Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2803461, 12 pages

https://doi.org/10.1155/2017/2803461

## Model for Selection of the Best Location Based on Fuzzy AHP and Hurwitz Methods

^{1}Faculty of Engineering, University of Kragujevac, Jovana Cvijića bb, 34000 Kragujevac, Serbia^{2}Public Company “Parking Servis Kragujevac”, Vojislava Kalanovica bb, 34000 Kragujevac, Serbia^{3}Faculty of Economics, University of Kragujevac, Djure Pucara Starog 3, 34000 Kragujevac, Serbia^{4}American University in the Emirates, Academic City, Dubai, UAE

Correspondence should be addressed to Aleksandar Aleksic

Received 27 December 2016; Revised 20 May 2017; Accepted 25 July 2017; Published 24 September 2017

Academic Editor: Anna M. Gil-Lafuente

Copyright © 2017 Slavko Arsovski et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of evaluation and selection of parking lots is a part of significant issues of public transport management in cities. As population expands as well as urban areas, solving the mentioned issues affects employees, security and safety of citizens, and quality of life in long-time period. The aim of this paper is to propose a multicriteria decision model which includes both quantitative and qualitative criteria, which may be of either benefit or cost type, to evaluate locations. The criteria values and the importance of criteria are either precise or linguistic expressions defined by trapezoidal fuzzy numbers. The human judgments of the relative importance of evaluation criteria and uncertain criteria values are often vague and cannot be expressed by exact precise values. The ranking of locations with respect to all criteria and their weights is performed for various degrees of pessimistic-optimistic index. The proposed model is tested through an illustrative example with real life data, where it shows the practical implications in public communal enterprises.

#### 1. Introduction

Rapid and constant increase of the number of city residents all over the world, in the developed as well as in the developing countries, and enhanced dynamics of living result in issues emerging in communal services, especially in the domain of stationary transport and parking [1]. Nonadequate strategies of management may cause pollution of the environment, increased load of traffic flow, reduction of the safety of traffic, and so forth.

According to the results of good practice, it is known that the zonal billing system is one of the parking strategies whose implementation is the best way to solve the mentioned issue. This parking strategy may be improved by building above ground parking garages. Construction of these demands certain investment costs but at the same time, it impacts competitiveness and prosperity of the city in the long run. With respect to these facts, it can be said that the problem of assessment and ranking of possible sites for construction of above ground parking garages may be one of the major tasks of the local government. The solution of this problem may be propagated into economic and political way of local government’s functioning.

Motivation for this research comes from the fact that there are no research papers which treat selection of land suitable for building above ground parking garages. In practice, the solution for the treated problem is obtained by using appropriate methods, such as Delphi technique, experts’ opinion, panel discussion, and tree analysis [2]. However, each solution obtained by applying mentioned methods is burdened by subjective opinions of decision makers in some degree.

The selection of locations for building of above ground parking garages depends on economic, environmental, and legislative issues as well as other demands by different stakeholders. It can be assumed that the considered problem can be stated as multicriteria decision making (MCDM) problem with some uncertain data.

By using methodology based on fuzzy sets, it is hard to believe that it is possible to obtain the solution of the treated problem by relying on single criterion [3]. Consideration of the problem situations as single criterion decision making problem presents merely an oversimplification of the actual nature of the problem; so consequently, it can lead to unrealistic decision. These authors suggest MCDM which is a powerful tool widely used for evaluating different problems with respect to multiple, usually conflicting criteria. There are no rules or suggestions on how the evaluation criteria and which MCDM method should be chosen.

