and redundant constraint(s) in multi-objective ... - Semantic Scholar

European Journal of Operational Research 201 (2010) 390–398

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Continuous Optimization

An approach to find redundant objective function(s) and redundant constraint(s) in multi-objective nonlinear stochastic fractional programming problems V. Charles a,*, A. Udhayakumar b, V. Rhymend Uthariaraj c a

CENTRUM Católica, Pontificia Universidad Católica del Perú, Santiago de Surco, Peru Department of Computer Applications, Hindustan College of Engineering, Chennai 603 103, India c Ramanujan Computing Center, Anna University, Chennai 600 025, India b

a r t i c l e

i n f o

Article history: Received 2 May 2007 Accepted 19 March 2009 Available online 25 March 2009 Keywords: Stochastic programming Fractional programming Multi-objective programming Redundancy

a b s t r a c t Structural redundancies in mathematical programming models are nothing uncommon and nonlinear programming problems are no exception. Over the past few decades numerous papers have been written on redundancy. Redundancy in constraints and variables are usually studied in a class of mathematical programming problems. However, main emphasis has so far been given only to linear programming problems. In this paper, an algorithm that identifies redundant objective function(s) and redundant constraint(s) simultaneously in multi-objective nonlinear stochastic fractional programming problems is provided. A solution procedure is also illustrated with numerical examples. The proposed algorithm reduces the number of nonlinear fractional objective functions and constraints in cases where redundancy exists. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Anything omitted without affecting the system of concern is redundant. In the light of this definition redundancy is something that permits reduction of a system to a smaller size having the same properties as the original system. Though it may appear a little vague, for nonlinear stochastic fractional programming (NLSFP), this description is found sufficient. Whenever a model builder models linear programming problems, inclusion of structural redundancies in constraints and variables due to inadvertency is common. Redundancy may occur in the formulation phase because of bad source data or to avoid the risk of omitting some relevant constraints while modeling a problem. Therefore the presence of redundancy in practical mathematical programming model is hardly disputed. These embedded redundant constraints/variables when present in the model can play havoc with linear programming (LP) solution procedures and greatly increase solution effort. Redundancy may have some favourable effects too, both in the problem formulating stage and in the problem solving stage. Charnes and Cooper (1954) added constraints and variables to a problem to transform the problem from a linear programming problem to a transportation problem as it is much more easier to solve than the general linear programming problem. Fractional programming deals with the optimization of one or more ratios of functions subject to constraints. Over the past four decades, fractional programming has become one of the planning tools. It is routinely applied in engineering, business, finance, economics and other disciplines. This acceptance may be due to (i) good algorithms (ii) practitioners’ understanding of the power and scope of fractional programming. When one or more of the data elements in a fractional program is represented by a random variable, a stochastic program results, which is defined a stochastic fractional programming problem. In various areas of real world, the problems are modeled as stochastic programming. For example, modeling of investment portfolio, modeling of strategic capacity investments, power systems and manpower planning in Jeeva et al. (2002, 2004) etc. Literatures and applications of stochastic programming as well as fractional programming are available in Minasian et al. (1976,1977,1987), Biswal et al. (1998) and Bajalinov (2003). Recently, Savitha Mishra (2007) used weighting method for solving bi-level linear fractional programming problems. Nembou and Murtagh (1996) presented a stochastic optimization model, which has been applied to an existing hydrothermal electricity generation planning problem in Port Moresby, Papua New Guinea. Further, an extension to more general form of combining stochastic fractional programming with sequential linear programming (SLP) is also presented. Literature of SLP is given in Cheney and Goldstein (1959) * Corresponding author. E-mail addresses: [email protected] (V. Charles), [email protected] (A. Udhayakumar), [email protected] (V. Rhymend Uthariaraj). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.03.020

