, n order to fnd an allocaton schedule that would allow the testng of the hypotheses. Once a set of schedules s found, the correspondng results need to be fltered to dsregard 73 KUTANOGLU, Erhan and WU, S. Davd. An Aucton-Theoretc Modelng of Producton Schedulng to Acheve Dstrbuted Decson Makng p 74 KOUTSOUPIAS Elas, and PAPADIMITRIOU Worst-case Equlbra. Document n PDF. Berkley Unversty and UCLA. 10p. 75 EYAL Even-Dar, KEESELMAN Alex, and YISHAY Mansour Convergence Tme to Nash Equlbra. School of Computer Scence. Tel Avv Unversty. March PDF Document.p17.

105 93 domnated ponts and fnally obtan a pont or a set of ponts to conform part of the Pareto Effcent Front obtaned from our approach SOLUTION METHODOLOGY In order to test the project s hypothess, that s, to fnd solutons for parallel machne schedulng through the means of Game Theory, a smulaton of a schedulng game wll be made. From ths smulaton, several schedules wll be attaned. Those schedules wll have the output for our research, snce dfferent systems varables can be analyzed. Snce two of these varables are conflctve, makespan and flow tme, a schedulng problem consderng both of these crtera can be solved by usng the mechansm desgn approach, see ARCHER, Aaron 76, where jobs are consdered agents, also see 77, and 1 KUTANOGLU, Erhan and WU, Davd 78 ; on the other hand Nsan also states the usefulness of a mechansm desgn for selfsh agents 79. Our model contans a dfferent type of nteracton and a modfcaton n the payoffs mplemented. Each job agent wll be playng wth a central agent n a sequental order. Each type of agent has an objectve that can nteract ratonally and lead overall system effcency, n order to meet both crtera through tradeoffs wthn a payoff matrx. 76 ARCHER, Aaron, TARDOS Eva Truthful Mechansms for One-Parameter Agents 77 KUTANOGLU, Erhan and WU, Davd. An Incentve Compatble Mechansm for Dstrbuted Resource Plannng. 78 RONEN, Amr. Solvng Optmzaton Problems Among Selfsh Agents. Jerusalem: Hebrew Unversty, NISAN, Noam. Algorthms for Selfsh Agents: Mechansm Desgn for Dstrbuted Computaton. Jerusalem: Insttute of Computer Scence. (PDF document) 17p

106 Elements and Assumptons The elements consdered n the game are as follow: Agent 0 (A 0 ): Ths agent s n charge of the overall system, t s the mechansm controller, ts nterest s to fnsh all jobs as fast as possble on the set of machnes, that s to mnmze the maxmum makespan. Job Agents (A ): There wll be an agent that wll look after each job; ts man nterest wll be to seze the correspondng job as soon as possble; there are as many job agents as jobs n the system. Job agents are not wllng to wat too long for ts job to be processed; they want ther job out of the system so they do not want to wat n queue. Ths s why most job agents wll have conflctve objectves wth Agent 0, snce ths last one would rather have jobs wth larger processng tmes allocated frst on each machne so that the whole system wll fnsh up n the least amount of tme, resultng n the mnmum completon tme of the last job. These job agents also want ther jobs to move ahead n the schedule, so they can be processed frst and so, be taken out of the system. Partal flow tme * wll gve an ndcator of how long does each specfc job on a partcular machne on each tme slot wll have to wat, so each agent wll want the partal flow tme to be as short as possble. Set of Machnes (M j ): The system wll be consdered to have n dentcal parallel machnes, j=1, 2, n. Jobs wll be allocated on the machnes accordng * Flow tme on the machne untl a correspondng job, that s flow tme untl a specfc tme slot on a partcular machne.

107 95 to what Agent 0 beleves s best for the overall system takng nto account what the job agent wll do, snce both are ntellgent and both know that they are actng ratonally and wll choose ther strategy accordng to what the other thnks the other wll do. Strateges for job agents (S ): Each job agent wll have a correspondng strategy vector S = {A, B}, accordng to ts selfsh nature and thus, wll have an nterest to act upon t, snce gong for ths strategy s what benefts hm the most. Strateges for Agent 0 (S 0 ): The controllng agent, A 0, wll have a correspondng vector S 0 = {C, D}, accordng to the overall system effcency. Payoff Matrx (P 2x2 ): Space matrx where decsons on the partcpatng agents are made, accordng to trade-offs among crtera. Payoffs (a kl ): Assocated cost that an agent wll face f he chooses a specfed strategy k, gven that the opponent agent chooses another strategy l. Thus, each agent wll want to mnmze the assocated cost n ts payoff matrx. A payoff matrx wll generate changes n the allocaton of jobs n the machnes, accordng to the preferences of the agents and assocated costs referrng thus, wll allow changes and teratons among jobs n the machnes. Throughout these teratons, new payoff matrces wll be generated and new swaps n jobs wll be made untl the model can not accomplsh better solutons, that s, the termnaton condton for the teratons wthn the model.

108 96 Assumptons: In order to smulate a schedulng game on two parallel machnes wth a central agent or controller, A 0, and a set of job agents, A (one agent per job ), t s necessary to take a set of assumptons under consderaton: Accordng to the premses of game theory, agents are assumed to act ratonally and, thus, they wll choose ther strategy accordng to what the other thnks the other wll do. Jobs can only move from one machne to another because of the dfference n load of the machnes, n order to try to balance the system, allowng t to move only from the machne wth the largest load to some other machne randomly pcked wth a lower load. Jobs are allowed to move to the same poston n the same machne only once. If A0 s tryng to move a job allocated on a tmeslot where the other randomly chosen machne does not have a correspondng job assgned on the same tmeslot; that s, the machne wth hghest load has more jobs assgned than the other, then the swappng wll be done wth the job selected, along wth the empty tme slot avalable on the other machne. Job agents only have knowledge of ther partal flow tme after each possble move and the overall total flow tme of the system, they do not know the processng tme of other jobs.

109 97 A job that s postoned n the frst tmeslot s not motvated to partcpate at all n the game, snce t has ts lowest possble Partal Flow tme, but t can be moved, f other jobs are swapped to ts poston Defnton of the Game Consder a non-cooperatve repeated game of two players, where player one represents a job agent and player 2, the controllng agent; agent 0. Each one has two strateges that correspond to ther own nterests, whch are, for the job agent, to seze hs job frst, whch leads to an mprovement of hs assocated partal flow tme, and for agent 0, to reach a better makespan. Regardng the system mechansm to reduce not only maxmum makespan but also total flow tme, t s well known that each player wthn the game wll play by assumng some costs mposed by the system n order to allow the condtons acquanted to be reached; e.g., the flow tme of the system s mproved when a job wth a lower processng tme s allocated n a prevous poston where the pror job before had a longer processng tme. In ths means, the model must penalze movements accordng to what represents a system mprovement rather than a selfsh mprovement (what the job agent wants). Strateges for job agent, S 1 = { A,B }: A= Stay on the current poston (tme slot). B= Move to the prevous avalable poston.

110 98 Strateges for agent 0, S 2 ={ C, D }: C= Leave job on the current machne. D= Move the job to another machne. The payoffs on the players wll be represented as costs, so the nvolved players wll want to mnmze the assocated costs, n tme unts. These costs are functons of the makespan and flow tme, whch are the two conflctve crtera. The assumptons and the objectves for both players enable the desgn for the payoffs n the correspondng way, each par of chosen strateges represent an outcome cost (OC). There are four costs for each agent, creatng a non-zero sum game matrx Costs Costs For each Job agent n the system: a 11 OC assocated when A decdes not to move to the prevous avalable poston and A 0 decdes for the job to stay n the current machne, no change s done over the schedule. a 12 OC assocated when A decdes not to move to the prevous avalable poston but A 0 decdes to move the job to another machne. a 21 OC assocated when A decdes to move to the prevous avalable poston and A 0 decdes for the job to stay n the current machne.

