Selection of the best outsourcing firm for WEEE under hesitant fuzzy environment

3.1 Hesitant fuzzy sets

HFSs has been defined by Torra [45] as follows [5]: Let X be a fixed set, a HFS on X is in terms of a function that when applied to X returns a subset of [0, 1]. Mathematical expression for HFS is as follows [5]: E = { < x , hE ( x ) > | x ∈ X }(1)

where hE (x) is a set of some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set E.

Linguistic variable is a variable whose values are not precise numbers but words or sentences in a natural or artificial language that is closer to human beings’ cognitive processes. The definitions and about linguistic variables and hesitant fuzzy sets are given in the following [15, 41]:

Definition 1. [15, 40] The symbolic translation is a numerical value assessed in [– 0.5, 0.5] that supports the “difference of information” between a counting of information β assessed in the interval of granularity [0, g] of the term set S and the closest value in {0, …,  g} which indicates the index of the closest linguistic term in S. This concept was used to develop a linguistic representation model that represents the linguistic information by means of 2-tuples (s_i,  α), si ∈ S, and α _i  ∈ [-0.5,  0.5).

Definition 2. [15, 40] Let S = {s₀, …,  s_g} be a set of linguistic terms. The 2-tuple set associated with S is defined as 〈S〉 = S × [-0.5, 0.5). We define the function Δ : [0, g] → 〈S〉 given by Δ ( β ) = ( si , α ) , with { i = round ( β ) α = β - i(2)

where round assigns to β the integer number i ∈ {0, …,  g} closest to β.

Definition 3. [40] Let G_H be a context-free grammar and S = {s₀, …,  s _g } a linguistic term set. The elements of G_H = (V_H,  V_T,  I,  P) are defined as follows: V N = { < primary term > , < composite term > , < unary relation > , < binary relation > , < conjunction > } V T = { lower than , greater than , at least , at most , between , and , s 0 , s 1 , … , s g } I ∈ V N

Definition 4. [40, 41] An HFLTS is an ordered finite subset of consecutive linguistic terms of S, where S = {s₀, …,  s_g} is a linguistic term set.

The comparative linguistic expressions are transformed into HFLTS by means of the transformation function EGH with respect to comparative linguistic expressions generated by GH.

Definition 5. [39, 40] Let E _{G H} be a function that transforms linguistic expressions into HFLTS, H _s . G _H is a context-free grammar and uses the linguistic term set S. And let S _ll be the expression domain generated by G _H . This relation can be shown as E _GH  : S _ll  → H _S and can be have the following transformations: E GH ( s i ) = { s i | s i ∈ S }(3) E GH ( at most s i ) = { s j | s j ∈ S and s j ≤ s i }(4) E GH ( lower than s i ) = { s j | s j ∈ S and s j < s i }(5) E GH ( at least s i ) = { s j | s j ∈ S and s j ≥ s i }(6) E GH ( greater than s i ) = { s j | s j ∈ S and s j > s i }(7)

E GH ( between si and s j ) = { s k | s k ∈ S and s j ≤ s k ≤ s i }(8)

Definition 6. [40, 41] The envelope of an HFLTS, env(HS), is a linguistic interval whose limits are obtained by means of its upper bound (max) and its lower bound (min): env ( H s ) = [ Hs - , Hs + ] , Hs - ≤ Hs +(9)

where the upper bound and lower bound of HS are defined as: Hs - = min ( s i ) = s j , s i ∈ Hs and s i ≥ s j ∀ i(10) Hs + = max ( s i ) = s j , s i ∈ Hs and s i ≥ s j ∀ i(11)

The fuzzy sets are often applied to define the semantics of linguistic descriptors with using membership functions defined in the interval [0, 1] [17, 42]. Using triangular fuzzy numbers enables to represent the uncertainty and vagueness of such linguistic assessments in a better way and its linear membership function f A ˜ ( x ) can be defined asfollows [49]: f A ˜ ( x ) = { 1 x = b ( x - a ) / ( b - a ) a ≤ x < b ( c - x ) / ( c - b ) b < x ≤ c 0 otherwise(12)

In this paper, TFNs have been used to represent the envelopes of HFSs. With this way, we aim to achieve to reflect the hesitancy and uncertainty better with using membership functions for linguistic descriptors.

