A model for solving incompatible fuzzy goal programming: an application to portfolio selection

• DOI: http://doi.org/10.1111/itor.12405
• Author: Mariano Jiménez,Mar Arenas‐Parra,Amelia Bilbao‐Terol
• Date: 01 May 2018
• Published date: 01 May 2018

Intl. Trans. in Op. Res. 25 (2018) 887–912

DOI: 10.1111/itor.12405

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

A model for solving incompatible fuzzy goal programming: an

application to portfolio selection

Mariano Jim´

eneza, Amelia Bilbao-Terolband Mar Arenas-Parrab

aDepartment of Applied Economics I, University of Basque Country (UPV/EHU), Plaza O˜

nati 1,

20018 San Sebasti´

an, Spain

bDepartment of Quantitative Economics, University of Oviedo, Avda. del Cristo,s/n 33006, Oviedo, Asturias, Spain

E-mail: mariano.jimenez@ehu.es [Jim´

enez]; ameliab@uniovi.es [Bilbao-Terol]; mariamar@uniovi.es [Arenas-Parra]

Received 28 September 2016; receivedin revised form 7 February 2017; accepted 20 February 2017

Abstract

For many fuzzy goal programming (GP) approaches, in order to build the membership functions of fuzzy

aspiration levels, a tolerance threshold for each one of them should be determined. In this paper, we address

the case in which the decision maker proposes incompatible thresholds, which could lead to an infeasible

problem. We propose an alternative algebraic formulation of the membership functions, which allows us

to formulate models capable of providing solutions, although some tolerance thresholds are surpassed. The

objective values that do not violate their corresponding threshold are evaluated positively according to the

degree of achievement to their fuzzy target,and in turn those who violate the threshold are penalized according

to their unwanted deviation with respectto the threshold. Thus, our model jointly uses the fuzzy GP approach

and the standard GP approach,which also allows incorporating fuzzy and crisp targets into the same problem.

The proposed procedure is applied to socially responsible portfolio selection problems.

Keywords:fuzzy goal programming; standard goal programming; membership function; infeasibility; socially responsible

investing; portfolio selection; developmentsustainable indexes

1. Introduction

The pioneering model of goal programming (GP) was proposed by Charnes and Cooper (1955)

and further developed by several authors, Ijiri (1965), Lee (1972), Ignizio (1976), Romero (1991),

among others.

GP has become one of the most popular techniques within the field of multiple criteria decision

making (Caballero et al., 1997; Tamiz et al. 1998; Jones and Tamiz, 2002, 2010; Aouni et al., 2009)

and is one of the most widely used methods due to its applicabilityto real problems (see, e.g., Blancas

et al., 2010; Marcenaro-Guti´

errez et al., 2010; Voces et al., 2010; Buyukozkan and Berkol, 2011;

Bilbao-Terol et al., 2012; Zopounidis and Doumpos, 2013; D´

ıaz-Balteiro and Romero, 2016). GP

C

2017 The Authors.

International Transactionsin Operational Research C

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USA.

888 M. Jim´

enez et al. / Intl. Trans. in Op. Res. 25 (2018) 887–912

is based on the satisficing solution concept, introduced by the Nobel Prize in Economics, Herbert

Simon (1955). Simon conjectured that in complex decisional problems with conflicting objectives

where the decision maker (DM) is unable to optimize all objectives simultaneously, more interest

exists for reaching aspiration levels. That is to say, an unattainable optimization is replaced by a

practical attainable satisfaction.

GP assumes that the DM is able to determine aspiration levels for each one of the relevant

attributesof the problem in order to achieve a satisficing solution. GP models are classified according

to the achievement function used to aggregate the unwanted deviations (positive, negative, or both)

from these aspiration levels. The most used variants (see, e.g., Jones and Tamiz, 2010) are weighted

additive GP (WGP), where the weighted sum of unwanted deviation is minimized; MinMax GP

also known as Chebyshev GP, where the maximum deviation from aspiration levels is minimized;

and lexicographic GP (LGP), where the goals are grouped in priority levels. The achievement of

goals included at a higher level is infinitely more important than those situated at a lower level.

Some extensions that are formulated by hybridizing the aforementioned main approaches have been

proposed (Rodriguez-Ur´

ıa et al., 2002; Romero, 2004; Caballero et al., 2006; Ak¨

oz and Petrovic,

2007). In this paper we focus on additivevariants, that is to say, on WGPor on LGP with a weighted

additive approach for each priority level.

Assuming that the achievement function is a linear function, the general algebraic formulation

of a WGP model is as follows:

min

Q



q=1αqnq+βqpq

subject to fq(x)+nq−pq=bqq=1,...,Q

x∈F

np,pq≥0q=1,...,Q,

(M1)

where bqis target level for the qth goal, nq,pqare negative and positive deviations from target value

of qth goal, xis a vector of decision variables, Fis the set of original constraints, αq=uq/Nqif nq

is unwanted, otherwise, αq=0, βq=vq/Nqif pqis unwanted, otherwise βq=0.The parameters

uq(vq)andNqare the weights reflecting preferential and normalizing purposes attached to the

achievement of the qth goal.

The negative and positive deviational variables of the qth goal are defined as

nq=bq−fq(x)

2+bq−fq(x)

2

pq=fq(x)−bq

2+fq(x)−bq

2,

which is the only solution derived from the following set of equations:

nq×pq=0

nq−pq=bq−fq(x).

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2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation of OperationalResearch Societies