Cobb-Douglas Based Firm Production Model under Fuzzy Environment and its Solution using Geometric Programming

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1 Avalable at Appl. Appl. Math. ISSN: Vol. Issue (Jue 6) pp Applcatos ad Appled Mathematcs: A Iteratoal Joural (AAM) obb-douglas Based Frm Producto Model uder Fuzzy Evromet ad ts Soluto usg Geometrc Programmg Abstract Palash Madal Ardam Gara ad Tapa Kumar oy 3 Departmet of Mathematcs Ida Isttute of Egeerg Scece ad Techology Shbpur Ida P 73 palashmadalmbss@gmal.com Departmet of Mathematcs Soarpur Mahavdyalaya ajpur West Begal Ida P 749 fuzzy_ardam@yahoo.com 3 Departmet of Mathematcs Ida Isttute of Egeerg Scece ad Techology Shbpur Ida P 73 roy_t_k@yahoo.co. eceved: May 7 5; evsed: Jauary 6 6 I ths paper we cosder obb-douglas producto fucto based model a frm uder fuzzy evromet ad ts soluto techque by makg use of geometrc programmg. A frm may use may fte puts such as labour captal coal ro etc. to produce oe sgle output. It s well kow that the prmary teto of usg producto fucto s to determe maxmum output for ay gve combato of puts. Also the frm may ga compettve advatages f t ca buy ad sell ay quattes at exogeously gve prces depedet of tal producto decsos. O the other had realty costrats ad/or objectve fuctos a optmzato model may ot be crsp quattes. These are usually mprecse ature ad are better represeted by usg fuzzy sets. Aga geometrc programmg has may advatages over other optmzato techques. I ths paper obb- Douglas producto fucto based models are solved by applyg geometrc programmg techque uder fuzzy evromet. Illustratve umercal examples further demostrates the feasblty ad effcecy of proposed model uder fuzzy evromet. oclusos are draw at last. Keywords: obb-douglas producto fucto; frm producto model; geometrc programmg techque; fuzzy decso makg; fuzzy geometrc programmg; fuzzy mathematcal programmg MS No.: 97 9B3 9B38 3E7 469

2 47 Palash Madal et al.. Itroducto The obb-douglas producto fucto s wdely used to represet the relatoshp of a output to puts. Kut Wcksell (96) had tally proposed the producto fucto. It was tested agast statstcal evdece by obb et al. (98). obb et al. (98) publshed a study whch they modelled the growth of Amerca ecoomy durg the perod They cosdered a smplfed vew of the ecoomy whch producto output s determed by the amout of labour volved ad the amout of captal vested. Whle there are may other factors affectg the producto output ther model was remarkably accurate. The producto fucto used that model was as follows: P L K al K. Here P: Total moetary value of all producto (goods produced a year) L: Total labour put (umber of perso hours worked a year) K: Total captal put (the moetary worth of all machery equpmet ad buldgs) a : Total factor productvty : The output elastcty of labour ad captal respectvely. Avalable techology may determe these values ad they are usually costats. It may be oted that output elastcty measures the respose of output to chage level of ether labour or captal used producto e.g. for =.5 sgle % crease labour may lead to approxmately.5% crease output. Whe the producto fucto has costat returs to scale. Hece a crease of % both L ad K creases P by %. Here returs to scale s a techcal property of producto whch exames chages output subsequet to proportoal chage all puts where all puts crease by a costat factor. Aga for returs to scale are decreasg; ad for returs to scale are creasg. I the case of perfect competto ad are labours ad captals share of output. Aalogous to Shvaa et al. (3) ferece s vewed as a process of propagato of elastc costrats. Oe mportat modfcato or chage classcal set theory that guded a paradgm shft mathematcs s the cocept of fuzzy set theory. It was troduced by Lotf Asker Zadeh 965. Accordg to Bellma et al. (97) a fuzzy set s a better represetato of real lfe stuatos tha classcal crsp set. I producto plag obb-douglas producto fucto may also be cosdered uder fuzzy evromet. As ao () metoed t s well kow that geometrc programmg techque provdes us wth a systematc approach for solvg a class of o-lear optmzato problems by fdg the optmal value of the objectve fucto ad the the optmal values of decso varables are obtaed. osequetly as Guey et al. () suggested geometrc programmg techque ca be appled obb- Douglas based frm producto model uder fuzzy evromet. Ths paper s arraged as follows. I Secto obb-douglas based frm producto model s dscussed detal by applyg dfferet approaches uder fuzzy evromet. Next Secto 3 a umercal example usg these fuzzy optmzato techques s solved. We also compare the results Secto 3. Fally coclusos are draw Secto 4.