In this paper, we developed a hesitant linguistic group decision making model for determining the best WEEE outsource firm for district of Eyüp. For this aim, fuzzy envelopes in hesitant decision making have been constructed. The steps of the proposed algorithm are detailed as shown in follow:

Step 1. Define two linguistic terms sets for criteria and alternatives separately. Firstly, S₁1 is defined for criteria as; S 11 = { very low ( vl ) , low ( l ) , medium low ( ml ) ,

and for alternatives, S₂₂ is defined as; S 22 = { very poor ( vp ) , poor ( p ) , medium poor ( mp ) , fair ( f ) , medium good ( mg ) , good ( g ) , very good ( vg ) }

Step 2. Define the context-free grammar G _H , where G _H  = {V _N ,  V _T ,  I,  P}.

Step 3. Define the membership values for the linguistic terms sets. In this paper, the scales that are shown in Tables 1 and 2 are used, respectively. Tables 1 and 2 shows the criteria and alternative scales for using to evaluate the criteria and alternatives.

Step 4. Aggregate the preference relations p ˜ and p ˜ k provided by experts k ∈ {1,  2, …,  t} for both criteria weights and evaluations for criteria and alternatives with by using linguistic term sets. Let experts set be represented by E = {e1,  e2, … et} where t ≥ 2, criteria sets be X = {x1,  x2, …  xn} where n ≥ 2, and alternatives sets be Y = {y₁,  y₂, … ym} where m ≥ 2. Then the matrix composed of preference relations p ˜ k can be shown as in Equation (13). The preference relations p ˜ and p ˜ k , k ∈ { 1 , 2 , … t } can be constructed by using terms or comparative expressions as linguistically, μp : X × T → S₁ and μ _pk  : X × Y → S₂₂ : p ˜ k = ( p ˜ 11 k … p ˜ 1 n k : : : . : : : p ˜ m 1 k … p ˜ mn k )(13)

in which each assessment p ij k ∈ S 22 , represents the preference degree of the alternative y _j for criteria x _i according to expert e _k . These preferences are expressed in the expression domain S₁ and S₂, generated by G _H . All evaluations will be scored with linguistic terms set which have membership degrees separately.

Also the criteria weights matrix can be shown as: p ˜ = ( p ˜ 11 … p ˜ 1 t : : : . : : : p ˜ n 1 … p ˜ nt )

In which each assessment p _nt  ∈ S₁1, represents the preference degree of the criteria x_i for expert e_k. All evaluations again will be scored with linguistic terms set which have membership degrees separately.

Step 5. Transformation linguistic expressions into linguistic intervals [40]: The transformation function E_GH introduced in Definition 5 provides an initial basis for GDM problems: E GH ( p ˜ ij k ) = Hs ( p ˜ ij k ) E GH ( p ˜ it ) = Hs ( p ˜ it )

where i ∈ {1, …… ,  n} n is the number of criteria, j ∈ {1, …… ,  m} m is the number of alternatives and k ∈ {1, …,  t} is the number of experts.

Step 6. Choice an aggregation operator [41]: Obtain an envelope for criteria weights [ p ˜ it - , p ˜ it + ] and alternative evaluations [ p ˜ ij k - , p ˜ ij k + ] for each HFLTS. Each of envelopes is formed as triangular fuzzy numbers and they can be defined for criteria weights and alternative-criteria evaluations as follows: env ( Hs ( p ˜ it ) ) = [ p ˜ it - , p ˜ it + ](14) env ( Hs ( p ˜ ij k ) ) = [ p ˜ ij k - , p ˜ ij k + ](15)

The preference relations of criteria weights’ and criteria-alternatives evaluations can represented as follows: p ˜ = ( [ p ˜ 11 , p ˜ 11 ] … [ p ˜ 1 t , p ˜ 1 t ] : : : [ p ˜ n 1 , p ˜ n 1 ] … [ p ˜ nt , p ˜ nt ] )(16) p ˜ k = ( [ p ˜ 11 k - , p 11 k + ] … [ p ˜ 1 n k - , p ˜ 1 n k + ] : : : [ p ˜ n 1 k - , p n 1 k + ] … [ p ˜ nn k - , p ˜ nn k + ] )(17)

Step 7. Select two linguistic aggregation operators φ and φ. In this paper, we used the arithmetic mean aggregation operator based on 2-tuple defined as follows [6]: x ˜ mean = Δ ( 1 n ∑ i = 1 n Δ - 1 ( s ˜ i , α ˜ i ) ) = Δ ( 1 n ∑ i = 1 n β ˜ i )(18)

All that calculations are made with triangular fuzzy numbers to obtain the fuzzy envelopes.

