The prioritisation of provinces for public grants allocation by a decision-making methodology based on type-2 fuzzy sets

Abstract

Regional development agencies (RDA) are the units established for accelerating regional development and increasing local capacity. They aim at activating the regional dynamics and to reduce the intra-regional and inter-regional development gap. In a region, each province may have a different development level. This would pose a problem for socio-economic development. In order to reduce the disparities among the provinces, financial support mechanisms of development agencies would be a beneficial influence only if supports are used with to correct strategies. In order to play a vital role to reduce intra-regional disparities, it is necessary to consider many criteria to categorise settlements in terms of socio-economic development. Some of these criteria are generally subjective and extremely difficult to express in numbers. However, fuzzy sets are a great help to decision makers in a prioritisation of provinces for public grants allocation process with linguistic variables and measurement challenges. In this study, a new city-ranking model has been proposed for development agencies operating in Turkey. To address ambiguities and relativities in real-world scenarios more conveniently, type-2 fuzzy sets and crisp sets have been simultaneously used in multicriteria decision making (MCDM) process of grants allocation. To illustrate the proposed model better, an application with real case data has been performed in the Middle Black Sea Development Agency in Turkey.

Keywords

AHP city ranking fuzzy sets regional development TOPSIS type-2 fuzzy sets

Introduction

Companies continuously make investments for many reasons such as capacity building and technological innovation. Often, there are several alternatives that fit the purpose of the investment. Among these alternatives, the company must choose the best option. In recent years, many states have provided direct financial support to private-sector investments in increasing proportions. States promote private-sector investments for many reasons – economic, politic and social. Some of the expected outcomes of projects with public support are increase in employment, prevention of environment damage, national income growth and intra-regional development gap reduction. From this point, the answer to the question of which projects should be financed by public support is to choose projects that maximise the benefit criteria mentioned above with the lowest cost (Kilic and Kaya, 2015).

Even if there are different tools in each country to promote private-sector instruments, many countries support investment projects through regional development agencies (RDAs). Duties of these agencies are as follows (OKA (Middle Black Sea Development Agency), 2013):

Providing technical support for the planning activities and duties of local authorities.

Supporting activities and projects, ensuring the implementation of the regional plan and programmes; monitoring and evaluating the implementation process of activities and projects supported within this context; and presenting results to the Undersecretariat of State Planning Organization.

Contributing to the improvement of the capacity of the region, concerning rural and local development, in accordance with the regional plans and programmes, and supporting the projects within this extent as well.

Monitoring other projects implemented by the public sector, private sector and non-governmental organisations in the region and considered as important in terms of regional plan and programmes.

Improving cooperation between the public sector, private sector and non-governmental organisations to achieve regional development objectives.

Carrying out research, or having research carried out, concerning the determination of resources and opportunities of the region, acceleration of economic and social development and enhancement of competitiveness, and supporting other research carried out by other bodies, organisations and institutions.

In this paper, a new multi-criteria decision-making (MCDM) model based on type-2 fuzzy sets is proposed for the current grant allocation system of RDA in Turkey. The proposed model is applied on a real case problem by the Middle Black Sea Development Agency in Turkey.

Many national or international organisations have ranked provinces for different purposes, such as tourism potential and terms of livability of cities, etc.

The research by the quality of life group first at the University of Glasgow and then at Strathclyde University has in many respects become the benchmark of quality of life ratings in Britain (Findlay et al., 1988; Rogerson, 1999; Rogerson et al., 1989). Kahn (1995) presented a method for ranking city quality of life. The Economist Intelligence Unit calculates the Global Livability Report every year. The rankings take the following into account: stability, calculated via crime rates and other forms of civil unrest; access to and quality of healthcare; culture and environment; availability and quality of education and infrastructure.

In addition to this, many international organisations, such as the OECD, UNDP, World Bank, calculate their own development indices for cities and allocate resources for fixing the development gap among cities.

Associated with the globalising of the economy, cities are becoming more critical agents of economic development. Their greater ability to be flexible and responsive to the changing needs of the markets, technology and culture has provided them with the opportunity to engage with, and respond to, the needs of capital (Castells and Hall, 1993). Of course, not all locations can respond equally to such forces or opportunities arising from the global changes. The competitiveness of cities reflects not only their current capacity to engage with global capital, but also is a function of their heritage, resulting in a spatially differentiated pattern of competitiveness.

One of the basic aims of development agencies is to reduce the intra-regional development gap.

In order to decrease the disparities in the region, it is highly necessary to determine the socio-economic development levels of the provinces located in the region. Dimensions of development are represented in Figure 1.

Figure 1.

Classic dimensions of sustainable development.

Figure 2.

The footprint of uncertainty (FOU) of the type-2 fuzzy set.

The rankings take many economic, social and environmental aspects into account, such as crime rates, access to and quality of healthcare, availability and quality of education and infrastructure. For this reason, ranking of cities is indeed an MCDM process.

The purpose of MCDM is to choose a best candidate from a set of alternatives by means of evaluating multiple attributes of the alternatives (Chen and Lee, 2010a). The intent of MCDM methods is ranking of provinces for grants allocation involving multiple criteria by making choices more explicit, rational and efficient. MCDM methods have six basic functions that support this overall goal (Hobbs and Meier, 2000):

to structure the decision process,

to display tradeoff among criteria,

to help decision makers reflect upon, articulate and apply value judgements concerning acceptable tradeoffs, resulting in recommendations concerning alternatives,

to help people make more consistent and rational evaluations of risk and uncertainty,

to facilitate negotiation,

to document how decisions are made.

Assigning numerical values for evaluation criteria is not always possible. For this reason, in cases where assigning a numerical value is not possible, linguistic variables are used. The fuzzy set theory enables comparison of alternatives by digitising linguistic variables.

In the process of decision making, the information about attribute values is usually uncertain or fuzzy because of the increasing complexity of the socio-economic environment and the vagueness of the inherently subjective nature of human thinking. This fact has led many authors to apply the fuzzy set theory (Zadeh, 1965) to model the uncertainty and vagueness in decision processes (Kilic and Kaya, 2015; Wang et al., 2012).

Unlike the published papers, a new MCDM methodology based on type-2 fuzzy sets is improved in this paper by integrating crisp and linguistic evaluation together with a new defuzzification method. Membership functions of type-1 fuzzy sets are crisp sets. For this reason, in cases where the meanings of criteria are not clear, the evaluators do not hold the same opinions and the setting of evaluation is noisy, type-1 fuzzy sets cannot offer effective decision support. In such cases, type-2 fuzzy sets, whose membership functions are type-1 fuzzy sets too, enables convenient modelling of the problem. If we can use interval type-2 fuzzy sets (Mendel et al., 2006) for handling fuzzy group decision-making problems, then there is room for more flexibility because of the fact that interval type-2 fuzzy sets are more suitable to represent uncertainties than type-1 fuzzy sets (Chen and Lee, 2010a). Type-2 fuzzy sets have been successfully applied in the decision-making process. Kilic and Kaya (2015) proposed a fuzzy MCDM methodology based on type-2 fuzzy sets to evaluation of investment projects. Wang et al. (2012) presented a new method to handle fuzzy group decision-making (GDM) problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets (IT2 FSs) and used an example to illustrate the fuzzy GDM process of the proposed method. Wu and Mendel (2007) presented a method using the linguistic weighted average and IT2 FSs for handling fuzzy multiple criteria hierarchical GDM problems, Wu and Mendel’s fuzzy multiple criteria hierarchical GDM method was to make decisions by means of aggregating the opinions of DMs. Chen and Lee (2010a) presented an IT2 fuzzy TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method to handle fuzzy MAGDM problems based on IT2 FSs, and used some examples to illustrate the fuzzy MADM process of the proposed method. Chen and Lee (2010b) also presented a new method to handle fuzzy MAGDM problems based on the ranking values and the arithmetic operations of IT2 FSs.