3 AAM: Iter. J. Vol. Issue (Jue 6) 47. obb-douglas Based Frm Producto Model Fuzzy Evromet I ths paper we cosder a frm that uses puts (e.g. labour captal coal ro) to produce oe sgle output q. Suppose p s the cost / ut of output. The frm producto fucto may be expressed as q= f x x x. It gves output as a fucto of puts the followg form: f x x x ax. (.) Here deotes the output elastcty of put compoets x Therefore total reveue amout s of the form: pq pax. Aga f r are the prces of the puts x gve by the followg expresso: x x x r x.. total expedture cost s I ths paper we pla to maxmze total reveue uder total lmted expedture cost. osequetly as per Lu (6) obb-douglas based frm producto model uder crsp evromet may be take as follows: Maxmze x x x pax subject to the costrats: x x x r x c x. (.) Usg the method descrbed by Duff et al. (967) geometrc programmg (GP) techque ca be appled to solve model (.). Next we may cosder the obb Douglas producto model uder fuzzy evromet where costrats are fuzzy form as follows: x x x x x x Maxmze x x x pax subject to the costrats: x x x r x c wth maxmum allowable toleraces c x. (.3) Here membershp fucto of fuzzy costrat

4 47 Palash Madal et al. s of the form: x x x x x x x x x x x x x x x f x c c c c x f c x c c c f x x x c. Next we may apply dfferet fuzzy optmzato techques o model (.3). Method.. Verdegay s approach (98) Accordg to Verdegay s approach (98) o fuzzy optmzato techque model (.3) reduces to followg parametrc optmzato model: Maxmze x x x pax subject to the costrats: r x c c [] x. The prmal geometrc programmg problem (PGPP) of the above model s as follows: Mmze pa x subject to the costrats: rx c ( ) c x. (.4) Model (.4) s a posyomal geometrc programmg problem whose degree of dffculty (DD) s zero. Its dual geometrc programmg problem (DGPP) s as follows: δ Maxmze d δ δ δ δ paδ r ( c ( ) c ) subject to the costrats: δ δ. The optmal soluto of ths problem s obtaed as * * for. It may be oted that although software ca be used to fd optmal solutos we have used oly pe ad paper to fd optmal solutos. Aga from the prmal dual relatos we have:

5 AAM: Iter. J. Vol. Issue (Jue 6) 473 ad x δ d δ δ δ δ pa * * * * * * Hece optmal puts are rx c( ) c * *. The optmal reveue s as follows: * ( c( ) c ) x ( ). r ( ) c c * * * * ; x x x pa. r Method.. Werer s approach (987) Frst model (.3) s solved wthout tolerace by GP techque. The t s solved wth tolerace by GP techque. Suppose that reveue wthout tolerace ad wth tolerace s ad respectvely. Fally fuzzy o-lear programmg problem s obtaed as follows: x Maxmze x x pax [ ] subject to the costrats: x x x rx c wth maxmum allowable toleraces c x. Therefore our task s to fd: x subject to the costrats: x x x pax wth maxmum allowable tolerace ( ) x x x r x c wth maxmum allowable tolerace c x. The fuzzy goal objectve fucto s gve by ts lear membershp fucto s as follows: x x x x { x ( x )} ;