Step 8. Obtain the P c - (pessimistic preference relations) and P c + (optimistic preference relations) by using φ.

The collective preferences are represented by 2-tuple linguistic values for criteria weights preferences as follows: P ˜ + = ( ( sr ˜ , α ˜ ) 11 + ⋮ ( sr ˜ , α ˜ ) n 1 + ) P ˜ - = ( ( sr ˜ , α ˜ ) 11 - ⋮ ( sr ˜ , α ˜ ) n 1 - )(19) ( sr ˜ , α ˜ ) i + = Δ ( φ ( Δ - 1 ( p ˜ ik + ) ) ) ∀ k ∈ { 1 , … … t }(20) ( sr ˜ , α ˜ ) i - = Δ ( φ ( Δ - 1 ( p ˜ ik - ) ) ) ∀ k ∈ { 1 , … … t }(21)

and for criteria-alternative evaluations: P ˜ c + = ( ( sr ˜ , α ˜ ) 11 + … … ( sr ˜ , α ˜ ) 1 m + : : : ( sr ˜ , α ˜ ) n 1 + … … ( sr ˜ , α ˜ ) mn + )(22) P ˜ c - = ( ( sr ˜ , α ˜ ) 11 - … … ( sr ˜ , α ˜ ) 1 m - : : : ( sr ˜ , α ˜ ) n 1 - … … ( sr ˜ , α ˜ ) mn - )(23) ( sr ˜ , α ˜ ) ij + = Δ ( φ ( Δ - 1 ( p ˜ ij k + ) ) ) ∀ k ∈ { 1 , … … t }(24) ( sr ˜ , α ˜ ) ij - = Δ ( φ ( Δ - 1 ( p ˜ ij k - ) ) ) ∀ k ∈ { 1 , … … t }(25)

being i ∈ {1,  2, …,  n}, j ∈ {1,  2, …,  m} and s _r  ∈ S = {s₀, … s _g } .

First of all, these calculations are hold for criteria weights. The pessimistic and optimistic criteria weights are calculated separately for using the forthcoming steps of TOPSIS methodology.

Step 9. The pessimistic and optimistic criteria weights are normalized by using linear scale transformation. If a preference is shown with (a _ij , b _ij , c _ij ) as a TFN, the normalization procedure for optimistic criteria weights can be applied as follows: P ˜ + = [ s ˜ i + ] n × 1 , i = 1 , 2 , … n(26)

s ˜ i + = ( a i , b i , c i ) The normalized membership degrees of s ˜ i + - now we call it as w ˜ i + - can be calculated as: w ˜ i + = ( a i ' , b i ′ , c i ′ ) a i ′ = a i ∑ i = 1 n c i , b i ′ = b i ∑ i = 1 n b i , c i ′ = c i ∑ i = 1 n a i(27)

Step 10. The aggregated criteria – alternative evaluation hesitant fuzzy enveloped matrix is normalized with a different type of linear normalization. The aggregated normalized hesitant fuzzy weights are calculated as follows [14]:

If a preference is shown with (d _ij , e _ij , f _ij ) as TFN, the normalization procedure for optimistic evaluations can be applied as follows: P ˜ c + = [ s ˜ ij + ] n × m , i = 1 , 2 , … n j = 1 , 2 , … m [ s ˜ ij + ] n × m , = ( d ij , e ij , f ij )(28)

The normalized membership degrees of s ˜ ij + - now we call it as v ˜ ij + - can be calculated as: υ ˜ i j + = ( d i j ' , e i j ' , f i j ' )

where vij ˜ = ( dij f i * , eij f i * , fij f i * ) , i ∈ B ;(29) s ij = ( d i - f ij , d i - e ij , d i - d ij ) , i ∈ C ;(30) f i * = max j f ij i ∈ B ; d i - = min d ij i ∈ C ;

