Obtaining Weak Pareto Points for Multiobjective Linear Fractional Programming Problems*

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1 БЪЛГАРСКА АКАДЕМИЯ НА НАУКИТЕ. BLGARIAN ACADEMY OF SCIENCES ПРОБЛЕМИ НА ТЕХНИЧЕСКАТА КИБЕРНЕТИКА И РОБОТИКАТА, 47 PROBLEMS OF ENGINEERING CYBERNETICS AND ROBOTICS, 47 София Sofa Obtanng Weak Pareto Ponts for Multobjectve Lnear Fractonal Programmng Problems* Boyan Metev Insttute of Informaton Technologes, Sofa. Introducton Lnear fractonal crtera are frequently encountered n fnance, marne transportaton, water resources management, health care, etc. []. The real decson makng n these felds must take nto account lnear fractonal (rato) crtera very often. The lnear fractonal programmng (LFP) problem s defned as follows : () max f = p q s.t. x S R n, where p and q are lnear functons and the set S s defned n the followng way: S = { x Ax = b, x > 0 }. Here А s a real valued m n matrx, b R m. We suppose that S s a nonempty bounded polyhedron. The maxmal value of f on S s denoted by f max. Many authors have proposed algorthms for solvng problem (), for example: [5,, 7] and others. Comparatve nvestgatons of such algorthms can be found n [, ]. Addtonal nformaton concernng especally the bad ponts s gven n the paper of [4]. A pont x S s called a bad pont f f when x x. A complete smplex type algorthm for solvng problem () s presented n []. B a z a r a a and S h e t t y [] have shown that the goal functon n () has several mportant propertes t s (smultaneously): pseudo convex, pseudo concave, quas convex, quas concave, strct quas convex and strct quas concave. Ths means that the pont, that satsfes the Kuhn-Tucker condtons for the maxmzaton problem gves the global maxmum on the feasble set. In addton, each local maxmum s n the same tme a global *Ths research was supported by Natonal Scentfc Research Fund under the contract No И-66/996. 5

2 one on the feasble set. Ths maxmum s obtaned at an extreme pont of S. The multobjectve lnear fractonal programmng (MOLFP) problem can be wrtten as follows: p max p s.t. q max q. p h max q k x S. Here S R n s a nonempty bounded polyhedron (as n problem ()). All p and q are lnear functons. We denote f j = p j / q j ( ) and suppose that q > 0, x S, =,,..., k. А descrpton of these problems, some basc nformaton and many examples can be found n []. Nykowsk and Zolkewsk [] have proposed a replacng multobjectve lnear programmng problem and a compromse procedure for ts solvng. Several years later Dutta, Rao and Twar [6] have shown that computatonally some of these results can be mproved for the case when the denomnators are dentcal. Choo has shown that the weak effcent set for problem () s not always a unon of polyhedrons, t may contan some nonlnear parts []. The explct descrpton of the weak effcent set can be very useful but t s often a hard problem to get such descrpton. An advantage of the weak effcent set of problem () s that t s always a closed set. (The effcent set may not be closed.[]). A nonlnear programmng technque and the reference pont method are proposed here for obtanng weak effcent ponts for problem ().. Analyss of the MOLFP problem usng an auxlary nonlnear programmng problem Let us consder problem (). We can try to use the reference pont method for an analyss of ths problem, thus we formulate the followng nonlnear programmng problem mn D s.t. () D > b (r f ),, x S. Here b > 0 ( ), the numbers r are the reference pont components, they satsfy the followng nequaltes : r > max f,. It can be seen that the soluton of problem () determnes weak effcent ponts for problem (). Really suppose that x s a soluton of problem (), that gves the mnmal value D mn, but x s not a weak effcent pont. Then there exsts another pont 5 4

