an operator theory for a class of linear fractional programming problems ii

M. LATA: Operator Theory for a Class of Linear Fractional Programming Problems - I1 75

ZAMM 66, 75 -88 (1976)

MANJU LATA

An Operator Theory for a Class of Linear Fractional Programming Problems 11.

I n Teil I wurde fur eine Klasse von Transportproblemen mit gebrochen-linearer Zielfunktion die Modifikation der optimalen Losung untersucht, falls basiserhaltende RIM-Operatoren angewandt werden, um die RIM-Bedingungen als Eineare Funktion eines einzigen Parameters zu andern. I m vorliegenden Teil 11 setzen wir diese Untersuchungen fort und studieren verschiedene Transformationen, die als Ergebnis von Kosten- und Schranken-Operatoren auf- treten. Wie zuvor beschranken wir um auf basiserhaltende Operatoren und entwickeln Algorithmen, um jeden Kosten- Operator anzuwenden und wm die aupersten Grenzen fur 6 zu f inden . Ein kleines numerisches Beispiel wurde gelost, um die verschiedenen Ergebnisse zu veranschaulichen.

In a recent paper on the same topic, we studied the problem of modifying the optimal solution to a class of transportation problems with linear fractional objective function when basis preserving rim operators are applied to vary the rim conditions as a linear function of a single parameter. We continue thzs study in the present paper and study various transformations arising as a result of a ~ ~ l ~ i n g cost and bound operators. A s before, we confine to basis preserving operators and develop algorithms for applying any of the cost operators and for f i n d i n g the maximum limit for 8. A small numerical example is solved at the end to illustrate various resu1t.s.

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1. Introduction

I n this paper we continue from [l] the development of an operator theory for a class of transportation problems, involving the minimization of a linear fractional objective function, of the form

subject to 2 xij = ai , i E I = { l , 2 ,... , m } , V J

,Z xij = bj ,

0 5 xij 5 uij , j E J = (1, 2, ... , n } ,

(i, j ) E I x J . (1.3)

(1.4) iOI

In [l] we studied operators which were basis preserving and applications of which resulted in continuous variation of rim conditions as a linear function of a single parameter. We shall now study various transformations arising as a result of applying basis preserving cost operators. We transform the optimal solution corresponding to changes in cij or dij or both and find the maximum extent to which the transformed solution remains optimal for the transformed problems without altering the optimal basis structure. We also study bound operators which arise when only the upper bounds are changed. The study of bound operators is shown to be equi- valent to the study of rim operators. We shall follow the same notation and terminology as in [l] and one should be familiar with the material in [ l ] before attempting a reading of this paper.

2. Basis Preserving Cost Operators

An operator 6T(P) is called a cost operator if at = /?, = vii = 0 in definition 2.1 [l]. A cost operator thus effects changes in cgi and dii only. Such an operator is denoted by 6CA or 6DA or

6SA according as i t indicates changes in cii or dci or both. The notation for a cell cost operator which increases or decreases the entry cPq in a single cell (p , q) only, is 6C’ Note that for the operator 6C&, yij = 0 for all (i, j ) except for (i, j ) = ( p , a). For the cell (p , q ) we have c~~ - cpq 6 6. Similarly we interpret the operators SO&. The cell operator which transforms both the entries cpq and dpq corresponding to the cell (p, q ) will be denoted by 6S& and the data for the transformed problem will be superscripted by ‘*’. This operator changes cpn and dpq t o cpq + 6yp, and dpa + 6ppa, respectively, where ypp and ppn are given scalars, thus subsuming all the diffe- rent possibilities of simultaneous increase, decrease or variation in opposite direction by equal or unequal amount in cPq and dpp.

In the following theorems we study the effect of applying basis preserving cost operators. An ‘operator’ will always mean a basis preserving operator.

+”“1

76 M. LATA: Operator Theory for a Class of Linear Fractional Programming Problems - I1

Theorem 2.1 (a): For the operator SCiq with ( p , q) d B, a n optimal solution for F f i s .f. - x.. - a7 f o r all (i, j )

with

The maximum value, that 6 can attain when ( p , q) E LB, is si = - 'x A < 0, w i w e A ' A = d2) + 2 dlkUlk .