In this study, a new MCDM technique based on type 2 fuzzy sets is used for the prioritisation of provinces for public grants allocation. For this aim, we use analytical hierarchy process (AHP) and TOPSIS methods based on type-2 fuzzy sets because of the nature of the problem which is the subject of our study. In the model, we used the fuzzy sets and crisp sets at the same time because some evaluation criteria were fuzzy and decision makers did not specify a definite value for some criteria.

The fuzzy sets and fuzzy numbers

To deal with vagueness of human thought, Zadeh (1965) first introduced the fuzzy set theory, which was based on the rationality of uncertainty due to imprecision or vagueness. A major contribution of fuzzy set theory is its capability of representing vague knowledge. The theory also allows mathematical operators and programming to apply to the fuzzy domain (Kahraman and Kaya, 2010).

The fuzzy set theory is an important method to provide measurement of the ambiguity of concepts that are associated with human beings’ subjective judgements, including linguistic terms, degree of satisfaction and degree of importance that are often vague. A linguistic variable is a variable whose values are not numbers but phrases in a natural language. The concept of a linguistic variable is very useful in dealing with situations that are too complex or not well defined to be reasonably described in conventional quantitative expressions (Zimmermann, 1991).

Membership functions of type-1 fuzzy sets are crisp sets. For this reason, in cases where the meanings of criteria are not clear, the evaluators do not hold the same opinions and the setting of the evaluation is noisy, type-1 fuzzy sets cannot offer effective decision support. In such cases, type-2 fuzzy sets, whose membership functions are type-1 fuzzy sets too, enables convenient modelling of the problem. Type-2 fuzzy sets can be regarded as an extension of type-1 fuzzy sets.

Interval Type-2 fuzzy sets

An interval type-2 fuzzy set is a special case of a generalised type-2 fuzzy set. Since generalised type-2 fuzzy sets require complex and immense computational burdensome operations, the widespread application of generalised type-2 fuzzy systems has not occurred. Interval type-2 fuzzy sets are the most commonly used type-2 fuzzy sets because of their simplicity and reduced computational effort with respect to general type-2 fuzzy sets (Kahraman et al., 2014).

In this section, the basic concepts and operations of interval type-2 fuzzy sets are introduced below (Chen and Lee, 2010a, 2010b; Chen et al., 2012; Lee and Chen, 2008; Mendel et al., 2006; Wang et al., 2012):

Definition 2.1. A type-2 fuzzy set $\tilde{\tilde{A}}$ in the universe of discourse X can be represented by a type-2 membership function $μ_{\tilde{\tilde{A}}}$ , shown as follows:

\overset{\approx}{A} = {((x, u), μ_{\overset{\approx}{A}} (x, u)) | \forall x \in X, \forall u \in J_{x} \subseteq [0, 1], 0 \leq μ_{\overset{\approx}{A}} (x, u) \leq 1}

(1)

where $J_{x}$ denotes an interval in [0, 1]. Moreover, the type-2 fuzzy set $\tilde{\tilde{A}}$ also can be represented as follows:

\tilde{\tilde{A}} = \int_{x \in X} \int_{u \in J_{x}} μ_{\tilde{\tilde{A}}} (x, u) / (x, u)

(2)

where $J_{x} \subseteq [0, 1]$ and ∫∫ denotes union over all admissible x and u.

Definition 2.2. Let $\tilde{\tilde{A}}$ be a type-2 fuzzy set in the universe of discourse X represented by the type-2 membership function $μ_{\tilde{\tilde{A}}}$ . If all $μ_{\tilde{\tilde{A}}} (x, u) = 1.00$ , then $\tilde{\tilde{A}}$ is called an interval type-2 fuzzy set. An interval type-2 fuzzy set $\tilde{\tilde{A}}$ can be regarded as a special case of a type-2 fuzzy set, represented as follows:

\tilde{\tilde{A}} = \int_{x \in X} \int_{u \in J_{x}} 1 / (x, u)

(3)

where $J_{x} \subseteq [0, 1] .$

Definition 2.3. The upper membership function and the lower membership function of an interval type-2 fuzzy set are type-1 membership functions, respectively.

A trapezoidal interval type-2 fuzzy set is shown in Figure 1. The FOU is shown as the shaded region. It is bounded by an upper membership function (UMF) and a lower membership function (LMF), both of which are type-1 fuzzy sets.

Operations on type-2 fuzzy sets

In this subsection, the main operations on type-2 fuzzy sets are introduced (Chen and Lee, 2010a, 2010b; Chen et al., 2012; Lee and Chen, 2008; Mendel et al., 2006; Wang et al., 2012):

Definition 2.4. The addition operation between the trapezoidal interval type-2 fuzzy set

\begin{array}{l} {\overset{\approx}{A}}_{1} = (Ã_{1}^{U}, Ã_{1}^{L}) = ((a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U},; H_{1} (Ã_{1}^{U}), H_{2} (Ã_{1}^{U})), (a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L},; H_{1} (Ã_{1}^{L}), H_{2} (Ã_{1}^{L}))) \\ and {\overset{\approx}{A}}_{2} = (Ã_{2}^{U}, Ã_{2}^{L}) = (a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U},; H_{1} (Ã_{2}^{U}), H_{2} (Ã_{2}^{U})), (a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L},; H_{1} (Ã_{2}^{L}), H_{2} (Ã_{2}^{L}))) \end{array}

is defined as follows:

\begin{array}{l} {\overset{\approx}{A}}_{1} \oplus {\overset{\approx}{A}}_{2} = (Ã_{1}^{U}, Ã_{1}^{L}) \oplus (Ã_{2}^{U}, Ã_{2}^{L}) \\ = ((((a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} + a_{23}^{U}, a_{14}^{U} + a_{24}^{U}; \\ \min (H_{1} (Ã_{1}^{U}), H_{1} (Ã_{2}^{U})), \min (H_{2} (Ã_{1}^{U}), H_{2} (Ã_{2}^{U}))) \\ (a_{11}^{L} + a_{21}^{L}, a_{12}^{L} + a_{22}^{L}, a_{13}^{L} + a_{23}^{L}, a_{14}^{L} + a_{24}^{L}; \\ \min (H_{1} (Ã_{1}^{L}), H_{1} (Ã_{2}^{L})), \min (H_{2} (Ã_{1}^{L}), H_{2} (Ã_{2}^{L}))) . \end{array}