3 x S, such that f ( x ) > f ( x ), (, ). Therefore t s obvously that for the correspondng value D we get D < D mn, and ths s a contradcton. Defnton. Consder the functon h: S E, where S s a nonempty convex set n Е n. The functon h s called strct quas convex, f for each two ponts x, x S, such that h(x ) h(x ), the followng nequalty holds: h ( x + ( ) x ) < max { h (x ), h (x ) } for all (0,). The functons g = b (r f ),, are strct quas convex because f are lnear fractonal []. It s obvous that n problem () the mnmum of the followng functon s searched = max [b (r f ) ] = max [ g ], =,,..., m. Theorem. Let S Е n be a nonempty convex set. Suppose that the functons g (, x S) are strct quas convex. Then the functon = max [ g ] s strct quas convex, too. Proof. Let 0 < <, x, x S,. Then ( x + ( ) x ) = max g (l x + ( l) x )< < max[max ( g ( x ), g (x )] = max [ maxg (x ), max g (x ) ]= = max [ ( x ), (x )] Therefore n problem () we have to mnmze a strct quas convex functon on the convex set S. Each local mnmum of a strct quas convex functon s n the same tme a global mnmum of ths functon on the feasble set S []. Ths means that we can solve problem () usng nonlnear programmng algorthms that gve local mnmum. The obtaned soluton wll gve a weak Pareto pont for problem () and a correspondng weak effcent pont.. Numercal example The Choo s example descrbed n [] wll be used here for llustraton purposes. Ths example s: max ( f = x / x ) max ( f = x ) s.t. max ( f = (x + x ) / ( + x ) ) x, x, x 4. The feasble set S s determned by the above gven constrants. The weak effcent set Е w s descrbed as follows []: where Е w = 4 5, 5 5

4 = { x S x = ( a, b, c ), a = bc }, = { x S x = ( 4, b, c ), bc 4 }, s the convex hull of the ponts (, 4, 4), (, 4, ), (4, 4, ), (4, 4, 4), 4 s the convex hull of the ponts (4,, 4), (,, 4), (, 4, 4), (4, 4, 4), 5 s the convex hull of the ponts (4,, ), (4,, 4), (,, ). In order to get weak effcent ponts for ths problem we use formulaton (). The computatons were made by program NELI. The feasble set s gven as shown above. The functons f are wrtten n general mode n the constrants contanng the varable D. These functons are determned explctly by separately wrtten constrants. On the other hand the reference pont components are numercally wrtten n the constrants contanng the varable D. In addton b for all. Table A contans data llustratng the behavour of the soluton. Table А r r r x x x f f f ,9999,00,999959, ,99959, ,75,00,75,75,75, ,00,758,845,845,845, ,4999,00,499959,4999,499959,50004 The components of four reference ponts: (5, 5, ), (6, 5, ), (5, 6, ), (5, 5, ) are wrtten n columns,, and n rows number, 4, 5, 6. The same rows and columns 4, 5, 6 contan the correspondng feasble ponts determned by the soluton of problem () obtaned wth the correspondng reference pont. The last three columns contan the correspondng crtera values. The comparson wth the gven explct descrpton of the weak effcent set shows that all feasble ponts wrtten n Table A are weak effcent. It must be ponted out that all used reference ponts domnate the deal pont for the problem. Table A very clearly llustrates the effects of ncreasng of one reference pont component keepng the rest unchanged. Row 4 contans a reference pont wth frst component ncreased wth respect to the reference pont n row. Ths leads to ncreasng the value of the frst crteron (rows and 4, column 7). The same effect can be seen for the second and the thrd reference pont component. Ths effect s generally descrbed n the paper [8]. It must be added here that the computatons made wth the nonlnear programmng formulaton () were compared and confrmed by computatons based on the usage of lnear programmng technque. 4. Some comments and concluson The paper [8] contans a result concernng the usage of reference ponts for the analyss of nonlnear multobjectve optmzaton problems. It s shown n the paper that the obtaned value of a gven crteron can be mproved by a correspondngly chosen reference pont. Ths result s vald for the consdered here multobjectve problems and s llustrated by the 5 6