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398

391

and Kelly (1960). Chen (2005) proposed algorithms based on a fractional programming method, which is efficient and compatible to existing algorithms to determine optimal values for the two control parameters of stochastic inventory models. Wide range of applications of stochastic programming can be seen in Charles (2005a,b,c,d). A heuristic algorithm for a chance constrained stochastic program is given in Concetta and Rader Jr. (2007). Zionts and Wallenius (1976) presented a man-machine interactive mathematical programming method for solving the multiple criteria problem involving a single decision maker. Teghem et al. (1986) has given an interactive method along with an illustrative example for multi-objective linear programming under uncertainty. To solve multi-objective stochastic linear programming problems Hulsurkar et al. (1997) has proposed a fuzzy programming approach. Abbas and Bellahcene (2006) provided a cutting plane technique to solve multi-objective stochastic linear programming. Hu et al. (2007) proposed a generalized varying-domain optimization method for fuzzy goal programming (FGP) incorporating multiple priorities, which balances the two conflict factors in goal programming problem, namely general equilibrium and optimization. The optimization of ratios of criteria gives more insight into the situation than the optimization of each criterion. Multi-objective fractional programming models for that reason have been of greater interest in recent times. In real life decision situation, a decision maker is sometimes required to evaluate economic activities like output/employee, profit/employee, etc with respect to some constraints. Literatures and applications to multi-objective fractional programming are available in Gulati and Nadia Talaat (1991), Dutta et al. (1993), Gulati and Izhar Ahmad (1998), Arora and Ritu Gupta (2003), Lalitha et al. (2003), Arora et al. (2005), Lalitha et al. (2005), Nykowski and Zolkiewski (1985), Weir et al. (1992) and Charles and Dutta (2006a,b). Duality for pseudolinear programming problems is studied and their application to certain multi-objective fractional programming is discussed in Bector et al. (1988). Mathematical programming problems that have been studied so far in relation to redundancy include linear, integer and nonlinear programming, but the main effort has been in linear programming. Gal (1975) presented a note on redundancy and linear parametric programming. Gal and Laberling (1977) proposed an algorithm to identify redundant objective function in linear vector maximization problem. Uthariaraj et al. (1999), and Jacob et al. (2002) proposed algorithms to identify redundant constraints a prior to the solution of LP problems. In this paper, an algorithm that identifies redundant objective functions and redundant constraints in multi-objective nonlinear stochastic fractional programming problems is developed. This helps in reducing the number of objective functions and constraints in cases where redundancy exists. NLSFP problem is one of the optimization problems that can be solved using a number of different techniques within the constraint satisfaction paradigm. The solution presented in this paper falls along the lines of Charnes andCooper, 1954, 1961, 1962, and Charles and Dutta (2006b). This Paper is divided into eight sections. Section 1 gives a brief introduction; Section 2 describes formulation of multi-objective nonlinear stochastic fractional programming problem. Section 3 deals with conversion of stochastic constraints into deterministic constraints and Section 4 gives the conversion technique that helps us to convert stochastic fractional objective functions into deterministic constraints. Section 5 provides the basic definitions of redundancy. Section 6 introduces the redundancy algorithm technique to find redundant objective functions and constraints that have been shown numerically with examples in Section 7 and conclusion has been drawn at the end. 2. Multi-objective nonlinear stochastic fractional programming (MONLSFP) A general format of the multi-objective linear fractional programming problem (MOLFPP) with identical denominator could be seen in Dutta et al. (1993). It is also shown that the general method of solving MOLFPP by Nykowski and Zolkiewski (1985) is computationally more involved than the method proposed by Dutta et al. (1993). A solution algorithm to fuzzy MOLFPP involving fuzzy parameters is suggested in Omar (2007). Baba and Morimoto (1993) proposed a stochastic approximation method for solving the stochastic multi-objective programming problem (SMOPP) and Caballero et al. (2001) provided efficient solution concepts in SMOPP. Fouad Ben Abdelaziz et al. (2007) presented multi-objective programming techniques to choose the portfolio best satisfying the decision makers’ aspirations and preferences. Here the concepts of MONLFPP and SMOPP are combined. A multi-objective nonlinear stochastic fractional programming problem in a criterion space is defined as follows:

Ny ðXÞ þ ay GðXÞ ¼ ½G1 ðXÞ; G2 ðXÞ; . . . ; Gk ðXÞ; where Gy ðXÞ ¼ ; y ¼ 1; 2; . . . ; k; Dy ðXÞ þ by " # n X ð1Þ ð1Þ P 1  pi i ¼ 1; 2; . . . ; p; p þ 1; . . . ; q; q þ 1; . . . ; m; aij xj 6 bi Subject to Pr Max

ð1Þ ð2Þ

j¼1 n X

ð2Þ

ð2Þ

aij xj 6 bi

i ¼ m þ 1; . . . ; h;

ð3Þ

j¼1 ð1Þ

ð1Þ

where, 0 6 X n1 ¼ jjxj jj  Rn is a feasible set, and R : Rn ! Rk ; Að1Þ jj; bm1 ¼ jjbi jj; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; ay ; by are scalars, mn ¼ jjaij P P ð2Þ ð2Þ ð2Þ Aðhmþ1Þn ¼ jjaij jj; bðhmþ1Þ1 ¼ jjbi jj; i ¼ m þ 1; . . . ; h; j ¼ 1; 2; . . . ; n; N y ðXÞ ¼ nj¼1 cyj x2j and Dy ðXÞ ¼ nj¼1 dyj x2j . ð1Þ ð1Þ In this model, out of N y ðXÞ; Dy ðXÞ; A and bi , at least one may be random variable. S ¼ fXj Eqs:ð2Þandð3Þ;X P 0; X  Rn g is non-empty, convex and compact set in Rn . 3. Deterministic equivalents of probabilistic constraints ð1Þ

Let aij be a random variable in Eq. (2) and it follows Nðuij ; s2ij Þ; i ¼ 1; 2; . . . ; p; j ¼ 1; 2; . . . ; n, where uij is the mean and s2ij is the variance. P ð1Þ Let li ¼ ni¼1 aij xj ; i ¼ 1; 2; . . . ; p.

Eðli Þ ¼

n X j¼1

uij xj ;

Vðli Þ ¼ X 0 V i X ¼

n X

s2ij x2j ;

j¼1 ð1Þ

where V i ith covariance matrix. When aij is independent, the covariance terms become zero. The ith deterministic constraint for Eq. (2) is obtained from Charles and Dutta (2001, 2003) as follows:

392

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398

Prðli 6 bi Þ P 1  pi ðorÞ PrðZ i 6 zi Þ P 1  pi ;

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð1Þ where Z i ¼ ðli  Eðli ÞÞ= Vðli Þ follows standard normal distribution and zi ¼ ðbi  Eðli ÞÞ= Vðli Þ. Thus, /ðzi Þ P /ðKqi Þ, where 1  pi ¼ qi ¼ /ðKqi Þ, is the cumulative distribution function of standard normal distribution. Clearly, /ð:Þ is a non-decreasing continuous function, hence zi P Kqi . Substituting in this the values of Eðli Þ and Vðli Þ, n X

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u n 2 2 ð1Þ uij xj þ Kqi t sij xj 6 bi :

j¼1

ð4Þ

j¼1

ð1Þ

ð1Þ

ð1Þ

ð1Þ

Suppose aij and bi are random variables in Eq. (2) i.e., aij  Nðuij ; s2ij Þ and bi  Nðubð1Þ ; s2ð1Þ Þ, i ¼ p þ 1; p þ 2; . . . ; q; j ¼ 1; 2; . . . ; n, where uij b i and ubð1Þ are means, and s2ij and s2ð1Þ are variances, respectively. With the similar argumenti that led to the inequality in (4), one can obtain bi i inequality (5), the ith deterministic constraint for Eq. (2) as follows: n X

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u nþ1 uX uij xj  Kpi t s2ij x2j 6 ubð1Þ ;

j¼1

j¼1

ð5Þ

i

where xnþ1 ¼ 1. ð1Þ ð1Þ If bi is a random variable in Eq. (2), i.e., bi  Nðubð1Þ , s2ð1Þ Þ, i ¼ q þ 1; q þ 2; . . . ; m, where ubð1Þ , s2ð1Þ are the mean and variance, respecb bi i i tively. With the similar argument that led to the inequalityi in (4), one can obtain inequality (6), the ith deterministic constraint for Eq. (2) as follows: n X j¼1

ð1Þ

ð1Þ

aij xj 6 ubi þ Kpi s2bð1Þ :

ð6Þ

i

4. Conversion of objective functions into constraints This Section considers all the objective functions in the form of constraints (Charles and Dutta, 2003, 2006b). The main feature of the model is that it takes into account the probability distribution of the objective functions by maximizing the lower allowable limit of the objective function under chance constraints where numerator and/or denominator coefficients being random. The unknown parameter ky , which is less than or equal to Gy ðXÞ is defined by,

Gy ðXÞ P ky i:e:;

Ny ðXÞ þ ay P ky Dy ðXÞ þ by

0 6 Ny ðXÞ þ ay  ky ½Dy ðXÞ þ by :

)

There are two cases in this problem. Case 1. ay > 0 P P Pn 2 4  Pn 2 4  n n 2 2 2 Case 1a: Assumption: N y ðXÞ  N and Dy ðXÞ  N j¼1 ucyj xj ; j¼1 scyj xj j¼1 udyj xj ; j¼1 sdyj xj , where ucyj and udyj are means, and scyj and s2dyj are variances.

Let f ðX; ky ; ay > 0Þ ¼ ky ½Dy ðXÞ þ by   Ny ðXÞ 6 ay h i E½f ðX; ky ; ay > 0Þ ¼ F E ðX; ky ; ay > 0Þ ¼ ky DEy ðXÞ þ by  NEy ðXÞ " # n n X X 2 udyj xj þ by  ucyj x2j ¼ ky j¼1

ð7Þ

j¼1

V½f ðX; ky ; ay > 0Þ ¼ F V ðX; ky ; ay > 0Þ ¼ k2y DVy ðXÞ þ N Vy ðXÞ ¼ k2y

n X

s2dyj x4j þ

j¼1

n X

s2cyj x4: j ¼

j¼1

n   X k2y s2dyj þ s2cyj x4j

ð8Þ

j¼1

Pr½f ðX; ky ; ay > 0Þ 6 ay  P 1  pð2Þ y    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2y DVy ðXÞ þ N Vy ðXÞ 6 ay ) ky DEy ðXÞ þ by  NEy ðXÞ þ /1 qð2Þ y vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # n n n   u  X X uX 1 2 2 ð2Þ t udyj xj þ by  ucyj xj þ / qy k2y s2dyj þ s2cyj x4j 6 ay ky j¼1

j¼1

Case 1b: Assumption: N y ðXÞ  N

P n

2 j¼1 ucyj xj ;

ð9Þ ð10Þ

j¼1

Pn

2 4 j¼1 scyj xj



and Dy ðXÞ  N

P

Pn 2 2 n j¼1 udyj xj ; j¼1 sdyj xj

 , where ucyj and udyj are means, and s2cyj

and s2dyj are variances.Similar to case 1a one can obtain the constraint given below:

Pr½f ðX; ky ; ay > 0Þ 6 ay  P 1 

pð2Þ y ky

" n X j¼1

# udyj xj þ by 

n X j¼1

ucyj x2j

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n   u  uX ð2Þ t þ/ qy k2y s2dyj x2j þ s2cyj x4j 6 ay 1

j¼1

ð11Þ

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398

Case 1c: Assumption: N y ðXÞ  N

P n

j¼1 ucyj xj ;

Pn

2 2 j¼1 scyj xj



and Dy ðXÞ  N

P

n 2 j¼1 udyj xj ;

393

 2 4 2 j¼1 sdyj xj , where ucyj and udyj are means, and scyj

Pn

and s2dyj are variances.Similar to case 1a one can obtain the constraint given below

Pr ½f ðX; ky ; ay > 0Þ 6 ay  P 1  pð2Þ y ky

" n X

#

n X

udyj x2j þ by 

j¼1

ucyj x2j þ /1



vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  u  uX t qð2Þ k2y s2dyj x4j þ s2cyj x2j 6 ay y

j¼1

ð12Þ

j¼1

Case 2. ay 6 0 Similar to cases 1a, 1b and 1c one can obtain the constraints 2a, 2b and 2c given below:

Case2a :

n X

ucyj x2j

 ky

" n X

j¼1

Case2b :

Case2c :

n X

# udyj x2j

þ by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n   u  uX ð2Þ t þ/ py k2y s2dyj þ s2cyj x4j P ay 1

j¼1

ucyj x2j

 ky

" n X

j¼1

j¼1

n X

" n X

ucyj xj  ky

j¼1

ð13Þ

j¼1

#

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n   u  uX ð2Þ t þ/ py k2y s2dyj þ s2cyj x2j x2j P ay 1

udyj xj þ by

ð14Þ

j¼1

# udyj x2j þ by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n   u  uX t þ/ pð2Þ k2y s2dyj x2j þ s2cyj x2j P ay y 1

j¼1

ð15Þ

j¼1

5. Definitions The following definitions are defined in consent of case 1a of Section 4. Similarly, one can define for Case 2 (2a, 2b, 2c) and for the constraints (3)–(6).

