Modified intuitionistic fuzzy SIR approach with an application to supplier selection

Abstract

In this paper, we propose a modified fuzzy intuitionistic multiple attribute decision (MAD) method retaining basic framework of the superiority and inferiority ranking (SIR) method. We utilize relative comparisons based on the advantage and disadvantage scores of intuitionistic fuzzy (IF) numbers so that the modified SIR method is better adapted to treat real-world MAD problems. The relative comparisons of the IF values use all the three parameters, namely membership degree (“more the better”), non-membership degree (“less the better”), and hesitancy degree (“less the better”) leading to trade-off values of the parameters. The modified IF-SIR flows, namely IF-superiority and IF-inferiority flows are calculated using the advantage and disadvantage score matrices, respectively, derived from the relative comparisons. These score matrices are based on positive contributions of all three parameters of the IF values, wherever applicable. The ranking of the alternatives is done using a consolidated score function derived from the modified IF-SIR flows. The methodology is validated through an application of a real-case of supplier selection problem.

Keywords

Multiple attribute decision making intuitionistic fuzzy numbers SIR method supplier selection

1 Introduction

Decision making has become a significant activity in the ultra-modern world despite being invaded with various updated scientific know-how and assisted decision tools. Technology alone sometimes fails to comprehend a decision without considering human cognitive and subjective ability. The real-world decision making requires a more elaborated framework in the form of multiple attribute decision making (MADM) when multiple criteria or attributes are taken into consideration [13]. In order to reinforce democracy and rationality of decision making, many real-world decision making processes take place in group settings. Multiple attribute group decision making (MAGDM) [14] is a key component of group decision making. The real-world decision making has been further developed in line with the development of uncertainty and chaos theory. The fuzzy set theory introduced by Zadeh [15] to deal with uncertain information has been overwhelmingly accepted by MADM researchers. Fuzzy-MADM and Fuzzy-MAGDM methods have been used in many practical applications of real-word decision making problems, see [5 , 20] and the references cited therein. However, the fusion has not come without critics and pessimistic view. In many practical fuzzy MAGDM problems, there may exist hesitation in eliciting preferences of the decision makers. The intuitionistic fuzzy sets (IFS) [16], an extension of fuzzy sets are more useful and flexible in dealing with uncertainty originating from vague or ambiguous information. The IFS theory is considered to solve the imprecision of cognitive thinking of humans due to its prominent characteristic that both attached and non-attached information can be taken into account in the decision making. From the large pool of real-world fuzzy decision making methods, there are four quite distinct families of such methods: (i) the outranking, (ii) the value and utility theory based, (iii) the multiple objective programming, and (iv) group decision and negotiation theory based methods.

Among these, the Superiority and Inferiority Ranking (SIR) method, developed by Xu [6] is a generalization of the well-known outranking method-PROMETHEE, which is an efficient approach for MADM. The SIR method combines together the strengths of many MADM methods in handling non-quantifiable, cardinal and/or ordinal data and deal with imprecise information through concepts like indifference and preference thresholds for each attribute. Furthermore, the SIR method is efficient in dealing with data in different units. In the past, Xu [6] explored the relationship between the SIR method and some of the classical MADM methods such as SAW, TOPSIS and PROMETHEE. Ma et al. [7] extended the SIR method wherein individual decision values and weights of the attributes and experts are provided as hesitant fuzzy information and in turn hesitant fuzzy SIR method and interval-valued hesitant fuzzy SIR method were proposed. Tam and Tong [8] presented a variant of SIR method called SIR-Grey for determining the location of large scale harbour-front project development. Marzouk [11] proposed a decision making model that utilizes SIR method together with SAW and TOPSIS methods to generate superiority and inferiority flows. Tam et al. [12] applied SIR method in facilitating the process of construction plant selection. Chai and Liu [9] proposed a novel intuitionistic fuzzy SIR method and illustrated the proposed approach in a simulation of group decision making problem related to supply chain management. Chai et al. [10] proposed a new SIR method under intuitionistic fuzzy environment in modelling uncertain information.

This paper proposes some new modifications in the SIR method, wherein relative comparisons of the IF values are based on all three parameters, namely membership degree (“more the better”), non-membership degree (“less the better”), and hesitancy degree (“less the better”). In return, we obtain trade-off values of these parameters that help to calculate the advantage and disadvantage score matrices, respectively. The score matrices are based on positive contributions of the parameters of the IF values, wherever applicable.

1.1 Major highlights of the proposed research

The distinct features of this research work are stated as under.

When compared with vast literature on intuitionistic fuzzy MADM problems, there is a limited literature on intuitionistic fuzzy MAGDM problems. Therefore, the development of a new MAGDM method utilizing intuitionistic fuzzy information particularly based on the relative comparisons may be of great interest among the researchers and practitioners for solving real-world applications of group decision making in management and science. In particular, when compared with existing literature on fuzzy SIR approach and its applications [8 –12], the proposed SIR method has sufficient novel contribution in terms of making modifications to better adapt the chosen approach for real-world decision making.

There exist significant differences between the proposed research and Chai et al. [10] wherein a new-rule based SIR method was proposed for the fuzzy MAGDM problems. They majorly relied on the TOPSIS approach wherein the relative closeness coefficients are used based on the distance from the positive ideal and negative ideal IF values. These ideal benchmarks against which the performance is evaluated are too impractical to be attained. On the other hand, we use the advantage and disadvantage scores based on measuring relative performance through all three parameters of the IF values leading to attainment of the ideal benchmarks, which are more realistic so that the decision maker knows not only about the best and worst performance but also relative comparison among the individuals. These benchmarks leads to the IF-SIR flows that takes into account both the facts that how good an alternative is when compared with the rest on multiple attributes (strength score) and how much preferable an alternative is over the rest in terms of its weakness on each attribute (weakness score).