5 example. The same result gves a way to move n the weak Pareto set. The paper [8] contans, n addton, a result about the Pareto ponts attanablty. Ths result s vald for the problems consdered here, too. The soluton of problem () determnes weak effcent ponts for problem () (and weak Pareto ponts, of course). In general, f the reference pont s close to a Pareto pont, then the soluton determnes a Pareto pont. Thus from practcal pont of vew t can be sad that weak Pareto ponts are attanable. It s worth notng that problem () must be fully solved n the followng sense. It s not suffcent to fnd (or to estmate) the needed mnmum only, wthout determnng the correspondng argument. Ths argument determnes the needed weak Pareto or weak effcent pont. The proposed way for obtanng weak Pareto (weak effcent) ponts seems to be more attractve n the cases, when the weak effcent set has a large number of extreme ponts and t s a hard problem to get a full descrpton of ths set. R e f e r e n c e s. A r s h a m, H., A. B. K a h n. A complete algorthm for lnear fractonal programs. Computers Math. Applc., 0, 990, No 7,.. B a z a r a a, M. S., C.M. S h e t t y. Nonlnear Programmng. Theory and Algorthms. John Wley and Sons, NY, B h a t t, S. K. Equvalence of varous lnearzaton algorthms for lnear fractonal programmng. ZOR Methods and Models of Operatons Research,, 989, C h a n k o n g, V., Y.Y. H a m e s. Multobjectve Decson Makng. Theory and Methodology. Amsterdam, North-Holland, C h a r n e s, A., W. W.C o o p e r. Programmng wth lnear fractonal functonals. Nav. Res. Logst. Q., 9, 96, D u t t a, D., J.P. R a o, R. N. T w a r. A restrcted class of multobjectve lnear fractonal programmng problems. Europ.J. of Oper.Research, 68, 99, No, M e t e v, B., I. Y o r d a n o v a. se of reference ponts for MOLP problems analyss. Europ. J. of Oper. Research, 68, 99, No, M e t e v, B. se of reference ponts for solvng MONLP problems. Europ. J. of Oper. Research, 80, 995, M e t e v, B. Applcatons of reference pont method for the analyss of lnear fractonal programmng problems. Problems of Engneerng Cybernetcs and Robotcs, 997, No M e t e v, B., D. G u e o r g u e v a. A feasble pont method for solvng lnear fractonal programmng problems. Submtted n 997 for publcaton n EJOR.. M a r t o s, B. Nonlnear Programmng. Theory and Methods. Amsterdam, North-Holland, N y k o w s k, I., Z. Z o l k e w s k. A compromse procedure for the multple objectve lnear fractonal programmng problem. Europ. J. of Oper. Research, 9, 985, S t e u e r, R. Multple Crtera Optmzaton Theory, Computaton and Applcaton. NY, John Wley and Sons, Chchester, V e r m a, V., S. K h a n n a, M. C. P u r. On Martos and Charnes-Cooper s approach vs-а-vs sngularponts. Optmzaton, 0, 989, No 4, W e r z b c k, A. A mathematcal bass for satsfcng decson makng. In: J.N.Morse (ed.). Organzatons: Multple Agents wth Multple crtera. Proceedngs, nversty of Delaware, Newark, 980; LNEMS, Sprnger-Verlag, Berln, 98, W e r z b c k, A. On the completeness and constructveness of parametrc characterzaton to vector optmzaton problems. OR Spectrum 8, 986, W o l f, H. A parametrc method for solvng the lnear fractonal programmng problems. Operatons Research,, 985,

6 Нахождение слабых точек Парето для задач многокритериального дробно-линейного программирования Боян Метев Институт информационных технологий, София (Р е з ю м е) Для анализа задачи многокритериального дробно-линейного программирования (все критерии максимизируются) предлагается использовать известную скалярную оптимизационную задачу, решение которой определяет слабые Паретовские точки (а также и слабоэффективные точки). В рассматриваемом случае в этой скалярной задаче минимизируется строго квазивыпуклая функция, что позволяет использовать алгоритмы нелинейного программирования, дающие локальный минимум. 5 8