Gen. Math. Notes, Vol. 4, No., June 0, pp. -9 ISSN 9-784; Copyright ICSRS Publiation, 0 www.i-srs.org Available free online at http://www.geman.in An Integer Solution of Frational Programming Problem S.C. Sharma an Abha ansal Department of Mathematis, University of Raasthan, Jaipur-0055, Inia. E-mail: sureshhan6@gmail.om Department of Mathematis, Dungurpur Engineering College & Tehnology, Dungurpur (Ra.)- 400, Inia E-mail: neelamhikusinghal00@gmail.om (Reeive: -6-/Aepte: -06-) Abstrat The present paper esribes a new metho for solving the problem in whih the obetive funtion is a frational funtion, an where the onstraint funtions are in the form of linear inequalities. The propose metho is base mainly upon simple metho, whih is very easy to unerstan an apply. This an be illustrate with the help of numerial eamples. Keywors: Frational programming, Simple metho. Introution Frational programming problem is that in whih the obetive funtion is the ratio of numerator an enominator. These types of problems have attrate onsierable researh an interest. Sine these are useful in proution planning, finanial an orporate planning, health are an hospital planning et. Algorithms for solving linear frational programming problems are well known by many authors [,, ]. Charnes-Copper [] replaes a linear frational program by one equivalent linear program, in whih one etra onstraint an one etra variable has been ae. The usual simple algorithm omputes the optimum solution. Isbell-Marlow an Martos [, 4] fin the solution of a sequene of linear programs. Wagner-Yuan [8] showe that, when the feasible set is boune. Chhas-Rahael [] solves a system of linear of inequalities in whih
S.C. Sharma et al. the obetive funtion is epresse as one of the onstraint along with the given set of linear onstraints of the problem. Resently Tantawy [7] has suggeste a feasible iretion approah an a uality approah to solve a linear frational programming problem. Here our aim is to fin the integer solution of frational programming problems (i.e. obetive funtion is the ratio of numerator an enominator of linear funtions). For it, we use simple metho an branh an boun metho. These methos are very easy to unerstan an apply. Preliminaries are given in the net setion. The steps of the propose algorithm are presente in setion. Numerial eamples have been worke out in setion 4 of the paper an finally in setion 5 we present the referenes. Preliminaries A maimization integer frational programming problem may be state as: Ma. Z (.) s.t. A b 0 an an integer where,, are n vetors, b is an m vetor,, enote transpose of vetors, an is an m n matri an α, β are salars onstants. It is assume that the onstraint. S { : A b, 0 an integer} is non- empty an boune. Algorithm Step : first, observes whether all the right sie onstants of the onstraints are non-negative. If not, it an be hange into positive value on multiplying both the sies of the onstraints by -. Step : net onverts the inequality onstraints to equations by introuing the non-negative slak or surplus variables. The oeffiients of slak or surplus variables are always taken zero in the obetive funtion. Step : onstruts the simple table by using the following notations. Let be the initial basi feasible solution of the given problem suh that b b where ( b, b,..., br, bs,..., bm ). Further suppose that Z Z where an are the vetors having their omponents as the oeffiients assoiate with the basi variables in the numerator an enominator of the obetive funtion respetively. Step 4: Now, ompute the net evaluation for eah variable (olumn vetor ) by the formula
An Integer Solution of Frational () Step 5: If all 0, the optimal solution is obtaine. Step 6: If optimal solution is an integer solution then we get the require solution otherwise use ranh an oun metho. 4 Numerial Eample Eample. Fin the integer solution of following frational programming problem:- ma. Z + + s.t. + + 4 Solution: 0 an an integer., After aing slak variables an 4, the onstraints beome + + + + 4 0 4,,, 4 Table- 0 0 Min. ratio - 0 0 i y 4 0 0-0 / 0 0 4 4 0 4/ i Z 0 Z Z 0 () Z 0 0 Z - 0 0 4 0 0
4 S.C. Sharma et al. Where Z 0 0 4 Z 0 + 0 4 0 () Sine Z 0 an Z, therefore ( ) as shown in table. () Z Table- 0 0-0 0 4 - -/ / 0 0 0 4 / 0 -/ Z Z Z Z 0-0 Z / 0 / 0-0 - 0 Z 0 ) Z ( + 0
An Integer Solution of Frational 5 () () - (/) - 0 (-) - (/) - 4 0 Here 0, an Z Z / Z. Sine all 0, therefore the urrent solution is the optimal basi feasible solution. Eample. Fin the integer solution of following frational programming problem:- s.t. 5 + 7 + ma. Z 6 6 + + + 6, 0 an an integer. Solution: After aing slak variables an 4, the onstraints beome 5 + + 6 7 + + 6 0 4,,, 4 Table- 0 0 Min. ratio 0 0 4 0 0 6 5 0 6/ 5 0 0 4 6 0 6/7 7 i y i Z 0 Z 6 Z 0
6 S.C. Sharma et al. () Z 0 0 Z 0 0 6 0 0 Where Z 0 6 0 6 Z 0 6 + 6 0 6 0 6 () Sine Z 0 an Z 6, therefore 6( ) as shown in table. () Z Table-4 0 0 Min. ratio 0 0 i y 4 0 0 /7 0 6/7-5/7 /4 6/7 /7 0 /7 6 i Z /7 Z 60/7 Z /5 Z 0 5/7 0 -/7 Z 0 4/7 0 -/7 0 5/49 0 -/7
An Integer Solution of Frational 7 Z 0 / 7 6 / 7 /7 Z 0 / 7 + 6 6 / 7 60/7 () () 0 () 60 5 4 5 7 7 7 7 49 () 0 () 4 60 + 7 7 7 7 84 49 7 Table-5 0 0 0 0 4 /4 0 7/6-5/6 /4 0 -/6 /6 Z 9/4 Z/4
8 S.C. Sharma et al. Z 9 Z 0 0-5/6 -/6 Z 0 0 -/4 -/4 0 0-9/4 0 Z () / 4 / 4 9/4 Z () / 4 + 6 / 4 9 () () 0 () 0 Here /4, /4 an Z Z / Z /4. Sine all 0, therefore the urrent solution is the optimal basi feasible solution. ut an are not integer value so make them integer, we use ranh an oun metho
An Integer Solution of Frational 9 Z /4 /4 /4 Z /4 0 Not feasible Hene we get the integer solution of the given problem. The optimal solution is 0, an ma. Z /4. Referenes [] Carnes an W.W. Cooper, Programming with linear frational funtional, Naval Researh Logistis Quartely, 9 (96), 8-86. [] S.S. Chaha, V. Chaha an R. Calie, A stuy of linear inequalities appliations an algorithms, Presente at the International Conferene on Operation Researh for Development [ICORD], Anna University, Chennai, Inia, Deember (00), 7-0. [] J.R. Isbell an W.H. Marlow, Atrition games, Naval Researh Logistis Quartely, (956), -99. [4] Martos, Hyperboli programming, Naval Researh Logistis Quartely, (964), 5-55. [5] J.K. Sharma, A.K. Gupta an M.P. Gupta, Etension of simple tehnique for solving frational programming problems, Inian J. Pure appl. Math., (8) (980), 96-968. [6] K. Swarup, Linear frational funtional programming, Operation Researh, (965), 09-06. [7] S.F. Tantawy, Using feasible iretions to solve linear frational programming problems, Australian Journal of asi an Applie Sienes, (007), 09-4. [8] H.M. Wagner an J.S.C. Yuan, Algorithm equivalene in linear frational programming, Management Siene, 4 (968), 0-06.