When compared with Ma et al. [7], the proposed research may be considered a novel contribution because of the understated reasons. The similarity measure used to analyse similarity between the hesitant fuzzy element (HFE), the positive hesitant fuzzy element (PHFE), and the negative hesitant fuzzy element (NHFE) in their work comprises the distances between HFE and PHFE and between HFE and NHFE, which are inappropriate in the sense that both the PHFE and NHFE are treated as ideal benchmarks against which the performance is evaluated; however, these benchmarks are too impractical to be attained. As explained above, the advantage and disadvantage scores used in the proposed research are more realistic benchmarks.

Additionally, when compared with [7], wherein the superiority and inferiority indexes are defined using the preference intensity of an alternative x_i over x_j with respect to each attribute. This requires the decision maker to specify intensity of the preferences using a threshold function, which is a non-decreasing function in the interval [0,1]. Such a choice brings lots of uncertainty in terms of getting different ranking results corresponding to different threshold values. In our approach, the decision making is not constrained by such choices and depends only on available intuitionistic fuzzy information.

When compared with Tam and Tong [8], the SIR approach developed in this paper utilizes all three parameters of the IF values. The method developed in [8] is based on crisp information, which is quite a restriction on decision making in fuzzy environment.

Additionally, the mathematical model of grey relational grades in [8] depend on comparisons with respect to the virtual perfect, and virtual worst alternatives. Creating a virtual perfect and worst alternative, which contains all the best and worst combinations, respectively, of all the compared alternatives is too unrealistic as far as setting a benchmark is concerned. Practically, this is too ideal to be implemented in an uncertain environment. As explained above, the advantage and disadvantage scores used in the proposed research are more realistic benchmarks.

1.2 Organization of the paper

This paper is organized as follows. Section 2 describes IFS in brief and contents of the classical SIR method. Section 3 discusses the modified IF-SIR approach. Section 4 presents a numerical application of the proposed approach to supplier selection problem. In Section 5, comparison analysis is made followed by concluding remarks in Section 6.

2 Preliminaries

In this section, we present concepts of IFS, score and accuracy functions, the aggregation operators, and a review on classical SIR method.

Definition 1. [1] Let Z be the universe of discourse. Then an intuitionistic fuzzy set A in Z is given by $\begin{matrix} A = {〈 z, μ_{A} (z), ν_{A} (z) 〉 | z \in Z} \end{matrix}$ where μ_A : Z → [0, 1] and ν_A : Z → [0, 1] are degrees of membership and non-membership of an element z ∈ Z, respectively, with the condition 0 ≤ μ_A (z) + ν_A (z) ≤1. For each A in Z, π_A (z) =1 - μ_A (z) - ν_A (z), z ∈ Z, is called the intuitionistic index of z to A. It represents the degree of indeterminacy or hesitation of z to A. For each z ∈ Z, 0 ≤ π_A (z) ≤1. Furthermore, an IF value is denoted as a = (μ_a, ν_a, π_a), where μ_a ∈ [0, 1], ν_a ∈ [0, 1], π_a ∈ [0, 1], and μ_a + ν_a + π_a = 1.

The next definition utilizes the score and accuracy functions defined in [2, 3].

Definition 2. The score function is denoted by S (a) = μ_a - ν_a and the accuracy function is denoted by H (a) = μ_a + ν_a. In order to compare any two IF values a₁, a₂, the following comparison laws [4] may be used.

If S (a₁) > S (a₂), then a₁ > a₂;

If S (a₁) < S (a₂), then a₁ < a₂;

If S (a₁) = S (a₂), then

If H (a₁) > H (a₂), then a₁ > a₂;

If H (a₁) < H (a₂), then a₁ < a₂;

If H (a₁) = H (a₂), then a₁ = a₂.

We next define two operators, namely intuitionistic fuzzy weighted average (IFWA) and intuitionistic fuzzy weighted geometric average (IFWG) used for aggregating intuitionistic fuzzy information [5]. It may be noted that the aggregated value based on membership and non-membership values using IFWA or IFWG operator is also a IF value.

Definition 3. Let a_i = (μ_{a
_i}, ν_{a
_i}) (i = 1, …, n) be a set of IF values. Then IFWA: Θⁿ → Θ is defined as $\begin{matrix} {IFWA}_{ω} (a_{1}, a_{2}, \dots, a_{n}) = \sum_{i = 1}^{n} ω_{i} a_{i} \\ = (1 - \prod_{i = 1}^{n} (1 - μ_{a_{i}})^{ω_{i}}, \prod_{i = 1}^{n} (ν_{a_{i}})^{ω_{i}}) \end{matrix}$ where ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector with ω_i ∈ [0, 1] and $\sum_{i = 1}^{n} ω_{i} = 1$ .

Definition 4. Let a_i = (μ_{a
_i}, ν_{a
_i}) (i = 1, …, n) be a set of IF values. Then IFWG: Θⁿ → Θ is defined as $\begin{matrix} {IFWG}_{ω} (a_{1}, a_{2}, \dots, a_{n}) = \sum_{i = 1}^{n} a_{i}^{ω_{i}} \\ = (\prod_{i = 1}^{n} (μ_{a_{i}})^{ω_{i}}, 1 - \prod_{i = 1}^{n} (1 - ν_{a_{i}})^{ω_{i}}) \end{matrix}$ where ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector with ω_i ∈ [0, 1] and $\sum_{i = 1}^{n} ω_{i} = 1$ .

2.1 The classical SIR method

We outline here some basic concepts of the SIR method originally proposed by Xu [6] for multiple criteria decision making problems.

