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Constrained integer linear fractional programmingproblemVanita Verma aa Dept. of Mathematics , Indian Institute of Technology , Hauz Khas, New Delhi, 110016, IndiaPublished online: 27 Jun 2007.

To cite this article: Vanita Verma (1990) Constrained integer linear fractional programming problem, Optimization: A Journal ofMathematical Programming and Operations Research, 21:5, 749-757, DOI: 10.1080/02331939008843602

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optimization 21 (1990) 5, 719-757 -4kademie-Vcriag Berlin

Constrained Tntegrr Linear Fractional Programming Problem

Sunlmary: An integer linear fractional programming problem, whose int,egcr sr 11.tt:on is required t o stlt>isfy any iL out of given TL sets of constraints has bee11 clisuussecl in this: paper. -Method for ranking and scanning all integer points has also been tleveloped and a numer- ical iilustranion is incintirtl in sr~pport of ~ , I > H O L . ~ .

AMS 1980 Subject Classification:

Primary: 90 C 32; Secondary: 90 C 101

Key words: Fractional programming, intogor progra~nnling.

1. Introduction

Integer linear fractional programming problem has been solved by many authors, see [ I , 2, 8, 91. This problem has applications in various fields like travelling sales~nan problem, job shop scheduling problem etc. Integer linear fractional programming can also be extended to study Bicriteria linear fractional pro- gramming problems [3] and Multicriteria linear fractional programming prob- lems [ 4 , 4 .

In many practical situations, integer solutions are required which in addition to satisfying primary constraints also satisfy some secondary constraints. For example, consider the case of distribution of scholarships to students in a school. Thus being a student of that school corresponds to primary constraint. Scholar- ship is awarded to students on means or merit basis or to SCJST securing a t least C grade. This corresponds to secondary constraints. The constrained inte- ger linear fractional programming problem intended to be studied can be mathe- matically stated as :

c'X + cr (p) Maximize Z = - d'X + B

subject t o X E S and

where X={X / A X = b , X z o } , A e R m x n , b ~ R m , d , c , X E R ? ) , E , PER.

Fi = {-x / gi(X) S , = , 2 0 , XZEO is an integer point)

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'i 50 optimization 21 (1990) 5

for all i i I= ( I , ? , ..., n), gi (X) 5 , =, S O is the set of secondary constraints in Pi.

the set of all subsets of I taking h of its elements at a time. where h is a fixed integer less than or eqnal. to n.

I t is sufficient to solve the relaxed problem (PI). c'x- - a.

(PI) Xasimize Z = ------ Cl'X - p subject to X E S

\ X z 0, X is an'integer point.

But if S $ F , then the integer points of S have to be scanned in a graded manner by repeated applications of various cuts discussed leter in the text, till an inte- r r n v -r\;y.+ fno&hin Cnr. P i o v n o n h n A TAnvn r o n l 4 n n +Lo ;n+---n - ~ : n + n *C O m n n - A + & l r ~ JIVLIIV IVWVIVIU 1-1 1 10 I Y W C I I U U . I T L U I V I C U l l n l l l & L I I V I U U G e U I p U I Y U U W I U U a i J l l W U

suffice as infeasibility in F of one jth best integer point of S may not necessarily imply a!! jth best integer points of S are infeasible for F. Far this reaso:: nnt mIy the b-teger points of 8 are ranked, hvt also the various slternative in ten~r 3--

p ~ i n t ere determjned ti!! feasibility in P is achievsd. In this process .9 is mc- cessively truncated.

