A Parametric Approach to Solve Bounded-Variable LFP by Converting into LP

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1 Iteratioal Joural of Operatios Research Iteratioal Joural of Operatios Research Vol., No., 0707 (06 A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP Saal Chakroborty ad M. abul Hasa Departmet of Electroics ad Commuicatios Egieerig, East West Uiversity, Dhaka, agladesh Departmet of Mathematics, Uiversity of Dhaka, Dhaka, agladesh Received April 06; Revised Jue 06; Accepted Jue 06 Abstract I this paper, we have developed a ew techique for solvig bouded-variable liear fractioal programmig problems by covertig ito liear programmig problems. I this techique, we have proposed to covert fractioal obective fuctio ito liear obective fuctio by makig a relatioship betwee the umerator ad deomiator with a parameter. A umber of umerical eamples are illustrated to demostrate our techique. We have also developed a computer code by usig a mathematical programmig laguage AMPL ad the preset a compariso of our method with eistig relevat methods. Keywords Liear Programmig, ouded variable, AMPL, Liear Fractioal Programmig, Obective Fuctio.. INTRODUCTION Liear fractioal programmig (LFP with bouded-variables is a special kid of mathematical programmig. It cosists of a obective fuctio which is the ratio of two liear fuctios ad some liear costraits with bouded variables. From last three decades LFP problems are gettig lots of attetio due to its importace i modelig various decisio processes i ecoomics, maagemet sciece, umerical aalysis, stochastic programmig, ad decompositio algorithms (Stachu-Miasia, 997. Hugaria mathematicia ela Martos (96 first formulated LFP. Mesiter ad Oettli (967, Aggrawal ad Sharma (970 applied the idea of fractioal programmig to calculate the maimum trasmissio rate i a iformatio chael. ereau (96 applied the idea of LFP i stochastic programmig problem. Uder certai ecoomic assumptios, Ziemba, rooks-hill ad Parkace (97 developed a fractioal programmig model for a ivestmet portfolio. There are may eistig techiques for solvig LFP problems. A. Chares ad W. W. Cooper (97 developed a trasformatio techique for solvig LFP problems by covertig it ito sigle liear programmig problems (LP. W. Dikelbach (aaliov, 00 developed a parametric approach to solve LFP. itra ad Novaes (97 developed a method called updated obective fuctio method to solve LFP by solvig a sequece of liear programs oly re-computig the local gradiet of the obective fuctio. Hasa ad Acharee (0 proposed a techique for solvig LFP by covertig ito LP. Das ad Hasa (0 developed a techique for solvig bouded-variable LFP problems. Das, Hasa ad Islam (0 proposed aother techique to solve bouded-variable LFP problems by covertig ito LP problems. ouded-variable LFP problems are more difficult to solve tha LFP. I this paper, we develop a ew techique for solvig bouded-variable LFP. We use the idea of Dikelbach s parametric trasformatio for solvig LFP ad the use the idea of the bouded valued simple algorithm for solvig LP problems to develop our techique. We use a mathematical programmig laguage AMPL to develop a computer code accordig to our algorithm. There are may eistig computer techiques that are developed by usig MATHEMATICA. ut our developed computer techique is easier to use ad ru. It is less time-cosumig tha other eistig methods. To show that i this paper, we have made a compariso betwee other eistig methods ad our method. The rest of the paper, we have orgaized as follows. I sectio, we discuss defiitio of LFP, bouded-variable LFP ad a relatio betwee LP, LFP ad bouded-variable LFP. I sectio, we discuss briefly about some eistig techiques. I sectio, we preset our proposed method. I sectio, we preset a computer code that we have developed by usig a mathematical programmig laguage AMPL. I sectio 6, we preset some umerical eamples ad solve these by usig our method. I sectio 7, we show a parallel represetatio of maual output ad the AMPL output of the umerical eamples. I sectio 8, we preset a compariso betwee our method ad eistig methods. Fially, i sectio 9, we have draw a coclusio about our work. Correspodig author s scb@ewubd.edu 8-7X Copyright 06 ORSTW