Suppose a decision maker provides the real-valued performance function g_j (Y_i) to m alternatives A_i (i = 1, …, m) under n criteria g_j (j = 1, …, n). Let f_j be the threshold function for the criteria g_j, which is a non-decreasing function and can be decided by the decision maker. For each pair (A_i, A_k) i, k = 1, …, m, P_j (A_i, A_k) = f_j (g_j (A_i) - g_j (A_k)) is called the preference intensity, which represents the superiority of A_i over A_k, and also the inferiority of A_k over A_i, with respect to the jth criterion. Then main principles of the classical SIR method are summarized as follows:

The SIR indexes: For alternative A_i, the superiority index S_j (A_i) and the inferiority index I_j (A_i) with respect to the jth criterion are defined as $S_{j} (A_{i}) = \sum_{k = 1}^{m} P_{j} (A_{i}, A_{k})$ and $I_{j} (A_{i}) = \sum_{k = 1}^{m} P_{j} (A_{k}, A_{i}),$ where P_j is the preference intensity function; j = 1, …, n, i, k = 1, …, m.

The SIR flows: Using the superiority matrix S = [S_j (A_i)] _m×n and inferiority matrix I = [I_j (A_i)] _m×n, the superiority flow δ^> (A_i) and inferiority flow δ^< (A_i) are defined as δ^> (A_i) = V [S₁ (A_i) , …, S_n (A_i)] and δ^< (A_i) = V [I₁ (A_i) , …, I_n (A_i)] where V is an appropriate aggregation function. Clearly, the higher δ^> (A_i) and the lower δ^< (A_i), the better the alternative A_i.

The SIR ranking: This ranking considers three relations, namely the preference intensity relation P, the indifference relation I, and the incomparability relation R. Using descending order of the δ^> (A_i), the superiority ranking $R_{>}^{*} = {P_{>}, I_{>}}$ can be obtained as follows: $\begin{matrix} A_{i} P_{>} A_{k} iff δ^{>} (A_{i}) > δ^{>} (A_{k}) and A_{i} I_{>} A_{k} iff \\ δ^{>} (A_{i}) = δ^{>} (A_{k}) . \end{matrix}$

Similarly, using ascending order of the δ^< (A_i), the inferiority ranking $R_{<}^{*} = {P_{<}, I_{<}}$ can be obtained as follows: $\begin{matrix} A_{i} P_{<} A_{k} iff δ^{<} (A_{i}) < δ^{<} (A_{k}) and A_{i} I_{<} A_{k} iff \\ δ^{<} (A_{i}) = δ^{<} (A_{k}) . \end{matrix}$

The two partial ranks $R_{>}^{*} = {P_{>}, I_{>}}$ and $R_{<}^{*} = {P_{<}, I_{<}}$ are combined into the final ranks by means of $R^{*} = {P, I, R} = R_{>}^{*} \cap R_{<}^{*}$ .

3 The modified IF-SIR approach for MAGDM

In this section, we present the modified IF-SIR approach to select the most preferred alternative or to obtain a ranking order of the same in the setting of intuitionistic fuzzy MAGDM. For the purpose, we first determine the individual measure degree of each DM according to its importance. Then, we perform two kinds of group aggregated evaluations: decision values and criteria weights. After that, we successively calculate the advantage score matrix and the disadvantage score matrix instead of the superiority index matrix and inferiority index matrix, respectively. By calculating an advantage score matrix, we determine “how an alternative is advantageous relative to all other alternatives on each attribute”. In other words, by finding advantage score of an alternative on an attribute over the rest, we measure “how much the first parameter of the IF value is larger and the second, and third parameters of the IF value are smaller in comparison with others”. Analogously, by calculating the disadvantage score matrix, we try to figure out “how an alternative is at a disadvantage in comparison to all other alternatives”, that is, in terms of disadvantage of an alternative on an attribute over the rest, we measure “how much the first parameter of the IF value is smaller and the second, and third parameters of the IF value are larger in comparison with others”. The use of both the advantage and disadvantage score matrices help us to know both “how good an alternative is when compared with the rest (advantage score)” and “how much preferable an alternative is over the rest in terms of its weakness on each attribute (disadvantage score)”. The aggregated performance values of the alternatives over all the attributes are obtained by finding the modified IF-SIR flows. The ranking of the alternatives and selection of the most preferred alternative(s) is completed on the basis of some decision rules following a simplified Net Flow Score algorithm.

Suppose there exists a finite set of attributes g_j, j = 1, …, n, with ω_j, j = 1, …, n being the associated attribute weights. Let there be several experts (DMs) e_k, k = 1, …, l in the decision group that are required to make their own evaluations $d_{ij}^{(k)}$ to assess a finite set of alternatives Y_i, i = 1, …, m. Further, let the associated weights of the DMs be w_k, k = 1, …, l. It is assumed that following required inputs are provided in the form of linguistic terms and have been captured by means of the IF values.

Input 1: The kth DM’s weight: w_k = (μ_k, ν_k, π_k), k = 1, …, l.

Input 2: The attribute weights given by the kth DM: $ω_{j}^{(k)} = (μ_{j}^{(k)}, ν_{j}^{(k)}, π_{j}^{(k)})$ , j = 1, …, n.

Input 3: The performance ratings given by the kth DM: $d_{ij}^{(k)} = (μ_{ij}^{(k)}, ν_{ij}^{(k)}, π_{ij}^{(k)})$ , i = 1, …, m; j = 1, …, n.

The whole process of the modified IF-SIR method contain six steps, which are explained as under.

Step 1. Determine individual measure degree α_k.

In this step, we transform the kth DM’s weight w_k = (μ_k, ν_k, π_k), k = 1, …, l to crisp individual measure degree, denoted by α_k. We find the advantage score a_k and disadvantage score b_k and calculate relative closeness coefficient of the IF values w_k = (μ_k, ν_k, π_k) using these scores. As long as the obtained coefficient is a real number lying in the interval [0, 1], this coefficient may be regarded as the individual measure degree. The whole process is outlined as under.