2. Notations

cj = jth component of vector c

dj= jth component, of ~ e c t o r d

{x:) =set all Bth best integer solutions of problem (PI), r E {I, 2 , ..., pk) x:,~ = jth component of X:

= activity vector of

B: = basis corresponding to X: I: = (i I a:,; E B:) i q = G 1 BF)

where 2F1 = c'X: + a

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TERM a, VAXITA: Integer Fractional Programming 751

Theorem 1: Al l integer feasible solutions of problem ( P I ) alternate to X: o n an edge of region S (or truncated region of S ) emanating from i t , in the direction of a w c t ~ n t S j i j$ E J; lie in the open haZf space xi-= 1, where X = (xr).

i € 3 ; ; { j l f < 1

P r o o f Since Xi i i n feasible sol~ztiolr of problem (PI',. so A X ~ = S , where X! = (zf,,), j = 1 to n .

For some jl E Jf

TI-here Qj, is a nonzero positive scalar.

i "'3i For 0 L Qji s Min - i 01 , ~ C I ! (Y'i,ijt I I

X: is defined as

and is a new integer feasible solution of problem (PI), pkmided Qj, is a positive integer end @j!y:,iil &re integers for all i E 1:

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Now

Therefore Xz is an integer feasible solution of problem (PI), alternate to x:, lying on an edge Bit of the truncated region of S (or region S) through x:, defined

by r ' k x ~ = x l , i - q . l y ~ w; , Ei ,= ,X tS / xi, =@,, . -

( / x ,=o , forall v~.Nf\{jl).

r* L]earl.r 2 zt,j = 1, since st .= 6 fGr j ~ ~ ~ ~ \ { f l : ~ Thus, the point X: lie; ir, the J 9 - 9 7

2 - ark, <: 3 i C L v 1ilJli

open haLf spec xi -= 1. r LJ

r c ~ : \ ( j ~ ) Observe that all integer feasible solutions alternate tn x:: 1 y i ~ g on edge Bjl

of the truncated region of S (or the region S) are obtained by taking all possible integral values of @ j , within the specified range and they all lie in the open half space

x j < 1 . j<~!\{id

Corollary: An integer feasible solution of (PI) not on a n edge E,,. jl C J: of the truncated region of S (or region S ) through a n integer feasible point, say kth best feasibk solution X! oj the problem (PI) lies in closed half space

Proof : Let 9 = (2ij) be an integer feasible solution of probleni (PI), such that it does not satisfy (1). Then 2j= 0 for all j€N:\{jl}.

Now 2,, = 0 or 2,, 0. Case I : I f 2il = 0, then index set of nonbasic variables in the optimal table

corresponding to X is the same as N:, since iV:=iV~\{jl) U{jl), which implies that index set of basic variables in optimal table corresponding to $ and B: are the same, which further implies that X = X : , that is not the case. Hence 5, > 0 for a t least one j E N:\ {jl).

Case I I : If Zil =- 0, then X must lie in the direction of vector a:,,,, jl . i l ~ J:.

Further, if 5j, >Min (yy;:, - I y:,"'. =-0) = & (say), then i.,=x:,,-%y:o, c 0 imply-

ing that X is infeasible, which is not true.

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Thus. fjl is a positive integer. such that

iihlch in:p!ies thzt lie. or? a:: eclsf: E , . xhich is not the case. Hence f j -=-I), fnr a t least one c iV: \(jl;.

Thus, in both c a m . it is shown that for at least one j in N : \ ( ~ ~ ) . f , > O and hence 2, z 1. irnpl~ing that ff lies in the closed half space

Remark 1 : Cut ni th rebpect to jl; J"; defined b y ji). is a nivdified fuml uf the DANTZTG-cut, in the sense that it truncates an edge, whereas the DANTZIG- cut (for whlch, J! = 0 ) . defined hy

Remark 2: If corresponding to a feasible solution of problem (Pl), say x:, J: = @ , then X t is the unique ktb best feasible solution of problem (PI), i.e.. there does not exist any other integer feasible solution Xi11 A', such that B(Xj = =z(x?) .