In conventional AHP [4], the ratings of the values of the existing variables are described by crips numbers. The usage of discrete scale is simple and easy, but it is not sufficient considering uncertainty associated with the mapping of one’s perception to a number [5]. Decision makers express their judgments far better by using linguistic expressions than by representing them in terms of precise numbers. The more suitable way for human way of thinking is to proceed with using linguistic variables which are introduced by [6], instead of precise numbers. The fuzzy linguistic approach based on the fuzzy set theory has unconstrained boundary between true and false and it is widely used approach for modelling the linguistic variables [7, 8]. Most of these approaches provide a priori fixed predefined linguistic expressions that decision makers are constrained to use for expressing their preferences in a simple way. Using simple fuzzy linguistic approaches composed of a single term is not always suitable to represent the real preferences of the decision makers.

The objective of this research may be interpreted as the development of the new model which includes integration of the fuzzy Analytic Hierarchical Process (FAHP) and Hurwitz method. The decision makers may assess the relative importance in smooth and precise manner if they analyze each pair of criteria separately (by analogy to AHP) rather than to make an analysis of all criteria in the same time. Hurwitz method has the same mathematical base as many other MCDM methods. The procedure for finding the optimal solution which is suggested in Hurwitz method is significantly less complex compared to the procedures developed in other MCDM methods. As it may be noticed, by application of Hurwitz method, the optimal solutions may be obtained in short period. Those solutions are precise enough so it may be assumed that this method is suitable for solving problems that are generated in real life environment. By using the proposed model, the best location for building* above ground parking garage* is obtained. It may be assumed that the decision made in this way is less hampered by subjective opinions of the decision makers, so it is more precise compared to decisions derived from qualitative methods.

The main contribution of the paper is application of scientific approach in decision making process in public enterprises. In this way, effectiveness of stationary transport in urban areas is significantly increased which is further propagated to possible enhancement of local government’s effectiveness.

The paper is organized in the following way: a fuzzification of AHP method, which can be found in the literature, is summarized and analysed in Section 2. Section 3 presents the proposed model to deal with fuzzy data to support the decision making. The proposed model is illustrated by real-life data in Section 4. At last, a discussion of research, conclusions, and future steps are presented in Section 5.

#### 2. Materials and Methods

In the literature, there are many papers in which selecting alternative problem in presence of uncertainty is considered. Uncertainties in the relative importance of criteria, alternative values, at the same time are described by linguistic expressions which are modelled by type-1 fuzzy numbers such as TFNs [9, 10], TrFNs [11–13], or the intuitionistic fuzzy sets. Many authors suggest that the uncertainties can be modelled in a better way by using the type-2 fuzzy numbers. The domains of type-1 fuzzy numbers used are defined in real line into different intervals. For instance, common measurement can be used (Ćurčić, 2011). Kelemenis and Askounis [14] have used interval 1–10, and domain of fuzzy numbers is defined by interval 0–10 [10, 11, 15]. Modelling of different uncertainties by TFNs or TrFNS is performed since they have the advantage of simplicity over other types of fuzzy numbers. Curves of higher order of membership functions lead to increased complexness of computation but at the same time they do not result in increased correctness of solution.

Selecting the best location with respect to many criteria which are defined either as benefit criteria (i.e., the larger the criterion value, the greater the preference) or cost criteria (i.e., the smaller the criterion value, the greater preference) can be performed by using the proposed methods [15, 16] (Ćurčić, 2011).

In the papers [15, 16] (Ćurčić, 2011), calculation of the weights vector of evaluation criteria is often based on fuzzy AHP framework. FAHP enables mapping of human perception by a particular number or a ratio and we are also able to consider the vagueness in the decision making process. The elements of fuzzy pairwise comparison matrix should be described by linguistic expressions which are modelled by triangular fuzzy numbers (TFNs) [10, 17]. Also, in the literature there are papers where the elements of fuzzy pairwise comparison matrix of the criteria relative importance are modelled by trapezoidal fuzzy numbers (TrFNs) [18, 19]. Handling of FAHP can be performed by using two approaches which are proposed in the literature.