Let scalar k ¼ minfky ’ Gy ðXÞj X be the unit vector and y ¼ 1; 2; . . . ; kg

ð16Þ

Let the decision space be

8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n n  <  u  X X uX O 1 n 2 2 ð2Þ t S ¼ X 2 R jk udyj xj  ucyj xj þ / qy k2y s2dyj þ s2cyj x4j 6 ay  kby ; : j¼1 j¼1 j¼1  Eqs:ð3Þ—ð6Þ; i ¼ 1; 2; . . . ; h; y ¼ 1; 2; . . . ; k; xj P 0; j ¼ 1; 2; . . . ; n 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n n  <  u  X X uX t udyj x2j  ucyj x2j þ /1 qð2Þ k2y s2dyj þ s2cyj x4j 6 ay  kby ; and Sw ¼ X 2 Rn jk y : j¼1 j¼1 j¼1 

Eqs: 3; 3; 4; 6; i ¼ 1; 2; . . . ; h; xj P 0; j ¼ 1; 2; . . . ; n; y – w

Definition 5.1. The constraint form of objective function

k

n X

udwj x2j



j¼1

n X

ucwj x2j

1

þ/



qð2Þ w

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n   uX t k2 s2 þ s2 x4 6 aw  kbw cwj

dwj

j¼1

j

j¼1

is redundant in the system (1) if and only if So ¼ Sw . The equivalent definition is Sw ¼ So if and only if

k

n X j¼1

udwj x2j



n X

ucwj x2j

1

þ/



qð2Þ w

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n   uX t k2 s2 þ s2 x4 6 aw  kbw cwj

dwj

j¼1

j

ð17Þ

j¼1

for all X 2 Sw and hence,

sw ðXÞ ¼ aw  kbw  k

n X j¼1

udwj x2j

þ

n X j¼1

ucwj x2j

1

/



qð2Þ w

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n    uX t k2 s2dwj þ s2cwj x4j : j¼1

Definition 5.2. The constraint form of the wth objective function (17) is redundant in the system (1) if and only if ^sw ¼ minimum fsw ðXÞjX 2 Sw g P 0. Definition 5.3. The constraint form of the wth objective function (17) is strongly redundant in the system (1) if and only if ^sw > 0. However, the constraint can be redundant without strongly redundant. Definition 5.4. The constraint form of the wth objective function (17) is weakly redundant in the system (1) if and only if ^sw ¼ 0.

394

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398

6. Identification of redundant constraints for NLSFP In this section, an algorithm is provided that helps to identify redundant fractional objective functions and redundant constraints in multi-objective linear stochastic fractional programming problems. Charles and Dutta (2003) using sequential linear programming provided a method for linearzing the constraint version of fractional objective function given in Section 4 and the constraint version of stochastic constraint given in Section 3. Consider linearizing the constraint and constraint form of fractional objective function.

9 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   P  n n n  > > P P > > ð2Þ 1 2 2 2 2 2 4 > qy ky sdyj þ scyj xj  ay þ kby 6 0; ðbased on assumption in Case 1aÞ > > > k udyj xj  ucyj xj þ / > > > > > > j¼1 j¼1 j¼1 > > > > > > s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >   P  = < P n n n  P ð2Þ 1 2 2 2 2 2 2 qy ky sdyj þ scyj xj xj  ay þ kby 6 0; ðbased on assumption in Case 1bÞ k udyj xj  ucyj xj þ / > > j¼1 j¼1 > > > > j¼1 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > >     > > n n n P P 2 2 2 > > P ð2Þ 1 > > 2 2 2 > > x k u x  u x þ / q k s x þ s a þ kb 6 0; ðbased on assumption in Case 1cÞ  y cyj j dyj > y > y y dyj j j cyj j ; : j¼1 j¼1 j¼1 then; G^y ðX int Þ þ rG^y ðXÞT ðX  X int Þ 6 0; y ¼ 1; 2; . . . ; k; X P 0:

ð18Þ

Inequality (18) can be viewed as G^ð1Þ X 6 ay  kby , X P 0, y ¼ 1; 2; . . . ; k for the following steps: y The matrix form of the above inequality can be viewed as

G^ð1Þ X 6 a  kb; X P 0; where G^ð1Þ 2 Rkn and ða  kbÞ 2 Rk :

ð19Þ

Let

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n > P P > ð1Þ > uij xj þ Kqi s2ij x2j  bi 6 0; i ¼ 1; 2; . . . ; p; > > < j¼1 j¼1 ð1Þ T i ðXÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > n nþ1 > P P 2 2 > > sij xj  ubð1Þ 6 0; i ¼ p þ 1; . . . ; q; > uij xj  Kpi : i j¼1 j¼1 8 n P ð1Þ ð1Þ > 2 > > < aij xj  ubi þ Kpi s ð1Þ 6 0; i ¼ q þ 1; . . . ; m; bi

j¼1

ð2Þ

T i ðXÞ ¼

n >P

> > :

j¼1

ð2Þ

ð2Þ

aij xj  bi

6 0;

i ¼ m þ 1; . . . ; h; ð1Þ

ð2Þ

where q nonlinear constraints and h  q þ 1 linear constraints are represented by T i ðXÞ and T i ðXÞ, respectively. ð1Þ Linearize the constraint function T i ðXÞ about the point X int as