6 474 Palash Madal et al. f x x x x x x x x x x ( x ) f x f x x x. The costrat s also fuzzy ad s gve by x x x x x x. Here our task s to fd x so as to maxmze the mmum of x x x ad x x x ad x. Method.3. Zmmerma s approach (976) Next model (.3) s solved by usg max-m operator developed by Zmmerma (976). Suppose mmum ( x x x) x x x The the sgle objectve optmzato problem s as follows: Maxmze c c x x x x x x subject to the costrats: c x.... The takg the verse of the objectve fucto we obta the posyomal geometrc programmg problem whose DD s. We solve t by usg GP techque. The dual of the problem s obtaed as follows:. δ r Maxmze d δ δ δ δ δ δ = δ ( c c ) c ( c c ) pa pa

7 AAM: Iter. J. Vol. Issue (Jue 6) 475 subject to the costrats :. (.5) Aga by usg pe ad paper the optmal soluto of model (.5) s obtaed as follows: * ( ) ( ).. Now substtutg * ( ) model (.5) the dual fucto s obtaed as follows: ( ) r Maxmze d( ) ( cc )( ) ( ) c ( c c )( ) pa ( ) ( ) pa. (.6) To fd the optmal values of we have to maxmze the dual objectve fucto d. Takg logarthms o both sdes of model (.6) ad dfferetatg partally wth ( ) respect to.e. oe by oe ad ext equatg those to zero we obta: log d ad log d ad r log log ( c c )( ) pa log log

8 476 Palash Madal et al. Aga we get: r c log log ( c c )( ) ( c c )( ) log log pa d. log log log d log d log d Here we may observe that. log d. log d log d. Method.4. Sakawa s method (993) Next model (.3) s solved by usg Sakawa s (993) method. Assumg that x x x mmum ( x x x ) x x x model (.3) becomes: Maxmze x x x subject to the costrats: x x x x x x x for x x x x x x []..e.

9 AAM: Iter. J. Vol. Issue (Jue 6) 477 pax Maxmze x x x subject to the costrats: rx c c pac c c x x for. (.7) Here x x x ' ' x x x To solve model (.7) by geometrc programmg techque we rewrte the problem as follows:. Mmze x subject to the costrats: r x x x. (.8) where pac = pa c c c c We fd that model (.8) s a posyomal PGPP wth DD beg. Its DGPP form s as follows:. Maxmze d δ δ δ δ r subject to the costrats:. (.9) The optmal soluto to model (.9) s obtaed as follows: *. * Now substtutg for model (.9) the dual fucto s obtaed as follows: d r. (.)

10 478 Palash Madal et al. To fd optmal soluto we have to maxmze the dual fucto d. Takg logarthms o both sdes of equato (.) ad dfferetatg wth respect to ad the equatg to zero we get: that s d d log d log log r log. Aga sce as d d logd Although Sakawa s (993) approach ad Zmmerma s (976) approach are smlar oe ma dsadvatage of Zmmerma s (976) method over Sakawa s (993) method s the crease degree of dffculty of the model Zmmerma s (976) method. It makes the model dffcult to solve GP techque uder fuzzy evromet. O the other had the advatage of Zmmerma s (976) method over Sakawa s (993) method s that oly oe problem eeds to be solved Zmmerma s (976) method but two problems eed to be solved Sakawa s (993) method. I ths paper tetoally we have solved oly oe problem. The other problem of Sakawa s (993) method ca be solved smlarly. Method.5. Max-addtve operator (987) Next we solve model (.3) usg max-addtve operator (987) as follows: x x x x x x x x x x x x maxmze subject to the costrats: x for...e. pa maxmze x r x c subject to the costrats: x.

11 AAM: Iter. J. Vol. Issue (Jue 6) 479 Now f pa c x r x x our problem becomes: maxmze subject to the costrats: x pa c x r x x x for. (.) ewrtg model (.) as PGPP form we get: mmze x subject to the costrats: x r x x x c pa pa x for. (.) Model (.) s a posyomal PGPP wth DD beg zero. Its DGPP s as follows: Maxmze d δ δ δ δ r c pa pa subject to the costrats: j j. The optmal solutos to the problem are * * *. Therefore d* δ * δ* δ* δ * r. pa c From prmal dual relatos we get:

12 48 Palash Madal et al. x δ* d* δ * δ* δ* δ * ad Hece * c x r x. pa * * x x pa. * Here the optmal puts are obtaed as follows: * pa x c for. r r ad the optmal reveue s obtaed as * * * * pa pa c. r r x x x Method.6. Max-product operator (978) Next we solve model (.3) usg max-product operator (978). Applyg max-product operator (978) the model becomes:.e. x x x x x x maxmze. subject to the costrats: x x x x x x x for.

13 AAM: Iter. J. Vol. Issue (Jue 6) 48 pax c c r x Maxmze. c pax c c r x subject to the costrats: c x for. Suppose that pax c c r x x x. c osequetly the above model becomes: Maxmze x. x pax c c rx x x c subject to the costrats: x for. (.3) Equato (.3) ca be wrtte PGPP form as follows: mmze x x subject to the costrats: c r x x c c c c x x x pa pa x for. (.4) Model (.4) s a posyomal PGPP wth DD beg uty. Therefore ts DGPP s as follows:

14 48 Palash Madal et al. Maxmze d δ δ δ δ δ δ r c c c pa pa c c subject to the costrats:. (.5) Usg pe ad paper optmal solutos of the model (.5) are obtaed as: * * *. Substtutg * * * for (.5) the dual fucto s obtaed as follows: ( ) r c d cc ( ) c c pa pa ( ) ( ) ( ) ( ). (.6) Next to fd optmal value of we maxmze the dual fucto: d. Takg logarthms o both sdes of model (.6) ad dfferetatg wth respect to ad ext equatg to zero we fd:.e. Hece we have: r d d l( d( )) l l l ( ) l. a ( c c )( ) pa

15 AAM: Iter. J. Vol. Issue (Jue 6) 483 d l d d ( ) ( ) ( ) ( ) ( ) ( ) We fd that the secod order dervatve s always egatve.. 3. Numercal Examples Now we cosder umercal examples o whch we may apply these optmzato techques ad solve obb-douglas based frm producto model uder fuzzy evromet. Aalogous to reese () the put data are take as gve Table. The output data obtaed by usg crsp optmzato techque to solve obb-douglas based frm producto model are gve Table. No. of Iputs Table. Iput data of obb-douglas based frm producto model Output elastcty of the Iput compoets Prces of the put compoets Sellg prce of a ut product Total productvty Avalable cost α α α 3 r r r 3 p a c Table. Output data usg crsp optmzato techque Dual Varables Prmal Varables eveue δ δ δ δ 3 x x x Next suppose that the put data uder fuzzy evromet s gve Table 3. No. of Iputs Table 3. Iput data of obb-douglas based frm producto model fuzzy evromet Output elastcty of the Iput compoets Prces of the put compoets (s.) Sellg Prce of a ut product (s.) Total productvty Avalable cost (s.) Avalable tolerace (s.) α α α 3 r r r 3 p a c c O solvg the model uder fuzzy evromet by Verdegay s approach (98) output data correspodg to dfferet values of asprato level are obtaed as gve Table 4.