The crisp advantage score a_k of the DM e_k relative to all other DMs e_s is obtained as follows:

$\begin{matrix} a_{k} & = & \frac{1}{2} [\sum_{\binom{k \neq s}{s = 1}}^{l} max (μ_{k} - μ_{s}, 0) + \sum_{\binom{k \neq s}{s = 1}}^{l} max (ν_{s} - ν_{k}, 0) \\ + \sum_{\binom{k \neq s}{s = 1}}^{l} max (π_{s} - π_{k}, 0)] . \end{matrix}$ (1)

Similarly, the crisp disadvantage score b_k of the DM e_k relative to all other DMs e_s is obtained as follows:

$\begin{matrix} b_{k} & = & \frac{1}{2} [\sum_{\binom{k \neq s}{s = 1}}^{l} max (μ_{s} - μ_{k}, 0) + \sum_{\binom{k \neq s}{s = 1}}^{l} max (ν_{k} - ν_{s}, 0) \\ + \sum_{\binom{k \neq s}{s = 1}}^{l} max (π_{k} - π_{s}, 0)] . \end{matrix}$ (2)

Combining (1) and (2), we find relative closeness coefficient α_k of the kth DM as

$α_{k} = \frac{a_{k}}{a_{k} + b_{k}} .$ (3)

From (3), it follows that α_k ∈ [0, 1]. Furthermore, larger the value a_k, and smaller the value b_k, the higher the weight w_k of the DM e_k in crisp form.

Step 2. Calculate the group aggregated decision information.

Here, we calculate the group aggregated decision values ${\bar{d}}_{ij} = ({\bar{μ}}_{ij}, {\bar{ν}}_{ij})$ and attribute weights ${\bar{ω}}_{j} = ({\bar{μ}}_{j}, {\bar{ν}}_{j})$ using standard intuitionistic fuzzy aggregation operators.

Aggregate the decision values using IFWA operator and α_k.

$\begin{matrix} {\bar{d}}_{ij} & = & {IFWA}_{α_{k}} (d_{ij}^{(1)}, d_{ij}^{(2)}, \dots, d_{ij}^{(l)}) \\ = & α_{1} d_{ij}^{(1)} \oplus α_{2} d_{ij}^{(2)} \oplus \dots \oplus α_{l} d_{ij}^{(l)} \\ = & (1 - \prod_{k = 1}^{l} (1 - μ_{ij}^{(k)})^{α_{k}}, \prod_{k = 1}^{l} (ν_{ij}^{(k)})^{α_{k}}) \\ = & ({\bar{μ}}_{ij}, {\bar{ν}}_{ij}) . \end{matrix}$ (4)

Aggregate the attribute weights using IFWG operator and α_k.

$\begin{matrix} {\bar{ω}}_{j} & = & {IFWG}_{α_{k}} (ω_{j}^{(1)}, ω_{j}^{(2)}, \dots, ω_{j}^{(l)}) \\ = & (ω_{j}^{(1)})^{α_{1}} \otimes (ω_{j}^{(2)})^{α_{2}} \otimes \dots \otimes (ω_{j}^{(l)})^{α_{l}} \\ = & (\prod_{k = 1}^{l} (μ_{j}^{(k)})^{α_{k}}, 1 - \prod_{k = 1}^{l} (1 - ν_{j}^{(k)})^{α_{k}}) \\ = & ({\bar{μ}}_{j}, {\bar{ν}}_{j}) . \end{matrix}$ (5)

Note that we can choose either of the aggregation operators for the purpose of aggregation. When compared with IFWG, the IFWA operator is more biased to the extremum IF values and hence the larger values have more influence than smaller values in the aggregation. Thus, the IFWG operator is recommended in aggregation of criteria weights so that the subjective influence of dominated decision makers is neutralized to some extent. The IFWA operator is recommended in aggregation of decision values to preserve the subjective judgments of the decision makers.

Step 3. Determine the crisp performance ratings of the alternatives Y_i on the attributes g_j.

In this step, we transform the aggregated decision values ${\bar{d}}_{ij} = ({\bar{μ}}_{ij}, {\bar{ν}}_{ij})$ into the real valued performance ratings.

Find advantage score matrix [a_ij] _m×n where a_ij ascertain as to how an alternative Y_i with respect to the jth attribute g_j is superior to or better than rest of the alternatives Y_f, f ≠ i as follows: $\begin{matrix} a_{ij} & = & \frac{1}{2} [\sum_{\binom{f \neq i}{f = 1}}^{m} max ({\bar{μ}}_{ij} - {\bar{μ}}_{fj}, 0) \\ + \sum_{\binom{f \neq i}{f = 1}}^{m} max ({\bar{ν}}_{fj} - {\bar{ν}}_{ij}, 0) \\ + \sum_{\binom{f \neq i}{f = 1}}^{m} max ({\bar{π}}_{fj} - {\bar{π}}_{ij}, 0)] . \end{matrix}$

Similarly, calculate the disadvantage score matrix [b_ij] _m×n where b_ij ascertain as to how an alternative Y_i with respect to the jth attribute g_j is inferior to or disadvantageous than all other alternatives Y_f, f ≠ i as follows: $\begin{matrix} b_{ij} & = & \frac{1}{2} [\sum_{\binom{f \neq i}{f = 1}}^{m} max ({\bar{μ}}_{fj} - {\bar{μ}}_{ij}, 0) \\ + \sum_{\binom{f \neq i}{f = 1}}^{m} max ({\bar{ν}}_{ij} - {\bar{ν}}_{fj}, 0) \\ + \sum_{\binom{f \neq i}{f = 1}}^{m} max ({\bar{π}}_{ij} - {\bar{π}}_{fj}, 0)] . \end{matrix}$

Step 4. Determine the modified IF-SIR flows.