By using Chang’s extent analysis method [20], the synthetic extent value of the pairwise comparison is calculated. The obtained normalized weights vector of criteria is not a fuzzy number [21]. This method is widely used in the literature [10, 17]. It has been shown that the extent analysis method cannot estimate the true weights from a fuzzy comparison matrix and has led to quite a number of misapplications. However, in the literature, this approach has widely been used because it does not involve cumbersome mathematical operations and it has the ability to capture the vagueness of the human thinking style.

In the method for handling of FAHP proposed by Wu et al. [22], the criteria weights are derived from the fuzzy preference rations; thus the developed approach allows a more reasonable description of the decision making process and reflects the thinking style of a human.

In this paper, an approach for handling fuzzy pairwise comparison [22] is used, by analogy with Tadić et al. [18] and Macuzić et al. [19].

In the FAHP framework, the alternative values may be described as precise numbers [15], uncertain numbers [23], and either precise or imprecise numbers at the same time (Ćurčić, 2011).

Assessment of the criteria relative importance is performed by direct way and modelled by TFNs. Fuzzy rating of the relative importance of considered criteria is stated as fuzzy group decision making problem. The assessed values are modelled by type-2 fuzzy sets. Aggregation of different opinions of decision makers into group consensus is achieved by using fuzzy averaging method.

As a unique problem, the ranking of considered locations may be performed by using AHP method [24, 25], fuzzy AHP [23], fuzzy TOPSIS [15, 16], adopted Hurwitz method (Ćurčić, 2011), and so forth. Compared to other employed multicriteria decision making methods, it may be assumed that Hurwitz method is computationally attractive which is very important for practitioners. In the literature, papers that employ adopted Hurwitz criterion for ranking issues can be found [26–28].

The relative importance of criteria in these papers is described by discrete fuzzy numbers [26, 27]. The weights vector is given by using the procedure for calculating measure of belief that any discrete fuzzy number is greater than or equal to other discrete fuzzy numbers (Petrović and Petrović, 2001). Ćurčić et al. [28] suggested that the relative importance of evaluation criteria may be obtained by using fuzzy AHP [20].

In this paper, the relative importance of criteria is stated by fuzzy pairwise comparison matrix whose elements are modelled by TrFNs. The weights vector is calculated by using fuzzy geometric operator [22], and its elements are modelled by TrFNs.

Petrovic and Petrovic [27] suggested a new normalization procedure for uncertain criteria values. As uncertain criteria in the considered problem are of benefit type, authors suggest that domains of TrFNs should be defined by the interval 0-1 [12, 18]. In this manner, the calculation volume is decreased. Transformation of the linguistic criteria values is achieved by linear normalization procedure [29] in [26, 28].

In this paper, crisp criteria are normalized by using vector normalization procedure.

Also, in this paper, the weighted normalized fuzzy decision matrix is constructed. The value of elements of these matrices are described by TrFNs. By using the defuzzification procedure, the fuzzy decision matrix is mapped into decision matrix. Selecting the best alternative with respect to all criteria and their weights is performed by adopted Hurwitz method proposed by the analyzed papers. As it is known, one of the shortages of this method is the stability of the solution, which depends on the optimistic-pessimistic coefficient values. In this paper, the stability of the proposed solution is tested.

#### 3. The Proposed Model

Increasing population in urban areas, rapid development of industry, and the delivery of different business activities result in delays due to traffic congestion, pollution of the environment, increase in the number of traffic accidents, and so forth. Solving this issue may be overcome through the enhancement of traffic infrastructure. One of the improvement strategies is the construction of above ground parking garages on the existing parking lots. The application of this strategy demands certain economic investment by local government which expects the return of the investment in the reasonable period of time. Therefore, the ranking and selection of existing parking lots for construction of above ground parking garages is one of the significant tasks for the management in any public communal enterprise. In this paper, a new integrated fuzzy multicriteria model for ranking of parking lots that are suitable for construction of above ground parking garage is proposed (Figure 1). It can be assumed that the rank of parking lots obtained by using the proposed model increases the correctness of the solution.