(4)

Definition 2.5. The multiplication operation between the trapezoidal interval type-2 fuzzy sets

\begin{array}{l} {\overset{\approx}{A}}_{1} = (Ã_{1}^{U}, Ã_{1}^{L}) = ((a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U},; H_{1} (Ã_{1}^{U}), H_{2} (Ã_{1}^{U})), (a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L},; H_{1} (Ã_{1}^{L}), H_{2} (Ã_{1}^{L}))) \\ and {\overset{\approx}{A}}_{2} = (Ã_{2}^{U}, Ã_{2}^{L}) = ((a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U},; H_{1} (Ã_{2}^{U}), H_{2} (Ã_{2}^{U})), (a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L},; H_{1} (Ã_{2}^{L}), H_{2} (Ã_{2}^{L}))) \end{array}

is defined as follows :

\begin{array}{l} {\overset{\approx}{A}}_{1} \oplus {\overset{\approx}{A}}_{2} = (Ã_{1}^{U}, Ã_{1}^{L}) \oplus (Ã_{2}^{U}, Ã_{2}^{L}) \\ = ((a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} + a_{23}^{U}, a_{14}^{U} + a_{24}^{U}; \\ \min (H_{1} (Ã_{1}^{U}), H_{1} (Ã_{2}^{U})), \min (H_{2} (Ã_{1}^{U}), H_{2} (Ã_{2}^{U}))) \\ (a_{11}^{L} + a_{21}^{L}, a_{12}^{L} + a_{22}^{L}, a_{13}^{L} + a_{23}^{L}, a_{14}^{L} + a_{24}^{L}; \\ \min (H_{1} (Ã_{1}^{L}), H_{1} (Ã_{2}^{L})), \min (H_{2} (Ã_{1}^{L}), H_{2} (Ã_{2}^{L}))) \end{array}

(5)

Definition 2.6. The arithmetic operations between the trapezoidal interval type-2 fuzzy sets

\begin{array}{l} {\overset{\approx}{A}}_{1} = (Ã_{1}^{U}, Ã_{1}^{L}) = ((a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U},; \\ H_{1} (Ã_{1}^{U}), H_{2} (Ã_{1}^{U})), (a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L},; \\ H_{1} (Ã_{1}^{L}), H_{2} (Ã_{1}^{L}))) \end{array}

and the crisp value k is defined as follows :

\begin{array}{l} k x {\overset{\approx}{A}}_{1} = (({k \times a}_{11}^{U}, {k \times a}_{12}^{U}, {k \times a}_{13}^{U}, {k \times a}_{14}^{U}; H_{1} (Ã_{1}^{U}), H_{2} (Ã_{1}^{U})), \\ (({k \times a}_{11}^{L}, {k \times a}_{12}^{L}, {k \times a}_{13}^{L}, {k \times a}_{14}^{L}; H_{1} (Ã_{1}^{L}), H_{2} (Ã_{1}^{L})) \end{array}

(6)

The proposed methodology based on type-2 fuzzy sets

The proposed model to categorise the settlements in terms of socio-economic development, composed of type-2 fuzzy AHP and type-2 fuzzy TOPSIS methods, consists of three phases as shown in Figure 3: (1) identify the criteria to be used in the model, (2) type-2 fuzzy AHP computations, (3) categorise the settlements with type-2 fuzzy TOPSIS and determination of the final rank. In the first phase, the criteria which will be used in categorisation of the settlements are determined and the decision hierarchy is formed. After that, the decision hierarchy is approved by the decision makers. In the second phase, criteria are assigned weights using type-2 fuzzy AHP. In this phase, pair-wise comparison matrixes are formed by using linguistic variables, to determine the criteria weights. The experts from the decision-making team make individual evaluations using the scale provided in Table 1, to determine the values of the elements of pair-wise comparison matrixes. Computing the geometric mean of the values obtained from individual evaluations, a final pair-wise comparison matrix is found. After checking the consistency of comparison matrix, the type-2 fuzzy weights of the criteria are calculated (Kilic and Kaya, 2015). In the last step of second phase, type-2 fuzzy weights are defuzzified by using the DTraT method that is used in this paper. City ranks are determined by using type-2 fuzzy TOPSIS method in the third phase. For the categorisation of the settlements, linguistic variables in Table 2 and crisp numbers are simultaneously used. Before the normalisation step, the type-2 fuzzy decision matrix is defuzzified. Ranking of the cities is determined according to closeness coefficient value in a descending order schematic.

Figure 3.

The proposed model for prioritisation of provinces for public grants allocation.

Table 1.

Definition and the scale of the linguistic variables used (Ucal Sarı et al., 2013).

Linguistic variables	Interval type-2 fuzzy scales
Absolutely strong (AS)	((7,8,9,9;1,1), (7.2,8.2,8.8,9;0.8,0.8))
Very strong (VS)	((5,6,8,9;1,1), (5.2,6.2,7.8,8.8;0.8,0.8))
Fairly strong (FS)	((3,4,6,7;1,1), (3.2,4.2,5.8,6.8;0.8,0.8))
Slightly strong (SS)	((1,2,4,5;1,1), (1.2,2.2,3.8,4.8;0.8,0.8))
Exactly equal (E)	((1,1,1,1;1,1), (1,1,1,1;1,1))
If factor i has one of the above linguistic variables assigned to it when compared with factor j, then j has the reciprocal value when compared with i	Reciprocals of the above

Table 2.

Linguistic terms and their corresponding interval type-2 fuzzy sets (Chen and Lee, 2010a).

Linguistic terms	Interval type-2 fuzzy sets
Very low (VL)	((0, 0, 0, 0.1; 1, 1), (0, 0, 0, 0.05; 0.9, 0.9))
Low (L)	((0, 0.1, 0.1, 0.3; 1, 1), (0.05, 0.1, 0.1, 0.2; 0.9, 0.9))
Medium low (ML)	((0.1, 0.3, 0.3, 0.5; 1, 1), (0.2, 0.3, 0.3, 0.4; 0.9, 0.9))
Medium (M)	((0.3, 0.5, 0.5, 0.7; 1, 1), (0.4, 0.5, 0.5, 0.6; 0.9, 0.9))
Medium high (MH)	((0.5, 0.7, 0.7, 0.9; 1, 1), (0.6, 0.7, 0.7, 0.8; 0.9, 0.9))
High (H)	((0.7, 0.9, 0.9, 1; 1, 1), (0.8, 0.9, 0.9, 0.95; 0.9, 0.9))
Very high (VH)	((0.9, 1, 1, 1; 1, 1), (0.95, 1, 1, 1; 0.9, 0.9))

Type-2 fuzzy AHP

AHP is a structured approach to decision making developed by Saaty (1980). It is a weighted factor-scoring model and has the ability to detect and incorporate inconsistencies inherent in the decision-making process (Ucal Sarı et al., 2013). In a typical AHP method, experts have to give a definite number within a 1–9 scale to the pair-wise comparison so that the priority vector can be computed. However, factor comparisons often involve some amount of uncertainty and subjectivity. For example, an expert may know one factor is more important than another; however, the expert cannot give a definite scale to the comparison because the expert is not sure about the degree of one factor over another. Sometimes, experts cannot compare two factors because of the lack of adequate information (Kahraman et al., 2009). In this case, type-2 fuzzy sets that provide a mathematical strength to capture the uncertainties associated with the human cognitive process can be used. The steps of a type-2 AHP method are (Kilic and Kaya, 2015):

Step 1: Construct the type-2 fuzzy pair-wise comparison matrix $A_{k}$ of the k^th decision-maker and construct the average decision matrix $\overline{A}$ . Each element of the pair-wise comparison matrix is an interval type-2 fuzzy set, representing which is the more important of two criteria. The pair-wise comparison matrix is shown as follows:

A_{k} = {\tilde{(a_{i j}^{k}})}_{n \times n} = [\begin{matrix} 1 & \overset{\approx}{a_{12}^{k}} & … & \overset{\approx}{a_{1 n}^{k}} \\ \overset{\approx}{a_{21}^{k}} & 1 & … & \overset{\approx}{a_{2 n}^{k}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \overset{\approx}{a_{n 1}^{k}} & \overset{\approx}{a_{n 2}^{k}} & … & 1 \end{matrix}] = [\begin{matrix} 1 & \overset{\approx}{a_{12}^{k}} & … & \overset{\approx}{a_{1 n}^{k}} \\ 1 / \overset{\approx}{a_{12}^{k}} & 1 & … & \overset{\approx}{a_{2 n}^{k}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 1 / \overset{\approx}{a_{1 n}^{k}} & 1 / \overset{\approx}{a_{2 n}^{k}} & … & 1 \end{matrix}]

(7)

\overline{A} = {(\overset{\approx}{a_{i j}})}_{n \times n},

(8)

where ${\tilde{\tilde{a}}}_{ij} = \sqrt[k]{{\tilde{\tilde{a}}}_{ij}^{1} \otimes {\tilde{\tilde{a}}}_{ij}^{2} \otimes ⋮ \otimes {\tilde{\tilde{a}}}_{ij}^{k}}$ , ${\tilde{\tilde{a}}}_{ij}$ is an interval type-2 fuzzy set, $1 \leq i \leq n$ , $1 \leq j \leq n$ , $1 \leq k \leq q$ . k denotes the number of decision makers. For the evaluation procedure, the linguistic terms given in Table 1 are used. The geometric mean is used to aggregate expert opinions.

Step 2: Examine the consistency of the fuzzy pair-wise comparison matrices. Assume $A = [a_{ij}]$ is a positive reciprocal matrix and $\tilde{\tilde{A}} = [{\tilde{\tilde{a}}}_{ij}]$ is a fuzzy positive reciprocal matrix. If the result of the comparisons of $A = [a_{ij}]$ is consistent, then it can imply that the result of the comparisons of $\tilde{\tilde{A}} = [{\tilde{\tilde{a}}}_{ij}]$ is also consistent (Buckley, 1985).

Step 3: Compute the fuzzy geometric mean for each criterion. The geometric mean of each row of $\tilde{\tilde{A}} = [{\tilde{\tilde{a}}}_{ij}]$ is calculated as:

{\tilde{\tilde{r}}}_{i} = \sqrt[n]{{\tilde{\tilde{a}}}_{i 1} \otimes {\tilde{\tilde{a}}}_{i 2} \otimes ⋮ \otimes {\tilde{\tilde{a}}}_{in}}

(9)

Step 4: Compute the weights of the criteria by normalisation. The type-2 fuzzy weight of the i^th criterion is calculated as:

{\tilde{\tilde{w}}}_{i} = {\tilde{\tilde{r}}}_{i} \otimes {[{\tilde{\tilde{r}}}_{i} \oplus {\tilde{\tilde{r}}}_{i} \oplus ⋮ . . \oplus {\tilde{\tilde{r}}}_{i}]}^{- 1}

(10)

Step 5: Defuzzify the type-2 fuzzy weights of the criteria. In this study, the Defuzzified Trapezoidal Type-2 Fuzzy Set (DTraT) approach developed by Kahraman et al. (2014) is utilised to defuzzify the type-2 fuzzy numbers as follows:

D T r a T = \frac{\frac{(u_{u} - l_{u}) + (β_{u} . m_{1 u} - l_{u}) + (α_{u} . m_{2 u} - l_{u})}{4} + l_{u} + [\frac{(u_{L} - l_{L}) + (β_{L} . m_{1 L} - l_{L}) + (α_{L} . m_{2 L} - l_{L})}{4} + l_{L}]}{2}

(11)

Interval type-2 fuzzy TOPSIS methodology

The TOPSIS is a multi-criteria decision analysis method, which was originally developed by Hwang and Yoon in 1981. TOPSIS is based on the concept that the chosen alternative should have the shortest geometric distance from the positive ideal solution and the longest geometric distance from the negative-ideal solution. In the following, we extend the TOPSIS method (Hwang and Yoon, 1981) to present a new method for handling fuzzy MCDM problems based on crisp numbers and interval type-2 fuzzy sets by revising the method proposed by Chen and Lee (2010a, 2010b). The details of the type-2 fuzzy TOPSIS method are presented as follows (Kilic and Kaya, 2015):

Step 1: Construct the decision matrix $D_{k}$ of the k^th decision-maker and construct the average decision matrix $\bar{D}$ , respectively, shown as follows:

D_{k} = {\tilde{(a_{i j}^{k}})}_{m \times n} = \begin{matrix} {\tilde{\tilde{a}}}_{1} \\ {\tilde{\tilde{a}}}_{2} \\ ⋮ \\ {\tilde{\tilde{a}}}_{m} \end{matrix} [\begin{matrix} \overset{\approx}{a_{11}^{k}} & \overset{\approx}{a_{12}^{k}} & \dots & \overset{\approx}{a_{1 n}^{k}} \\ \overset{\approx}{a_{21}^{k}} & \overset{\approx}{a_{22}^{k}} & \dots & \overset{\approx}{a_{2 n}^{k}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \overset{\approx}{a_{m 1}^{k}} & \overset{\approx}{a_{m 2}^{k}} & \dots & \overset{\approx}{a_{m n}^{k}} \end{matrix}]

(13)

\tilde{\tilde{D}} = {({\tilde{\tilde{a}}}_{ij}^{k})}_{m \times n}

(14)

where ${\tilde{\tilde{a}}}_{ij} = (\frac{{\tilde{\tilde{a}}}_{ij}^{1} \oplus {\tilde{\tilde{a}}}_{ij}^{2} \oplus ⋮ . \oplus {\tilde{\tilde{a}}}_{ij}^{k}}{q})$ , ${\tilde{\tilde{a}}}_{ij}$ is an interval type-2 fuzzy set or crisp number, $1 \leq i \leq m$ , $1 \leq j \leq n$ , $1 \leq k \leq q$ . k denotes the number of decision makers.

Step 2: Based on equation (11), defuzzify the interval type-2 fuzzy set ${\tilde{\tilde{a}}}_{ij}$ . Construct the defuzzified decision matrix $\overline{D^{*}}$ .