Using the advantage score matrix [a_ij] _m×n and the disadvantage score matrix [b_ij] _m×n, the IF-SIR flows can be calculated using intuitionistic fuzzy aggregation operators. The IF-superiority flow is obtained as

$\begin{matrix} δ^{>} (Y_{i}) & = & \sum_{j = 1}^{n} {\bar{ω}}_{j} a_{ij} = {IFWA}_{a_{ij}} ({\bar{ω}}_{1}, {\bar{ω}}_{2}, \dots, {\bar{ω}}_{m}) \\ = & (1 - \prod_{j = 1}^{n} (1 - {\bar{μ}}_{j})^{a_{ij}}, \prod_{j = 1}^{n} ({\bar{ν}}_{j})^{a_{ij}}) \\ = & (μ^{>} (Y_{i}), ν^{>} (Y_{i})) . \end{matrix}$ (6)

The IF-inferiority flow is obtained as

$\begin{matrix} δ^{<} (Y_{i}) & = & \sum_{j = 1}^{n} {\bar{ω}}_{j} b_{ij} = {IFWA}_{b_{ij}} ({\bar{ω}}_{1}, {\bar{ω}}_{2}, \dots, {\bar{ω}}_{m}) \\ = & (1 - \prod_{j = 1}^{n} (1 - {\bar{μ}}_{j})^{b_{ij}}, \prod_{j = 1}^{n} ({\bar{ν}}_{j})^{b_{ij}}) \\ = & (μ^{<} (Y_{i}), ν^{<} (Y_{i})) . \end{matrix}$ (7)

In the above stated IF-SIR flows, δ^> (Y_i) assesses how much Y_i is superior to all other alternatives and δ^< (Y_i) assesses how much Y_i is inferior to all other alternatives based on their relative advantage and disadvantage scores over all the attributes. Clearly, the higher IF-superiority flow δ^> (Y_i) and the lower IF-inferiority flow δ^< (Y_i), the better the alternative Y_i.

Step 5. Induce decision rules based on outranking relations.

Through pairwise comparison, we can obtain outranking relations of the alternatives pairwise. Using obtained IF-SIR flows, we compare the alternative Y_i (δ^> (Y_i) , δ^< (Y_i)) with other alternatives Y_t (δ^> (Y_t) , δ^< (Y_t)) where i, t = 1, …, m, t ≠ i. The possible outranking relations are shown as follows:

Comparing δ^> (Y_i) and δ^> (Y_t), we have δ^> (Y_i) > δ^> (Y_t), δ^> (Y_i) = δ^> (Y_t) or δ^> (Y_i) < δ^> (Y_t).

Comparing δ^< (Y_i) and δ^< (Y_t), we have δ^< (Y_i) > δ^< (Y_t), δ^< (Y_i) = δ^< (Y_t) or δ^< (Y_i) < δ^< (Y_t).

The simultaneous consideration of both the IF-superiority, and IF-inferiority flows for construction of the decision rules leads to nine such rules; however, only three of these are able to affirm the outranking relations between Y_i and Y_t. For alternative Y_i, if Y_i ≻ Y_t, we say Y_i is superior than Y_t with respect to the considered attribute, which is affirmed by the following superior rules.

[S-Rule. 1] If δ^> (Y_i) > δ^> (Y_t) and δ^< (Y_i) < δ^< (Y_t), then Y_i ≻ Y_t.

[S-Rule. 2] If δ^> (Y_i) > δ^> (Y_t) and δ^< (Y_i) = δ^< (Y_t), then Y_i ≻ Y_t.

[S-Rule. 3] If δ^> (Y_i) = δ^> (Y_t) and δ^< (Y_i) < δ^< (Y_t), then Y_i ≻ Y_t.

Similarly, for alternative Y_i, if Y_i ≺ Y_t, we say Y_i is inferior than Y_t with respect to the considered attribute, which is affirmed by the following inferior rules.

[I-Rule. 1] If δ^> (Y_i) < δ^> (Y_t) and δ^< (Y_i) > δ^< (Y_t), then Y_i ≺ Y_t.

[I-Rule. 2] If δ^> (Y_i) < δ^> (Y_t) and δ^< (Y_i) = δ^< (Y_t), then Y_i ≺ Y_t.

[I-Rule. 3] If δ^> (Y_i) = δ^> (Y_t) and δ^< (Y_i) > δ^< (Y_t), then Y_i ≺ Y_t.

Furthermore, if the pair (Y_i, Y_t) cannot be affirmed by any of the above rules, we say the alternatives Y_i and Y_t are incomparable under the given decision environment. In such cases, the comparison law [4] may be used here, which is based on the score and accuracy functions.

Step 6. Provide the decision recommendation.

Using the induced decision rules, we calculate a specific score for each alternative Y_i obtained as

$Score (Y_{i}) = Sup (Y_{i}) - Inf (Y_{i}),$ (8) where $\begin{matrix} Sup (Y_{i}) & = & card ({Y_{t} : at least one superior rule \\ affirms Y_{i} ≻ Y_{t}}), \\ Inf (Y_{i}) & = & card ({Y_{t} : at least one inferior rule \\ affirms Y_{i} ≺ Y_{t}}), \end{matrix}$ for i, t = 1, …, m, i ≠ t.

Here, Sup (Y_i) counts the number of alternatives to which the alternative Y_i is superior. Similarly, Inf (Y_i) counts the number of alternatives to which the alternative Y_i is inferior. Furthermore, the Score (Y_i) is the quantitative measure used to identify priorities in the final ranking. Using the calculated scores, a clear decision recommendation can be provided as to which alternative is most suitable.

4 An application to supplier selection

The modified IF-SIR approach proposed in the previous section is used here for facilitating the process of supplier selection. Supplier selection, a MAGDM problem, requires trade-off between multiple attributes exhibiting vagueness and imprecision with the involvement of a group of experts. The overall objective of supplier selection process is to reduce purchase risk, maximize overall value to the purchaser, and build the closeness and long term relationships between buyers and suppliers [17] and it includes both tangible and intangible factors. In today’s highly competitive environment, it is impossible for a company to successfully produce low-cost, high-quality products without satisfactory suppliers. The selection of appropriate suppliers has long been one of the most important functions of any company’s purchasing department. Boer et al. [18] gave a good review and classification of the MADM approaches for supplier selection.