T i ðX int Þ þ rT i ðXÞT ðX  X int Þ 6 0; ð1Þ

ð1Þ

X P 0;

i ¼ 1; 2; . . . ; p; p þ 1; . . . ; q:

ð20Þ

The matrix form of the above inequality can be viewed as

T ð1Þ X 6 bT ð1Þ X P 0; where T ð1Þ 2 Rqn and bT ð1Þ 2 Rq : The matrix form of h  q + 1 linear inequalities can be viewed as

T ð2Þ X 6 bT ð2Þ X P 0; where T ð2Þ 2 Rðhqþ1Þn and bT ð2Þ 2 Rhqþ1 : Hence the matrix form of h inequalities can be viewed as

" TX ¼

T ð1Þ

#

T ð2Þ

X6b¼



bT ð1Þ bT ð2Þ

; where T 2 Rhn and b 2 Rh1 :

ð21Þ

The matrix form of k þ h inequalities can be viewed as

a  kb 3 6   7 7 6 6 7 AX ¼ 4 ——— 5X 6 b ¼ 6 7 where A 2 RðkþhÞn and b 2 RðkþhÞ1 : 4 bT ð1Þ 5 T bT ð2Þ 2

b ð1Þ G

3

2

ð22Þ

Adding slack variables to the k constraints form of objective functions and to the h constraints, pre multiplying by the inverse of an approB priate basis and redefining the variables (both slacks and structural variables) as xNB j (or) xj according to their status (NB for non-basic, and B for basic), yields an equivalence system

h

A1 NB I

i xNB xB

¼ g;

xB ; xNB P 0:

395

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398

1 The matrix A1 NB is usually referred to as the Contracted Simplex Tableau Dantzig (1963). Let us refer to the elements of ANB as cij , g is the ‘‘updated right hand side”.

Theorem 6.1. A regular constraint or constraint form of objective function is redundant if and only if its associated slack variable sw has the property sw ¼ xBf in a basic solution in which cfj 6 0; j ¼ 1; 2; . . . ; n and gf P 0. 0

Proof. IF: In a basic solution xBf ¼ gf 

Pn

B j¼1 fj xj ,

c

since in any feasible solution the value of the xNB will be at least zero, the sum is atleast j

zero and hence, sw ¼ xBf P gf P 0. Therefore ^sw P 0. ONLY IF: Let us consider the f th row of tableau as the objective function for the sequential linear programming minimum fsw ðXÞjX 2 Sw g; then if ^sw P 0, it follows that in the optimal solution cfj 6 0 for all j ¼ 1; 2; . . . ; n with gf P 0. Since this optimal solution is a feasible extreme point of Sw , it is a basic feasible solution for the original set of constraints and constraint form of objective functions. h Since, in the theorem above ^sw ¼ gf , the regular constraint or constraint form of objective function is strongly redundant if gf < 0 and weakly redundant if gf ¼ 0. Redundancy algorithm: 1. Construct a matrix-of-intercept with decision variables and slack variables as rows and columns, respectively. ^ð1Þ ^ð1Þ 1.1 For finding redundant objective function(s)if ay 6 0 then hjy ¼ ðay  kby Þ=Gyj ; Gyj P 0, y ¼ 1; 2; . . . ; k, j ¼ 1; 2; . . . ; n. else ^ð1Þ ^ð1Þ hjy ¼ ðay  kby Þ=Gyj ; Gyj < 0, y ¼ 1; 2; . . . ; k, j ¼ 1; 2; . . . ; n. 1.2 For finding redundant constraint(s)

hji ¼ bi =T ij ;

i ¼ 1; 2; . . . ; h;

j ¼ 1; 2; . . . ; n:

2. Identify the pivot element in each row ¼ maxi fhji g else Wobj ¼ mini fhji g, for all j while the objective is max2.1 For finding redundant objective function(s): if ay 6 0 then Wobj j j imum, vice versa. ¼ maxi fhji =hji > 0g for all j, while the objective is minimization and constraints are lower 2.2 For finding redundant constraint(s): Wconst j ¼ mini fhji =hji > 0g for all j, while the objective is maximization and constraints are upper bound. bound; Wconst j 3. Score out the row and column corresponding to the entering and leaving variables. If a column has more than one maximum/minimum, score out those rows also. 4. The constraints corresponding to the slack variables in the unscored column, if any, ab initio are assumed and predicted to be redundant. 5. Remove these redundant constraint forms of fractional objective functions and redundant constraints tentatively from the original model.

6.1. General algorithm 1. Convert the stochastic fractional objective functions into constraint form using Section 4. 2. Convert the stochastic constraints into deterministic constraints using Section 3. 3. Adopting the technique of SLP Rao (2000) or Jeeva et al. (2002) or Charles and Dutta (2003), linearise 3.1 the constraint form of the objective functions. 3.2 the nonlinear constraints. 4. Apply the algorithm under Section 6 to identify the redundant objective function(s) and redundant constraint(s) and ignore them from the system. 5. Solve the reduced multi-objective stochastic fractional programming to get the optimal solution as in Charles and Dutta (2001, 2003, 2006b) or by using any other stochastic programming solvers such as SLP-IOR and AMPL.

7. Numerical examples 7.1. Example

 Max

RðXÞ ¼

c11 x21 þ c12 x22 þ a1 c21 x21 þ c22 x22 þ a2 ; d11 x21 þ d12 x22 þ b1 d21 x21 þ d22 x22 þ b2



Subject to 3x1 þ 5x2 6 15; 5x1 þ 2x2 6 10; a1 x1 þ a2 x2 6 5;

ð23Þ

where a1 ¼ a2 ¼ 0; b1 ¼ b2 ¼ 1, x1; x2 P 0. Let the third constraint satisfy atleast 90%. The mean and variance of the random variables are given in Table 1.