Step 3: The defuzzified decision matrix $\overline{D^{*}}$ is normalised by using equation (15).

r_{ij} = {\begin{matrix} \frac{r_{ij}}{r_{i}^{+}}, a_{i} \in A_{1} \\ \frac{r_{i}^{-}}{r_{ij}}, a_{i} \in A_{2} \end{matrix}

(15)

where $A_{1}$ denotes the set of benefit criteria, $A_{2}$ denotes the set of cost criteria, and $1 \leq i \leq m$ .

Step 4: Weighted normalised decision matrix is obtained by multiplying normalised matrix with the weights of the criteria calculated by the type-2 fuzzy AHP method:

v_{ij} = w_{i} \times r_{ij}

(16)

where, $1 \leq i \leq m$ , $1 \leq j \leq n$ .

Step 5: Positive ideal solution $v^{+}$ = ( $v_{1}^{+}$ , $v_{2}^{+}$ , …, $v_{m}^{+}$ ) and the negative-ideal solution $v^{-}$ = ( $v_{1}^{-}$ , $v_{2}^{-}$ , …, $v_{m}^{-}$ ) are determined as:

v_{i}^{+} = {\begin{matrix} max_{1 \leq j \leq n} v_{j} if a_{i} \in A_{1} \\ min_{1 \leq j \leq n} v_{j} if a_{i} \in A_{2} \end{matrix}

(17)

and

v_{i}^{-} = {\begin{matrix} max_{1 \leq j \leq n} v_{j} if a_{i} \in A_{2} \\ min_{1 \leq j \leq n} v_{j} if a_{i} \in A_{1} \end{matrix}

(18)

where $A_{1}$ denotes the set of benefit attributes, $A_{2}$ denotes the set of cost attributes, and $1 \leq i \leq m$ .

Step 6: The distance of each alternative from positive ideal solution $v^{+}$ and negative-ideal solution $v^{-}$ is calculated as:

d^{+} (p_{j}) = \sqrt{\sum_{i = 1}^{m} {(v_{ij} - v_{i}^{+})}^{2}},

(19)

d^{-} (p_{j}) = \sqrt{\sum_{i = 1}^{m} {(v_{ij} - v_{i}^{-})}^{2}},

(20)

where $1 \leq j \leq n$ .

Step 7: The closeness coefficient of each alternative $C (p_{j})$ is calculated as:

C (p_{j}) = \frac{d^{-} (p_{j})}{d^{+} (p_{j}) + d^{-} (p_{j})},

(21)

where $1 \leq j \leq n$ .

Step 8: Sort the values of $C (p_{j})$ in a descending sequence, where $1 \leq j \leq n$ . The larger the value of $C (p_{j})$ , the higher the preference of the alternative $p_{j}$ , where $1 \leq j \leq n$ .

A real case application

The Middle Black Sea Development Agency (OKA) was established by decision of the Council of Ministers on 10 November 2008 in order to support economic, social and cultural development of four cities located in the Middle Black Sea Region of Turkey. These cities are Samsun, Amasya, Tokat and Çorum, among which Samsun has been chosen as the centre of the Agency as shown in Figure 4. OKA is one of the RDA which was founded by the Turkish Government throughout the whole country in the NUTS-2 regions to coordinate regional development, to introduce strategies to enable regions to use their capacities to the maximum benefit of the region, to supply the region with the means to improve their competitiveness and reduce the imbalance existing within and between the regions. Objectives of the Agency are to improve cooperation between public sector, private sector and NGOs and to promote the effective and efficient use of resources in order to accelerate the sustainability of regional development by evoking local potential (OKA, 2013).

Figure 4.

The location of Middle Black Sea Development Agency in Turkey.

Since the budget allocated for support is not sufficient for supporting all investment projects in all cities, the OKA needs to be able to use its resources in the most efficient and effective way possible. For this reason, determination of which city should be supported is greatly a significant decision-making problem.

Evaluating projects without making a classification on the basis of cities constitutes some problems. Every city has a different level of socio-economic development. There are fewer SMEs in less-developed cities. In addition, the institutional capacity, economic structure and project management skill levels in these SMEs are lower than that of other SMEs. Without a policy of positive discrimination towards the SMEs in less-developed cities, the possibility for these SMEs to be successful in project competitions organised by RDA is lower than that of the other SMEs. Consequently, regional development agency funds flow to developed cities and thus intra-regional disparities increase. In order to play a vital role to reduce intra-regional disparities, it is necessary to categorise the settlements in terms of socio-economic development.

In the application phase, four provinces Amasya ( $p_{1}$ ), Çorum ( $p_{2}$ ), Samsun ( $p_{3}$ ), Tokat ( $p_{4}$ ) have been ranked using the linguistic variables in Table 2 and crisp numbers by three decision makers $D M_{1}$ , $D M_{2}$ , $D M_{3}$ that are considered as expert.

Similar studies in the literature, similar organisations providing financial support such as the OECD, World Bank, UNDP have been examined. Next, the criteria have been finalised following meetings with OKA staff, experts and from the literature review. In this study, 26 criteria have been utilised to categorise provinces as follows:

Demographic (C₁)

Net Migration Rate (c₁₁): the number of net migration per thousand persons who are able to migrate. If out-migration is more than in-migration, net migration is negative.

Fertility Rate (15–49 age) (c₁₂): number of births to mothers of age 15–49 to the average female population of age 15–49.

Green Card Holder Population Rate(c₁₃): the rate of green card holder citizens who do not have social security to total population of a city.

Rate of Urbanisation (c₁₄): the rate of province and district centre’s population to total population in a specific area.

High School Graduation Rate Among Females (c₁₅): rhe rate of high school graduate females to total female population over the age 15 in a city.

Social Inclusion (C₂),

vi. Schooling Rate (c₂₁): the rate of the number of students to school age population.

vii. The Average Score of University Entrance Exam (c₂₂): the average score of people in a city who take the University Entrance Exam.

viii. The Number of Doctors per 10,000 People (c₂₃): the number of physicians per 1000 persons in a city.

ix. Gender Equality Score (c₂₄): a gender equality index for 81 provinces developed by the NGO ‘Economic Policy Research Foundation of Turkey’. It was developed on basis indicators such as the level of education, services provided for women, etc.

x. Unemployment Rate (c₂₅): the rate of job seekers to total number of workforce.

Quality of Life (C₃)

Particulate Matter in the Atmosphere (c₃₁): the level of particles in the air detected by the air quality measurement stations in a city.

The Crime Rate (c₃₂): the ratio of convicts to total population in a city.

Rentable Mall Space per 1000 People (c₃₃): the size of rentable mall space per 1000 people in a city.

Moviegoer Rate (c₃₄): the rate of the number of movie tickets sold to total population of a city.

Rate of Theatre Audiences (c₃₅): the rate of the number of theatre tickets sold to total population of a city.

Green Areas (c₃₆): a measure for the sufficiency of green areas such as parks and botanic gardens in a city. Because of measurement difficulties and non-objectivities of the criteria, we use type-2 fuzzy sets for calculation.