( I , * ) € U B

If ( p , q ) E U B , st i s computed from')

llii (i, j ) E LB , > 0 , 1 - 2; A, > 0 where A, = d2) + 2 d;kUZk. ( 1 , k)E U B ( l , k ) # ( P , 9 )

If the sets over which the minimum is taken, are empty, then Kt = M , M a large number. Proof: Since the operator is basis preserving, we have Bt = B, U B f = U B , LBt = LB. Also at, b,

do not change for a cost operator, hence a t = a$ for all i and b/ = b, for all j . Therefore, x$ = xii for all (i, j ) . Since the cell ( p , q) B B, we have c$ = cii for all (i, j ) E B, therefore, the multipliers u:, t$, u:, v; remain un- changed. The quantities d;i are unaltered for all (i, j ) . c& do not change except for the cell ( p , q) , for which

Two cases arise according as the cell ( p , q) E LB or ( p , q ) E U B .

can be easily verified that 74 = q i j for all ( 2 , j ) # ( p , q )

c;; = c& - (u; + v;) = C i q + 6 .

Case 1: (p , q) E LB. This implies zgq 7 xpq = 0 :. Zf = 2 in this case. Also z+( I ) = z+@) = d2) . It

qgq = z(2)(Ciq + 6) - z(l)diq 4- 2 (dk(ciq 8) - Ckdiq) u l k = qpq + 6 4 9

( 1 , k)E UB where

A = d2) + 2 dikUtk . ( 1 , k)s U B

&will be 2 0 if 6 2 - % for A < 0 . A

Hence 6f = - rpp, A < O . -

A z p + sup,

z(2) - Case 2: ( p , q ) E U B . This implies xp'p = xpq = Upq, :. Zf = 0

Again z+(V = ,&I, 2+(2) = ~ ( 2 )

'&q = z(2)(Ciq + 6) - z(')diq f 2 (1,k)E UB (1 , w # (P, I)

(&(C;p f 8) - Ckdkq) UZk

= qpq + 6Al where A, = d2) + 2 d;kUtk . ( I , k)s U B ( 1 , a) #(a, 9)

The cell (p , q) will satisfy the optimality condition, if

Ss-k, A , > O . (2.3) A1 Now consider the value of the expression 7; for (i, j ) # ( p , q)

9' 13 - - d2)c;j - z(')d& + (d&$j - &d&) Ulk + (d&& - ( C i q + S) d;i) Up, = vii - 6diiUpq . ( 1 , k)s U B ( J , W # ( i , j )

# ( P > P ) The conditions of optimality will be satisfied if

From (2.3) and (2.4) we obtain the desired bound for 6. -

l) Throughout this paper, the quantities cij, d i j correspond t o the quantities ct j , d:j in [l].


A result analogous to Theorem 2.1 (a) can now be stated for the operator SC& Theorem 2.l(b): For the operator SCFrl with ( p , q ) B B, an optimal solution to P- is

- 2.. - 2. 23 - r j for all (i, j)

A > 0 where A = d2) + dikU1) 6- - r P q

A ' ( I , k)E U n and for ( p , q ) E U B

Theorem 2.2 (a) : For the operator SO;, with ( p , q ) B U

x+ 2) - - xil for all (i, j)

P roof : The proof of theorem 2.l(a) can be applied here with obvious modifications.

Theorem 2.2(b): For the operator SO& with ( p , q) B B - x.. - x. r7 - for all (i, j ) ,

and for ( p , q ) E U B

(2.6)

(2 .7)

(2.1.0)

E , where E i s a n arbitrary small poSitive number . and xpq > 0.

78 11. LATA: Operator Theory for a Class of Linear Fractional Programming Problems -- I1

P r o o f : Since the algorithm for linear fractional functionals programming problems requires the denominator to be positive for every feasible solution, we have to impose the additional restriction z r ) - 6x PP > 0

i.e. 6 <-, x p q > O . XP!?

the problem P" in this case has the same data except that czq = cpq + 6yp, and and ,up* are given quantities.