Table 1 Mean and variance of random variables. Random variables

c11

c12

c21

c22

d11

d12

d21

d22

a1

a2

Mean Variance

6 2

3 1

15 1

10 1

5 2

2 1

1 1

1 1

2 1

3 1

396

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398 ð2Þ

ð2Þ

Take p1 ¼ 0:10 and p2 ¼ 0:90. The deterministic equivalent of constraint form of fractional objective functions is given below:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      ffi 6x21 þ 3x22  k1 5x21 þ 2x22 þ 1 þ 1:28 2k21 þ 2 x41 þ k21 þ 1 x42 P 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi       15x21 þ 10x22  k2 x21 þ x22 þ 1  1:28 k22 þ 1 x41 þ k22 þ 1 x42 P 0:

ð24Þ ð25Þ

Let k ¼ minf1:125; 8:333g ¼ 1:125 at ðx1 ; x2 Þ ¼ ð1; 1Þ from Eq. (16) Therefore, inequalities 24,25 reduces to 26,27

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:375x21 þ 0:750x22 þ 1:28 4:531x41 þ 2:266x42 P 1:125; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13:875x21 þ 8:875x22  1:28 2:266x41 þ 2:266x42 P 1:125:

ð26Þ ð27Þ

Using the SLP Charles and Dutta (2003) the following linear constraints are obtained:

5:200x1 þ 3:725x2 P 5:587; 25:025x1 þ 15:026x2 P 21:121: The deterministic equivalent of stochastic constraint in system (23) is given below:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2x1 þ 3x2 þ 1:28 x21 þ x22 6 5: Using the SLP Charles and Dutta (2003) the following linear constraint is obtained:

2:910x1 þ 3:910x2 P 5:010: Solution to the system (23) with deterministic constraints is x1 ¼ 1:718, x2 ¼ 0:000, k1 ¼ 1:653, k2 ¼ 8:858 and the corresponding value of objective functions in system (23) is [1.124,11.204]. Solution using Redundancy algorithm is given below. From the Table 2, it can be inferred that the constraint due to second objective function is strongly redundant and first, second constraints are strongly redundant. Therefore, ignoring the second objective function and the first two constraints from the original problem, the problem is then solved. The bi-objective stochastic fractional programming problem reduces to SFP problem as follows:

Max

0:90k1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      ffi Subject to 6x21 þ 3x22  k1 5x21 þ 2x22 þ 1 þ 1:28 2k21 þ 2 x41 þ k21 þ 1 x42 P 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2x1 þ 3x2 þ 1:28 x21 þ x22 6 5

ð28Þ

x1 ; x2 ; k1 P 0: The solution is obtained as x1 ¼ 1:718, x2 ¼ 0:000, k1 ¼ 1:653. The corresponding value of objective functions in Eq. (23) is [1.124,11.204]. Here, it is to be noted that 90% of importance is given to the first objective function. 7.2. Example

Max

RðXÞ ¼



c11 x21 þ c12 x22 þ a1 c21 x21 þ c22 x22 þ a2 ; d11 x21 þ d12 x22 þ b1 d21 x21 þ d22 x22 þ b2

ð29Þ

Subject to a11 x1 þ a12 x2 6 1; a21 x1 þ a22 x2 6 b2 ; 16x1 þ x2 6 4; where a1 ¼ 7; a2 ¼ 2; b1 ¼ 4; b2 ¼ 4, x1 , x2 P 0. Let the first constraint be satisfied at least 90% and the second constraint be satisfied at least 80%. The mean and variance of the random variables are given in Table 3. ð2Þ ð2Þ Take q1 ¼ 0:10 and q2 ¼ 0:90. The deterministic equivalent of constraint form of fractional objective functions is given below:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi       k1 2x21 þ 2x22 þ 4  x21  2x22  1:28 k21 þ 1 x41 þ k21 þ 0:5 x42 6 7; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q       k2 2x21 þ 3x22 þ 4  x21  x22 þ 1:28 k22 þ 0 x41 þ k22 þ 0:5 x42 6 2:

ð30Þ ð31Þ

Let k ¼ minf1:250; 0:444g ¼ 0:444 at ðx1 ; x2 Þ ¼ ð1; 1Þ form (16)

Table 2 Matrix-of-intercept for Example 7.1. Slack/Decision variables

x1 x2

Objective functions

Constraints

S1

S2

Wobj j

S3

S4

S5

Wconst j

1.074 1.500

0.845 1.408

1.074 1.500

5 3

2 5

1.722 1.281

1.722 1.281

397

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398 Table 3 Mean and variance of random variables. Random variables

c11

c12

c21

c22

d11

d12

d21

d22

a11

a12

a21

a21

b2

Mean Variance

1 1

2 0.5

1 0

1 0.5

2 1

2 1

2 1

3 1

2 1

1 1

3 2

4 3

3 2

Table 4 Matrix-of-intercept for Example 7.2. Slack/Decision variables

x1 x2

Objective functions

Constraints

S1

S2

Wobj j

S3

S4

S5

Wconst j

0.913 0.636

5.350 0.648

0.913 0.636

0.316 0.462

0.651 0.478

0.250 4

0.250 0.462

Therefore, inequalities 30,31 reduces to 32,33

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0:112x21  1:112x22  1:28 1:197x41 þ 0:697x42 6 5:224; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0:112x21 þ 0:332x22 þ 1:28 0:198x41 þ 0:697x42 6 0:224:

ð32Þ ð33Þ

Using the SLP Charles and Dutta (2003) the following linear constraints are obtained:

 2:451x1  3:521x2 6 2:238; 0:309x1 þ 2:550x2 6 1:653: The deterministic equivalent of stochastic constraints is given below:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2x1 þ x2 þ 1:645 x21 þ x22 6 1; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3x1 þ 4x2 þ 0:84 2x21 þ 3x22 þ 2 6 3: Using the SLP Charles and Dutta (2003) the following linear constraints are obtained:

3:163x1 þ 2:163x2 6 1; 3:635x1 þ 4:952x2 6 2:365; 16x1 þ x2 6 4: Solution to the system (29) with deterministic constraints is x1 ¼ 0:000, x2 ¼ 0:462, k1 ¼ 1:545, k2 ¼ 0:419 and the corresponding value of objective functions in system (29) is [1.678,0.477]. Solution using Redundancy algorithm is given below. From the Table 4, it can be inferred that the constraint due to second objective function is strongly redundant and second constraints is strongly redundant. Therefore, ignoring the second objective function and second constraint from the original problem, the problem is then solved. The biobjective stochastic fractional programming problem reduces to SFP problem as follows:

Max

0:90k1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi       Subject to k1 2x21 þ 2x22 þ 4  x21  2x22  1:28 k21 þ 1 x41 þ k21 þ 0:5 x42 6 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2x1 þ x2 þ 1:645 x21 þ x22 6 1

ð34Þ

16x1 þ x2 6 4 x1 ; x2 ; k1 P 0: The solution is obtained as x1 ¼ 0:000, x2 ¼ 0:462, k1 ¼ 1:545. The corresponding value of objective functions in Eq. (29) is [1.678,0.477]. Here, it is to be noted that 90% of importance is given to the first objective function.

8. Conclusion The method presented here identifies redundant stochastic fractional objective functions and redundant constraints. The method has been developed with the intention of solving multi-objective nonlinear stochastic fractional programming problems and incidentally it identifies and eliminates both redundant stochastic fractional objective functions and redundant constraints, provided redundancy exist. The main motive for reducing a model is to solve a given problem quickly. It should be realized that any reduction in the model has ramification for the situations being modeled. These ramifications should hopefully lead to greater insight and development of a new method, which also identifies redundancy among decision variables.