2 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 (06 8. SOME PREREQUISITE I the curret sectio, we preset some basic defiitios relevat to our work. First, we discuss about liear fractioal programmig (LFP. The bouded-variable LFP ad fially discuss a relatio betwee liear programmig (LP, LFP ad bouded-variable LFP. We have discussed these as follows.. Liear Fractioal Programmig (LFP Problems of LFP arise whe it becomes importat to optimize the efficiecy of some activities. Mathematically LFP problems ca be defied as follows (Stachu-Miasia, 997. Maimize (or Miimize subect to z p d + α + β ai (,, bi, i,..., m ( 0,,..., ( m m Where, p, d R, ai R R is a m matri bi R ad α, β R. It is assumed that d + β 0. Equatio ( is represetig the obective fuctio which has to be maimized or miimized. It is the ratio of two liear fuctios. Equatio ( is represetig the liear costraits ad equatio ( is represetig o-egative restrictios. I the et sectio, we have discussed about bouded-variable liear fractioal programmig problem.. ouded-variable LFP ouded-variable LFP is a special kid of LFP. Here decisio variables are bouded above ad below. Mathematically bouded-variable LFP is beig formulated as follows (Stachu-Miasia, 997. Maimize (or Miimize subect to z p d + α + β ai (,, bi, i,..., m ( l u,,..., (6 From equatio (6, we observe that decisio variables are bouded above by u ad below by l. LFP ad bouded-variable LFP problems have a close relatio to LP problems. ouded-variable LFP problems ca be coverted ito LP problems. I the et sectio, we have discussed briefly the relatioship betwee LFP ad bouded-variable LFP with LP.. Relatio etwee LP, LFP ad ouded-variable LFP LFP problems ad bouded-variable LFP problems ca be solved by covertig ito LP problems. I the curret sectio, we have discussed some coversio procedures of LFP ad bouded-variable LFP ito LP (aaliov, 00 as follows. Case-: Cosider the obective fuctio preseted i equatio ( or (. If d 0, β i either of these equatios the the fractioal obective fuctio will be coverted ito a liear obective fuctio i.e. equatio ( ad ( will be coverted ito a liear fuctio as (7 z p + α Case-: If d 0, β the equatio ( ad ( will be coverted ito liear obective fuctio as follows. ( ( 8-7X Copyright 06 ORSTW

3 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 (06 9 p α z + Case-: If p 0 the equatio ( ad ( will become (8 β β z α. Here d + β is a liear fuctio. d + β Therefore equatio ( ad ( ca be coverted ito liear fuctio i this way. I the et sectio, we have discussed some solutio procedures of LFP ad bouded-variable LFP problems.. LITERATURE REVIEW There are may eistig techiques to solve bouded-variable LFP ad LFP problems. I the curret sectio, we discuss about some eistig techiques. We discuss about Das ad Hasa s method, Chares ad Cooper s method, Hasa ad Acharee s method, itra ad Novae s method ad Das, Hasa, Islam s method briefly.. Das ad Hasa s Method H. K. Das ad M. abul hasa (0 developed a techique for solvig bouded-variable LFP problems. They eteded the idea of bouded-variable simple algorithm for solvig LP problems to solve bouded-variable LFP problems. They also developed a computer code by usig MATHEMATICA accordig to their algorithm. Their techique has bee discussed briefly as follows. Step-: Coverted bouded-variable LFP problem ito stadard form. Step-: The performed ( ( z p z z d z where z p + α, z d + β util 0. Step-: Applied idea of bouded variable simple algorithm. Their techique is very similar to Swarup s method (96 for solvig LFP problems.. Chares ad Cooper s Method Chares ad Cooper (97 developed this method. They developed this techique for solvig LFP problems by covertig ito LP problems. Their techique has bee discussed briefly as follows. Step-: Cosidered the trasformatio y t where t 0 Step-: The coverted LFP ito two differet LP problems. Step-: Optimal solutio of LFP will be foud if two of these LP problems have a optimal solutio. If oe of them is icosistet or ubouded the LFP will have o optimal solutio. Demerits This method is oly applicable for solvig LFP problems ot for bouded variable LFP problems. This techique is also very time-cosumig because oe has to solve two differet LP problems.. Hasa ad Acharee s Method M. abul Hasa ad Sumi Acharee (0 developed this techique. They proposed a trasformatio system to covert LFP ito LP. We have discussed their techique as follows. Step-: I equatio (, deomiator fuctio must have to be positive i.e. d + β > 0. Step-: They cosidered t p d g, 8-7X Copyright 06 ORSTW y α ad g. d + β β Step-: The the obective fuctio of the LFP becomes z py + g. They calculated variable values of LFP by β y. d y Demerits Like Chares ad Coopers method, this techique is applicable for solvig LFP problems oly. Although this techique is very iterestig but it is laborious ad time cosumig because oe has to do a lots of calculatios to covert LFP problem ito sigle LP problem.