2.p

Now we consider simultaneous variation of cPP and dpP. Such an operator is denoted by SS&. Recall that = dpq + bpPq where ?.'pP

Theorem 2.3: For the operator SAS;~ with ( p , q) B B,

(i, j) E I x J , * XQ = xij ,

Z" Z P _ _ + __ 6YPCXP:Pp - - Z f ) + b,uP'1XP, *

(2.11)

where

and for ( p , q ) E U B

(2.12)

2.p - E , in case ,uPg . xPq < 0, E being an arbitrarily small positive number, I - PPnxPP

Proof : In this ease we also can easily verify that z$ = xti is a basic feasible solution for P* and the quantities dl), d2) , ut, vi, u:, vf, cij, dlj do not change except c& and dkn, which become

1 c;; = c;, - ( a p + u;) = c;* + 6Y,, , dPP - PY (4 + $1 = 4 J P + 8PPP *' - d" -

The remaining calculations for finding6" can be carried out cxactly on the saiiie lines as in the proof of Theorem 2.1 (a), and hence will be omitted.

To study cell cost operators with ( p , q ) E B, we partition the sets I and J into mutually exclusive sets I*, I,, JP, Jp with the help of the 'scanning routine' described in [23, such that

I, n I, = 0 ,

J , n J, = 0. I = I , u I , J = J, u J ,

and

and

The sets I,, I,, J,, Jq are characterized as follows:

I , = { p ) u {i E I : i is connected to p in Q) , J, = { j E J : j is connected to p in a> , I, = {i E I: i is connected to q in Q} , J, = { q } u ( j E J: j is connected to q in Q} ,

where SZ = { B - (p, q ) } . Note that

{ B - b n)} c { ( I , x JP) u (1, x J,)} with (2% 4 ) E (1, x J q ) *


Theorem 2.4 (a) : For the operator Sf& with (p , q) E B :

x f . 27 - - Xij 9 (i, i) E I x J ,

- 6+ = min

where

(2.13)

Proof : x$ = xij for all (i, i) is a basic feasible solution for P+. We define the multipliers ui+l, $l, ut2, vi2 as follows:

uil ~ {$,+ 6 3 i E IP,

U t 2 I ui" , i E I , i E I , , (2.14)

.i'2 = V j 2 9 i E J . It is easily to verify that the multipliers satisfy the relations

c$ = utl + ui' for all (i, j ) E B . Also we have

(2.15)

(2.16)

80

q$ will satisfy the optimality conditions, if

M. LATA: Operator Theory for a CIass of Linear Fractional Programming Problems - I1

(2.17)

For (i9 j ) E ( I p x J p ) , we have

(2.18)

For optimality,

6 IriillEil 3 (i, j) E {(I, x J p ) n U B J , &j < 0 , %l/Et, 9 ( i t i ) E ((1, x J p ) n -w 9 Eij > 0 .

Fro111 (2.17), (2.18) and (2.19), we obtain (2.13).

Theorem 2.4(b): For the operator SC;;, with ( p , a) F: B : -

x.. 21 - - xij 9 (i, j ) E I x J ,

(2.19)

(2.20)

where Aii, B,, have the same meaning as i n Theorem 2.4(a).

M. LATA: Operator Theory for a Class of Linear Fractional Programming Problems - IT 81

P r o o f : The proof is similar t o the proof of Theorem 2.4(a) except that we define the multipliers with the help of (2.21) and (2.22) below and make the necessary associated changes.

(2.21)

(2.22)

(,2.23)

where

P r o o f : The proof of this theorem is similar to that of Theorem 2.4(a) except that we define uif’, v;’, uif2, v;z as follows:

u ~ l = u i , 1 i E I ,

and make corresponding changes.

Theorem 2.5(b): For the operator SO;, with ( p , q) E 3:

(i, j ) E I x J , %: - a j - Xij 2

(2.24)

(2.25)

82 M. LATA: Operator Theory for a Class of Linertr Practional Programming Problems - If

(2.26)

(2.27)

(2.28)

and proceed as in the proof of Theorein 2.4 (a) making necessary changes. The requirement that the denominator remains positive throughout leads to the restriction

(2.29)


where

(2.31)

and proceed as in the proof of Theorem 2.4(a). The calculations are straightforward, but cumbersome and hence omitted.