111 99 a 22 OC assocated when A decdes to move to the prevous avalable poston and A 0 decdes for the job to move to another machne. Costs For Agent 0: a0 11 OC assocated when A 0 decdes for the job to stay n the current machne and A decdes not to move to the prevous avalable poston. a0 12 OC assocated when A 0 decdes to move the job to another machne assocated but A decdes not to move to the prevous avalable poston. a0 21 OC assocated when A 0 decdes for the job to stay n the current machne assocated and A decdes to move the job a prevous avalable poston. a0 22 OC assocated when A 0 decdes for the job to move to another machne and A decdes to move to the prevous avalable poston. These OC s * represent dfferent costs for each agent makng the game non-zero sum. The costs assocated are functons of the maxmum makespan, the partal completon tme **, partal flow tme *** and total flow tme. So the job agent wll be better off, f hs assocated partal flow tme were smaller, snce he wll have to wat less to be sezed. On the other hand, for agent 0, the cost s represented by the assocated makespan under gven condtons that have to do wth hs decson of swappng one job from one machne to the other. * OC: Outcome Costs ** Partal Completon tme represents the amount of tme untl a set of jobs are processed, untl the last job that s beng takng under account. *** A partal flow tme tself represents the amount of tme the job has to wat for t to be sezed, that s the flow tme that represents all jobs before t.

112 100 If the job agent has decded for the job to stay on ts place, then hs outcome depends on what agent 0 decdes for t to do, to swap t or not to swap t. Equatons Defntons for the costs wthn the game matrx Job Agent The assocated costs functons for the job agent depend manly on flow tme varables, snce ths s what major concerns the agent. So there s an ncentve for hm to move forward n the schedule, but as prevously stated there s the need wthn the model to place a payment over the opportunty loss the system wll face f the gven job s moved or f t s not. In order to acheve all those stated condtons the proposed equaton s as follows: (1) Job Cost TotFT Fm As the job agent s tryng to mnmze ths cost, he would want the second term n the equaton to be as hgh as possble n order to decrease the value of the job cost. Equaton Term by Term: Tot FT represents the total flow tme assocated from the system, t ndcates the maxmum value f the rest of the equaton tends to zero, that s, f the movement s the least convenent, then ths term wll represent almost all the cost; the cost assocated to a non convenent strategy, whch n terms of job agents and flow

113 101 tme s, to allocate a job that has a greater processng tme compared to the one that was formerly on that tmeslot. F job represents the assocated flow tme for that machne, that s the machne where was fnally allocated. n terms of agents t represents the fracton of the processng tme of the job on the tme slot where agent0 decded to move or leave, wth respect to the partal flow tme for that job on the analyzed machne and poston. Together wth F job, they represent the opportunty cost, that s, what the system and the job agent gve up n order for the movement assocated from the swap among machnes, or movement ahead n the schedule nto a prevous tmeslot to take place. Assumng that the agent decdes to move ahead to a prevous tmeslot, then not makng the movement represents the opportunty lost, and ths s taken nto account by consderng the processng tme of the job that was formerly n the prevous slot. If J 1 s pcked by the system to play the game of swtchng allocatons then the assocated job agent would want to swtch to the tme slot where J s. It s convenent for the system f and only f P 1 P. Fgure 11. Job agent s decsón to swtch. Ths way t wll be an mprovement for the overall flow tme. If ths condton s not true, then the system must penalze the job agent n order to encourage hm not to

114 102 move from hs frst poston. As these parameters depend on P 1 and P, then the rato of the ncentve can be defned as the rato between P (processng tme of job that was chosen, and the partal flow tme assocated wth a specfc machne, the current one where the job s, n case agent 0 * decdes for the job to stay on the same machne, or the second machne, n case agent 0 decdes for the job to move to an alternate machne. So ths assocated flow tme on ths machne s taken nto account n order to obtan as: PartalFT P job_ selected on_ machne _ selected _ by _ agent0 Ths rato shows a relatonshp for how much of the flow tme s absorbed by the job chosen, up to where t s located, snce t s partal. The smaller the processng tme, the less t wll cost to agent 0 whch s the controller. Notce that (1) mples a negatve sgn; that s, that agents wll receve a bonus for ther job dependng on the partal flow tme on the correspondng machne and the processng tme. Whenever they gan, agent 0 loses. That way the system balances tself. ** Ths way no better completon tme can be acheved, then flow tme can be compensated, so overall performance can be acheved. * Recall that ths game s based on suppostons as how the other player can react, snce both players are assumed. ** Mechansm desgned s based on ncentves whch mply losses for one sde and gans for others n order to mantan balance.

