Multiobjective De Novo Linear Programming *

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1 Acta Unv. Palack. Olomuc., Fac. rer. nat., Mathematca 50, 2 (2011) Multobjectve De Novo Lnear Programmng * Petr FIALA Unversty of Economcs, W. Churchll Sq. 4, Prague 3, Czech Republc e-mal: pfala@vse.cz Dedcated to Lubomír Kubáček on the occason of hs 80th brthday (Receved March 31, 2011) Abstract Mathematcal programmng under multple objectves has emerged as a powerful tool to assst n the process of searchng for decsons whch best satsfy a multtude of conflctng objectves. In multobjectve lnear programmng problems t s usually mpossble to optmze all objectves n a gven system. Trade-offs are propertes of nadequately desgned system a thus can be elmnated through desgnng better one. Multobjectve De Novo lnear programmng s problem for desgnng optmal system by reshapng the feasble set. The paper presents approaches for solvng the MODNLP problem, extensons of the problem, examples, and applcatons. Key words: De Novo programmng, multple objectves, lnear programmng, trade-offs 2010 Mathematcs Subject Classfcaton: 90C29 1 Introducton Tradtonal concepts of optmalty focus on valuaton of already gven systems. New concept of desgnng optmal systems s appled (Zeleny [7]). Multobjectve lnear programmng (MOLP) s a model of optmzng a gven system by multple objectves. In MOLP problems t s usually mpossble to optmze all objectves together n a gven system. Trade-off means that one cannot ncrease the level of satsfacton for an objectve wthout decreasng ths for another objectve. Trade-offs are propertes of nadequately desgned system and * The research project was supported by Grant No. P402/10/0197 Revenue management models and analyses from the Grant Agency of the Czech Republc and by Grant No. IGA F4/16/2011, Faculty of Informatcs and Statstcs, Unversty of Economcs, Prague. 29

2 30 Petr Fala thus can be elmnated through desgnng better one. The purpose s not to measure and evaluate tradeoffs, but to mnmze or even elmnate them. An optmal system should be tradeoff-free. As a methodology of optmal system desgn can be employed De Novo programmng for reshapng feasble sets n lnear systems. De Novo concept was ntroduced by Mlan Zelený (see [5]). Basc concepts of the De Novo optmzaton are summarzed. The paper presents approaches for solvng the Mult-objectve De Novo lnear programmng (MODNLP) problem, possble extensons, methodologcal and real applcatons, and an llustratve example. The approach s based on reformulaton of MOLP problem by gven prces of resources and the gven budget. Searchng for meta-optmum wth a mnmal budget s used. The nstrument of optmum-path rato s used for achevng the best performance for a gven budget. Searchng for a better portfolo of resources leads to a contnuous reconfguraton and reshapng of systems boundares. Innovatons brng mprovements to the desred objectves and the better utlzaton of avalable resources. 2 Mult-objectve lnear programmng problem Mult-objectve lnear programmng (MOLP) problem can be descrbed as follows Max z = Cx s.t. Ax b (1) where C s a (k, n)-matrx of objectve coeffcents, A s a (m, n)-matrx of structural coeffcents, b s an m-vector of known resource restrctons, x s an n-vector of decson varables. In MOLP problems t s usually mpossble to optmze all objectves n a gven system. Trade-off means that one cannot ncrease the level of satsfacton for an objectve wthout decreasng ths for another objectve. For mult-objectve programmng problems the concept of non-domnated solutons s used. (see for example Fala [2]). A compromse soluton s selected from the set of non-domnated solutons. Two subjects Decson Maker and Analyst are ntroduced. Classfcaton of methods for soluton of MOLP problems accordng to nformaton mode: Methods wth a pror nformaton. Decson Maker provdes global preference nformaton (weghts, utlty, goal values,...). Analyst solves a sngle objectve problem. Methods wth progressve nformaton nteractve methods. Decson Maker provdes local preference nformaton. Analyst solves local problems and provdes current solutons. Methods wth a posteror nformaton. Analyst provdes a non-domnated set. Decson Maker provdes global preference nformaton on the non-domnated set. Analyst solves a sngle objectve problem.