Rr~nark 3 : If for all j5 F J : ( + 0). Min -= 1, corresponding to ~ C I :

bolution Xk of (Pl). then either X! is the unique kt" best feasible ~olutlon of problem (PI) or (PI) has integer solitions alternate to XZ, not lying on edges of truncated region of 5' (or region S ) emanating from x:. In either of the cases, apl'lication of (1) -1elds x:"' ~f X: 1s unique or x;. 1~ (2, 3, .... p k ) , if Xr has an alternate solution In S

3. Methodology for Scanning Integer Feasible Solutions

After obtaining an integer feasible solution of problem (PI), say the ut" best Xy ZE {I , 2, .. . , p,,). and con~puting all integer points alternate to Xf , if they exist, along an edge Ejr for some jrcJ:, edge EjT is truncated by adding cut (1) with respect to j, E Jy.

Optin~al feasible solution over the truncated region will be either an integer feasible solution of (PI) alternate to Xy or next best integer solution X;+l or a non-integer point.

In case that X: is the unique ut" best integer feasible solution of (PI). i t is truncated by adding c u t (2 ) . yielding again next best integer point xYC1 or a non- integer point.

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- - I a4 optimization 21 (1990) 5

Repeated applications of (1) (or ( 2 ) ) as the case may be, followed by applica- tion of GOMORY-cut to the corresponding solution yields all integer feasible solu- tions of poblem (PI).

Solve problem (PI) and find Xi. If X:E F, then Xi is an o p t i ~ a l feasible solution of problem (P).

If X ? @ F , then compute ( X t ) , l € ( 2 , 3, ..., pl ) . I f X t e F , for some I € ( 2 . 3. ...: pl). then X: is an optimal feasible solution of (PI. If X t e F for any I E (1, 2 , ..., pl), then compute X i for r z 2 and proceed as above till either of the follov-ing situations occurs.

i. For some positive integer, say q,

and Xlf €3'. for some Z E { I , 2 , ..., p,;, yielding XP, as an optimal feasible solution of problem (P).

2. Infeasibility appears, (i.e. current truncated region contains no feasible point), indicating that all integer points of S have keen obtained and none of them is a feasible point of F.

Remark 4: In particular, when F= F1= ( X / D X = d , X s O , X is an integer point), where D is an ml X n matrix, X E kn, d Rml, then problem (I?) reduces to

c'X + cc (PI) Maximize Z = ---

d ' S f ,8

subject to AX= b

X z O and X E P

Optima! feasible ~ ~ l u t i o i i of (I?") stated as

P1) c'X + cc

Maximize Z = - d'X - tB

subject to HX = e

X z 0 and integer

yields an optimal feasible solution of problem (PI ) where H =

problem ( P j can be solved by using any integer programming technique.

Theorem 2: An integer feasible solution of problem ( P ) obtained by the algoritlun i s a n optimal feasible solution of problem ( P ) and conversely, a n optimal feasible solution of problem (P) is obtained by the proposed algorithm.

P r o o f : Direct part is obyious. Conversely, let -if possible- X* be an optimal feasible solution of problem (P)

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- - F VERMA, VaXTii: Integer Frt tc t iu~ial Programming 133

and skipped by the procedure. Let fi ( i X * ) , say the pth best feasible solution of problem (PI) is an optimal feasible solution of problem (P), pielded by the procedure. Then

since otherwise. X* is one of the recorded solutions of problem (PI). Thus, there exists an u E ( 1 , 2 , .. . , p), such that

TI-hich implies that X* lies in the direction reserved by impiementation of cut (I) , while yielding Z ( X y ) from Z(XY- l ) , thereby implying that Z ( X * ) = Z ( x : - l ) , a contradiction to (3). Thus X* :is not skipped by the procedure.

Remark 5: Since the feasible region is bounded and is truncated at each step by repeated applications of (1) corresponding to the respective integer feasible solution, followed by application of GGMORY cut, whole of S is scanned in such a way that points and/or edges once scanned and truncated cannot reappear, leading to convergence of the procedure. As after truncating a point or an edge, GOMORS cuts are applied, efficiency of the proposed algorithm depends upon that of the GOMORY cutting plane method. Any cut which becomes inactive at inter- mediate stage is dropped.

5 . Numerical Illustration

(P)

where

2x3 - x2 :1- 23 Maximize Z =

x i + x , + x 3 + 2

subject to 3x1 + 4x2 + 2x3 s I 1

x17 x?, x3 z 0, and integer, and x = ( X I , x2 , X Q ) E F = ( P I n p2) u ( s 2 n p 3 ) u ( p 3 n m

The optimal basic feasible solution of the relaxed problem ( P I )

2xi + 2, + 2 3

( P I Maximize Z = zi +z2 + x 3 + 2

subject to 3x1+ 4x2 + 2x3 s 11

x i , x z , x 3 Z O , andintegers,

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is given by X: =

But Xi6 F. X: is given by X: is given by

(3, 0, 0) yielding value of the objective function as Z(X:) = 615.

Table 1 .

Alternate integer solutions are yielded by moving x? to values 1 and 2, giving rise to Xi=(2, 0, 1) and Xi=(2, 0, 2). But again XifjP and Xl4F.

Adcling the cut 2 xf :in 1, we get f€h'j \{7>

Xi= (2, 1, 0)fj F .

Adding the cut 2 xi z 1, next integer solution obtained is X f = (1, 0, 41, with J € N ~ , \ (7)

z(x,?) = 617, such that {I, 0, 4) 8 F. X,? is given by X: = (I, 1, 2), yielding value Z(Xf) = 516.

Table 2 showing this solution is the following.

But x:$ P. Its alternate solution Xi = (1,O, Z), is yielded by moving to value I.

X: E Pl fl F ~ , where Pl n F 2 c F. Thus, X: = (1, 0, 3) is an optimal feasible solu- tion of problem (P), with Z(Xz) = 516 as optimal value.

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References

[ I ] CsalrDEU, S.; &I. (:HAXDRA%TOHAX: An Improved Branch and Bound Method f'or Mixed Integer Linear Fractiomi Programs. Zeitsehrift fiir Xngewandte Mtttllomatik grid Mechn.nik. 59 (1979), pp. 575-577.

* - 7 j:j C~asl;ea, S . ; 11. C Z A S ~ X A ~ \ Z ; ? E . ~ ~ ; : A x c . t e ai; Linear frnctinn:tl Pmqramrning. ??a- val Research Logistics Quarterly, 27 (1980), pp. 171-171.

[ 3 ] C'soo, E. r.; D. I%. ATKITS: Ricritpria Linet~r Fractinnd l'rogranuning. Journal of Optimization Theory and Applications, 36 (19S2) 2, pp. 203-220.

[i] KOILXBLCTH. J. Q . H.; R. E. STEIJER: Multiple Objecti.i-e Linear Fractional P~.ogrtlrrj- ming. Management Sciel~ce. 27 (1981) 9, pp. 1024-1039.

[5] MTLRTY, K. G.: Linear Programming. John Wiley and Sons, Inc . 1983. (61 XI-KOWSKI, 1. ; Z. ZOLKIEWSKI: A Conipr~uLise P~i?~ei j i i~ .e for the Multiple Objective

Linear Fractional Programming Problem. European Jourr~al of Operational Rese- arch, 19 (1985) 1, pp. 91-97.

[7] SALKIN. H.: Integer Programming. Addison-Wesley, Reading Mass, 1975. [8] SESHAN, C'. 8.; V. G. TIBEKAR: ,4lgorit.llrns for Integer Fractional Programming.

Journal of the Indian 1nstitut.e of Science. Section-B: Physical and Chernicai Series, 6% (1989;, PP. 9-15.

[a] Sw.&x~?, X.: S o z e Aspects of Line.? Fractioaal E'unctionnls Programming, Austra- liar? Journal of Statktics; 7 (1965), pp. - - 90-104,

Received August i989, revised Eecember 1989

Vanita Verma Indian Institute of Technology Dept. of Mathematics Hauz Klms New Delhi - llOOlG India

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