Primal-Dual Approach to Solve Linear Fractional Programming Problem

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1 Journal of the Applied Mathematics, Statistics and Informatics (JAMSI), 4 (2008), No. Primal-Dual Approach to Solve Linear Fractional Programming Problem VISHWAS DEEP JOSHI, EKA SINGH AND NILAMA GUPA Abstract In this paper a new method is suggested for solving the problem in which the objective function is a linear fractional function, and where the constraint functions are in the form of linear inequalities. he proposed method is based mainly upon revised primal dual simple algorithm (RPDSA).he algorithm can be combined with interior-point methods to move from an interior point to a basic optimal solution. he advantages of RPDSA algorithm are simplicity of implementation and less computational effort. Mathematics Subject Classification 2000: 90c2, 90c5 General erms: linear fractional programming, RPDSA algorithm. Additional Key Words and Phrases: primal-dual simple method, interior-point method.. INRODUCION Linear fraction maimum problems (i.e. ratio objective that have numerator and denominator) have attracted considerable research and interest, since they are useful in production planning, financial and corporate planning, health care and hospital planning. he field of LFP, largely developed by Hungarian mathematician. Martos and his associates in the 960 s, is concerned with problem of optimization. Several methods to solve this problem are proposed in (962), Charnes and Cooper have proposed their method depends on transforming this (LFP) to an equivalent linear program. Another method is called up dated objective function method derived from itran and Novaes (97) is used to solve this linear fractional program by solving a sequence of linear programs only re-computing the local gradient of the objective function.also some aspects concerning duality and sensitivity analysis in linear fraction program was discussed by itran and Magnant I (976), and Singh.C. (98) in his paper made a useful study about the optimality condition in fractional programming. he suggested method in this paper depends mainly on the solving linear fractional functions, and where the constraint functions are in the form of linear inequalities, the proposed method is based mainly upon RPDSA method given by Paparrizos et. al. in 200 [9]. We use the concept of duality to solve this problem.. Since the earlier methods based on the verte information may have difficulties as the problem 6

2 V. D. JOSHI, E. SINGH AND N. GUPA size increases, our method appears to be less sensitive to the problem size. An eample is given to clarify the developed theory and the proposed method. In section 2 proposed algorithm and definitions of the (LFP) problem is given while in section a numerical eample is given to clarify the proposed algorithm. In section 4 we give a conclusion remarks about this proposed method and in section 5 we given the references. 2. Proposed Algorithm: he optimal solution to a general Linear Fractional Programming Problem, if it eists, can be obtained by using the following steps. Step 0: LFP problem can be formulated mathematically as follows c u+ α Minimize Fu ( ) = d u+ Subject to Au b u 0...( i) n where u U( R ), A is (m n) matri, also c and d are n-vectors, b R and α, are scalars. U is compact set. Moreover d u+ > 0 everywhere in U m Step I: In (i) we shall assume that 0, than an equivalent form of (i) can be formulated as α u α Minimize F( ) = ( c d ) + d + 62

3 PRIMAL-DUAL APPROACH O SOLVE LINEAR FRACIONAL PROGRAMMING PROLEM Step II: b u b Subject to ( A+ d )...( ii) d + u If we define y = 0 than (ii) can be written in the form d + of Linear Program Minimize Subject to Where and Z A c b α c = ( c d ) b A = ( A+ d ) b = b = + = α 0...(iii) ( Interior - point method & RPDSA Algorithm) Step 0 (initialization): Start with a primal feasible solution y for problem (LP) and a dual feasible basic partition (,N) to problem (LP). Set ( ) =, = ( ) ( ), ( N) = ( N) N A b w c A s c w A Step (general step): λ { } i () While ( i, 2,... m : < 0) r ( ) r ( ) = = i () < dr ( ) dr ( ) y = + λd rn = ( ) r N /*computat H A, d = y ma : 0 /*computation of the leaving variable*/ /*computation for the net feasible point*/ ion of pivoting row*/ s N() t s N( j) µ = = min : H rn ( j) < 0 HrN () t HrN ( j) /*choice of the entering variable = */ k = (), r p = N(), t () r = p, N() t = k /*updatesetsandp*/ = A b w = c A s = c w A ( ), ( ) ( ),( N) ( N) N d = y /* computation of the net direction*/ end N(j) p 6

$V. D. JOSHI, E. SINGH AND N. GUPA Linear fractional program Minimize Graphical representation of algorithm.$

4 V. D. JOSHI, E. SINGH AND N. GUPA Linear fractional program Minimize Graphical representation of algorithm. NUMERICAL EXAMPLE Subject to: , 2 0 y transformation formula we can change linear fractional programming problem to linear problem. 64

5 PRIMAL-DUAL APPROACH O SOLVE LINEAR FRACIONAL PROGRAMMING PROLEM After transformation we got following LPP Primal 2 Minimize 2 Subject to: + + = = 4,,, Dual w + w Maimize 2 Subject to: w + 2w + s = w s = w + s = w + s = w, w, s, s, s, s We can see that for primal problem 0 A =, c =,, 0, 0, b = (, ) =,4, N =,2. Initialization with dual feasible basic partition { } { } Let We get sn w = ( 0,0 ), s =,,0, y =,,, 2 4 0, 0, 0,,, = = = 0 65 ( ) 0 w = ( 0,0) = ( 0,0) 0 =, ( 0,0) =,, s =,,0,0, 0 2 2