4 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 (06 0. itra ad Novae s Method I this sectio, we have summarized itra ad Novae s method. I (itra, 97, they developed this. Assumig that the costraits set is oempty ad bouded ad the deomiator proceeds as follows. (i Covert the LFP ito a sequece of LP. (ii The solve these LPs util two of them give idetical solutio.. Das, Hasa ad Islam d + β > 0 for all feasible solutios, the authors Das, Hasa ad Islam (0 proposed a techique to solve bouded-variable LFP by covertig ito LP problem. They used Has ad Acharee s (0 trasformatio idea to covert LFP ito LP problem. The they used idea of bouded-variable simple method for solvig LP problems. They developed a computer code by usig MATHEMATICA. They demostrated their techique by illustratig a umber of umerical eamples..6 Dikelbatch s Method This is the most popular techique for solvig LFP problems (aaliov, 00. I this sectio, we discuss this method briefly as follows. Step-: Covert obective fuctio ( ito a liear form as z ( p + α t( q + β 8-7X Copyright 06 ORSTW Step-: Solve it by usig the simple algorithm. Step-: Solve by usig simple algorithm util the modified obective fuctio gives zero value. I the et sectio, we preset algorithm of our proposed method. We have used the idea of Dikelbatch s techique ad the bouded-variable simple algorithm s (Hillier, 00 idea to develop this method.. PROPOSED METHOD I this sectio, we have preseted our proposed method. Cosider the bouded-variable LFP preseted i Sectio.. We first covert the obective fuctio ito a liear form. The substitute those variables whose lower bouds are ot zero by usig aother variable. The covert the whole problem ito stadard form. The apply the idea of bouded-variable simple method. We have discussed our method i the followig steps. Step-: Covert the obective fuctio preseted i equatio ( ito liear fuctio as follows. Where t ad k is represetig umber of iteratios. α (9 f ( p + t ( d + β Step-: Cosider l + y i equatio (6. The the bouded-variable LFP problem will be coverted ito the, followig form where g p l + α t ( d l + β Step-: Cosider t i i γ b a l, ad v u l. Maimize (or Miimize f p y t d y + g subect to p l + y + α ( ( (0 ai y (,, γ i, i,..., m ( 0 y v,,..., ( ad choose ( k y 0 iitially i.e. whe k. d l + y + β Step-: Covert the problem ito stadard form preseted by equatios (0, (, ad (.

5 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 (06 Step-: If ay variable is at a positive lower boud the it should be substituted at its lower boud. The apply idea of simple method. ( Step-6: Let y k be a o basic variable at zero level which is selected to eter the basic. Compute the followig ( y i ui ( y i quatities. θ mi, ai > 0, θ mi, ai < 0, ad θ mi( θ, θ, u. Here ( y i are basic variables. ai ai Step-7: Now we have to do ay oe of the followig three alteratives. Ifθ θ the ( y r leaves the solutio ad y eters by usig the regular row operatio of simple method. Ifθ θ the ( y r leaves the solutio ad y eters. The ( y r beig o-basic at its upper boud ad ' ' must be substituted by ( y u ( y,0 ( y u. r r r r r ' Ifθ u the y is substituted at its upper boud differece u y, while remaiig o-basic. Step-8: If f 0 ad all 0 the stop. Otherwise repeat steps to 7. Here E p ai ad p E.. COMPUTER CODE I the curret sectio, we preset a computer code. We develop this by usig a mathematical programmig laguage AMPL. Our code cosists of three differet parts. These are AMPL model file, AMPL data file ad AMPL ru file. We preset AMPL model file below. # #AMPL Model File # param ; param m; param alpha; param beta; param lembda; param c{i i..}; param d{i i..}; param a{i i..m, i..}; param b{i i..m}; param lw{i i..}; param up{i i..}; var {i i..}>0; # o of variables # o of rows # umerator costat # deomiator costat # a parameter # umerator's coefficiets #deomiator's coefficiets # costrait's coefficiets # r.h.s costats # lower bouds # upper bouds # o of variables maimize obv: sum{i i..} (c[i][i]+alpha-lembda(sum{i i..}(d[i][i]+beta; subect to cost{ i..m}: sum{i i..} a[,i][i]<b[]; subect to limit{i i..}: lw[i]<[i]<up[i]; # X Copyright 06 ORSTW