16 484 Palash Madal et al. Table 4. Output data of obb-douglas based frm producto model by Verdegay s approach (98) Asprato Level Dual Varables Prmal Varables ost (s.) eveue (s.) β δ δ δ 3 x x x O solvg the same model wth the same put data by max-m operator (Zmmerma 976) uder fuzzy evromet the output data s obtaed as gve Table 5. Table 5. Output data usg Zmmerma s approach (976) Dual Varables Prmal Varables Optmal eveue Optmal ost Asprato level δ δ δ δ 3 δ 4 δ δ x x x μ ((x x )) ad μ ((x x )).5 ad.5 O solvg the same model wth the same put data by Sakawa s (993) method uder fuzzy evromet the output data s obtaed as gve Table 6. Table 6. Output data usg Sakawa s (993) method Dual Varables Prmal Varables Optmal eveue Optmal ost Asprato level δ δ δ δ 3 δ 4 x x x μ ((x x )) ad μ ((x x )).57 ad.43

17 AAM: Iter. J. Vol. Issue (Jue 6) 485 O solvg the same model wth the same put data by max-addtve (987) operator uder fuzzy evromet the output data s obtaed as gve Table 7. δ δ Dual varables δ δ 3 Table 7. Output data usg max-addtve (987) operator δ 4 Prmal varables Optmal eveue (s.) Optmal ost (s.) x x x Asprato level μ ((x x )) ad μ ((x x )).5 ad.5 O solvg the same model wth the same put data by max-product (978) operator uder fuzzy evromet the output data s obtaed as gve Table 8. δ. δ δ Dual Varables δ 3 Table 8. Output data usg max-product (978) operator δ 4 δ δ Prmal varables Optmal eveue (s.) Optmal ost (s.) x x x Asprato Level μ ((x x )) ad μ ((x x )).5 ad.5 Fally we may compare the results obtaed by usg dfferet fuzzy optmzato techques to solve obb-douglas based frm producto model uder fuzzy evromet. Method Zmmerma s approach (976) Sakawa s method (993) max-addtve operator (987) max-product operator (978) Table 9. omparso of outcomes dfferet techques Optmal Optmal Iputs eveue Optmal ost x x x Hece the optmal reveue classcal optmzato techque s s wth optmal cost s. 85. But f the same model s cosdered uder fuzzy evromet ad solved by usg max-m operator Zmmerma s (976) techque the optmal reveue comes as s wth optmal cost beg s As maxmzg reveue s a prmary objectve to decso makers ths outcome s more acceptable tha the soluto uder crsp evromet.

18 486 Palash Madal et al. Aga f max-m operator Sakawa s (993) techque s used to solve the same model uder fuzzy evromet the optmal reveue s s a far more acceptable soluto tha the soluto uder crsp evromet. If max-addtve (987) operator s used to solve the same model uder fuzzy evromet the optmal reveue s s aother better optmal soluto tha crsp soluto. If max-product (978) operator s used to solve the same model uder fuzzy evromet the optmal reveue s s aga oe better optmal soluto tha crsp soluto. 4. ocluso I ths paper we have cosdered obb-douglas producto fucto based model a frm uder fuzzy evromet ad ts soluto techque by makg use of geometrc programmg. Here the objectve s to maxmze the reveue uder lmted total expedture cost ad to mmze the total expedture costs subject to target reveue. To match wth realty the model s cosdered uder fuzzy evromet ad solved usg dfferet fuzzy optmzato techques. I ths paper geometrc programmg s appled to solve the model obtaed by fuzzy optmzato techques. The advatage of geometrc programmg over other optmzato techques s that t provdes us wth a systematc approach for solvg a class of o-lear optmzato problems by fdg the optmal value of the objectve fucto ad the the optmal values of decso varables are obtaed. Moreover GP ofte reduces oe complex optmzato problem to set of smultaeous lear equatos. We kow that a decso maker s the kg ad hs decso s fal. Accordgly ths paper we collect formato from a decso maker; the based o such formato fuzzy optmzato approach s chose. The GP s used to fd optmal soluto. The optmal soluto s preseted to the decso maker. If he/she s satsfed wth the soluto stop. Otherwse aother fuzzy techque may be used. We stop whe the decso maker s satsfed. We have ot used ay software but oly pe ad paper to compute the optmal solutos by usg geometrc programmg techque. Software avalable o the market ca also be used to fd the optmal soluto. We further pla to develop a few terestg results o obb-douglas based frm producto model fuzzy evromet. We also pla to use the proposed techque to geerate optmal solutos agro-dustral sector ear future. Ackowledgemet: Ths research work s supported by Uversty Grats ommsso Ida vde mor research project (PSW-7/3-4 (W-3) (S.N. 963)). The secod author scerely ackowledges the cotrbutos ad s very grateful to them.

