A NOTE ON INTEGER LINEAR FRACTIONAL PROGRAMMING

Suresh Chandra and M. Chandramohan

Indian Institute of Technology Dellri, India

ABSTRACT

This note consists of developing a method for enforcing additional constraints lo linear fractional programs and showing its usefulness in solving integer linear fractional programs.

1 . INTRODUCTION

Fractional cutting plane methods for solving integer linear fractional programs have been proposed, for example, by Swarup [61, Grunspan and Thomas [31 and Granot and Granot [21. In contrast with these, the method proposed in the sequel does not impose any severe restric- tion on the problem as in Swarup [61, does not consist of solving many integer programs as in Grunspan and Thomas [31 or does not involve complicated computations in deriving the cuts as in Granot and Granot [2l. The results to be followed consist of developing a method for onforcing additional constraints to a linear fractional program and then using it for solving pure and mixed integer linear fractional programs by cutting plane methods.

2. METHOD OF ENFORCING ADDITIONAL CONSTRAINTS

Consider the following linear fractional program:

J'1 (PI : maximize i d,x, + P 1'1

n

subject to C aijxj = am, i = 1, 2, . . . , m,

j = l , 2 , . . . , n, J' l

XJ 2 0,

n

j -1

where it is assumed that d,x, + P > 0 for all feasible solutions.

Suppose that a basic feasible solution to (PI is known and with reference to this, let the constraints be

171

172 S. CHANDRA AND M. CHANDRAMOHAN

(1) xg, = b, + Y , ~ ( - x ~ ) , i = 1, 2, ..., m, / E N

x j a o , j = = l , 2 ) . . . ) n,

where xBi , i = 1, 2, . . . , m are the basic variables, N is the index set of nonbasic variables and the basic solution is obtained by putting xi = 0 for every j E N i n (1). Let z’, z2 be respective-

m m m m

d j ) ) / z 2 be calculated for all j EN . Let the additional constraint to be appended be

(2)

Substituting for x ~ , , i = 1, 2, . . . , m from (1) in (2) let (2) take the form

(3)

Let us denote by (PI) the new problem obtained from (PI by appending (3). Then

(4)

is a basic solution to (PI) where we designate n + 1 for Bm+l. Let c,+~ and d,+l be assigned the values zero.

The problem of interest here occurs when bm+l < 0. To establish the validity of the suc- cessive steps of the method to be followed in reoptimizing the problem (PI), we require the following lemmas, the proofs of which can be constructed with the help of Hadley 141.

LEMMA 1. If bm+l < 0 and if the set of feasible solutions to (PI) is nonempty, then there exists a j E N such that y m + l , j < 0.

LEMMA 2. For every j E N, there exists an i E (1, 2, . . . , m ) such that y,. > 0.

LEMMA 3 . If bm+l < 0 and if there exists a k E N satisfying ~ , , , + ~ , k < 0 and

then by a change of basis, i.e., by replacing xk by xEm+, in the set of basic variables a basic feasible solution to ( P I ) is obtained and .6

bm+l Z 2

Y m f 1 . k - - (22 - dk) < 0.

1

The method of reoptimizing (Pl> can now be given as follows:

ALGORITHM 1.

STEP 1. Set up a tableau giving xg,, b,, y,,, z; - cJ, z; - dJAJ for i = 1, 2, . . . , m and j E N. Compute z * , z 2 and z. Append the additional constraint in the form (3) .

NOTE ON LINEAR FRACTIONAL PROGRAMMING 173

STEP 2. If bm+l < 0, set J = 0 and go to step 3. Otherwise, set J = 1 and go to step 8.

STEP 3. Set M = ( j l j E N, ym+l,J < 0). If Mis empty, no feasible solution exists, stop.

for every j E N. Set MI = bdJ)

y r ( J ) , J , STEP 4. Compute - -

bm+ l < ””). If MI is empty go to step 6. Otherwise, set J = 1, r = m + 1 and Y m + l . j Y r ( i ) . j go to step 5.

l m + i , j < 0 where 6 + 1 . j = Y m + i . / - I STEP 5. Find k such that

b m + 1

z2 - (z; - dJ) and go to step 7.

STEP 6. Find k such that Y m f 1 . k = Min ym+l , j . If ym+l ,k 2 0 no feasible solution exists; stop. j € N

Otherwise, find r such that

STEP 7. Do a simplex pivoting to obtain a new basic solution by replacing xB, by x k in the set of basic variables. Modify N, calculate A j , z) - c ~ , z; - d, for all j E Nby modifying their definitions to take the summation from 1 to m + 1. Also calculate zl, z 2 and z. If J = 0 return to step 3.

STEP 8. If A , 2 0 for all j , stop; the optimal solution is xB, = b, for i = 1, 2, . . . , m + 1, and b r

xJ = 0 for j E N. Otherwise let A k = Min A j , - = Min j E N y r k I € { 1 , 2 . . . . . m + 1 1

return to step 7.

3. FRACTIONAL CUTTING PLANE METHOD FOR INTEGER LINEAR FRACTIONAL PROGRAMS

As an application of Algorithm 1 we present in this section a cutting plane method for solving integer linear fractional programs. The method for (mixed) integer linear fractional programs follows closely Gomory’s fractional cutting plane method for (mixed) integer linear programs and can briefly be described as follows.

Solve the problem obtained by omitting the integer restrictions. If the solution satisfies the integer restrictions, then it is optimal; otherwise, introduce a Gomory’s (mixed integer) fractional cut and reoptimize by using Algorithm 1 and repeat the process.

REMARK. Enforcing a single additional constraint to a linear fractional program can be done easily by using the dual simplex method after employing the Charnes and Cooper 111 transformation. But in the case of integer linear fractional programs, the transformation des- troys the integer nature of the variables and Gomory’s fractional cuts in their usual forms can- not be used. However, cuts can be obtained as given by Granot and Granot [2l, but it can be observed that such derivations of cuts involve more complicated computations than those in our case.

174 S. CHANDRA AND M. CHANDRAMOHAN

Since the objective function is not integer constrained, the proof of finiteness of the above method cannot possibly be given in a manner similar to that of Gomory's methods.

ACKNOWLEDGMENT

The authors wish to thank the referee for his suggestions for improvement of an earlier version of this paper.

REFERENCES

[ll Charnes, A., and W.W. Cooper, "Programming with Linear Fractional Functionals," Naval Research Logistics Quarterly, Vol. 9, pp. 181-186 (1962).

[2l Granot, D., and F. Granot, "On Integer and Mixed Integer Fractional Programming Prob- lems," Annals of Discrete Mathematics I , Studies in Integer Programming, pp. 221-231, eds. P.L. Hammer et al. (North Holland Publishing Company, 1977).

[31 Granspan, M., and M.E. Thomas, "Hyperbolic Integer Programming," Naval Research Logistics Quarterly," Vol. 20 (21, pp. 341-356 (1973).

[41 Hadley, G., "Linear Programming," (Addison Wesley, Reading, Massachusetts, 1962). [51 Swarup, K., "Linear Fractional Functional Programming," Operations Research, 13, pp.

[61 Swarup, K., "Some Aspects of Linear Fractional Functional Programming," Australian Jour- 1029-1036 (1 965).

nal of Statistics, Vol. 7, pp. 90-104 (1965).