6 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 (06 Due to the large volume of whole computer code we have preseted AMPL model file here. If readers are iterested the please cotact with the authors. 6. NUMERICAL EXAMPLES I this sectio, we preset some umerical eamples ad apply our proposed method to solve these. 6. Numerical Eample : This Eample has take from Erik. aaliov (00. Maimize subect to z + + ( + 0 ( + 60 ( (6 0 (7 Solutio: Suppose, + y ad + y where y 0, y 0. Substitutig these ito the equatios ( to (7 ad covertig the obective fuctio ito liear form ad covertig the problem ito stadard form we obtai the followigs. Maimize f ( y + y + t ( y + y + (8 subect to y y + s (9 Here s, s 0 are slack variables ad ( t t y + y + s 8 (0 y y 0 y 0 ( 0 y 6 ( y + y Iitially let y ( i 0, i,. Now we have for the et iteratio. The equatio (8 will be coverted ito the followig form. 6 + ( 7 ( f y y Solutio steps for this problem have bee preseted ito the followig tables. Table : Iitial Table for eample p p asis y y s s y θ θ 0 s E i s 0 8 p a p E ( f 0 Discussio: From Table-, we observe that maimum value of has obtaied at fourth colum. Therefore y is the 8 eterig variable ito the solutio. It has bee idicated by a vertical arrow i the table. Hereθ, θ, ad 8-7X Copyright 06 ORSTW

7 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 ( θ mi{ θ, θ, u} mi{,, 6}. Therefore, s will leave basis ad y will eter ito that place. Although ( f 0 here but sice ot all are egative so we have to go for the et iteratio. I the et table, we have preseted the optimal solutio. p p Table : Optimum table for eample asis y y s s y 0 s 0 8 y 0 8 E p a i p E F( y 6 ( Discussio: Here f y + y F( y. From Table-, we see that all are egative ad ( f 0. Therefore, optimal solutio has obtaied. Optimal solutio is y 0, y. Therefore, ,, zma Numerical Eample : This Eample has take from Erik. aaliov ( Maimize z + + subect to Solutio: Cosider, + y ad + y where y 0, y 0. Substitutig these ito the equatios ( to (0 ad covertig the obective fuctio ito liear form ad covertig the whole problem ito stadard form we obtai the followig epressios. Maimize f (y + y + t ( y + y + 7 ( subect to y + y + 6 ( y y + 0 ( 0 y ( 0 y 8 ( 0 (6 8-7X Copyright 06 ORSTW ( ( + (6 (7 (8 0 (9 0 8 (0

8 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 ( (7 y + y + Here t ad choose y ( i 0, i, iitially. The equatio ( will be, y + y + 7 have preseted the iitial calculatios for this problem ito the et table. p p 9 Table : Iitial Table for eample asis y y We 9 9 f y y y θ θ E p a i p E ( f 0 Discussio: From Table-, we observe that y is the eterig variable ito the basis ad is the departig variable from the basis. Sice ot all are egative so we have to go ito the et iteratio. We have preseted optimum solutio ito the et table. p 6 p Table : Optimum Table for eample asis y y y y E p a i F ( y p E 0 60 ( Discussio: Here f y y F( y. From Table-, we see that all are egative or equals to ( 7 7 zero ad f 0. Therefore, optimal solutio has obtaied. Therefore y 6, y 6 0, 0, ad optimal solutio of the bouded-variable LFP problem is 6, 6 0, 0,, zma X Copyright 06 ORSTW

