A modeling framework for incorporating DEA efficiency into set covering, packing, and partitioning formulations

• Author: I. Giannikos,S. K. Georgantzinos
• DOI: http://doi.org/10.1111/itor.12409
• Date: 01 November 2019
• Published date: 01 November 2019

Intl. Trans. in Op. Res. 26 (2019) 2387–2409

DOI: 10.1111/itor.12409

INTERNATIONAL

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IN OPERATIONAL

RESEARCH

A modeling framework for incorporating DEA efficiency

into set covering, packing, and partitioning formulations

S. K. Georgantzinos and I. Giannikos

Department of Business Administration, University of Patras, Patras, GR-26500, Greece

E-mail: sgeor@mech.upatras.gr [Georgantzinos]

Received 15 February2016; received in revised form 22 January 2017; accepted 27 February 2017

Abstract

The development of a modeling framework for efficient and cost-effective decisions considering the set

covering, packing, and partitioning problems is the main concern of this paper. To achieve this goal, the

simultaneous data envelopment analysis model is combined with the appropriate covering-type formulation

providing a series of multiobjective mathematical models. Simple numerical examples are implemented and

analyzed in order to validate and test the proposedmodels. The results demonstrate that the methods are able

to successfully find solutions for set covering,packing, and partitioning problems that also take into account

the efficiency of the coverage offered.

Keywords:DEA; set covering problem; set packing problem; multiple criteria analysis

1. Introduction

The essence of location analysis models is to select appropriate sites for a set of facilities (servers)

according to one or more decision criteria that may include cost, time, coverage, access, etc. These

criteria are typically defined on the basis of the spatial interaction between the facilities and cus-

tomers or demand points they serve. However, the facilities to be located utilize inputs such as

personnel, equipment, or other resources to produce outputs that constitute the service to the cus-

tomers. The location of each facility may influence its operational efficiency, namely its ability to

transform inputs into outputs due to differences in infrastructure, the labor market,etc. Differences

in operational efficiency between facilities may affect the customers’ perception of the service they

will receive and, consequently, their choice of the facility they will patronize. Customers may be

prepared to travel farther in order to be served by a more efficient facility. Hence, operational

efficiency must be taken into account as a criterion in order to develop more meaningful location

models.

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2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies

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2388 S.K. Georgantzinos and I. Giannikos / Intl. Trans. in Op. Res. 26 (2019) 2387–2409

The concept of efficiency is commonly described by data envelopment analysis (DEA), a non-

parametric technique introduced by Charnes, Cooper, and Rhodes (CCR) in 1978 for calculating

the efficiency of a set of decision-making units (DMUs) that utilize multiple inputs to produce

multiple outputs. DEA has been utilized to estimate efficiency in various fields such as health care

(Hollingsworth,2003; Worthington, 2004), university departments (Reichmannand Sommersguter-

Reichmann,2006; Kounetas et al., 2011), military operations (Bowlin, 2004), electric utilities (Santos

et al., 2011), banks (Tsolas, 2010; Holod and Lewis, 2011), mining operations (Tsolas, 2011), and

manufacturing productivity (Gattoufi et al., 2004). For a survey of major research efforts in DEA

over the 30 years since the seminal paper by Charnes et al (1978), see the review paper by Cook and

Seiford (2009). Furthermore, for a review of research activities related to DEA in the period 2000–

2014, see the survey by Liu et al. (2016). In this particular study, a network clustering method has

been applied to group papers through a citation network established from the DEA literature over

the period 2000–2014. Narb´

on-Perpi˜

n´

a and De Witte (2018a, 2018b) presented a detailed overview

of the studies investigating publicsector efficiency across various countries, comparing the data and

samples employed, and the main results obtained. They described which techniques have been used

for measuring efficiency in the contextof local governments and summarized the inputs and outputs

used. Recently, Song and Liu (2018) improved a newly proposed DEA cross-efficiency aggregation

method based on the Shannon entropy. The weights for determining cross-efficiency derived from

minimizing the square distance of weighted cross-efficiency and weighted CCR efficiency.

In the location analysis literature, there have been some attempts for incorporating efficiency

into location models. Thomas et al. (2002) proposed a combined location/DEA stepwise model for

locating obnoxious facilities that must be at some distance from the population centers they serve.

The location component fits into the general categoryof p-anticenter models where the objective is to

place p facilities at maximum distance from a set of customers. The authors employ the combined

location/DEA model in an iterative process that constructs the efficient frontier by running the

location and the DEA component iteratively. Klimberg and Ratick (2008) developed formulations

combining the uncapacitatedand capacitated facility location problem with DEA. They then applied

these models to a small hypothetical dataset and presented the results, concluding that this modeling

frameworkprovides a promising approach to multiobjectivelocation problems.The proposed model

formulations address the problem of optimization of both spatial interaction between facilities and

the customers they serve and the efficiency of facilities atthe selected locations. The authors conclude

that their models may be most useful when site attributes will differentially change the ability of

facilities to serve customers. Furthermore, Moheb-Alizadeh et al. (2011) incorporated the concept

of efficiency defined by DEA into location-allocationmodels in a fuzzy environment, demonstrating

how this can influence the pattern of facility location and the allocationof demands. They proposed

a fuzzy multiobjective nonlinear programming model and presented a solution procedure based on

a modification of fuzzy parametric programming and minimum deviation method. In order to test

their model, they provided an example to illustrate how considering the efficiency of facilities can

influence the location pattern and demand assignment. The example involved the location of three

uncapacitated facilities in five candidate locations and the allocation of eight customer nodes to the

located facilities.

On a more practical note, Shroff et al. (1998) described a multiple criteria location assessment

model, incorporating efficiency measurement methodologies based on DEA to estimate the rela-

tive siting efficiency of 26 potential sites for a major health care provider. Mitropoulos et al. (2013)

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2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies