Applied Mathematics and Computation

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1 Appled Matheatcs and Coputaton 27 (20) Contents lsts avalable at ScenceDrect Appled Matheatcs and Coputaton journal hoepage The explct soluton of the proft axzaton proble wth box-constraned nputs L. Bayón, J.M. Grau, M.M. Ruz, P.M. Suárez Departent of Matheatcs, Unversty of Ovedo, Gjón, Span artcle nfo abstract Keywords Optzaton theory Proft axzaton Cost nzaton Cobb Douglas Box constrants Non-ear prograng Infal convoluton In ths paper we study the proft-axzaton proble, consderng axu constrants for the general case of -nputs and usng the Cobb Douglas odel for the producton functon. To do so, we prevously study the fr s cost nzaton proble, proposng an equvalent nfal convoluton proble for exponental-type functons. Ths study provdes an analytcal expresson of the producton cost functon, whch s found to be a pece-wse potental. Moreover, we prove that ths soluton belongs to class C. Usng ths cost functon, we obtan the explct expresson of axu proft. Fnally, we llustrate the results obtaned n ths paper wth an exaple. Ó 20 Elsever Inc. All rghts reserved.. Introducton One of the ost portant ssues for frs n the feld of croeconocs [] s the proft-axzaton proble. In ths paper we consder a fr that operates under perfect copetton,.e. ts prces are ndependent of the fr s nput and output decsons. Consder a fr eployng a vector of nputs x 2 R þ to produce an output y 2 R þ, where R þ ; R þ are non-negatve - and -densonal Eucldean spaces, respectvely. Let P(x) be the feasble output set for the gven nput vector x and L(y) the nput requreent set for a gven output y. Now, the technology set [2] s defned as T ¼ ðx; yþ 2R þ þ ; x 2 LðyÞ; y 2 PðxÞ We assue that ths set satsfes the followng well-nown regularty propertes closedness, non-eptness, scarcty, and no free lunch. Only on a few occasons have addtonal constrants been eployed n the lterature; for exaple, an expendture constrant s consdered n [3]. Most classcal studes, however, splfy resource utlzaton wthout consderng constrants on nput usage. In ths paper we establsh, for the frst te, a box-constraned proft-axzaton proble, consderng axu constrants for the nputs. Generally, the proft axzaton proble can be forulated n the followng way the fr chooses nputs and output n order to axze profts p (where profts are revenue nus costs), subject to technology constrants (.e. the relatonshp between nputs and output) st pðp; wþ ¼ax py wxþ; x;y y ¼ f ðxþ; ðx; yþ 2T; 0 6 x 6 M ; ¼ ;...; ; ðþ Correspondng author. Address EPI, Departent of Matheatcs, Unversty of Ovedo, Capus of Vesques, Gjón, Span. E-al address bayon@unov.es (L. Bayón) /$ - see front atter Ó 20 Elsever Inc. All rghts reserved. do0.06/j.ac

2 8706 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) where p s the prce of the output, w 2 R are the vector prces of the nputs, M the axu constrants for the nputs, and f(x) s the producton functon, whch s a contnuous, strctly ncreasng and strctly quasconcave functon. In ths paper we consder an -nput Cobb Douglas producton functon [4], [5] y ¼ f ðxþ ¼A Y x a There are two ways of solvng ths proble () we can ether forulate the proble axzng over the nput quanttes, wth plan Lagrange/Kuhn Tucer or substtutng the constrant n the objectve functon; or () we forulate the proble usng a nu cost functon and then axze over the output quantty. In ths paper we use ths short-cut-va-cost nzaton. Note that f a fr s axzng ts profts and decdes to offer a level of producton y, t ust be nzng the cost of producng ths output. Otherwse, a cheaper way of obtanng y producton unts would exst, whch would ean that the fr s not be axzng ts profts. Naely proft ax ples cost n. On the other hand, the proft-axzaton proble has tradtonally been solved by dfferentatng the varable x. Nevertheless, soe authors avod usng total dfferentaton of frst-order condtons, ndcatng that ths gves rse to coplcated equatons whch are dffcult to handle. For exaple, [6] and [7] eploy geoetrc prograng to derve the axal proft for the proft functon. In the present paper we obtan the analytcal and explct forulas usng the classcal ethod of calculaton. The followng are coon probles than can arse () The producton functon ay not be dfferentable, n whch case we cannot tae frst-order condtons. () The frst-order condtons gven above assue an nteror soluton, but we ust also consder boundary solutons. () A proft axzng plan ght not exst. (v) The proft axzng producton plan ght not be unque. In ths paper we prove, under certan assuptons, the exstence and unqueness of the soluton and that t belongs to class C. The paper s organzed as follows. Our box-constraned proft-axzaton proble s solved n two stages we frst deterne how to nze the costs of producng each aount y and then what aount of producton actually axzes profts. In the next secton we present the box-constraned cost-nzaton proble. By changng certan varables, we then transfor t nto a non-ear (exponental) separable prograng proble [8], whch we state as a constraned nfal convoluton proble [9]. In Secton 3, we provde a nuber of basc defntons and develop all the atheatcal results necessary for the soluton of the nfal convoluton proble. Secton 4 presents the optal soluton of the box-constraned cost-nzaton proble. In Secton 5, we obtan the optal soluton of the box-constraned proft-axzaton proble to then dscuss the results of a nuercal exaple n Secton 6. Fnally, Secton 7 suarzes the an conclusons of our research. 2. Cost-nzaton proble In ths secton we frst present the classc fr s cost-nzaton proble. Ths proble can be expressed as follows produce a gven output y, and choose nputs to nze ts cost st cðw; yþ ¼n wx; xp0 f ðxþ ¼y; ð2þ where x 2 R are the nputs and w 2 R are the factor prces. There are a nuber of dfferent ways to atheatcally express how nputs are transfored nto output. In ths paper we consder the general Cobb Douglas producton functon y ¼ f ðxþ ¼A Y x a and we shall usually easure unts so that the total factor productvty A =. The su of a deternes the returns to scale. The forulas for the correspondng cost functon c(w, y) are well nown [0] when the producton functon follows the Cobb Douglas odel Y cðw; yþ ¼aya w a a ; a wth a ¼ X a These forulas, whch can be obtaned sply usng the Lagrange ultplers ethod, present the drawbac that they are not applcable when upper lt constrants are consdered for the dfferent nputs. In ths paper we establsh the analytcal expresson for the cost functon c(w, y) usng the Cobb Douglas odel, consderng axu constrants for the nputs. Our cost-nzaton proble wll be

3 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) st cðw; yþ ¼n P w x ; y ¼ Q x a ; 0 6 x 6 M ; ¼ ;...; ð3þ Probles of ths nd, wth box constrants, becoe coplcated n the presence of boundary solutons. There s a vast array of software pacages for nuercally solvng nonear optzaton probles []. These ethods only obtan an approxate soluton for specfc values of the output y, but do not provde the analytcal expresson of the cost functon c(w, y). It s thus not possble to now the argnal cost y)/@y needed to solve the proft axzaton proble. We shall address ths proble n an exact way n ths paper, transforng t nto a non-ear (exponental) separable prograng proble, whch we state as a constraned nfal convoluton proble. Tang nto account the followng changes n the varables y ¼ q; a x ¼ z ; ¼ ;...; ; the cost-nzaton proble (3) s equvalent to the nfal convoluton proble st ~cðw; qþ ¼n P a w e z ; P z ¼ q; < z 6 a M ¼ ; ¼ ;...; The functon ~cðw; Þ s, n fact, the nfal convoluton of the exponental functons ð4þ F ðz Þ ¼ w e a z The case of quadratc F functons s well nown and has been studed by the authors n [2] wthn the fraewor of hydrotheral optzaton. However, the sae nd of study s unnown for exponental functons. In ths paper we develop the necessary atheatcal tools to justfy the proposed ethod for solvng the stated proble. 3. Infal convoluton proble Let us calculate the nfal convoluton of the convex functons F (z ) consderng ther doan to be constraned to ð ; Š. Let us assue throughout the paper, wthout loss of generalty, that 6 F 0 ; 8 ¼... ð5þ F 0 þ Pax þ Let the functon F ð ; Šð ; Š!R gven by Fðz ;...; z Þ ¼ X F ðz Þ Let C q be the set (, ) X C q ¼ ðz ; ; z Þ2 ; ; P ax z ¼ q The nfal convoluton of ff g s ðf F ÞðqÞ ¼ n C q X F ðz Þ Let us now see the defntons of the eleents that are present n our proble. Defnton. Let us call the functon W ð ; P W ðqþ ¼z ; 8 ¼ ;...; ; j¼ Pax j where (z,...,z ) s the unque nu of F on the set C q,.e. X W ðqþ ¼q and X F ðw ðqþþ ¼ ðf F ÞðqÞ Š! ð ; Š the th dstrbuton functon, defned by

4 8708 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) The followng lea guarantees that f z reaches ts axu value, all those z for whch the dervatve of F at ts axu value s less than or equal to the dervatve correspondng to F wll lewse have already reached ther axu. Lea. If the functon F reaches at (a,...,a ) the nu on the set C q, and for a certan 2f;...; g; a ¼ 6 F 0 ) a ¼ 8 2f;...; g=f 0 Pax, then Proof. We shall argue by contradcton. Let us assue that for a certan 2f;...; g; a ¼ F 0 j j 6 F 0 Consder the functon gðeþ ¼Fða ;...; a j þ e;...; a e;...; a Þ Fða ;...; a Þ; gðeþ ¼F j ða j þ eþþf ða eþ F j ða j Þ F ða Þ It s clear that f (a,...,a ) 2 C q, then (a,...,a j + e,...,a e,...,a ) 2 C q for 0 6 e < j a j and that a j < j, beng Let us show the exstence of an e such that g(e) < 0, whch contradcts the fact that F has a nu n (a,...,a ) wthn C q. We have that g s contnuous and dervable at zero wth g(0) = 0; therefore t suffces to observe that g 0 (0) < 0. In fact, g 0 ðeþ ¼F 0 j ða j þ eþ F 0 ða eþ ¼F 0 j ða j þ eþ F 0 e ; g 0 ð0þ ¼F 0 j ða jþ F 0 < F 0 j j F The followng lea establshes the order of the ponts at whch the varables reach ther axu value. Lea 2. The paraeters satsfy h ¼ X a ¼ P a ax þ X ¼ a a w þ X a w ; h 6 h h ¼ X Proof h ¼ X ¼ þ X a P a ax ¼þ þ X a w ¼ a þ X a a w þ X a w ¼ X ¼þ ¼ X ¼þ a a a þ þ þ X w þ Pax þ X a a a a w þ þ X a þ w ¼þ ¼þ a w a þ X ¼ h þ 6 X ¼þ a w þ þ Pax þ a þ a þ The followng theore establshes a necessary and suffcent condton to obtan the nteror soluton. Theore. The functon F attans ts nu value on the set C q at the pont ða ;...; a Þ2C o q ff q < X a a þ X a a w ¼ h a w Proof. NecesstyIf (a,...,a ) s an nteror pont where F attans ts nu value, t s a pont of relatve nu of F on the set ( ) ðz ;...; z Þ 2 ð ; Þ... ð ; Þ X z ¼ q

5 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) It follows that for soe 2 R; ða ;...; a Þ s a crtcal pont of F ðz ;...; z Þ¼Fðz ;...; z Þ ðz þþz qþ Usng the Lagrange ultplers ethod, we have that 9 h 9 w a e z a ¼ 0 z ¼ a w a h w 2 a 2 e z2 a2 ¼ 0 >= z 2 ¼ a 2 w 2 a 2 >=. )... h w a e z a ¼ 0 z ¼ a >; w a >; z þ z 2 þþz ¼ q z þ z 2 þþz ¼ q Hence and ¼ P q þ a ¼ P a " # e q Y a P w a a X w a a Let us consder W (q) to be a functon of the unnown z W ðqþ ¼z ¼ a " X P q þ a # a w ; a a w W ðqþ ¼ a " X P q þ a # a w ¼ P a a ax ; w a ¼ q Lettng D () X ¼ X P a ax þ X a a w a w P a ax þ X a a w a a w and, bearng n nd (5), we see that h ¼ D 6 D D It s evdent that for every, the soluton W (q) s strctly ncreasng as a functon of q. Thus, q P h ) W ðqþ ¼a P W ðh Þ¼ or, conversely, W ðqþ ¼a < ) q < h (Suffcency). Snce C q s copact, the nu of F clearly exsts. Let us now consder ða ;...; a Þ¼ðW ðqþ;...; W ðqþþ; a crtcal pont of the convex functonal where F ðz ;...; z Þ¼Fðz ;...; z Þ ðz þþz qþ; ¼ " # e q Y a P w a a We have that (a,...,a ) delvers the nu value to F and, hence, t s also the nu of F under the constrant ( ) X ðz ;...; z Þ z ¼ q

6 870 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) Moreover, t s evdent that for every, the soluton W (q) s strctly ncreasng as a functon of q. Thus, q < h ) q < D ; o so that ða ;...; a Þ2C q. 8 ¼ ;...; ) W ðqþ ¼a < ; 8 ¼ ;...; ; h Havng proven the above results, we are now n a poston to obtan the dstrbuton functons Theore 2. For every =,...,, the th dstrbuton functon s 8 a P q þ P a w a a a f q < h w >< " # W ðqþ ¼ P a a q þ P a w a a ¼jþ Pj w f h j 6 q < h jþ 6 h ¼jþ > f q P h wth the coeffcents h ¼ X ¼ a P a ax þ X ¼ a a w þ X a w Proof. In vew of Theore, fq < h, then the dstrbuton functons W ðqþ < for all and t reans to derve the expresson for z.ifh 6 q < h 2, then the nu of P F ðz Þ cannot be attaned n the nteror. Accordng to Lea, at least z ¼. Thus, W ðqþ ¼. The sae arguent apples to the reanng proble of denson. W ðqþ ¼ a P ¼2a " # q þ X a ¼2 a w a w If h 2 6 q < h 3, then W ðqþ ¼ and, argung as above, W 2 ðqþ ¼ 2, and for > 2, we have that " # W ðqþ ¼ a P ¼3a q 2 þ X ¼3 a w a w a Fnally, repeatng the arguent once agan, we have that f h j 6 q < h j+, then the th dstrbuton functon s equal to q P h, and f h > q ( =,...,j + ), W ðqþ ¼ a P a ¼jþ " X j q þ X a # a w a ¼jþ w We shall also prove that the nfal convoluton of the functons ff g belongs to class C. Let us see the followng lea frst. f Lea 3. Let ff g 2 C ðrþ be two convex functons satsfyng F 0 ðm Þ 6 F 0 2 ðm 2Þ. Let us consder Then ðf F 2 ÞðnÞ ¼ nff ðxþþf 2 ðyþg; D wth D ¼fðx; yþ 2 ð ; M Š ð ; M 2 Šjx þ y ¼ ng ðf F 2 Þ2C ð ; M þ M 2 Š Proof. Let ^g and ^g 2 be the functons of class C that satsfy the followng equalty n ff ðxþþf 2 ðyþg ¼ F ð^g ðnþþ þ F 2 ð^g 2 ðnþþ xþy¼n wth ^g ðnþþ^g 2 ðnþ ¼n and F 0 ð^g ðnþþ ¼ F 0 2 ð^g 2ðnÞÞ8n 2 R. We now have that the nfal convoluton of the functons F constraned to ther respectve doans (,M ] wll be ðf F 2 ÞðnÞ ¼F ðg ðnþþ þ F 2 ðg 2 ðnþþ

7 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) wth g 2 ðnþ ¼ n g ðnþ; g ðnþ ¼ M f ^g ðnþ > M ; ^g ðnþ f ^g ðnþ 6 M Let d be such that ^g ðdþ ¼M (note that F 0 ðm Þ¼F 0 2 ð^g 2ðdÞÞ ¼ F 0 2 ðd M ÞÞ ðf F 2 ÞðnÞ ¼ F ðg ðnþþ þ F 2 ðg 2 ðnþþ f n 6 d; F ðm ÞþF 2 ðn M Þ f d < n 6 M þ M 2 In (,d), the functon (F F 2 ) obvously belongs to class C and also n (d,m + M 2 ], snce (F F 2 )(n)=f (M )+F 2 (n M ). The only conflctng pont s d. Let us study the contnuty of (F F 2 )nd ðf F 2 Þðd Þ ¼ F ð^g ðdþþ þ F 2 ðd ^g ðdþþ ¼ F ðm ÞþF 2 ðd M Þ¼ðF F 2 ÞðdþÞ Let us lewse study the contnuty of the dervatve n d ðf F 2 Þ 0 ðd Þ ¼ F 0 ð^g ðdþþ^g 0 ðdþþf0 2 ðd ^g ðdþþð ^g 0 ðdþþ ¼ F0 ð^g ðdþþ ¼ F 0 ðm Þ; ðf F 2 Þ 0 ðdþþ ¼ F 0 2 ðd M Þ¼F 0 ðm Þ Therefore, (F F 2 ) 2 C. h Theore 3. Let ff g C ðrþ. Let us consder Then X ðf F 2 F ÞðnÞ ¼ n F ðx Þ; D ( wth D ¼ ðx ;...x Þ2 Y X ð ; M Š ) x ¼ n ðf F 2 F Þ2C ; X M # Proof. It suffces to reason by nducton, bearng n nd, due to the assocatvty of, that ðf F 2 F Þ¼ðF F ÞF We ay now also obtan the analytcal expresson of (F F 2 F ). Theore 4. The nfal convoluton of the exponental functons F (z ) s an exponental (plus constant) pecewse functon 8 < ðf F 2 F ÞðqÞ ¼ ~w ~a e q f q < h wth the coeffcents ~l ¼ X w e a ; ~a ¼ X a ¼ ; "! # 2 ~w ¼ exp,~a X Y 4~a Moreover, t belongs to class C. ~l þ ~w e q ~a f h 6 q < h j¼ w j a j a j ~a 3 5 Proof. Fro Theore 3, t s evdent that ðf F 2 F Þ2C ; X #