Competitiveness (C₄)

The Rate of Doctoral Graduates (c₄₁): the rate of people having a Phd degree to total population over the age of 30.

Number of Patent Applications (c₄₂): the rate of the number of patent applications in the last 5 years to total population of a city.

Bank Deposits per Capita (c₄₃): the rate of the total amount of bank deposits in a city to total population.

Value of Agricultural Production per Capita (c₄₄): the ratio of the monetary value of agricultural production in a province to rural population.

Infrastructure (C₅)

Length of Rail Lines (c₅₁): the ratio of total railway length in a province to its acreage.

Concrete or Asphalt Village Road Rate (c₅₂): the ratio of the length asphalt or concrete roads to total length of village roads.

Number of Hospital Beds per 100,000 Population (c₅₃): the number of total hospital beds per 100,000 people in a province.

Number of Parcels Started Production in Organised Industrial Zones (c₅₄): the number of parcels in organised industrial zones that have an active production facility in a province.

Number of Broadband Internet Subscriptions (c₅₅): the ratio of the number of broadband internet subscriptions to total number of dwellings.

Percentage of Rural Population Who Receive Water or Wastewater Services (c₅₆): the rate of rural population who receive water or wastewater services to total rural population.

In the decision making process the criteria c₁₁, c₁₃, c₂₅, c₃₁ and c₃₂ are regarded as cost criteria whereas the other criteria are considered as benefit criteria. Decision makers have used the type-2 fuzzy set for evaluation criteria of c₃₆; however they have used crisp number without any scale for the rest of the evaluation criteria. The hierarchical structure of the province ranking methodology is shown in Figure 5.

Figure 5.

The hierarchical structure of the province ranking methodology.

The three decision makers $D M_{1}$ , $D M_{2}$ and $D M_{3}$ use the linguistic terms shown in Table 2 to represent the evaluating values of the alternatives with respect to different criteria, as shown in Table 3.

Table 3.

Decision matrix.

Criteria		Amasya	Çorum	Samsun	Tokat
Net Migration Rate (−)		7.05	8.21	3.05	32.99
Fertility Rate (15–49 age)		66.65	72.27	66.65	69.3
Green Card Holder Population Rate		11.81	15.93	14.69	16.75
Rate of Urbanisation		65.62	68.97	67.14	58.39
High School Graduation Rate Among Females		15.56	11.98	16.03	13.45
Vocational Schooling Rate		49.45	38.06	37.50	40.93
The Average Score of University Entrance Exam		204.69	209.65	210.57	209.72
The Number of Doctors per 10,000 People		6.33	5.94	6.79	9.94
Gender Equality Score		0.49	0.51	0.52	0.56
Unemployment Rate		5.6	5.1	5.7	5.9
Particulate Matter in the Atmosphere		35	64	51	44
Crime Rate		143.62	121.03	88.36	105.70
Rentable Mall Space per 1000 People		0	0	129.06	46.77
Rate of Cinema Audiences		0.08	0.18	0.38	0.22
Rate of Theatre Audiences		0.01	0.07	0.11	0.01
Green Areas	DM1	ML	ML	ML	ML
	DM2	M	M	M	M
	DM3	L	L	L	L
Rate of Doctoral Graduates		1.63	1.63	3.27	2.42
Number of Patent Applications		3.73	10.15	10.06	5.68
Bank Deposits per Capita		3.05	3.63	3.93	2.05
Value of Agricultural Production per Capita		12.46	12.55	9.85	8.92
Length of Rail Lines		12.10	0	15.83	13.60
Concrete or Asphalt Village Road Rate		46.18	48.22	39.52	48.83
Number of Hospital Beds per 100,000 Population		270	277	331	286
Number of Parcels Started Production in Organised Industrial Zones		96	192	220	271
Number of Broadband Internet Subscriptions		1.75	1.41	1.81	1.49
Percentage of Rural Population Who Receive Water or Wastewater Services		96	98	83	96

After taking the ratings of three decision-makers’ evaluation of the criteria defined by linguistic variables based on equation (11), the defuzzified values of the interval type-2 fuzzy set is calculated. Normalisation is performed after the defuzzification of the decision matrix.

If we used merely the type-2 set, we could have continued the process without the need for the normalisation step. But, different from other studies in the literature, our model allows use of both type-2 fuzzy set and crisp numbers. In addition to this, the unit free nature of crisp numbers made the use of normalisation a must to be able to continue the process.

Type-2 AHP method is used on determination of weights of the city-ranking criteria. The decision makers make their evaluations by using the linguistic terms defined in Table 2. Table 4 presents the linguistic variables of the pair-wise comparison matrix which are defined by three different experts for the criteria.

Table 4.

Linguistic variables of the pair-wise comparison matrix.

	C1	C2	C3	C4	C5
C1	E,E,E	W,W,W	W,W,W	SW,SW,SW	SW,SW,SW
C2	S,S,S	E,E,E	SW,SW,SS	SS,SS,SS	SS,E,SS
C3	S,S,S	SS,SS,SW	E,E,E	SS,SS,SS	SS,SS,SS
C4	SS,SS,SS	SW,SW,SW	SW,SW,SW	E,E,E	SW,E,SW
C5	SS,SS,SS	SW,E,SW	SW,SW,SW	SS,E,SS	E,E,E

The elements of the pair-wise comparison matrices are calculated by using the geometric mean method as follows:

\begin{matrix} \overset{\approx}{a_{i j}} = \sqrt[k]{\overset{\approx}{a_{i j}^{1}} \otimes \overset{\approx}{a_{i j}^{2}} \otimes \dots \otimes \overset{\approx}{a_{i j}^{k}}} \\ \overset{\approx}{a_{23}} = ((0.2, 0.25, 0.5, 1; 1, 1) \\ (0.21, 0.26, 0.45, 0.83; 0.8, 0.8) \\ \otimes (0.2, 0.25, 0.5, 1; 1, 1) \\ (0.21, 0.26, 0.45, 0.83; 0.8, 0.8) \\ \otimes (1, 2, 45; 1, 1) {(1.2, 2.2, 3.8, 4.8; 0.8, 0.8))}^{\frac{1}{3}} \\ = (0.34, 0.50, 1.00, 1.71; 1, 1) (0.37, 0.53, 0.92, 1.49; 0.8, 0.8) \end{matrix}

(22)

In the same way we calculate the rest of the pair-wise comparison matrix shown in Table 5.

Table 5.

Type-2 fuzzy pair-wise comparison matrix for the criteria.

