Multicriteria Decision Making

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1 Multcrtera Decson Makng Andrés Ramos Pedro Sánchez Sonja Wogrn

2 Contents 1. Basc concepts 2. Contnuous methods 3. Dscrete methods Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 2

3 Basc concepts Contnuous methods Dscrete methods 1 Basc concepts

4 Decson theory Decson: choosng the best of possble To state best and possble Possble solutons or feasble decsons: Fnte set: alternatves can be enumerated Contnuous set: alternatves are defned through constrants The Best: One crteron (classc Optmzaton and classc Decson Theory) Multple crtera o multple decson makers (Game Theory and Multcrtera Decson Makng) Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 4

5 Whch s the best anmal of the nature n runnng, flyng and swmmng smultaneously? The fastest runner? Cheetah s the fastest runnng anmal n the world The fastest flyng? Peregrne falcon s the fastest brd n the world The fastest swmmer? Salfsh s the fastest fsh n the world Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 5

6 And the wnner s the DUCK Is able to run although less than the cheetah Is able to fly although less than the peregrne falcon Is able to swm although less than the salfsh Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 6

7 Methods Contnuous decsons Multobjectve optmzaton Compromse programmng Satsfyng methods (goals) Dscrete decsons Analytcal Herarchc Process (AHP) Outrankng methods (Electre, Promethee,...) General statement: opt z = ( z ( x),..., z ( x)) x F n F : feasble decson space (contnuous: feasble regon, F ) p z( F) : crteron space (numercal crtera: z( F) ) 1 p Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 7

8 Basc concepts Attrbute: observable "value" (measurable) of an alternatve, ndependent of the decson maker Objectve: drecton to mprove an attrbute (max. o mn. f numercal; otherwse, preferences) Target: an acceptable level of achevement for an attrbute Goal: combnaton of an attrbute wth ts target Crteron: relevant attrbutes, objectves or goals to a decson problem Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 8

9 Soluton concept Effcency or Pareto optmalty crteron One feasble soluton s effcent or Pareto optmal f no other feasble soluton can yeld an mprovement n one objectve wthout causng a degradaton n at least another objectve Domnated or non effcent alternatve: there s another one wth better attrbutes max ( z, z ) Crtera space Effcent set or Pareto fronter Domnated alternatves 1 2 z 2 Numercal attrbutes, objectves maxmsaton Effcent set: { x F : / x F wth zk ( x ) zk ( x) k and t { 1,..., p} wth zt ( x ) zt ( x) } ε = > Best compromse soluton: effcent soluton chosen by the decson maker z 1 Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 9

10 Basc concepts Contnuous methods Dscrete methods 2 Contnuous methods

11 Multobjectve optmzaton max z = ( z ( x),..., z ( x)) x F Pay-off matrx: optmum value for an objectve (wthout the others), and values of the other objectves for ths soluton. Conflct level E A 1 D B 2 Qualty 9 11 p C z ( x): mathematcal expresson for attrbute x F : decsonal varables vector : constrants defnng feasble solutons Trade-offs: degradaton of one objectve to mprove another one n one unt ("cost" of one objectve n terms of another one). Effcent set slopes Pay-off matrx: Qualty Trade-offs: ' ' = = ' ' = = T A B T B C Cost Qualty Cost Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 11

12 Multobjectve optmzaton Weghted-Sum Method (Zadeh, 1963) Multplyng any objectve functon wth a weght or non negatve factor and addng n a sngle composte objectve functon Parametrc programmng (changng weghts the effcent set s obtaned) λ > 0 p max λ z ( x) = 1 P( λ) x F λ 0 Theorem: If then any optmal soluton of P(λ) s effcent. Converse of theorem s true under some assumptons (convexty, lnearty) Normalzed crtera (unts) Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 12

13 Multobjectve optmzaton Epslon-Constrant Method Optmsaton of one objectve ncludng the rest of objectves as parametrc constrants Parametrc programmng (changng rght hand sde, effcent set) max zl ( x) x F Pl ( ε) zk ( x) εk k = 1,..., l 1, l + 1,..., p Theorem: If there s only one soluton, t s effcent. Theorem: If x * s effcent, l ε such thatx * k s optmum of P( ε) Multcrteron smplex method (Zeleny, 1974) Only for lnear objectves and constrants Each teraton, check effcency of solutons (extreme ponts or vertces) All the effcent vertces are obtaned Effcent set: lnear combnaton of adjacent vertces Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 13 l

14 Compromse programmng Ideal pont: optmum values of each objectve subject to problem constrants * * * * * * z = ( z1,..., z,..., zp) z = max z ( x) z : anchor value x F Optmum element or best-compromse soluton: effcent soluton closest to the deal pont (Zeleny s axom of choce, 1976) Degree of closeness between attrbute -th and ts anchor value, normalzed * z z ( x) d ( x) = z z * mn L x F π z * * ant-deal or nadr pont (worst value crtera n effcent set) p * π z z ( x) = w * = 1 z z * Weghts, mportance of dscrepancy of crtera (subjectve orderng, Saaty,...) π= Tchebychev or mnmsng maxmum dstance (lnear) π=1 lneal addton of weghted dstances (lnear) π [ ] L L Compromse set: varyng π (usually ) 1, 1/ π z 2 z 1 Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 14

15 Goal programmng Satsfyng logc (Smon, 1955): decson-makng behavour where, nstead of attemptng to optmze system performance, a level of aspraton s set ether subjectvely or heurstcally and no further effort s expended to exceed that level of performance Goal programmng (Charnes y Cooper(61), Lee (72) e Ignzo (76)) Attrbute mathematcal expresson: z ( x) Target or level of aspraton: acceptable level of achevement Goal: z ( x) zˆ Wth devaton varables: Devaton varables to be mnmzed: f goal s "at least" one value, ; f t s "at most", Model (f at least): z ( x) + n p = zˆ n p mn n x F goal constrants = 1 p zˆ Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 15

16 Goal programmng: varatons Weghted GoalpProgrammng: mn ( α n + β p ) = 1 x F n 0, p 0 = 1,..., p z ( x) + n p = zˆ = 1,..., p MINIMAX or Tchebychev Goal Programmng: (balanced soluton) mn D α n + β p D = 1,..., p z ( x) + n p = zˆ = 1,..., p x F í (dvdng by the target: %) n 0, p 0 = 1,..., p Lexcographc Goal Programmng: Prorty levels goals (pre-emptve). To solve: sequentally each level, keepng values of unwanted devaton varables prevously acheved Lex mn a= [ g ( n, p) + g ( n, p), g ( n, p), g ( n, p) + g ( n, p) + g ( n, p) ] í Departamento de Organzacón Industral Escuela Técnca Superor de Ingenería ICAI Multcrtera Decson Makng 16

17 Andrés Ramos Pedro Sánchez Sonja Wogrn Departamento de Organzacón Industral Alberto Agulera, Madrd, Span Tel Fax nfo-do@do.ca.upcomllas.es

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