2.1. Dealing with imprecise and hesitant information

Uncertain information is one of the remarkable features of EEs. It is very important to deal with such type of information to cope with EEs successfully. Therefore, information domains utilized by experts to provide their opinions/assessments in quantitative and qualitative contexts are revised.

(1) Information domain for quantitative contexts

In real world problems, it is difficult for experts to provide their assessments using numerical values, when the EE information is uncertain, such as people affected, property losses, or costs of alternatives. However, in such situations, interval values^15,22,31 are suitable for experts to provide their assessments due to their useful and simple technique for representing uncertainty. Thus, interval values are utilized as the information domain for quantitative contexts in our proposal.

(2) Information domain for qualitative contexts

A fairly common approach to model qualitative information is the fuzzy linguistic approach³⁹ based on the fuzzy set theory. Different linguistic models have been discussed in different approaches^20,21,26. In our proposal, linguistic term sets are utilized to model the uncertain information in qualitative contexts (see Fig. 2).

Fig. 2.

Linguistic term set

Usually the information of MAGEDM problems is uncertain; experts involved in such problems are bounded by cognition ² and under pressure because of the urgent time constraints in an emergency response. Moreover, their decision might provoke potentially serious results ¹⁶. Hence, in such situations, it is common for experts to hesitate when they provide their assessments. Therefore, it seems necessary to deal with experts’ hesitation in MAGEDM problems.

To model the hesitant information in qualitative contexts, the concept of HFLTS ²⁷ was introduced, drawing increased attention recently ^25,27.

HFLTS is a powerful and useful tool to model experts’ hesitation; the use of context-free grammars ²⁷ allows generation of complex linguistic expressions close to the natural language utilized by human beings in the real world ^27,28, which can be modeled by HFLTS. This approach has been widely applied to deal with different decision problems ^1,33,34.

S_ll denotes the expression domain generated by G_H , which might be either complex linguistic expressions or single linguistic terms.

Taking into account that experts can provide their assessments by utilizing quantitative and qualitative information in order to make computations with different types of information, it is necessary to unify them into a unique domain. The process of unifying different types of information is presented in section 3.3.

2.2. Dynamic decision making

Some existing dynamic MADM methods ^3,4, which have the following remarkable features, are revised:

1. (i)

   The alternatives are changeable because they might be deemed non-available or removed; meanwhile new alternatives might be considered and added.

2. (ii)

   The criteria are not immobilized, since their values might change along with time, and also, the current criteria might be removed or new criteria might be taken into account.

3. (iii)

   The temporal profile of an alternative matters for comparison with other alternatives. This point is referred as the notion of feedback ^3,42.

According to these three features, dynamic MADM methods should be capable of managing interdependent decisions in a changing environment, wherein not only alternatives, but criteria might also change (non-available, removed or added new ones, etc.) and the final decisions at each decision moment must consider the feedback from previous ones. Due to the dynamic evolution of EEs, a reasonable and effective MAGEDM method should consider not only the three aforementioned features, but also the changes of experts because they might give up the decision process or new experts might be invited to join the decision process in real world situations.

To make the proposed MAGEDM method understandable, some necessary concepts are first given, and then the dynamic MADM method ^3,4,42 is briefly revised.

Let T = {1, 2, …} set of discrete decision moments (possibly infinite), and P_t be the set of alternatives that are usable at each decision moment, t, t ∈ T. Suppose that a static MADM method is being utilized at each decision moment, t ∈ T, to compute ratings for each available alternative, p ∈ P_t, concerning the assessments of all criteria, C_t = {c₁, c₂, …, c_m}. The ratings obtained by the static MADM method are called static ratings or non-dynamic ratings, denoted by R_t(p). The dynamic rating of alternatives is computed based on its static rating obtained in the previous stages to which it belonged.

The dynamic decision process deals with a feedback mechanism from previous ones. For any alternative, p, its dynamic rating function, E_t(p),is defined as ^3,4,42:

where Փ is an aggregation function (operator).

For each alternative, p, either belonging to the existing set of alternatives, P_t, or carried over from the previous one by means of the historical set, H_t−1, there are three different situations.

1. (i)

   if the alternative, p, belongs only to the current set of alternatives, P_t, but not to the historical set, H_t−1, that is, p ∈ P_t\H_t−1, its dynamic rating, E_t(p), is equal to its static rating, R_t(p);

2. (ii)

   if the alternative, p, belongs not only to the current, but also the historical set of alternatives, that is, p ∈ P_t ∩H_t−1, its dynamic rating, E_t(p), is calculated by aggregating its static rating, R_t(p), with its dynamic rating, E_t−1(p), at the former decision moment; and

3. (iii)

   if the alternative, p, belongs to the historical set of alternatives only, that is, p ∈ H_t−1\P_t, its dynamic rating, E_t(p), is equal to E_t−1(p).

The dynamic decision process can be conducted for several decision moments. The moment wherein the process is stopped depends on the problem and the DM’s assessments.