To validate the applicability of the proposed modified IF-SIR method in the setting of MAGDM, we consider the supplier selection case of a multinational company located in India specialized in designing and manufacturing various kinds of automobile parts (e.g., steering wheels, armrests, audio/video systems, and charging system). In the recent years, the automobile industry in India has undergone major changes because many reputed global automobile manufacturers have established their base in the country. To meet the regularity conditions such as ISO 14001, REACH, WEEE, ROHS, etc. of its target market in Europe, the company is interested in incorporating sustainable initiatives into its vendor evaluation and selection to follow the worldwide popular sustainability practices. The task force of the company (hereafter referred to as the decision-makers) consists of three qualified experts (e_k, k = 1, 2, 3) from departments such as purchasing, marketing, quality control, production, engineering and research. There are five potential alternative suppliers (Y_i, i = 1, 2, 3, 4, 5) that have been identified and the company wishes to select the most favorable supplier(s) according to the following four attributes (g_j, j = 1, 2, 3, 4) for assigning order quantity: g₁ - Financial situation; g₂ - Technology performance; g₃ - Sustainability performance; g₄ - Service performance.

The decision organizers assess these experts and identify their weights according to their importance, denoted by w_k. In addition, each DM needs to give his/her evaluation in two aspects: (i) the weights of given attributes according to their importance, denoted by ω_j; (ii) the decision values of alternative suppliers according to their performance under each attribute, denoted by d_ij. All the decision information is represented in linguistic terms. The Table 1 gives the IF values that measure linguistic terms on “Importance” and “Performance”, comprising nine levels.

Table 1
IF-measures of linguistic terms on “Importance” and “Performance”

Levels Importance terms Performance terms IFVs

L1 Extremely Important (EI) Extremely Positive (EP) (1.00, 0.00)

L2 Absolutely Important (AI) Absolutely Positive (AP) (0.90, 0.10)

L3 Very Very Important (VVI) Very Very Positive (VVP) (0.80, 0.10)

L4 Very Important (VI) Very Positive (VP) (0.70, 0.20)

L5 Important (I) Positive (P) (0.60, 0.30)

L6 Medium (M) Medium (M) (0.50, 0.40)

L7 Less Important (LI) Negative (N) (0.40, 0.50)

L8 Not important (NI) Very Negative (VN) (0.05, 0.80)

L9 Unconsidered (UC) Extremely Negative (EN) (0.00, 1.00)

Levels	Importance terms	Performance terms	IFVs
L1	Extremely Important (EI)	Extremely Positive (EP)	(1.00, 0.00)
L2	Absolutely Important (AI)	Absolutely Positive (AP)	(0.90, 0.10)
L3	Very Very Important (VVI)	Very Very Positive (VVP)	(0.80, 0.10)
L4	Very Important (VI)	Very Positive (VP)	(0.70, 0.20)
L5	Important (I)	Positive (P)	(0.60, 0.30)
L6	Medium (M)	Medium (M)	(0.50, 0.40)
L7	Less Important (LI)	Negative (N)	(0.40, 0.50)
L8	Not important (NI)	Very Negative (VN)	(0.05, 0.80)
L9	Unconsidered (UC)	Extremely Negative (EN)	(0.00, 1.00)

Based on Table 1, the Table 2 presents weights of the experts, and weights of the attributes provided by the corresponding experts. Furthermore, the Table 3 presents decision values of the alternative suppliers under each attribute provided by the experts.

Table 2

The weights using the linguistic terms on “Importance”

Experts (e_k)	Weights of	Weights of attributes (ω_j)
	experts (w_k)	ω ₁	ω ₂	ω ₃	ω ₄
e ₁	EI	AI	VI	VVI	I
e ₂	VVI	VVI	I	VI	VI
e ₃	VI	VI	VI	I	I

Table 3

The decision values using the linguistic terms on “Performance”

Experts	Suppliers	Attributes (g_j)
(e_k)	(Y_i)	g ₁	g ₂	g ₃	g ₄
e ₁	Y ₁	VP	P	VVP	VP
	Y ₂	P	M	VP	P
	Y ₃	AP	VVP	VVP	VVP
	Y ₄	P	M	VVP	VP
	Y ₅	M	N	VP	M
e ₂	Y ₁	VVP	VP	VP	VP
	Y ₂	VP	P	P	M
	Y ₃	VVP	VP	VVP	VVP
	Y ₄	VP	M	VP	P
	Y ₅	P	M	VP	P
e ₃	Y ₁	VP	P	VVP	VP
	Y ₂	M	VP	P	P
	Y ₃	VVP	VVP	VP	P
	Y ₄	VP	P	VP	P
	Y ₅	P	M	P	M

Step 1. We determine individual measure degree α_k, see Table 4.

Table 4

Individual measure degree

Experts (e_k)	Advantage	Disadvantage	Relative
	Score (a_k)	Score (b_k)	Closeness
e ₁	0.5	0	1
e ₂	0.1	0.2	0.333
e ₃	0	0.4	0

Step 2. We calculate group aggregated decision information using the inputs from Table 1, Table 2, and the obtained α_k. Using Equation (4), the aggregated attribute weights are obtained as $\begin{matrix} {\bar{ω}}_{j} & = & ({\bar{μ}}_{j}, {\bar{ν}}_{j}, {\bar{π}}_{j}) \\ = & {(0.835, 0.131, 0.033), (0.59, 0.29, 0.12), \\ (0.71, 0.165, 0.125), (0.533, 0.35, 0.117)} . \end{matrix}$

Furthermore, the inputs from Table 1, Table 3, and the obtained α_k are used in Equation (5) to obtain the aggregated decision values that provided in Table 5.