115 103 Agent 0: The assocated costs functons for Agent 0 manly depend on the completon tme, snce ths s of major concern for hm. So there s an ncentve for hm not to allow low processng tmes frst, snce hs nterest s to fnsh all jobs as soon as possble. In order to acheve all those stated condtons, and as sad before, balance both sdes of the game, then the ncentves gven to the job agents have to be pad by the agent controller, and then the proposed equaton follows wth a postve sgn, n other words ths s an opportunty cost agent 0 wll face to compensate the job agents: (2) Agent 0 Cost Cmax 0 C As Agent 0 s tryng to mnmze ts cost he would want to choose ts cost as low as possble. Equaton Term by Term: C max s the maxmum completon tme calculated for the machne wth the hghest load on the set, whch agent zero wants to have as low as possble. 0 for agent 0 t represents the opportunty cost assocated to the movement that s about to be done. As agent 0 s concerned about the machnes and not one sngle job on them, ths wll consder rato of partal completon tmes. More specfcally t compares the partal completon tme for the job that has been selected to the partal completon tme on the second machne, for the same 0 stands for agent 0, whle stands for the job agent.

116 104 assocated tmeslot. If <1, then the second machne has a hgher partal completon tme for that same tmeslot. For ths reason a swap among jobs can be worth the cost. On the other hand f >1, then a hgher cost for agent 0 s mpled, snce the partal completon tme on the second machne s lower than the partal completon tme on the current machne analyzed. Of course, ths does not guarantee that a swap between jobs may mprove total makespan. There may be tmes when even wth >1, the swap results n a better Schedule. Stll the model needs to penalze these costs n a sound way, such as the one chosen. 0 C C current machne other machne Fgure12. Analyzng for the agent as an opportunty cost of affectng overall flow tme

117 105 By assumng ths cost and addng t to the equaton of the assocated costs for the job agent, a coherent functon wll be declared, regardng how much wll be taken nto account about the other parameter C snce ths would now mean the percentage that the model wll take from ths other parameter. C ths parameter concerns the completon tme for the chosen job wthn the machne that contans the chosen job, dependng on the gven condtons of whether the job stays on the same poston or f t changes to another machne. So f t changes, the completon tme accounted wll be where the job fnally gets to. The two sceneraros can be represented as follows: 1. Fgure 13. Agent 0 decdes to swtch the job to the other machne affectng Cj of second machne

118 Fgure 14. Agent 0 decdes not to swtch job, from the machne concerned, t wll stay that way untl another condton s reached. As the costs concernng the job agent depend on the decsons made by agent 0, and vce versa, an extensve form of the game can be analyzed and ths vsualzes the whole perspectve n order to obtan the stated solutons n terms of the decsons made by the other player (to be thnkng ratonally about what the other agent s thnkng). An extensve perspectve can be appled to show how the decsons among agents react for the duraton of the game by the followng tree:

119 107 Fgure 15. Decson Tree for Job Agent In Fgure 15, the job agent wll enter the game, knowng already what agent 0 mght do, so the gray part represents gven condtons where the job agent has no control.

120 108 Splt Out Decsons for Job Agent: 1. If Job agent decdes, gven that Agent 0 has decded not to move that job nto another machne, to stay on the current machne and assume a cost of OC= (1)-(2)*(3), accordng to the numbers gven n the tree to dentfy all elements wthn the outcome costs, for any decson made. 2. If Job Agent decdes to stay n the current poston gven that Agent 0 has decded to swtch jobs the outcome costs would be: OC= (4) - (5)*(6). 3. If Job Agent decdes to move to a prevous poston gven that Agent 0 has decded not to move to another machne, then the OC= (7) - (8)*(9). 4. If Job Agent decdes to move to a prevous poston gven that Agent 0 has decded to move the job to another machne, then OC = (10) (11)*(12). Same analyss can be made for Agent 0 n Fgure 16, whch shows how agent 0 wll enter the game, knowng already what the correspondng job agent mght do, so the gray part represents gven condtons where agent 0 has no control of.

121 109 Fgure 16. Decson Tree for Agent 0 Splt Out Decsons for Agent 0: 1. If agent 0 decdes, gven that the job agent has decded not to move the job to a prevous poston, to let the job stay on the current machne, then he would assume

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