3 Multobjectve De Novo lnear programmng 31 There are proposed many methods from these categores. Most of the methods are based on trade-offs. The next part s devoted to the trade-off free approach. 3 Mult-objectve De Novo lnear programmng problem Mult-objectve De Novo lnear programmng (MODNLP) s problem for desgnng optmal system by reshapng the feasble set. By gven prces of resources and the gven budget the MOLP problem (1) s reformulated n the MODNLP problem (2) Max z = Cx s.t. Ax b 0 (2) pb B where b s an m-vector of unknown resource restrctons, p s an m-vector of resource prces, and B s the gven total avalable budget. From (2) follows pax pb B. Defnng n-vector of unt cost v = pa we can rewrte problem (2) as Solvng sngle objectve problems Max z = Cx s.t. vx B (3) Max z = C x =1, 2,...,k s.t. vx B (4) z s k-vector of objectve values for the deal system wth respect to B. The problems (4) are contnuous knapsack problems, the solutons are { 0 j x j = j B/v j j = j where j {j (1,...,n) max j (c j /v j)}. The meta-optmum problem can be formulated as follows Mn f = vx s.t. Cx z (5)

4 32 Petr Fala Solvng problem (5) provdes soluton: x B = vx b = Ax The value B dentfes the mnmum budget to acheve z through soluton x and b. 4 Optmum-path ratos The gven budget level B B. The optmumpath rato for achevng the best performance for a gven budget B s defned as r 1 = B B. The optmum-path rato provdes an effectve and fast tool for the effcent optmal redesgn of large-scale lnear systems. Optmal system desgn for the budget B: x = r 1 x, b = r 1 b, z = r 1 z If the number of crtera k s less than that of varables n, we can ndvdually solve the problem and obtan synthetc solutons. Sh [4] defned the synthetc optmal soluton as follows x =(x 1 j 1,...,x k j k, 0,...,0) R n,wherex q j q s the optmal soluton of (4). For the synthetc optmal soluton a budget s used. There s possble defne sx types of optmum-path ratos (Sh [4]): r 1 = B B, r 2 = B B, r 2 = B B, r 4 = λ B j, r 5 = λ B j B B, r 6 = λ B j B. Optmum-path ratos are dfferent. There s possble to establsh dfferent optmal system desgn as optons for decson maker. 5 Extensons There are extenson possbltes of De Novo programmng (DNP): Fuzzy DNP. Interval DNP. Complex types of objectve functons. Contnuous nnovatons. Fuzzy De Novo Programmng (FDNP) uses nstruments as fuzzy parameters, fuzzy goals, fuzzy relatons, and fuzzy approaches (L and Lee [3]). Inexact De Novo programmng (IDNP) ncorporates the nterval programmng and de Novo programmng, allowng uncertantes represented as ntervals

5 Multobjectve De Novo lnear programmng 33 wthn the optmzaton framework. The IDNP approach has the advantages n constructng optmal system desgn va an deal system by ntroducng the flexblty toward the avalable resources n the system constrants (Zhang et al. [8]). Complex types of objectve functons are defned. The mult-objectve form of Max (cx - pb) appears to be the rght functon to be maxmzed n a globally compettve economy (Zeleny [6]). Searchng for a better portfolo of resources leads to contnuous reconfguraton and reshapng of systems boundares. Innovatons brng mprovements to the desred objectves and the better utlzaton of avalable resources. The technologcal nnovaton matrx T =(t j ) s ntroduced. The elements n the structural matrx A should be reduced by technologcal progress. T should be contnuously explored. The problem (2) s reformulated n to nnovaton MODNLP problem (6) Max z = Cx s.t. TAx b 0 (6) pb B The mult-objectve optmzaton can be then seen as a dynamc process n three tme horzons: 1. short term equlbrum: trade-off, operatonal thnkng. 2. md term equlbrum: trade-off free, tactcal thnkng. 3. long term equlbrum: beyond trade-off free, strategc thnkng. 6 Applcatons The tradeoffs-free decson makng has a sgnfcant number of methodologcal applcatons. All such applcatons have the tradeoffs-free alternatve n common: Compromse programmng mnmze dstance from the deal pont. Rsk management portfolo selecton tradeoffs between nvestment returns and nvestment rsk. Game theory wn-wn solutons. Added value value for the producer and value for the customer both must beneft. There are real applcatons of De Novo approach. For example producton plan for a real producton system s defned takng nto account fnancal constrants and gven objectve functons (Babc and Pavc [1]). The paper

