AN ALGORITHM FOR RESTRICTED NORMAL FORM TO SOLVE DUAL TYPE NON-CANONICAL LINEAR FRACTIONAL PROGRAMMING PROBLEM

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1 RAC Univerity Journal, Vol IV, No, 7, pp 87-9 AN ALGORITHM FOR RESTRICTED NORMAL FORM TO SOLVE DUAL TYPE NON-CANONICAL LINEAR FRACTIONAL PROGRAMMING PROLEM Mozzem Hoain Department of Mathematic Ghior Govt College, Ghior, Manikgon and Monur Morhed Department of Mathematic Darul ihan Univerity, Dhaka, angladeh and Mohammed Forhad Uddin Department of Mathematic UET, Dhaka, angladeh ASTRACT In thi paper, an algorithm i preented to olve non-canonical linear fractional programming (LFP) problem, conidering the retricted normal form It provide a new way to olve all type of LFP problem When the LFP problem i only in canonical form, Forhad et al [] derived an algorithm conidering the retricted normal form In thi paper, the algorithm of Forhad et al [] ha been generalized to olve the LFP problem which i non-canonical form alo Thi algorithm required neither tranformation nor the iterative calculation of imple method ut it require only algebraic elimination Keyword: Linear Fractional Programming (LFP), Canonical and Non-canonical form, Tranformation, Simple method I INTRODUCTION To olve LFP problem Charne-Cooper [] developed a tranformation technique which tranform the LFP into two Linear Programming (LP) problem, itran-novae [] developed an algorithm which tranform the obective function of LFP problem into a linear obective function and olve a equence of LP problem, o, it take more time and labor On the other hand, Swarup [] developed an algorithm that ha fewer tep than previou technique but cannot avoid the iterative calculation of imple method of Dantzig [, 4] To overcome the compleitie of LP problem William et al [9] uggeted the retricted normal form Further, Forhad et al [] modified Swarup [] primal imple type method for olving LFP problem baed on primal imple method [ 4 ] for olving LP problem, which etend the cope of the method Swarup [] primal imple type method can be applied only when the contraint et i in canonical form Latter on, Swarup ugget to apply the dual imple type method in the cae where the et of contraint i not in canonical form ut Swarup [] dual imple type method cannot be applied in the cae where the dual feaible bai i not obtained To over come the compleitie of thee method, Forhad et al [] uggeted a modified approach to olve any type of LFP problem Moreover, when the LFP problem i only in canonical form, Forhad et al [] derived an algorithm conidering the retricted normal form [9] In thi paper, the algorithm of Forhad et al [] ha been generalized to olve the LFP problem which i non-canonical form alo

2 Mozzem Hoain, Monur Morhed and Mohammed Forhad Uddin II LINEAR FRACTIONAL PROGRAMMING (LFP) PROLEMS The LFP problem can be defined a follow: c + α (LFP) Maimize F( ) () d + β Subect to primary contraint,,,, N () and imultaneouly ubect to M m + m + m additional contraint, m of them are of the form ai+ ai+ ai + + ainn bi, () i,,, m m of them are of the form a + a + a + + a b, (4) N N m+,, m+ m and m of them are of the form ak+ ak+ ak+ + aknn bk, k m+ m+,, m+ m+ m The variou a i () can have either ign, or be zero The fact that b mut all be non-negative i the matter of convention only, ince one can multiply any contrary inequality by - There i no particular ignificance in the number of the contraint m being le than, equal to, or greater than the number of unknown N III DIFFERENT TYPES OF METHODS FOR SOLVING LFPP III Swarup dual imple type method Swarup [ ] primal imple type method howed that the baic feaible olution will be optimal if, where z c + α z d + β z c a z d b a z (c -z )-z (d -z ),,,,n The above obervation preent the following intereting poibility, if one can tart with ome baic but not feaible olution to a given LFP problem with all and remove from thi baic olution to another by changing one vector at a time in uch a way that he keep all provided no baic olution i to be repeated, an optimal olution to LFP problem will be obtained in a finite number of iteration That i the fact that thi algorithm maintain all at each iteration and i not concerned about the feaibility of the baic olution III The modified approach of Swarup primal imple type method Swarup[ ] firt developed a method for olving LFP problem However, the method can be applied only when the ytem A b i in a canonical form, that i, all contraint are le than or equal form ( ) The problem that i not in canonical form, one can olve by uing Swarup [ ] dual imple type method Likewie, LP problem, dual imple type method alo cannot be applied in the cae where the dual feaible bai i not obtained To overcome the above limitation of Swarup [ &] method, Forhad et al [] uggeted a modification baed on Dantiz [] two phae method for olving linear programming problem III Numerical eample: Eample (LFP) Maimize + 9 Subect to + 4, + + Now, introducing urplu and lack variable and to t, nd and rd contraint repectively to make the LFP problem in the tandard form a follow: (LFPI) Maimize + 9 Subect to + 4, ,,, 88