398

V. Charles et al. / European Journal of Operational Research 201 (2010) 390–398

Acknowledgement The authors would like to thank the referees and the editors for their helpful comments, which have improved the presentation of this research article. References Abbas, M., Bellahcene, F., 2006. Cutting plane method for multi-objective stochastic integer linear programming. European Journal of Operational Research 168, 967–984. Arora, S.R., Ritu Gupta, 2003. A hybrid fuzzy-goal programming approach to multiple objective linear fractional programming problems. Opsearch 40, 202–218. Arora, S.R., Ravi Shanker, Malhotra, N., 2005. A goal programming approach to solve linear fractional multiobjective set covering problem. Opsearch 42, 112–125. Baba, N., Morimoto, A., 1993. Stochastic approximations methods for solving the stochastic multi-objective programming problem. International Journal of Systems Sciences 24, 789–796. Bajalinov, E.B., 2003. Linear-Fractional Programming: Theory, Methods, Applications and Software. Kluwer Academic Publishers, The Netherlands. Bector, C.R., Chandra, S., Durgaprasad, M.V., 1988. Duality in pseudolinear multiobjective Programming. Asia-Pacific Journal of Operational Research 5, 134–139. Biswal, M.P., Biswal, N.P., Duan, Li., 1998. Probabilistic linear programming problems with exponential random variables: A technical note. European Journal Operational Research 111 (3), 589–597. Caballero, R., Cerdá, E., Munoz, M.M., Rey, L., Stancu-Minasian, I.M., 2001. Efficient solution concepts and their relations in stochastic multi-objective programming. Journal of Optimization Theory and Applications 110, 53–74. Charles, V., 2005a. Application of stochastic programming models to finance, Management of Cost, Finance and Human Resource for Good Governance. Karnataka Law Society’s, IMER. pp. 106–115. Charles, V., 2005b. A financial model for ELSS mutual fund schemes in India, Management of Cost, Finance and Human Resource for Good Governance. Karnataka Law Society’s, IMER. pp. 127–136. Charles, V., 2005c. A stochastic goal programming model for capital rationing – carry forward of cash problem with mixed constraints. ICFAI Journal of Applied Finance 11 (11 & 12), 37–48. Charles, V., 2005d. Reliability stochastic optimization – An application of stochastic integer programming-n stage series system with m chance constraints. In: Janat Shah (Ed.), International Federation of Operational Research Society, International Conference on Operations Research for Development II, pp. 267–271. Charles, V., Dutta, D., 2001. Linear stochastic fractional programming with branch-and-bound technique. In: Nadarajan, R., Kandasamy, P.R. (Eds.), National Conference on Mathematical and Computational Models. Allied Publishers, Chennai. Charles, V., Dutta, D., 2003. Bi-weighted multi-objective stochastic fractional programming problem with mixed constraints. In: Nadarajan, R., Arulmozhi, G. (Eds.), Second National Conference on Mathematical and Computational Models. Allied Publishers, Chennai. Charles, V., Dutta, D., 2006a. Extremization of multiobjective stochastic fractional programming problem. Annals of Operation Research 143, 297–304. Charles, V., Dutta, D., 2006b. Identification of redundant objective functions in multi-objective stochastic fractional programming problems. Asia-Pacific Journal of Operational Research 23, 155–170. Charnes, A., Cooper, W.W., 1954. The stepping stone method of explaining linear programming calculations in transportation problems. Management Science 1 (1), 49–69. Charnes, A., Cooper, W.W., 1961. Management Models and Industrial Applications of Linear Programming I and II. Wiley, New York. Charnes, A., Cooper, W.W., 1962. Programming with linear fractional functions. Naval Research Logistic Quarterly 9, 181–186. Chen, Y.F., 2005. Fractional programming approach to two stochastic inventory problems. European Journal of Operational Research 160, 63–71. Cheney, E.W., Goldstein, A.A., 1959. Newton’s method of convex programming and tchebycheff approximation. Numerische Methematik 1, 253–268. Concetta, A.D., Rader Jr., D.J., 2007. A heuristic algorithm for a chance constrained stochastic program. European Journal of Operational Research 176, 27–45. Dantzig, G.B., 1963. Linear Programming and Extensions. Princeton University Press, Princeton. Dutta, D., Rao, J.R., Tiwari, R.N., 1993. A restricted class of multi-objective linear fractional programming problems. European Journal of Operational Research 68, 352–355. Fouad Ben, Abdelaziz, Belaid, Aouni, Rimeh EI, Fayedh, 2007. Multiobjective programming for portfolio selection. European Journal Operational Research 177 (3), 1811–1823. Gal, T., 1975. A note on redundancy and linear parametric programming. Operational Research Quarterly 26, 735–742. Gal, T., Laberling, H., 1977. Redundant objective functions in linear vector maximum problems and their determination. European Journal of Operational Research 1, 176–184. Gulati, T.R., Izhar Ahmad, 1998. Multiobjective duality using Fritz John conditions. Asia-Pacific Journal of Operational Research 15, 63–74. Gulati, T.R., Nadia Talaat, 1991. Duality in non-convex multiobjective programming. Asia-Pacific Journal of Operational Research 8, 62–69. Hu, C.-F., Teng, C.-J., Li, S.-Y., 2007. A fuzzy goal programming approach to multi-objective optimization problem with priorities. European Journal of Operational Research 176, 1319–1333. Hulsurkar, S., Biswal, M.D., Biswas, 1997. Fuzzy programming approach to multiobjective stochastic linear programming problems. Fuzzy Sets and System 88, 173–181. Jacob, V.C., Natesan, T.R., Rhymend Uthariaraj, V., 2002. Analysis of hybrid heuristic model reduction algorithms for solving linear programming problems. Opsearch 39, 202– 214. Jeeva, M., Rajalakshmi, R., Charles, V., 2002. Stochastic programming in manpower planning – cluster based optimum allocation of recruitments. In: Artalejo, J.R., Krishnamoorthy, A. (Eds.), Advances in stochastic modeling. Notable Publications Inc., New Jersey, USA. Jeeva, M., Rajalakshmi, R., Charles, V., Yadavalli, V.S.S., 2004. An application of stochastic programming with Weibull distribution – cluster based optimum allocation of recruitments in manpower planning. Journal of Stochastic Analysis and Applications 22, 801–812. Kelly, J.E., 1960. The cutting plane for solving convex programs. Journal of SIAM – VIII 4, 703–712. Lalitha, C.S., Suneja, S.K., Khurana, S., 2003. Symmetric duality involving invexity in multiobjective fractional programming. Asia-Pacific Journal of Operational Research 20, 57–72. Lalitha, C.S., Garg, P.K., Khurana, S., 2005. Duality for multiobjective fractional programming via linearization and scalarization approaches. Opsearch 42, 37–54. Minasian, S., 1977. Bibliography of fractional programming 1960–1976, Preprint Number 3. Department of Economic Cybemetics, ASE, Bucharest. Minasian, S., Tigan, St., 1987. The stochastic linear-fractional max–min problem. University Babes-Bolyai, Cluj-Napoca, Research Seminars, Preprint Number 5, pp. 275–280. Minasian, S., Wets, M.J., 1976. A Research Bibliography in Stochastic Programming 1955–1975, CORE Discussion Papers 7603. University Catholique de Louvain, Belgium. Nembou, C.S., Murtagh, B.A., 1996. A chance-constrained programming approach to modeling hydrothermal electricity generation in Papua–New Guinea. Asia-Pacific Journal of Operational Research 13, 105–114. Nykowski, I., Zolkiewski, Z., 1985. A compromised procedure for the multiple objective linear fractional programming problems. European Journal of Operational Research 19, 91–97. Omar, M. Saad, 2007. On stability of proper efficient solutions in multiobjective fractional programming problems under fuzziness. Mathematical and Computer Modelling 45 (1–2), 221–231. Rao, S.S., 2000. Engineering Optimization Theory and Practice, third ed. New Age, New Delhi. Savitha, Mishra, 2007. Weighting method for bi-level linear fractional programming problems. European Journal of Operational Research 183 (1), 296–302. Teghem Jr., Dufrane, J., Thauvoye, M., Kunsch, P., 1986. STRANGE: An interactive method for multiobjective linear programming under uncertainty. European Journal of Operational Research 26, 65–82. Uthariaraj, V.R., Natesan, T.R., Sankaranarayanan, V., 1999. Algorithms for identification of redundancies in linear programming problems. In: 19th IFIP International Conference on Systems Modeling and Optimization, Cambridge, London. Weir, T., Mond, B., Egudo, R.R., 1992. Duality without constraint qualification for multiobjective fractional programming. Asia-Pacific Journal of Operational Research 9, 195– 206. Zionts, S., Wallenius, J., 1976. An interactive programming method for solving the multiple criteria problem. Management Science 22, 652–663.