6 34 Petr Fala (Zhang et al. [8]) presents an Inexact DNP approach for the desgn of optmal water-resources-management systems under uncertanty. Optmal supples of good-qualty water are obtaned n consderng dfferent revenue targets of muncpalndustralagrcultural competton under a gven budget. 7 Illustratve example The MOLP problem s formulated: Max z 1 = x 1 + x 2 Max z 2 = x 1 +4x 2 3x 1 +4x 2 60, x 1 +3x 2 30, x 1 0, x 2 0. The MODNLP problem s formulated: Input: p =(0.5, 0.4) B =42, unt costs v = pa =(1.9, 3.2). Max z = C x =1, 2,...,k z 1 =22.11, z 2 =52.50, s.t. vx B r 1 = B B =0.761 Mn f = vx x 1 =11.98, x 2 =10.13 s.t. Cx z B = vx =55.17 b = Ax, b 1 =76.48, b 2 =42.39 Optmal system desgn for B: x = r 1 x, b = r 1 b, z = r 1 z, x 1 =9.12, x 2 =7.71, b 1 =58.23, b 2 =32.25, z 1 =16.83, z 2 = The nnovaton MODNLP problem s formulated: Input: p =(0.5, 0.4) B =42, technologcal nnovaton matrx T = unt costs v = pt A =(1.48, 2.44), z 1 =28.38, z 2 =68.85, x 1 =14.89, x 2 =13.49, B = vx =54.95, r 1 =0.764, x 1 =11.38, x 2 =10.31, z 1 =21.69, z 2 = [ ] 0.8 0, 0 0.7

7 Multobjectve De Novo lnear programmng 35 The solutons n dfferent tme horzons are represented n Fg. 1. z Conclusons z 1 Fg. 1. Solutons for the llustratve example Tradtonal concepts of optmalty focus on valuaton of already gven system. New concepts of optmalty are orented on desgnng optmal systems. The purpose s not to measure and evaluate tradeoffs, but to mnmze or even elmnate them. An optmal system should be tradeoff-free. De Novo programmng s used as a methodology of optmal system desgn for reshapng feasble sets n lnear systems. MOLP problem s reformulated by gven prces of resources and the gven budget. Searchng for a better portfolo of resources leads to a contnuous reconfguraton and reshapng of systems boundares. Innovatons brng mprovements to the desred objectves and the better utlzaton of avalable resources. These changes can lead to beyond tradeoff-free solutons. Multobjectve optmzaton can be taken as a dynamc process. De Novo programmng approach s open for further extensons and applcatons. References [1] Babc, Z., Pavc, I.: Multcrteral producton plannng by De Novo programmng approach. Internatonal Journal of Producton Economcs 43 (1996), [2] Fala, P.: Modely a metody rozhodování. Economa, Praha, [3] L, R. J., Lee, E. S.: Fuzzy Approaches to Multcrtera De Novo Programs. Journal of Mathematcal Analyss and Applcatons 153 (1990), [4] Sh, Y.: Studes on Optmum-Path Ratos n De Novo Programmng Problems. Computers and Mathematcs wth Applcatons 29 (1995),

8 36 Petr Fala [5] Zelený, M.: De Novo Programmng. Ekonomcko-matematcký obzor 26 (1990), [6] Zeleny, M.: Multobjectve Optmzaton, Systems Desgn and De Novo Programmng. In: Zopounds C., Pardalos P. M. (eds.): Handbook of Multcrtera Analyss, Sprnger, Berln, [7] Zeleny, M.: Optmal Gven System vs. Desgnng Optmal System: The De Novo Programmng Approach. Internatonal Journal of General System 17 (1990), [8] Zhang, Y. M., Huang, G. H., Zhang, X. D.: Inexact de Novo programmng for water resources systems plannng. European Journal of Operatonal Research 199 (2009),