9 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 (06 7. AMPL OUTPUT I this sectio, we solve umerical eamples preseted i Sctio-6., ad 6. by usig our developed AMPL code. I Table-, we preset maual output ad AMPL output of umerical eamples. We also preset CPU time used by our AMPL code. We have used _ampl_time commad to determie these times. Numerical Eamples Table : Maual ad AMPL outputs Maual Output AMPL Output Time Used (secods Eample 0 Variables,, 6 Obective, zma 7 [] : Obective, z ma Eample Variables,, 6 0, 0 Obective, zma [] :. 0. Obective, z ma Discussio: From parallel represetatio of maual output ad AMPL output of eamples, we observe that both the algorithm we developed ad computer code we developed are givig same output. I AMPL output, some variable s value ad obective fuctio value have bee preseted i decimal form. For eample, i Numerical eample- fourth variable value is.. I the et sectio, we preset a compariso betwee our method with other methods for solvig bouded-variable LFP problems. 8. COMPARISON I the curret sectio, we make a compariso betwee our method with other two differet methods for solvig bouded-variable LFP problems. I Sectio., we preset Das ad Hasa s method (0 ad i Sectio. we preset aother method developed by Das, Hasa ad Islam (0. They used MATHEMATICA to develop their computer code ad used Timeused[ ] commad to determie CPU time. To fid ru time we use _ampl_time_ commad. We have used Itel(R Petium(R Dual CPU Memory(RAM:.00G, System type: -bit operatig system. We develop a graphical compariso betwee these two methods ad our developed method o the base of CPU time used by computer code. We preset this compariso below. 8-7X Copyright 06 ORSTW

10 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 ( X Copyright 06 ORSTW Figure : Compariso betwee eistig methods ad our method I the above figure, we cosidered Das ad Hasa s (0 method as Method- ad Das, Hasa ad Islam s method (0 as Method-. We observe that Method- ad Method- used 0.9 ad 0. secods respectively to solve eample- ad it takes secods to solve by our developed method. For eample-, Method- ad Method- take.06 ad 0. secods respectively ad secods by our method. Therefore we ca coclude that our developed techique is easier ad less time cosumig tha other techiques. 9. CONCLUSION There are may eistig techiques to solve bouded-variable LFP problems. I this paper, we developed a ew techique to solve bouded-variable LFP. We used a parametric trasformatio to covert fractioal obective fuctio ito liear form. We developed a computer code usig a mathematical programmig laguage AMPL. We solved umerical eamples by our developed method. The compared the results obtaied by computer code. Fially, we made a compariso betwee our method with two other eistig methods. Fially, foud that our techique is easier ad took less time tha the other techiques. REFERENCES. Aggarwal, S. P., Sharma, I. C. (970. Maimizatio of the Trasmissio Rate of a Discrete, Costat Chael, Uterehmesforschug,, -.. ereau,. (96. Programmig De Risqué Miimal E Programmatio Lieaire Stochastique, C. R. Acad. Sci (Paris, 9 (, aaliov, E.. (00. Liear Fractioal Programmig: Theory, Methods, Applicatios ad Software, osto: Kluwer Academic Publishers.. itra, G. R., ad Novaes, A. G. (97. Liear Programmig with Fractioal Obective Fuctio, Uiversity of Sao Paulo, razil,, -9.. Chares, A., Cooper, W.W. (97. A Eplicit Geeral Solutio i Liear Fractioal Programmig, Naval Research Logistics Quarterly, 0(, Das, H. K., Hasa, M.. (0, A Proposed Techique for Solvig Liear Fractioal ouded Variable Problems, The Dhaka Uiversity Joural of Sciece, 60(, Das, H. K., Hasa, M.., Islam, A. (0. Solvig ouded Variable LFP Problems by Covertig it ito a Sigle LP Problem, Iteratioal Joural of Data Aalysis ad Iformatio System, (, Hasa, M.., Acharee, S. (0. Solvig LFP by covertig it ito a Sigle LP, Iteratioal Joural of Operatios Research, 8(, Hillier, F. S., Lieberma, G. J. (00. Itroductio to Operatios Research, MacGraw-Hill series i Idustrial Egieerig ad Maagemet Sciece. 0. Martos,. (96. Hyperbolic Programmig, Naval Research ad Logistics Quarterly,, -.

11 Saal Chakroborty ad M. abul Hasa: A Parametric Approach to Solve ouded-variable LFP by Covertig ito LP IJOR Vol., No., 0707 (06 7. Meister,., Oettli, W. (967. O the Capacity of a Discrete, Costat Chael, Iformatio ad Cotrol, (, -.. Stachu-Miasia, I. M. (997. Fractioal Programmig: Theory, Method, ad Applicatios, Kluwer Academic Publishers.. Swarup, K. (96. Liear Fractioal Fuctioal Programmig, Operatios Research, (6, Ziemba, W. T., rooks-hill, F. J., Parkace (97. Calculatios of Ivestmet Portfolios with Risk Free orrowig ad Ledig, Maagemet Sciece, (, X Copyright 06 ORSTW