We now turn to the area cost operator 6CA, for which the cost entry cij becomes cij + 6yij for all (i, j ) , where the yij are given scalars. We define quantities &:, G:, such that

yij = &: + $1 , (i, i) B , (2.32)

i.e. &:, $ are determined from yij in the same manner as u:, and vi are determined from cij, (i, j ) E B. Theorem 2.7(a): For the operator 6CA, an optimal solution to PA is given b y

(i, j ) E I x J , A x.. - 5.. aj - zj 9

z p + 6 z yijxij (i,j)cr X J

2(2) ZA = - ,

0

(2.33)

where

y& i n this expression represents the quantity yij - 2; - $a. relation (2.32) and define

P roof : x$ = xcj, (i, j ) E I x J , is a basic feasible solution for P A . Determine &t, z j with the help of

(2.34)

6"

84 M. LATA: Operator Theory for a Class of Linear Fractional Prorrramminr! Problems - I1

The conditions of optimality are satisfied, if

Hence the result.

In this case define the quantities &f, 6;, such that Now we consider the area cost operator 6DA, which changes dij to dij + 6,uij, the cij’s remaining the sarrle.

p.. 11 - - $$ + G; ,

,ii - 5. .

(i, j ) E B . Theorem 2.7(b): For the operator 6DA:

23 - Z l J (i, i) I x $

Mij < 0 7

_ _ - E , in case z pijxij < 0, zb2’

2 pijxij ( i , j ) s I x J l- ( i , j ) s I x J E being a n arbitrary small positive number, where

(2.35)

(2.36)

and proceed as in the proof of theorem 2.7 (a). The restriction, that the denominator remains positive, leads to the condition

Now we consider the area cost operator, for which cij becomes cij + 6yij and dij becomes dij + dpi,. We shall study only a particular class of such operators, denoted by 6SAJ which transform cij and dij by the same amount i.e. yij = pij for all (i, j ) . The general case, when yij f p i j for some or all (i, j ) , leads to complicated situations. Again we define quantities G+, 4 by the relations

(2.37) yii = f, + 5,s (i, j ) E B . Theorem 2.8: For the operator &’A,

(i, j ) E I x J x4 - x . . a) - a9 J

M. LATA: Operator Theory for a Class of Linear Fractional Programming Problems - II 85

2 p - E , in caae 2 yiixfj < 0,

Z yiyiixij ( i , j ) c I x J l- ( i , j ) c I x J E being an arbitrary small positive number,

Proof : We define ufl, up’, up2, u t 2 as follows

} ~ E J , $1 = vi’ + 66, up” = vi” + 66, ’

(2.38)

(2.39)

(2.40)

and make the remaining calculations as in the proof of Theorem 2.7(a).

for 6 and the other for applying any of the cost operators.

(a) To determine S A associated with 6CA, 6DA or 6SA, go to (e). If 6’ or 6* is to be determined corresponding

(b) If (p , q) E 23, go to (d). I f (p , q) E UB, go to (c). Otherwise proceed as follows:

The above results are now summarized in the form of two algorithms, one for finding the maximum limit

A lgor i thm 1: For finding 8 for a basis preserving cost operator.

to SC:q, 6D& or Ski’&, go to (b).

(b.1) For the operator SCiq, determine 6+ or% from (2.1) or (2.5) as the case may be. (b.2) For the operator SO&, determine 8+ or 8- from (2.7) or (2.9). (b.3) For the operator AS&, f* is determined from (2.11).

STOP. (c) (c.1) If it is the operator 6Cgq,-8+ or 8- is determined from (2.2) or (2.6).

6+ or 8- is determined from (2.8) or (2.10). (c.2) If it is the operator (c.3) For SS,*,, determine 8* using (2.12).

STOP. (d) Find the sets I,, I , , J_p , J , using the scanning routine and proceed to (d.l), (d.2) or (d.3) as the case may be.

(d.1) For SCzq, find 6+ or &- from (2.13) or (2.20). (d.2) For 6DZq, find 8. or 8- from (2.23) or (2.26). (d.3) For 6S&, f inds* using (2.29).

(e) (e.1) For 6CA, determine S A from (2.33). (e.2) For 6DA, determine aA from (2.35). (e.3) For 6SA, determine from (2.38). STOP.