19 AAM: Iter. J. Vol. Issue (Jue 6) 487 EFEENES Bellma. E. ad Zadeh L. A. (97). Decso makg a fuzzy evromet Maagemet Scece Vol.7. ao B. Y. (). Optmal Models ad Methods wth Fuzzy Quattes Studes Fuzzess ad Soft omputg Vol. 48. Sprger. reese.. (). Geometrc Programmg for Desg ad ost Optmzato wth Illustratve ase Study Problems ad solutos Secod Edto Morga & laypool Publshers. Duff. J. Peterso E. L. ad Zeer. (967). Geometrc Programmg: Theory ad Applcato New York: Wley. Guey I. ad Oz E. (). A Applcato of Geometrc Programmg Iteratoal Joural of Electrocs; Mechacal ad Mechatrocs Egeerg Vol. No.. Gara A. Madal P. ad oy T. K. (6). Itutostc fuzzy T-sets based optmzato techque for producto-dstrbuto plag supply cha maagemet OPSEAH (Accepted). Gara A. Madal P. ad oy T. K. (5). Itutostc fuzzy T-sets based soluto techque for multple objectve lear programmg problems uder mprecse evromet Notes o Itutostc Fuzzy Sets Vol. No. 4. Gara A. Madal P. ad oy T. K. (5). Iteractve tutostc fuzzy techque mult-objectve optmzato Iteratoal Joural of Fuzzy omputato ad Modellg (Accepted). Lu S. T. (6). A Geometrc Programmg Approach to Proft Maxmzato Appled Mathematcs ad omputato Vol.8. Sakawa M. Yao H. ad Nshzak I. (3). Lear ad Mult-objectve Programmg wth Fuzzy Stochastc Extesos. Iteratoal Seres Operato esearch & Maagemet Scece Vol. 3 Sprger. Shvaa E. Keshtkar M. ad Khorram E. (). Geometrc Programmg Subject to System of Fuzzy elato Iequaltes Applcatos ad Appled mathematcs Vol. 7 No.. Werer B. (987). Iteractve multple objectve programmg subject to flexble costrats Europea Joural of Operatoal esearch Vol. 3. Werer B. (987). A teractve fuzzy programmg system Fuzzy Sets ad Systems Vol. 3. Zadeh L. A. (965). Fuzzy sets Iformato ad otrol Vol. 8 No Zmmerma H- J. (976). Descrpto ad Optmzato Of Fuzzy Systems Iteratoal Joural of Geeral Systems Vol. No..

20 488 Palash Madal et al. Authors bography Palash Madal was bor Kachukhal South 4 Pargaas West Begal Ida. urretly he s Juor esearch Fellow the Departmet of Mathematcs Ida Isttute of Egeerg Scece ad Techology Shbpur Howrah West Begal. Hs research terests clude multobjectve optmzato optmzato mprecse evromet vetory models ad fuzzy set theory. Ardam Gara was bor ad brought up Burdwa. urretly he s Assstat Professor ad Head Departmet of Mathematcs Soarpur Mahavdyalaya West Begal Ida. Hs research terests clude fuzzy set theory fuzzy optmzato mult-objectve optmzato portfolo optmzato stochastc optmzato etc. Tapa Kumar oy receved Ph. D. from Vdyasagar Uversty W.B. Ida. urretly he s wth Departmet of Mathematcs Ida Isttute of Egeerg Scece ad Techology Shbpur WB Ida. He has more tha hudred publcatos ad has guded may studets to Ph.D. degree tll date. Hs research terests clude fuzzy set theory mult-objectve optmzato vetory problems fuzzy relablty optmzato portfolo optmzato stochastc optmzato etc.