8 872 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) Furtherore, the nfal convoluton expresson for the exponental functons ff g s easly obtaned sply tang nto account the defnton of (F F 2 F ), Theore 2 and the fact that W (q)=z, " =,...,. h 4. Soluton of the cost-nzaton proble Consderng the fact that cðw; yþ ¼~cðw; yþ ¼ðF F 2 F Þð yþ, the followng theore s verfed Theore 5. The condtonal deand functon for the th nput s 8 Q >< exp x ðw; yþ ¼ Q > a a w ~a a y w ¼jþ e a Pj a w a w ~a f y < e h ; =~a jþ a ~a jþ y ~a jþ f e h j 6 y < e h jþ 6 e h ; f y P e h and the cost functon s a pecewse potental (plus constant) 8 < cðw; yþ ¼ ~w ~a y f y < e h ; ~a ~l þ ~w y f e h 6 y < e h ; where ~l ; ~a and ~w are the coeffcents defned n Theore 4. Proof. Tang nto consderaton the changes n the varable y ¼ q; z ¼ a x ¼ W ðqþ and Theore 2, we obtan the expresson of the condtonal deand functon for the th nput, x (w,y). Slarly, as cðw; y ¼ ~cðw; yþ, fro Theore 3 we obtan the cost functon expresson c(w,y). h 5. Soluton of the proft-axzaton proble Havng calculated the cost functon C(y) ¼ c(w,y) and havng establshed ts character, C, the proft-axzaton proble pðp; wþ ¼axðpy cðw; yþþ ¼ axðpy CðyÞÞ y y translates nto the deternaton of the optu level of output y for whch the argnal cost concdes wth the prce p. Naturally, ths consderaton s only vald for output levels for whch the argnal cost s ncreasng (C(y) convex). p ¼ C 0 ðy Þ^ convexty of C ) pðp; wþ ¼py Cðy Þ Bearng n nd that the cost functon s pecewse potental, the correct calculaton of the output level requres pror nvestgaton of the nterval ½e h ; e h Š for whch C 0 ðe h Þ 6 p 6 C 0 ðe h Þ Ths queston s trval, as we already have the analytcal expresson of C(y). 6. Exaple We shall now present a proft axzaton proble wth a Cobb Douglas type producton functon whch, wthout consderng techncal constrants for the nputs, would be totally unreal, as t would present ncreasng returns to scale at all levels of producton (and hence a concave cost functon). By consderng nputs to be lted, the proble becoes totally real and the resultng cost functon presents a regon of concavty (ncreasng returns to scale) and another of convexty (decreasng returns to scale) where the soluton to the proble s to be found the level of producton at whch the argnal cost and output prce concde.