STOP.

Algor i thm 2: For applying any of the basis-preserving cost operators, (a) If the operator a t hand is an area operator doA, 6DA or 6SA, go to (d). If i t is a cell operator, SC&, SDig,

(b) If ( p , q) E B, go to (c), otherwise proceed to (b.l), (b.2) or (b.3) as the case may be. 6S& go to (b).

(b.1) For 6Cig, set the data of P’ the same as P, except that cp”p = cpq f 6 . Set B’ = B, LB’ = LB, z p f 8xpq

2 p lJB* = U B , Z’ = and X* = X.

(b.2) For 6Diy, P’ has the same data as P except that d;y = dPq f 6. Set B’ = B, LB’ = LB, ZC)

z p f 6xpg UB’ = UB, 2’ = and Xk = X.

(b.3) For 6S&, set c& = cpq + 6yPq and d?rp = dPy + 6ppq. Also set B* = B, LB* = LB, UB* = U B , * - ’$I1’ + ‘yPqxpg and X * = X . z -

STOP. z p + 4-+*XP*

86 M. LATA: Operator Theory for a Class of Linear Fractional Programming Problems - I1

(c) Proceed as in (b) except that the sets I,, I,, J,, Jq are to be determined corresponding to ( p , q ) E B.

(d) (d.1) For the operator 6CA, P A has the same data as P except that c$ = cij + 6yij for all (i, j ) . Set BA = B STOP.

(l) + 6 2 yijxij

- ( i , j ) t l X J 20

2(2) LBA = LB, UBA = UB, X A = X , Z A =

2 0 )

28' + 6 2 Pijxij'

0 (d.2) For the operator 6DA, set d$ = dij + 6,uij for all (i, j ) , B A = B, LBA = LB, U B A = UB, X A = X

and ZA = __

( i , j ) ~ I x J

(d.3) For 6XA, set c$ = cij + dye, d$ = dij + 6yij for all (i, j ) , BA = B, LBA = LB, UBA = UB, X A = X , zp) + 6 yijxq

x p + 6 2 yijxij *

(i,j)cZx J

( i , j ) c Z x J

2-4 =

STOP.

3. Basis Preserving Bound Operators

An operator 6T(P), for which at = /3, = yij = = 0 for all (i, j ) in definition 2.1 [l] is called a bound operator and is denoted by 6L(P). As usual, bound operators are classified into area and cell bound operators denoted by 6LA and SL& respectively.

Theorem 3.l(a): For the operator SL;, with ( p , q) 4 U B : -

x; = xij for all (i, j ) , Z+ = z , 6+ = M (arbitrary large number) ,

Proof : Since the cell (I), q) e UB, any change in Up, does not effect the constraints 2 xii = a( , i E I ,

and 2 xii = b,, j E J . Nor the quantities u:, vj, ut, ~ 7 , cij, dij, qij are effected. Hence the optimaljty conditions

remain satisfied for any value of 6.

j c J

i s I

Theorem 3.l(b): Por the operator 6L& with ( p , q ) e UB, 6 - = U - -

xij = xij for all (i, j ) , z-=z, PP XP9 '

Theorem 3.2(a): For the operator 6Liq with (p , q ) E UB, X + i s obtained by applying 8RiP to P and then setting xgq = xpq + 6. The value of the objective function for P+ turns out to be

The maximum value that 6 can attain, is

(A . I

[ the minimum obtained in ([I], 2.18), where tij = d&i - c&-3ij.

P roof : To preserve the optimal basis structure, we have to set xig = xpq + 6 = Up, + 8. This change does not effect the basic variables, nor are the quantities ui, v;, uj, v?, clj, dij, dl), d2) effected. The quantities qij undergo slight changes :

= qii + atij where t.. - d' c:. - c' d:. 21 - PP $7 p q 21 *

The solution with x:$ = xpq + 6 remains optimal, if

(3.2)


Changing xp, to xpq + 6, however, violates the relations xij = b, for j = q. To

counteract this effect we apply the rim operator 6R& (see Theorem 2.2 (b) in [l]) and accordingly modify the values of basic variables. Also we obtain the limit

xij = ai for i = p and j h J i€I

8- (see 2.18 in [l]) (3.3: (3.2), and (3.3) yield the desired bound for 6.

