Some remarks about the transformation of Charnes and Cooper by Ezio Marchi *)

Transcription

1 Some remars about the transformaton of Charnes an Cooper b Eo Marh * Abstrat In ths paper we eten n a smple wa the transformaton of Charnes an Cooper to the ase where the funtonal rato to be onsere are of smlar polnomal. Kewors: programmng, transformaton, Charnes an Cooper, fratonal programmng. Introuton * ouner an rst Dretor of the Insttuto e Matemáta Aplaa San Lus, CONICE-UNSL, Argentna.

2 Charnes an Cooper n [3] have ntroue an mportant transformaton. hs regars the equvalene about programmng wth lnear fratonal funtonal an lnear ones. Members of ths lass have been enountere n a varet of ontets. One suh ourrene [3] nvolve stuatons n whh the more usual senstvt analses were etene to problems nvolvng plans for optmal ata hanges. In these nstanes, lnear programmng nequaltes were to be onsere relatve to a funtonal formulate as a rato of two varables wheren one varable, n the numerator, represente the volume hanges that mght atten the possble varatons of a partular ost oeffent. Another eample was ealt wth b M. Klen n [8]. o hanle the problem of the fratonal funton, Klen apple a square-root transformaton whh be attrbute to C. Derman [5] n orer to effet a reuton to an equvalent lnear programmng problem. nall, a speal nstane of our general ase was treate b J. R. Isbell an W. H. Marlow n ther artle on Attrton Games, [7]. In onserng a rato, Isbell an Marlow were able to establsh a onvergent teratve proess whh nvolve replang the rato b the problem of optmng a sequene of fferent lnear funtonals. he lnear funtonal at an stage n the teratons was etermne b optmaton of the lnear funtonal at the preeng stage. he objetve of the present short paper s to eten the Charnes an Cooper transformaton to some other problems where the funtonal nvolves are quotent of lnear ones. As a partular ase of t we get the result that a general non-lnear problem s equvalent to a lnear one. General ratonal Moels Conser a general lass of fratonal moels arsng n programmng, whh are renere n the followng form:. A b 0 where A s an mn matr an b s an m vetor so that the two sets of onstants for the onstrans are relate b the n vetor of varables,. 2

3 It s assume, unless otherwse note, that the onstrants of. are regular, so that the soluton set n nonempt an boune. form P Here R nates the reals. n { R : A b, 0 } he general lass of fratonal moels to be onsere n ths paper s of the subjet to.: Where,,, mn f A b 0., are transpose of the n vetors of oeffents, where : 0, L,. If 0 we have the ase stue b Charnes an Cooper [4]. B reason of smplt we assume that for eah, we have the last part of the enomnator 0. > hus sne f s a ontnuous funton efne on a non-empt ompat set, t reahes ts mnmum. hat s to sa the problem P s solvable. ollowng the eas presente b Charnes an Cooper n [4], we wll prove that the guarantee the equvalene between P an the programmng program wth n varables P 2 subjet to, K, n, an the feasble set 2 : mn f, 3

4 2 A b 0 0, 0. We now proee to prove: Lemma : If, 2, then > 0. we obtan Proof: Assume that 0, then A 0. ae ˆ an µ > 0 an arbtrar, A But ˆ 0 an 0, ˆ A ˆ b µ. ontans the half straght lne { ˆ / λ 0} 0. hs s ontrator wth the fat that s boune. hen 0 λ f. But ths last fat proves a ontraton wth the seon restrton of P 2. Sne 0 an 0, we have > 0. Q. E. D.. As an mmeate onsequene of the prevous fat, we have that, f, 2 then :. Lemma 2: If, let us efne then, 2. : an : Proof: Sne > 0, we get 0 an 0 for eah. On the other han, f we ve the nequalt A b b we obtan A b whh s equvalent to A b 0. nall q hene, 2. Q. E. D.. 4

5 5 heorem 3: P an 2 P are solvable. Moreover, f s an optmal soluton of P,, s an optmal soluton of 2 P, an reproall, f, s an optmal soluton of 2 P, then s an optmal soluton of P. Proof: We now that P s solvable. Let an optmal soluton of P. Gvng a par 2, then 0 > an b Lemma an the omment after t. Sne we assume that f f for eah, f we now assoate to the vetor 2,,, we have, f, f hen we have prove that, s an optmal soluton of 2 P. herefore 2 P s also solvable.

6 Assume now that, s an optmal soluton of P 2. Conser a vetor an let the transformaton an. Applng the result after the Lemma, we have f f, f, whh t sas that s a soluton of P. Q. E. D.. Our result t turns out to be of mportant nterest n the ase of 0. In ths nstane t reues to the mportant transformaton of Charnes an Cooper [4]. In more general ases, t reues the egree of omplet of the polnomal. f Conlusons Here we have obtane an etenson of the Charnes an Cooper transformaton to the ase when we have a polnomal form n the numerator an the enomnator. he ase when some an are equal t reues the omplet n more egrees n the enomnator. urthermore, there est some varetes of stuatons that the smplt of the new programs appears naturall. or eample the ase where?for S N. hen n suh a ase we obtan goo reuton. Moreover f all 0 an the,, K, we have obtane an nterestng result, sne the transforme program s lnear. or further omments we subjet the reaer to refer to the orgnal paper [4]. here are also further etensons whh we shall treat elsewhere. Anowlegement Part of the researh unerlng ths paper was supporte b grants of the CONICE Consejo Naonal e Investgaones Centífas énas an b the Consejo e Investgaones of the ault of. M. N. S. of UNSL. 6

7 Bblograph [] Barlow, R. E., an L. C. Hunter: Mathematal Moels for Sstem Relablt, he Slvana ehnologst XIII, an [2] Charnes, A., an W. W. Cooper: Management Moels an Inustral Applatons of Lnear Programmng John Wle an Sons, In., New Yor, 96. [3] Charnes, A., an W. W. Cooper: Sstems Evaluaton an Reprng heorems, O.N.R. Memoranum No. 3, September 960. [4] Charnes, A., an W. W. Cooper: Programmng wth ratonal untonals: I, Lnear ratonal Programmng, Sstems Researh Group, he ehnologal Insttute, Northwestern Unverst, ONR Researh Memoranum No. 50, ebruar 962. [5] Derman, C.: On Sequental Desons an Marov Chans, Management Sene, Vol. 9, No., Otober 962. [6] Goberna, M. A., V. Jornet an R. Puente: Optmaón Lneal, eoría, Métoos Moelos In Spansh. [7] Isbell, J. R., an W. H. Marlow: Attrton Games, Naval Researh Logsts Quartel, 3, an 2, [8] Klen, M.: Inspeton-Mantenane-Replaement Sheule uner Marovan Deteroraton, Management Sene, Vol. 9, No., Otober 962. [9] Shable, S.: Nonlnear fratonal programmng, Sehste Oberwolfah-agung über Operatons Researh 973, el II, pp Operatons Researh Verfahren, Ban XIX, Han, Mesenhem am Glan,