	C1	C2	C3	C4	C5
C1	(1, 1, 1, 1; 1,1)(1, 1, 1, 1; 1,1)	(0.14, 0.17, 0.25, 0.33; 1,1)(0.15, 0.17, 0.24, 0.31; 0.8,0.8)	(0.14, 0.17, 0.25, 0.33; 1,1)(0.15, 0.17, 0.24, 0.31; 0.8,0.8)	(0.20, 0.25, 0.50, 1.00; 1,1)(0.21, 0.26, 0.45, 0.83; 0.8,0.8)	(0.20, 0.25, 0.50, 1.00; 1,1)(0.21, 0.26, 0.45, 0.83; 0.8,0.8)
C2	(3, 4, 6, 7; 1,1)(3.2, 4.2, 5.8, 6.8; 0.8,0.8)	(1, 1, 1, 1; 1,1)(1, 1, 1, 1; 1,1)	(0.34, 0.50, 1.00, 1.71; 1, 1)(0.37, 0.53, 0.92, 1.49; 0.8, 0.8)	(1, 2, 4, 5; 1,1)(1.2, 2.2, 3.8, 4.8; 0.8,0.8)	(1, 1.59, 2.52, 2.92; 1,1)(1.13, 1.69, 2.44, 2.85; 0.8,0.8)
C3	(3, 4, 6, 7; 1,1)(3.2, 4.2, 5.8, 6.8; 0.8,0.8)	(0.58, 1, 2, 2.92; 1,1)(0.67, 1.08, 1.87, 2.68; 0.8,0.8)	(1, 1, 1, 1; 1,1)(1, 1, 1, 1; 1,1)	(1, 2, 4, 5; 1,1)(1.2, 2.2, 3.8, 4.8; 0.8,0.8)	(1, 2, 4, 5; 1,1)(1.2, 2.2, 3.8, 4.8; 0.8,0.8)
C4	(1, 2, 4, 5; 1,1)(1.2, 2.2, 3.8, 4.8; 0.8,0.8)	(0.20, 0.25, 0.50, 1.00; 1,1)(0.21, 0.26, 0.45, 0.83; 0.8,0.8)	(0.20, 0.25, 0.50, 1.00; 1,1)(0.21, 0.26, 0.45, 0.83; 0.8,0.8)	(1, 1, 1, 1; 1,1)(1, 1, 1, 1; 1,1)	(0.34, 0.40, 0.63, 1.00; 1,1)(0.35, 0.41, 0.59, 0.89; 0.8,0.8)
C5	(1, 2, 4, 5; 1,1)(1.2, 2.2, 3.8, 4.8; 0.8,0.8)	(0.34, 0.40, 0.63, 1.00; 1,1)(0.35, 0.41, 0.59, 0.89; 0.8,0.8)	(0.20, 0.25, 0.50, 1.00; 1,1)(0.21, 0.26, 0.45, 0.83; 0.8,0.8)	(1, 1.59, 2.52, 2.92; 1,1)(1.13, 1.69, 2.44, 2.85; 0.8,0.8)	(1, 1, 1, 1; 1,1)(1, 1, 1, 1; 1,1)

According to the type-2 fuzzy pair-wise comparison matrix, the type-2 fuzzy weights of criteria are obtained by the following computational procedures:

\begin{matrix} \overset{\approx}{r_{i}} = \sqrt[n]{\overset{\approx}{a_{i 1}} \otimes \overset{\approx}{a_{i 2}} \otimes \dots \otimes \overset{\approx}{a_{i n}}} \\ \overset{\approx}{r_{1}} = ((1, 1, 1, 1; 1, 1) (1, 1, 1, 1; 1, 1) \\ \otimes (0.14, 0.17, 0.25, 0.33; 1, 1) \\ (0.15, 0.17, 0.24, 0.31; 0.8, 0.8) \\ \otimes (0.14, 0.17, 0.25, 0.33; 1, 1) \\ (0.15, 0.17, 0.24, 0.31; 0.8, 0.8) \\ \otimes (0.20, 0.25, 0.50, 1.00; 1, 1) \\ (0.21, 0.26, 0.45, 0.83; 0.8, 0.8) \\ \otimes (0.20, 0.25, 0.50, 1.00; 1, 1) \\ {(0.21, 0.26, 0.45, 0.83; 0.8, 0.8))}^{\frac{1}{5}} \\ = (0.24, 0.28, 0.44, 0.64; 1, 1) \\ (0.25, 0.29, 0.41, 0.58; 0.8, 0.8) \end{matrix}

(23)

In the same way we calculate the remaining ${\tilde{\tilde{r}}}_{i}$ as follows:

\begin{matrix} {\tilde{\tilde{r}}}_{2} = (1.01, 1.45, 2.27, 2.81; 1, 1) \\ (1.10, 1.53, 2.18, 2.68; 0.8, 0.8) \\ {\tilde{\tilde{r}}}_{3} = (1.12, 1.74, 2.86, 3.48; 1, 1) \\ (1.25, 1.86, 2.75, 3.35; 0.8, 0.8) \\ {\tilde{\tilde{r}}}_{4} = (0.42, 0.55, 0.91, 1.38; 1, 1) \\ (0.45, 0.57, 0.86, 1.24; 0.8, 0.8) \\ {\tilde{\tilde{r}}}_{5} = (0.58, 0.79, 1.26, 1.71; 1, 1) \\ (0.63, 0.83, 1.20, 1.59; 0.8, 0.8) \end{matrix}

(24)

The type-2 fuzzy weights are computed by normalisation as follows:

\begin{matrix} \overset{\approx}{w_{i}} = \overset{\approx}{r_{i}} \otimes {[\overset{\approx}{r_{i}} \oplus \overset{\approx}{r_{i}} \oplus ⋮ ⋮ \oplus \overset{\approx}{r_{n}}]}^{- 1} \\ \overset{\approx}{w_{1}} = (0.24, 0.28, 0.44, 0.64; 1, 1) \\ (0.25, 0.29, 0.41, 0.58; 0.8, 0.8) \\ \otimes [(0.24, 0.28, 0.44, 0.64; 1, 1) \\ (0.25, 0.29, 0.41, 0.58; 0.8, 0.8) \\ \oplus (1.01, 1.45, 2.27, 2.81; 1, 1) \\ (1.10, 1.53, 2.18, 2.68; 0.8, 0.8) \\ \oplus (1.12, 1.74, 2.86, 3.48; 1, 1) \\ (1.25, 1.86, 2.75, 3.35; 0.8, 0.8) \\ \oplus (0.42, 0.55, 0.91, 1.38; 1, 1) \\ (0.45, 0.57, 0.86, 1.24; 0.8, 0.8) \\ \oplus (0.58, 0.79, 1.26, 1.71; 1, 1) \\ {(0.63, 0.83, 1.20, 1.59; 0.8, 0.8)]}^{- 1} \\ = (0.02, 0.04, 0.09, 0.19; 1, 1) \\ (0.03, 0.04, 0.08, 0.16; 0.8, 0.8) \end{matrix}

(25)

In the same way, we can obtain the remaining ${\tilde{\tilde{w}}}_{i}$ as follows:

\begin{matrix} {\tilde{\tilde{w}}}_{2} = (0.10, 0.19, 0.47, 0.83; 1, 1) \\ (0.12, 0.21, 0.43, 0.73; 0.8, 0.8) \\ {\tilde{\tilde{w}}}_{3} = (0.11, 0.22, 0.59, 1.03; 1, 1) \\ (0.13, 0.25, 0.54, 0.91; 0.8, 0.8) \\ {\tilde{\tilde{w}}}_{4} = (0.04, 0.07, 0.19, 0.41; 1, 1) \\ (0.05, 0.08, 0.17, 0.34; 0.8, 0.8) \\ {\tilde{\tilde{w}}}_{5} = (0.06, 0.10, 0.26, 0.51; 1, 1) \\ (0.07, 0.11, 0.24, 0.43; 0.8, 0.8) \end{matrix}

(26)

In order to check the consistency of pair-wise comparison matrices, we calculated the consistency index for each comparison matrix as follows:

\begin{matrix} CI = \frac{λ_{max} - n}{n - 1} \\ C I_{1} = \frac{6.24 - 5}{5 - 1} = 0.035 \end{matrix}

(27)

In the same way, we calculated $C I_{2} = 0.084$ , $C I_{3} = 0.056$ , $C I_{4} = 0.089$ , $C I_{5} = 0.050$ , $C I_{6} = 0.071$ .