6 V. D. JOSHI, E. SINGH AND N. GUPA d =,,, ( 0, 0,, ) =,,, Choice of the leaving variable 4 λ = = = d d 5 d 4 ma,, Computation for net feasible point y = + λd = ( 0,0,, ) +,,, =,, 0, Computation for the pivoting row H rn = (,0) = 2 0 Choice of the entering variable s s s µ = = = H H 2 H min,, So leave the basis and 2 enter ~ = N () ~ 2 = N (2) rn () rn (2) rn (2) Second iteration We get ( 2,4 ), N (,) = = 2 0 w =,0 =,0, = =, = ( 0,, 0, ) 0 2 sn = (,0 ),0 =,, s =,0,,0, d = y =,, 0, Choice of leaving variable λ = = 5 4 ma, d4 6 Computation for the net feasible point 66

7 PRIMAL-DUAL APPROACH O SOLVE LINEAR FRACIONAL PROGRAMMING PROLEM y = + d =,, 0, 0, 6 Computation for pivoting row H rn 2 = ( 0, ) = Choice of the new entering variable s s s µ = = = HrN () H H min, 0 So leave the basis and 2 enters rn (2) rn (2) ~ = N () ~ = N (2) hird iteration 2 =,,, 0, = = 2 After third iteration is feasible point of primal problem so we got the answer =, = 2 2 =0 2 = * y 2 =0 4 =0 y =0 Geometric illustration of the problem 67

8 V. D. JOSHI, E. SINGH AND N. GUPA CONCLUSION he proposed algorithm is developed to solve linear fractional programming problem and restriction is 0 and denominator value in objective function must be grater than zero. his algorithm may be etended to solve linear fractional programming problem whose objective function of the numerator and denominator value may be either positive or negative. Also this may be etended to solve linear fractional programming problems with bounded variables. REFERENCES ANDERSEN, E.D., YE, Y Combining interior-point and pivoting algorithms for linear programming. Management Science 42(2),79. ANSREICHER, G., ERLAKY, A monotonic build-up simple algorithm for linear programming. Operation Research 42, CHEN, H., PARDALOS, P. AND SAUNDERS, M he simple algorithm with a new primal and dual pivot rule. Operations Research Letters 6, 2 7. DOSIOS, K., PAPARRIZOS, K Resolution of the problem of degeneracy in a primal and dual simple algorithm. Operations Research Letters 20, HILLIER, F. S. AND LIEERMAN, G.J. 2005, Introduction to operation research, ata McGraw- Hill KAMO, N.S. 99, Mathematical Programming echniques. East-West Press PV LD MEGIDDO, N. 99. On finding primal and dual optimal bases. ORSA Journal on Computing (),6 5. MURY, K.G., FAHI, Y A feasible direction method for linear programming. Operations Research Letters (),2 7. PAPARRIZOS, K., E.AL A new efficient primal dual simple algorithm. Computer and operation research 0,8-99 PAPARRIZOS, K. 99. An eterior point simple algorithm for general linear problems. Annals of Operation Research 2, ROLAND, W. F., FLORIAN JARRE.994. An interior-point method for fractional programs with conve constraints. Mathematical Programming 67, SAKHIVEL, S., RAMRAJ, E A new approach to solve linear fractional programming problems. he Mathematics Education XXX X, -8 SCHAILE, S ibliography in fractional programming. Zeitschrift fur Operation Research 26, 2-24 ERLAKY,., ZHANG, S. 99. Pivot rules for linear programming A survey. Annals of Operations Research 46, 20. WOLF, H A Parametric Method for Solving the Linear Fractional Programming Problem. Operation Research,

9 PRIMAL-DUAL APPROACH O SOLVE LINEAR FRACIONAL PROGRAMMING PROLEM VISHWAS DEEP JOSHI Research Scholar Department of Mathematics Malaviya national institute of technology, Jaipur, India EKA SINGH Research scholar Department of Mathematics Malaviya national institute of technology, Jaipur, India NILAMA GUPA Department of Mathematics Malaviya national institute of technology, Jaipur, India Received October

AN ITERATIVE METHOD FOR SOLVING LINEAR FRACTION PROGRAMMING (LFP) PROBLEM WITH SENSITIVITY ANALYSIS

AN ITERATIVE METHOD FOR SOLVING LINEAR FRACTION PROGRAMMING (LFP) PROBLEM WITH SENSITIVITY ANALYSIS Mathematical and Computational Applications, Vol. 13, No. 3, pp. 147-151, 2008. Association for Scientific Research AN IERAIVE MEHOD FOR SOLVING LINEAR FRACION PROGRAMMING (LFP) PROLEM WIH SENSIIVIY ANALYSIS