Table 5

Aggregated decision values

	g ₁	g ₂
Y ₁	(0.825, 0.093, 0.083)	(0.732, 0.175, 0.092)
Y ₂	(0.732, 0.175, 0.092)	(0.632, 0.268, 0.101)
Y ₃	(0.942, 0.046, 0.012)	(0.866, 0.058, 0.075)
Y ₄	(0.732, 0.175, 0.092)	(0.603, 0.295, 0.102)
Y ₅	(0.632, 0.268, 0.101)	(0.524, 0.368, 0.108)
	g ₃	g ₄
Y ₁	(0.866, 0.058, 0.075)	(0.799, 0.117, 0.084)
Y ₂	(0.779, 0.134, 0.087)	(0.683, 0.221, 0.096)
Y ₃	(0.883, 0.046, 0.071)	(0.883, 0.046, 0.071)
Y ₄	(0.866, 0.058, 0.075)	(0.779, 0.134, 0.087)
Y ₅	(0.799, 0.117, 0.084)	(0.632, 0.268, 0.101)

Step 3. We next determine the performance ratings of the alternatives on the attributes using their advantage and disadvantage scores, see Table 6.

Table 6

Performance ratings

a_ij: Advantage Score
	g ₁	g ₂	g ₃	g ₄
Y ₁	0.378	0.438	0.154	0.304
Y ₂	0.101	0.136	0	0.051
Y ₃	0.845	0.974	0.222	0.640
Y ₄	0.101	0.079	0.154	0.244
Y ₅	0	0	0.02	0
b_ij: Disadvantage Score
	g ₁	g ₂	g ₃	g ₄
Y ₁	0.117	0.134	0.169	0.084
Y ₂	0.302	0.335	0.299	0.414
Y ₃	0	0	0.0	0
Y ₄	0.302	0.42	0.017	0.124
Y ₅	0.704	0.738	0.218	0.617

Step 4. Based on the advantage score matrix, we calculate the IF-superiority flow δ^> (Y_i) according to Equation (6). The Table 7 provides these values along with S^> (Y_i) (the score function) and H^> (Y_i) (the accuracy function) values.

Similarly, based on the disadvantage score matrix, we calculate the IF-inferiority flow δ^< (Y_i) according to Equation (7). The Table 8 provides these values along with S^< (Y_i) (the score function) and H^< (Y_i) (the accuracy function) values.

Clearly, the higher δ^> (Y_i) and the lower δ^< (Y_i), the better the supplier Y_i.

Table 7

IF-Superiority Flow & the associated Score & Accuracy functions

IF-Superiority Flow
Y ₁	(0.776, 0.148, 0.076)
Y ₂	(0.29, 0.653, 0.058)
Y ₃	(0.957, 0.018, 0.024)
Y ₄	(0.467, 0.433, 0.1)
Y ₅	(0.025, 0.964, 0.011)
Score Function	Accuracy Function
0.627	0.924
–0.363	0.942
0.939	0.976
0.034	0.899
–0.939	0.989

Table 8

IF-Inferiority Flow & the associated Score & Accuracy functions

IF-Inferiority Flow
Y ₁	(0.34, 0.593, 0.067)
Y ₂	(0.783, 0.135, 0.082)
Y ₃	(0, 1, 0)
Y ₄	(0.645, 0.274, 0.081)
Y ₅	(0.931, 0.034, 0.035)
Score Function	Accuracy Function
–0.253	0.933
0.648	0.918
–1	1
0.371	0.919
0.897	0.965

Steps 5-6. Finally, we provide the decision recommendation aided by the induced decision rules. Following the induced rules and Equation (8), we compare each pair of suppliers and calculate the net flow score as shown in Table 9. Note that ‘_’ represents inexistent items.

Table 9

Pairwise comparison with Net flow score

	Y ₁	Y ₂	Y ₃	Y ₄	Y ₅
Y ₁	_	>	<	>	>
Y ₂	<	_	<	<	>
Y ₃	>	>	_	>	>
Y ₄	<	>	<	_	>
Y ₅	<	<	<	<	_
Sup (Y_i)		Inf (Y_i)		Score
3		1		2
1		3		–2
4		0		4
2		2		0
0		4		–4

Using the values in Table 9, the supplier 3 selected as the most appropriate supplier with maximum score. The suppliers are ranked from best to worst in a preference order as: Y₃ ≻ Y₁ ≻ Y₄ ≻ Y₂ ≻ Y₅.

In what follows next, we present numerical comparison with some closely related works to highlight the outperforming points of our approach.

4.1 Comparison with [7]

We solve the supplier selection problem using interval-valued hesitant fuzzy SIR (IVHF-SIR) approach presented in [7] with appropriate modifications. For the purpose, the IF information is converted into interval-valued hesitant fuzzy information by taking h (z) = [μ_A (z) , 1 - ν_A (z)], where h (z) represents the interval-valued hesitant fuzzy set. The rank of the alternatives obtained is: Y₃ ≻ Y₂ ≻ Y₁ ≻ Y₄ ≻ Y₅, which is different from Y₃ ≻ Y₁ ≻ Y₄ ≻ Y₂ ≻ Y₅ obtained using the proposed approach. Although, the ranks are different but the most suitable alternative is Y₃ in both the methods. Thus, the proposed methodology can be adapted to a particular situation such as the one used in [7] but the same is not true about the latter. Furthermore, we also solve the outstanding teacher selection problem presented in [7] using our methodology with appropriate modifications. For the purpose, the interval-valued hesitant fuzzy information is converted into the IF information by taking μ_A (z) = min h (z) and ν_A (z) =1 - max h (z). The rank of the alternatives obtained is: Y₂ ≻ Y₃ ≻ Y₁ ≻ Y₅ ≻ Y₄, which is same as obtained in [7]. Thus, the proposed methodology can be adapted to situations such as the one used in [7].

4.2 Comparison with [8]

In order to compare with Tam and Tong [8], we solve the facility location selection problem presented therein using our methodology with appropriate modifications. The rank of the alternatives obtained is: Y₇ ≻ Y₃ ≻ Y₆ ≻ Y₁ ≻ Y₄ ≻ Y₅ ≻ Y₈ ≻ Y₂, which is different from Y₇ ≻ Y₆ ≻ Y₃ ≻ Y₁ ≻ Y₈ ≻ Y₅ ≻ Y₄ ≻ Y₂ obtained in [8] using either the conservative or aggressive approach. Although, the ranks are different but the most suitable alternative is Y₇ in both the methods. Thus, the proposed methodology can be adapted to yet another particular situation such as the one used in [8]; however, the same is not true about the latter.