Step 11. Weighted normalized decision matrices are obtained with multiplying the normalized criteria weights which are formed as pessimistic and optimistic values separately and the decision matrix for alternatives which are also formed separately. For example, for optimistic evaluations, weighted normalized decision matrix can be obtained as: V ˜ + = [ v ˜ ij + ] n × m , i = 1 , … n ; j = 1 , … m W ˜ + = [ w ˜ i + ] n × 1 , i = 1 , … n rij ˜ = w ˜ i + ⊗ v ˜ ij +(31)

where w _i j represents the importance of criterion C _i .

Step 12. The Euclidean distance from the best solution (1, 1, 1) and the worst solution (0, 0, 0) for each optimistic and pessimistic triangular fuzzy numbers enveloped preferences are calculated. So, we have the distances both from pessimistic and optimistic preferences. The distances of each criterion for every alternative are summed. In this way, we calculate the single values for the envelopes of non-aggregated preferences. The distances are calculated for the best solution and the worst solution. For example, for the optimistic solutions, the distances from positive and negative ideal solutions can be calculated as: r ˜ i * = ( 1 , 1 , 1 ) ; r ˜ i * - = ( 0 , 0 , 0 ) , i = 1 , … n d j * = ∑ j = 1 m ( ∑ i = 1 n ( r ˜ i * - r ˜ ij + ) ) j = 1 , … m(32) d j - = ∑ j = 1 m ( ∑ i = 1 n ( r ˜ i * - - r ˜ ij + ) ) j = 1 , … m(33)

Step 13. The distance to ideal solution is obtained with for pessimistic and optimistic evaluations can be calculated as: CCi = d j * d j * + d j - j = 1 , … , m(34)

Step 14. A collective linguistic interval for alternative y_j has been computed. The collective linguistic preferences based on perceptions P C + and P C - are aggregated by using an aggregation operator φ, which may or may not be the same as φ. In this step, same operator with φ (arithmetic average) is applied for the aggregation of pessimistic and optimistic preferences.

Step 15. Finally, the best alternative according to the evaluations provided by experts which is constituted by the aggregation of pessimistic and optimistic distances is obtained.

4.1 Problem description and Objective

Our aim is to select the best outsourcing firm for Eyüp Municipality to manage all of the process for WEEE. Unlike other studies, we used HFSs for inclusion of uncertainty inherent in the process to the decision procedure precisely. The use of hesitant fuzzy linguistic term set (HFLTS) enables formal and flexible way to deal with uncertainty in comparative linguistic expressions [19]. HFSs are considered as a strong tool to express ambiguous information in the process of multi-attribute decision making problems [48]. Therefore, evaluation of experts is modeled with the help of HFSs and more objective and realistic results have been investigated. Unlike other studies used HFSs, this paper expresses the envelopes of HFSs in the form of triangular fuzzy sets. Thus, it is aimed better transfer of uncertainties arising from the evaluation of decision-makers to the calculations. It can be argued that better results will be obtained in these strategic decision processes where decisions are made taking into consideration multiple criteria such as WEEE management with the advantages provided by the hesitant fuzzy sets. It is also claimed that closer results to the reality can be obtained by the extension that we contribute to the regular hesitant fuzzy sets.

4.2 Hierarchical structure

The criteria used for the ranking of alternative outsourcing firms are examined in detail primarily with the review of relevant literature and ideas of experts who are working in the EEE sector. As a result of study, the criteria used in the evaluation of alternatives are determined as follows: (i) Organization (C1), (ii) Cost (C2), (iii) Environment (C3), (iv) Technology (C4), (v) Quality (C5). The names of firms which are also alternatives are not included due to confidentiality rules. Instead of sharing the names of firms, we call firms as “A, B, C, D, E” throughout the paper. The hierarchical structure regarding to the problem is shown in Fig. 1. In the scope of this paper, five alternative firms have been evaluated with respect to the five criteria as seen in Fig. 1. The aim is clarified as finding the most appropriate firm for WEEE management in Eyüp, İstanbul.

OPEN IN VIEWER

Fig.1

Hierarchical representation of problem.