Weighted sum of maximum regrets in an interval MOLP problem

• Author: S. Rivaz,M.A. Yaghoobi
• Date: 01 September 2018
• DOI: http://doi.org/10.1111/itor.12216
• Published date: 01 September 2018

Intl. Trans. in Op. Res. 25 (2018) 1659–1676

DOI: 10.1111/itor.12216

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

Weighted sum of maximum regrets in an interval MOLP

problem

S. Rivaz and M.A. Yaghoobi

Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman,

Kerman, Iran

E-mail: sanazrivaz@math.uk.ac.ir; sanazrivaz@gmail.com [Rivaz]; yaghoobi@uk.ac.ir [Yaghoobi]

Received 21 January 2014; received in revised form15 September 2015; accepted 21 September 2015

Abstract

This paper presents a multiobjectivelinear programming problem with interval objective function coefficients.

Considering the concept of maximum regret, the weighted sum problem of maximum regrets is introduced

and its properties are investigated. It is proved that an optimal solution of the weighted sum problem of

maximum regrets is at least possibly weakly efficient. Further, the circumstances under which the optimal

solution is necessarily efficient (necessarily weakly efficient or possibly efficient) are discussed. Moreover,

using a relaxation procedure, an algorithm is proposed, which for a given set of weights finds one feasible

solution that minimizes the weighted sum of maximum regrets. A numerical example is provided to illustrate

the proposed algorithm.

Keywords:multiple objective programming; linear programming; interval programming;uncertainty modeling

1. Introduction

Optimization problems with multiple, and usually conflicting, criteria constitute a significant and

active research area. In this context, multiobjective linear programming (MOLP) has been an

important topic of research since the 1960s (Ehrgott et al., 2007). It has been applied to many

fields of science, including engineering, economics, finance, logistics, and mathematics. Different

researchers have tried to solve MOLP problems in various ways (Ehrgott, 2005). Some difficulties

may occur in formulatingreal-world problems to mathematical programming problems. One source

of difficulty is lack of certainty in knowledge and information. In this regard, statements such

as “the profit rate will be almost 10 ($/min)” or “we want the profit substantially larger than 10

million” are two examples of such uncertainty (Inuiguchiand Sakawa, 1996a). In addition to having

multiple and conflicting objective functions,MOLP models must explicitly deal with the uncertainty

inherent to the model parameters. Uncertainty in MOLP has usually been handled by stochastic

or fuzzy programming (Stancu-Minasian, 1984; Sakawa, 1993; Lai and Hwang, 1996). Stochastic

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1660 S. Rivaz and M.A. Yaghoobi/ Intl. Trans. in Op. Res. 25 (2018) 1659–1676

programming assumes the coefficientsof the problem as random variables with knowndistributions,

and fuzzy programming considers the coefficients of the problem as fuzzy numbers with known

membership functions. However, some subjectivity is involved in determining the distributions of

random variables and membership functions of fuzzy numbers. For instance, Gaussian (normal)

distribution with different parameters and the bell-shaped or S-shaped membership functions have

been used in stochastic programming and fuzzy programming, respectively, by many researchers.

Nevertheless,there might be a mismatch between these specifications and real situations (Wu, 2009).

In this sense, interval programming providesan alternative choice for tackling uncertainty in MOLP

problems.In interval programming, uncertain quantities are modeled by closed intervals.In spite of

the potential subjectivity in specifying closed intervals, one might arguethat it is easier to determine

closed intervals for uncertain parameters than to specify distributions and membership functions in

stochastic programming and fuzzy programming, respectively (Wu, 2009). Interval programming

has attracted the attention of many researchers (Steuer, 1976; Ida, 1996; Inuiguchi and Sakawa,

1996a, 1996b, 1997; Chinneck and Ramadan, 2000; Benjamin, 2002; Allahdadi and Mishmast Nehi,

2013). Moreover, it has been applied extensively to solve portfolio selection problems (Lai et al.,

2002; Ida, 2003; Giove et al., 2006; Hlad´

ık, 2008).

Interval programminghas been used by some authors for solving MOLP problems with uncertain

coefficients. Bitran (1980) discussed MOLP problems with interval objective function coefficients

and introduced twotypes of solutions: possibly efficient and necessarily efficient solutions.Inuiguchi

and Kume (1991) considered the optimistic and pessimistic attitudes of a decision maker to find

a compromise solution via the goal programming approach. In this context, they formulated and

solved four types of goal programming problems with interval coefficients in which the target val-

ues were also assumed to be closed intervals. Urli and Nadeau (1992) used an interactive method

for solving MOLP problems with interval coefficients. Oliveira and Antunes (2007) provided an

overview of MOLP problems with interval coefficients by illustrating some numerical examples.

Also, Oliveira and Antunes (2009) presented an interactive method to solve such problems. Wu

(2009) proposed some solution concepts for a multiobjective programming problem with interval

objective function coefficients. In fact, these solution concepts follow from some ordering relation-

ships between two closed intervals in Rand the efficiency concept in conventional multiobjective

programming. Under these settings,Wu derived Karush–Kuhn–Tucker optimality conditions. Nec-

essarily efficient solutions are the most important solutions for an MOLP problem with interval

objective function coefficients. However, recognizing necessarily efficient solutions may be compu-

tationally expensive. Hlad´

ık (2010) stated some necessary and sufficient conditions to find such

solutions. Recently, Rivaz and Yaghoobi (2013) suggested a new solution concept, the minimax

regret solution, for MOLP problems with interval objective function coefficients.

The concept of maximum regret has long been used as a criterion for making decisions under

uncertainty (Mausser and Laguna, 1998). Inuiguchi and Kume (1994) and Inuiguchi and Sakawa

(1995) applied the concept of maximum regret to introduce the minimax regret approach to a

linear programming problem with a single interval objective function. Other scholars, also, used

the concept of maximum regret for solving real-world problems (Loulou and Kanudia, 1999; Dong

et al., 2011). In this paper, we attempted to investigate MOLP problems with interval objective

function coefficients.For simplicity,these problems are called interval MOLP problems.The current

research tries to apply the concept of maximum regret to obtain a convenient method for dealing

with interval MOLP problems. Indeed, necessarily (weakly)and possibly (weakly) efficient solutions

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2015 The Authors.

International Transactionsin Operational Research C

2015 International Federation of OperationalResearch Societies