4.3 Comparison with [10]

The ranking of suppliers by the proposed approach and the one developed in [10] are same given as Y₃ ≻ Y₁ ≻ Y₄ ≻ Y₂ ≻ Y₅. The proposed method is based on a more realistic scenario than [10] due to the following reasons. The distance calculated in [10] between the DM weight and positive ideal-IF (PIIF) value (1, 0, 0) and negative ideal-IF (NIIF) value (0, 1, 0) in step 1 as well as distance between the aggregated decision values and PIIF and NIIF values in step 3 to define relative closeness is inappropriate in the sense that the PIIF and NIIF values may be treated as benchmarks against which the performance is evaluated. These benchmarks are too unrealistic to be achieved in practice. On the other hand, the advantage and disadvantage scores are obtained by measuring relative performances based on all the three parameters of IF values. Thus, in the process, the benchmarks are obtained both on the advantage and disadvantage evaluations, which are more realistic because the DM knows not only about the best and worst performances but also relative comparison of the individual performances.

4.4 Comparison with [19]

For comparison, the ranking of suppliers using the method underlined in [19] is to be carried out as follows: Using the advantage score matrix [a_ij] _m×n and the disadvantage score matrix [b_ij] _m×n obtained in step 3, the superiority flow or the strength score, and the inferiority flow or the weakness score is calculated whereby the aggregation is done over all the attributes as provided in Tables 7 and 8 of the proposed method. Afterwards, the strength index score S_i and the weakness index score W_i is calculated and results are provided in Table 10. Furthermore, using the strength and weakness index scores, we calculate total performance score of the alternatives given in Table 10. Based on total performance scores of the alternatives, they are ranked as Y₃ ≻ Y₁ ≻ Y₄ ≻ Y₂ ≻ Y₅.

Table 10
Strength Index Score S_i and Weakness Index Score W_i

Suppliers S _i W _i Performance score

Y ₁ 2.975 0.611 0.830

Y ₂ 0.515 2.798 0.155

Y ₃ 5.11 0.139 0.973

Y ₄ 1.157 1.863 0.383

Y ₅ 0.124 4.47 0.027

Suppliers	S _i	W _i	Performance score
Y ₁	2.975	0.611	0.830
Y ₂	0.515	2.798	0.155
Y ₃	5.11	0.139	0.973
Y ₄	1.157	1.863	0.383
Y ₅	0.124	4.47	0.027

5 Concluding remarks and scope for future work

We developed a new decision method retaining basic framework of the superiority and inferiority ranking (SIR) method for fuzzy MAGDM problems. The work has been accomplished under the assumption that weights of the decision makers, and performance ratings of the alternatives on the given attributes are IF values. The advantage and disadvantage scores of the alternatives on the given attributes were used for defining the relative closeness coefficients. The score matrices derived from the relative comparisons were used for obtaining the modified IF-SIR flows, namely IF-superiority and IF-inferiority flows. The ranking of the alternatives was suggested on the basis of a consolidated score function using both the IF-SIR flows. To demonstrate the proposed methodology, we considered a practical example of supplier selection problem and compared our results with existing approaches based on SIR-method and its applications. The main advantage of the modified SIR-method is that the selection of best alternative is based on relative comparison of performances of the alternatives among them rather than using some hypothetical benchmarks or peers. The comparisons are based on all three parameters of the IF values where the first parameter (membership) is treated as “more the better”, the second parameter (non-membership) is treated as “less the better”, and the third parameter (indeterminacy) is also treated as “less the better”. The ranking of the alternatives is done through the trade-off values used in the modified IF-SIR flows. The proposed approach do not lose any major advantages of the existing SIR-methods and makes sufficient contribution to the literature in terms of using additional information obtained from the relative comparisons to better rank the alternatives.

The proposed method has few shortcomings in terms of not being able to treat intuitionistic fuzzy MAGDM problems with uncertainty in importance weights of the decision makers, e.g. weights partially known, completely unknown. For the future research, we would like to extend the proposed MAGDM approach under interval-valued intuitionistic fuzzy information, MAGDM methods based on granular computing techniques in different granular contexts such as rough sets, multi-granularity linguistic context. Furthermore, the application of intuitionistic fuzzy MAGDM in many important fields of investigation such as portfolio selection, personnel examination, medical diagnosis, water environment assessment, threat evaluation and missile weapon system selection, and warship combat plan evaluation could be of further research interests.

Footnotes

Acknowledgments

We are thankful to the Editor and the reviewers for their valuable comments and detailed suggestions to improve presentation of the paper. The first and second authors also acknowledge support through Research and Development Grant received from University of Delhi, Delhi, India.

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Modified intuitionistic fuzzy SIR approach with an application to supplier selection

Abstract

Keywords

1 Introduction

1.1 Major highlights of the proposed research

1.2 Organization of the paper

2 Preliminaries

2.1 The classical SIR method

3 The modified IF-SIR approach for MAGDM

4.2 Comparison with [8]

4.3 Comparison with [10]

4.4 Comparison with [19]

Table 10 Strength Index Score S i and Weakness Index Score W i Suppliers S i W i Performance score Y 1 2.975 0.611 0.830 Y 2 0.515 2.798 0.155 Y 3 5.11 0.139 0.973 Y 4 1.157 1.863 0.383 Y 5 0.124 4.47 0.027

Footnotes

Acknowledgments

References

Table 10
Strength Index Score S_i and Weakness Index Score W_i

Suppliers S _i W _i Performance score

Y ₁ 2.975 0.611 0.830

Y ₂ 0.515 2.798 0.155

Y ₃ 5.11 0.139 0.973

Y ₄ 1.157 1.863 0.383

Y ₅ 0.124 4.47 0.027