9 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) We consder the followng exaple st pðp; wþ ¼axðpy wxþ; y ¼ Y x a ; x;y 0 6 x 6 M ; ¼ ;...; wth = 20 nputs, output prce p = 20, and wth the data presented n Table. Table Exaple data a M w a M w c(w,y) Fg.. The cost functon. y Table 2 The pecewse cost functon. c(w,y) y 2 [a,b) y [0,0.30] y [0.30,0.934) y [0.934, ] y [0.3066,0.572] y [0.572, ] y [0.8529, ] y [0.9383,.9077] y [.9077, ] y [2.7568,2.7648] y [2.7648,3.23] y [3.23,3.2972] y [3.2972,3.8202] y [3.8202, ] y.0989 [3.8726,4.924] y.4925 [4.924,5.5063] y.859 [5.5063, ] y [5.7495,6.098] y [6.098, 6.44] y 6.25 [6.44,6.687] y 20. [6.687,6.89]

10 874 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) c( w, y)/ y p=20 Increasng y y Fg. 2. The argnal cost functon. Table 3 Soluton for y = x x Fg. shows the graph of the cost functon c(w,y), together wth the resultng graph n the case of not havng consdered constrants for the nputs; note that both concde n the nterval ½0; e h ¼ 030Š. Fro ths pont onward, not consderng constrants leads to a very dfferent cost functon to the correct one. In addton, the area n whch the producton functon presents decreasng returns to scale, and hence a convex cost functon, s hghlghted n grey. The values fe h g 20 ¼ ¼f030; 0934; 03066; 0572; 08529; 09383; 9077; 27568; 27648; 323; 32972; 38202; 38726; 4924; 55063; 57495; 6098; 644; 6687; 689g consttute the dfferent levels of output at whch the paraeters of the cost functon expresson change. These correspond to the levels at whch the dfferent nputs acheve ther axu value, whch, accordng to the theoretcal developent (5), they do so n ths exaple n the followng order f; 0; 7; 2; 8; 8; 5; 2; 9; 9; 3; 6; 3; 7; 5; ; 20; 6; 4; 4g. The analytcal expresson of the pecewse cost functon s presented n Table 2, beng obtaned as shown n Theore 5 Fg. 2 shows the graph of the argnal cost functon, whch, as has already been establshed, s contnuous (.e. the cost functon belongs to C ). It can be seen that there are two ponts for whch the argnal cost s p = 20. Naturally, however, the area represented n whte does not provde the axu value, as t s located n an area of decreasng argnal cost the only axu s obtaned for the output value y = Fnally, n Table 3 we present the condtonal deand functon for the th nput. 7. Conclusons In ths paper we have establshed the analytcal soluton for the classc fr s proft axzaton proble n the general case wth nputs. We have used the Cobb Douglas odel for the producton functon and have consdered, for the frst te, axu constrants for the nputs. Our study has a nuber of advantages over other ethods the exact boundary soluton s obtaned and the ethod s not affected by the sze of the proble. Acnowledgeent Ths wor was supported by the Spansh Governent (MICINN, project MTM ). References [] H.R. Varan, Interedate Mcroeconocs A odern approach, Seventh ed., W.W. Norton & Copany, New Yor, [2] R. Fare, D. Pront, Mult-Output Producton and Dualty Theory and Applcatons, Kluwer Acadec Publshers, Netherlands, 995.

11 L. Bayón et al. / Appled Matheatcs and Coputaton 27 (20) [3] R.E. Just, D.L. Hueth, A. Schtz, The Welfare Econocs of Publc Polcy a Practcal Approach to Project and Polcy Evaluaton, Edward Elgar Pub.,, USA, [4] G.A. Jehle, P.J. Reny, Advanced Mcroeconoc Theory, Second ed., Addson-Wesley, Boston, 200. [5] D.G. Luenberger, Mcroeconoc Theory, McGraw-Hll, 995. [6] S.T. Lu, A geoetrc prograng approach to proft axzaton, Appled Matheatcs and Coputaton 82 (2006) [7] S.T. Lu, Proft axzaton wth quantty dscount an applcaton of geoetrc progra, Appled Matheatcs and Coputaton 90 (2007) [8] I. Grva, S.G. Nash, A. Sofer, Lnear and Nonear Optzaton, SIAM, Phladelpha, [9] J.B. Hrart-Urruty, C. Learéchal, Convex Analyss and Mnzaton Algorths I, Sprnger-Verlag, Ber, 996. [0] D.S. Haeresh, Labor Deand, Prnceton Unversty Press, 996. [] Nonear Prograng Software, <http//neos.cs.anl.gov/>. [2] L. Bayón, J.M. Grau, P.M. Suárez, A new forulaton of the equvalent theral n optzaton of hydrotheral systes, Matheatcal Probles n Engneerng 8 (3) (2002) 8 96.

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