After calculating the consistency index of each comparison matrix, we calculated the consistency ratio of each comparison matrix as follows:

\begin{matrix} CR = \frac{CI}{RI} \\ C R_{1} = \frac{0.035}{1.12} = 0.031 \end{matrix}

(28)

In the same way, we calculated $C R_{2} = 0.075$ , $C R_{3} = 0.050$ , $C R_{4} = 0.072$ , $C R_{5} = 0.055$ , $C I_{6} = 0.058$ .

Since all CR values are smaller than 10%, all of the pair-wise comparison matrices are consistent.

Next, we calculated the weight of each criterion with the DTraT method and then we multiplied each variable of the decision matrix with weight of each criterion. Table 6 shows the normalised defuzzified weighted decision matrix.

Table 6.

Weighted normalised defuzzified decision matrix.

	Amasya	Çorum	Samsun	Tokat	$v^{+}$	$v^{-}$
$f_{1}$	0.011	0.010	0.026	0.002	0.002	0.026
$f_{2}$	0.003	0.003	0.003	0.003	0.003	0.003
$f_{3}$	0.008	0.006	0.007	0.006	0.006	0.008
$f_{4}$	0.016	0.017	0.016	0.014	0.017	0.014
$f_{5}$	0.007	0.006	0.007	0.006	0.007	0.006
$f_{6}$	0.047	0.036	0.036	0.039	0.047	0.036
$f_{7}$	0.129	0.132	0.133	0.132	0.133	0.129
$f_{8}$	0.042	0.039	0.045	0.065	0.065	0.039
$f_{9}$	0.014	0.015	0.015	0.016	0.016	0.014
$f_{10}$	0.025	0.027	0.024	0.024	0.024	0.027
$f_{11}$	0.058	0.032	0.040	0.046	0.032	0.058
$f_{12}$	0.070	0.083	0.113	0.095	0.070	0.113
$f_{13}$	0.000	0.000	0.021	0.008	0.021	0.000
$f_{14}$	0.007	0.016	0.033	0.019	0.033	0.007
$f_{15}$	0.004	0.027	0.043	0.004	0.043	0.004
$f_{16}$	0.089	0.089	0.089	0.089	0.089	0.089
$f_{17}$	0.009	0.009	0.018	0.013	0.018	0.009
$f_{18}$	0.004	0.011	0.011	0.006	0.011	0.004
$f_{19}$	0.027	0.033	0.035	0.018	0.035	0.018
$f_{20}$	0.062	0.063	0.049	0.045	0.063	0.045
$f_{21}$	0.008	0.000	0.010	0.009	0.010	0.000
$f_{22}$	0.052	0.055	0.045	0.055	0.055	0.045
$f_{23}$	0.035	0.036	0.043	0.037	0.043	0.035
$f_{24}$	0.003	0.006	0.007	0.008	0.008	0.003
$f_{25}$	0.018	0.014	0.018	0.015	0.018	0.014
$f_{26}$	0.031	0.032	0.027	0.031	0.032	0.027

Next step, we got the positive ideal solution and the negative-ideal solution. After that, we calculated the distance $d^{+} (p_{j})$ between each alternative $p_{j}$ and the ideal solution $v^{+}$ and we can calculate the distance $d^{-} (p_{j})$ between each alternative $p_{j}$ and the negative-ideal solution $v^{-}$ , respectively, where $1 \leq j \leq 4$ .

\begin{matrix} d^{+} (p_{1}) = \sqrt{\sum_{i = 1}^{m} {(v_{ij} - v_{i}^{+})}^{2}} = 0.07 \\ d^{-} (p_{1}) = \sqrt{\sum_{i = 1}^{m} {(v_{ij} - v_{i}^{-})}^{2}} = 0.05 \\ d^{+} = (0.07, 0.05, 0.06, 0.06) \\ d^{-} = (0.05, 0.06, 0.06, 0.05) \end{matrix}

(29)

In the last step we calculated the relative degree of closeness $C (p_{j})$ with respect to the positive ideal solution $v^{+}$ , where $1 \leq j \leq 4$ as shown in Table 7.

Table 7.

Relative degree of closeness of alternatives.

Alternatives	Scores $C (p_{j})$
Amasya	0.45
Çorum	0.54
Samsun	0.51
Tokat	0.44

Since $C (p_{4})$ < $C (p_{1})$ < $C (p_{3})$ < $C (p_{2})$ the suggested ranking of province is:

Çorum-Samsun-Amasya-Tokat.

When this ordering is considered, the most livable cities are Çorum and Samsun alternatives. Going into the details of the provinces, we examined which characteristics made them superior to other provinces. As a result, it was found that the economic structures of these provinces are strong. For this reason, the unemployment rate and net migration rate of these provinces are relatively low. In order to reduce disparities among the provinces, priority should be given to first and fourth provinces in the grant allocation process.

Conclusion

Fixed investments are an extremely important indicator from the aspect of economic development. In conditions where the socio-economic conjuncture is not conducive for fixed investment, the public sector encourages the private sector to make investments in order to ensure regional development. In some situations, the policy of relinquishing tax responsibility to the investors is used; in others, the investors are provided with direct financial support by public institutions. Although investment projects supported by the state are extremely important in terms of national policy, the inter-regional and intra-regional development gap should not be overlooked (Kilic and Kaya, 2015).

In this paper a decision making methodology based on type-2 fuzzy sets is proposed for the prioritisation of provinces for public grants allocation. A highly transparent and comprehensive prioritisation model is required to allocate the grant to the provinces. It is necessary to consider many criteria to categorise provinces in terms of socio-economic development. Some of these criteria are subjective and extremely difficult to express in numbers. However, using fuzzy sets provides huge facilities to decision makers in a city-ranking process with linguistic variables and measurement challenges. Type-2 fuzzy sets whose membership functions are type-1 fuzzy sets, provide us with additional degrees of freedom to represent the uncertainty and the fuzziness of the real world.

In the application phase, four provinces have been categorised in terms of socio-economic development using linguistic variables and crisp numbers by three agency specialists. After four different provinces were evaluated for 26 criteria, the provinces with the lowest scores have been given priority in the grant allocation process. As a consequence of this application, it has been observed that the proposed model has proved effective in he categorisation of provinces in MCDM problems in a broader perspective and flexible fashion.

In future studies, the proposed methodology based on fuzzy sets could be used to cover all provinces for all country and could also be a base for a state subsidy system.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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