rem 2.2 (a) in El]) and then setting xi‘ = zp, - 6, Theorem 3.2(b): For the operator SL, with (p, q) E U B , X - i s obtained by applying 6R:¶ to P (Theo-

(3.4)

I U p , , the minimum obtained in (El], (2.13)). Proof: The proof is similar to that of Theorem 3.2(a) except that Up, cannot be decreased below zero;

therefore, we have the additional restriction 6 5 Upp The area operator 6LA can be applied similarly. Again we have t o set x$ = xii + 6vij, (i, j ) E UB, in

order to preserve the optimal basis. To counteract the effect of this operator on rim conditions, we apply the area rim operator &RA with

and / ? I = - v i j , ~ E J .

{ i f I : ( i , j ) c U B }

The upper bound for 6 can be calculated as in Theorem 3.2 (a).

4. Numerical Example

We consider the same problem as in [l] and reproduce here the optimal tableau a1 u: u;

1

1 0

1 7

1 2 1 3 !q -1 0

4

2 4 1 1 1

V; 4 0 3 1 vi” 1 0 1 1

First consider the operator SC& with (2 , l ) E UB. Using (2.2) we obtain 8+ = g. If 6 is increased beyond $, the quantity q12 - 6d;zU2, tends to become negative. Thus the cell (1,2) restricts the - value of 6 at -$.

For the operator as$, (2,4) E LB with y2, = - 1 and A, = 1, we obtain 6* = $ = from (2.11), the value of the expression A being -44.

For the operator SC& with (3,2) E B, the sets 13, I,, J3, J, turn out to be 1, = ( 1 , 2 , 3 } , I , = 0 , J3 = { 1 , 3 , 4 } , Ja = (2) .

Also A,, = 0, A,, = 0, A,, = - 4, A,, = - 4, B12 = 22, B,, = 26. Using (2.13), we obtain 20

- 6 + = m i n { g , % } = A .

The value of 6 is limited by the cell (1,2).

88 M. LATA: Operator Theory for a Class of Linear Fractional Programming Problems - 11

Now consider the operator 6 C A . The quantities yij are recorded in the following table and Gi, G i are determined with the help of relations (2.32).

6: - - . -

I 13

._

, 3 4 ; 2 1 2 1 0 4 4 1

1-1i - O i - 1 1 2 1 - 3

8 1 3 2 3 1

Making necessary calculations, we obtain L12 = 0, L,, = 36, L,, = - 18, L,, = - 18, = 36, L,, = 54. From (2.33), $A = min (2, $, E} = 1 limited by the cell (2, 1). The solution to the transformed problem for the operator 6CA (for 6 = 1) is shown below.

at u p u p

0

I ‘ A

1 _ _ -. _ ~ _ ~ -

4 9 4 6 1 3 1 5 3 5 4

$2 1 0 1 1

For the bound operator 6L&, (2 , l ) E UB, we have to increase x2, to x , ~ + 6. To counteract this we apply 6RTl and find that 8- for the operator 6RG comes out to be 1. Calculating the limit for 6Ll1, we find that this -

zL1) + 1.1 - 1.3 z p ) + 1.1 - 1.1 -24 ’

42 limit is also 1 with the final value of the objective function Z-F = - -

Acknowledgment I am deeply indebted to myresearch supervisor, Dr. R. N. KAUL, for helpful suggestions and valuable com- ments throughout the preparation of this work.

References 1 MANJU LATA, An operator theory for a class of linear fractional programming problems I, ZAMM 55, 133-140 (1975). 2 SRINIVASAN, V., and THOMPSON, G. L., An operator theory of parametric programming for the transportation problem.

3 SRINIVASAN, V., and THOMPSON, G. L., An operator theory of parametric programming for the transportation problem.

4 GASS, S. I., and SAATP, T., The computational algorithm for the parametric objective function, Naval Res. Logist. Quart.

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Eingereicht : 12.3. 1974

AwSchnft: MANU LATA, 16R Railway Colony, Tilak Bridge, New Delhi-110001, India

Montercy, California, 1973.

an operator theory for a class of linear fractional programming problems ii

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