3 An Algorithm for Retricted Thu the initial baic olution -,, and Since - <, it fail the feaibility, that i, Swarup [ ] primal imple type method fail to olve the LFP problem Now, we tart, Swarup [ ] dual imple type method to olve the above LFP problem Initial Table c d c d - - i z - z 9 -/9 c -z d -z [4] To obtain optimal olution it mut be maintained that all at each optimization tage ut in the initial table, it i oberved that 4 >, which indicate the failure of Swarup [ ] dual imple type method III 4 The algorithm of thi paper To overcome the above limitation of Swarup [ &] method, we ugget a modification on Forhad et al [] Finally, the Eample i olved by our derived retricted normal form a follow: PHASE : (ALP) Minimize L w w + Subect to 4 + (7),,,,, w Here, N 6 and M; the left-hand variable are w, and ; the right- hand variable are, and The obective function i written o a to depend only on the right-hand variable For any problem in retricted normal form, it can be intantly read off a feaible baic vector (although not necearily the optimal feaible baic vector) Simply et all right -hand variable equal to zero, and the equation (7) then give the value of the left-hand variable for which the contraint are atified The idea of the imple method i to proceed by erie of echange In each echange, righthand variable and a left-hand variable change the place At each tage we maintain a problem in retricted normal form that i equivalent to the original problem It i convenient to record the information contant of the equation (6) and (7) in a o-called tableau, a follow: Table: L - - w Step: The mot negative L row entry i o i the left hand variable Step: There are three negative entry below it The ratio are, 667,, the minimum value i, o, w i the right hand variable and o w i replaced by Step : Now, olving in favor of w, namely w+ w+ Then ubtitute thi value into the old obective function, L + + w + w 89

4 Mozzem Hoain, Monur Morhed and Mohammed Forhad Uddin and into all other old left hand variable, 4 w w w 7 + w In table form: Table: w L Since all c * and there i no artificial variable in the lat table, it yield a primal feaible olution, thi table give another ub optimal point, 4, and, with Min L Now, the Phae of the problem i a follow: PHASE : Now the initial baic olution i 4,, and the original obective function become Maimize z In table form : Table: c - d Algorithm of the Retricted Normal form: Here relative profit factor c, relative cot factor d and the ratio, Where c + α d c d a a + β and z ( c ) ( d Step I: To elect the pivot column, conider z (c -z ) - z (d -z ) Chooe ma > 4 Here,, So, i the new left-hand variable Step II: To chooe pivot element, the minimum ratio tet need to apply In our problem there are two negative entrie, namely, and The ratio are 4 7 and, the minimum value i and o right hand variable and i replaced by left hand variable ) 9

5 An Algorithm for Retricted Step III: Now, by olving the pivot-row equation for the new left-hand variable in favor of the old, namely, 7 7 then ubtitute thi value into the original obective function 7 + z and into all other the old left-hand variable row, ( 7 ) Step IV: Go back and repeat the firt tep, until all, ignaling that no further improve i poible Thu after firt iteration, Table: 4 c - - d Since all i not, it i needed to improve the reult and repeat the above tep Second iteration: Repeating tep I, and 4, that i firt column i the pivot column, and uing the minimum ratio tet of tep II, the ratio i 4, the minimum value i and o 7 right hand variable and replace by left hand variable Now, by olving the pivot-row equation for the new left-hand variable in favor of the old, namely, then ubtitute thi value into the old obective function + + z And into all other the old left-hand variable row, in thi cae, Hence after econd iteration, Table: c d

6 Mozzem Hoain, Monur Morhed and Mohammed Forhad Uddin Since all i, ignaling that no further improve i poible Thu the olution of the eample i, 4 with ma IV CONCLUSION It i oberved that to olve LFP problem by uing Swarup [] method, it i not needed any tranformation but it require the iterative calculation of imple method of Dantzig [, 4] Forhad et al [] derived retricted normal form to olve LFP problem, which i only in canonical form On the other hand, thi paper preent an algorithm on retricted normal form to olve LFP problem, which i not in canonical form Further, thi algorithm require neither tranformation nor the iterative calculation of imple method It require only algebraic elimination Finally, it i noted that to ue Swarup [ & ] method, it ha to conider the lack or urplu variable in each table For thi reaon, the number of variable i increaed; and etra calculation are needed ut the algorithm decribed in thi paper doe not require to conider non-baic variable in each table, that i why, it i needed le calculation, ave time and labor Hopefully the dicued algorithm help to olve all type of LFP problem eaily REFERENCES itran, GR and Novae, AG: Linear Programming with a Fractional Obective Function, Operation Reearch, Vol, pp -9, (97) Charne, A, & Cooper, WW: Programming with Fractional Functional, Naval Reearch Logitic Quarterly 9, pp 8-86,(96) Dantzig, G: Linear Programming and etenion, Princeton Univerity Pre, Princeton, N J, (96) 4 Dantzig, G: Inductive proof of the imple method, IM Journal of reearch and development, Vol4, No, (96) Gillet E: Introduction to Operation Reearch, McGraw Hill, Inc, New York, (998) 6 Ilam, MA, & Nath G: Invetigation on ome Algorithm for olving Linear Fractional Programming Problem, angladeh Sci Re, 4(), pp -, (996) 7 Lipchutz, S & Poe, A: Programming with FORTRAN including Structural Fortran, McGraw Hill, Inc, New York, (998) 8 Mayo, EW & Cwiakala,M: Programmng with FORTRAN 77, McGraw Hill, Inc, New York, (99 ) 9 Pre, HW, Teukolky, AS, Vetterling, TW & Flannery, P: Numerical Recipe in FORTRAN 77, Cambridge Univerity Pre, (99) Swarup, K: Linear Fractional Programming, Operation Reearch, Vol, No 6, pp 9-6 Swarup, K:, Some Apect of Linear Fractional Function Programming, Autralian Journal of Statitic, V(), pp 9-4 (96) Farhad, U M, Ahad,MA & Ilam, MA Modified approach of Swarup method, Ganit: J angladeh Math Soc Vol4, pp89-98(4) Farhad, U M, & obayedullah, M A New Approach to Solve LFPP, Ganit: J angladeh Math Soc Vol pp-() 9