2.3. Fuzzy TOPSIS method based on alpha-level sets

TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method was first proposed by Huwang and Yoon ¹²; it is a popular MADM method been widely applied to solve different decision problems ^5,6,12,32. To cope with complex problems and uncertain information in the real world, the TOPSIS method has been extended to deal with fuzzy MADM problems ^5,6,32.

The fuzzy TOPSIS method based on alpha-level sets ³² is a distinctive and powerful approach among other fuzzy TOPSIS versions ^5,6,8,9 due to its prominent advantages of keeping uncertain information in a better way. This is the significant difference between the fuzzy TOPSIS method based on alpha-level sets and other versions. Due to such advantages, the fuzzy TOPSIS method based on alpha-level sets will be used as the static MADM method in order to calculate the static rating of each alternative at different decision moments in our proposal.

In fuzzy MADM problems, criteria/attribute values and the relative weights are usually characterized by fuzzy numbers ^6,32. The most commonly used fuzzy numbers are trapezoidal fuzzy numbers, Ã = (a, b, c, d) or triangular fuzzy numbers, Ã = (a, b, d) with a degree of membership between 0 and 1. When b = c, the triangular fuzzy number is a special case of a trapezoidal fuzzy number.

According to Zadeh’s extension principle ⁴¹, a fuzzy number/set, Ã, can be also expressed by its intervals, that is,

where α-level sets or α-cuts of à denoted as A_α. µ_Ã(x) is the membership function of fuzzy number ó².

Based on the short revision of fuzzy numbers aforementioned, the fuzzy TOPSIS method based on alpha-level sets ³² is briefly introduced.

Let X˜=(x˜ij)n×m be a fuzzy decision matrix characterized by membership functions, μx˜ij(x)(i=1,…,n,   j=1,…,m), and W˜=(w˜1,…,w˜m) be the fuzzy weights characterized by μw˜j(x)(j=1,…,m). If all the criteria/attributes, {c₁,…, c_m}, are assessed by using linguistic term sets with the same syntax and semantics, then the fuzzy decision matrix, X˜ , has the same dimension, and therefore, it is not necessary any normalization. Otherwise, X˜ has to be normalized.

If x˜ij=(aij,bij,cij,dij)(i=1,…n,  j=1,…,m) are trapezoidal fuzzy numbers, then the normalization process can be carried out by (the same normalization process for triangular fuzzy numbers)

where where Ω_b and Ω_c denote the sets of benefit and cost criteria/attributes, respectively.

It can be seen that r˜ij belong to [0,1]; thus, positive and negative ideal solutions can be defined as P^* = {1, …, 1} and P⁻ = {0, …, 0}, respectively. For a fuzzy decision matrix, X˜=(x˜ij)n×m , without normalization, the positive and negative ideal solutions can be obtained as follows:

Let (rij)α=[(rij)αL,(rij)αU] and (wj)α=[(wj)αL,(wj)αU] be alpha-level sets of r˜ij and w˜j , respectively. Then, the relative closeness (RC), RC_i, of the alternative, p_i, with respect to P^* can be written as:

where

RC_i is an interval value based on Eq. (15); its upper and lower bounds can be calculated by utilizing the following simplified pair of fractional programming models (see ³² for further details):

When different alpha levels are set, then (RCi)α=[(RCi)αL,(RCi)αU] can be obtained by solving Eqs. (18) and (19), respectively. According to Eq. (7), RC˜i can be expressed as: where RC˜i represents the fuzzy RC of alternative, p_i, based on corresponding alpha levels from 0 to 1.

According to Eq. (6), the dynamic ratings of alternatives are related not only to their static ones, but also their performance in previous stages if it has one. In order to calculate the dynamic ratings of alternatives, it is firstly necessary to compute the static ratings of alternatives. The averaging level cuts ²⁴ are used in this paper for sake of simplicity to obtain the static ratings of alternatives.

Let α₁, …, α_K be different alpha levels; the static rating, m(RC˜i) , of alternative, p_i, can be determined by ²⁴

where K is the number of alpha levels.

3.1. Framework definition

The following notions and terminology will be used in the proposed dynamic MAGEDM method.

• •

  T = {1, 2, 3, …}: the set of discrete decision moments (possible infinite), for each decision moment, t ∈ T.

• •

  P_t = {p₁, p₂, …, p_n}: the set of available alternatives at decision moment, t, where p_i denotes the i-th alternative, i = 1, 2, …,n.

• •

  C_t = {c₁, c₂, …, c_m}: the set of criteria/attributes at decision moment, t, where c_j denotes the j-th criterion/attribute, j = 1, 2, …, m.

• •

  E_t = {e₁, e₂, …, e_H}: the set of experts at decision moment, t, where e_h denotes the h-th expert, h = 1, 2, …, H . In dynamic MAGEDM problems, the experts might leave or be added during the decision process according to expert’s willingness or decision problems.

• •

  Xth=(xijh)n×m: the information matrix provided by the expert, e_h, at decision moment, t, where xijh denotes the opinions/assessments provided by the h-th expert over the i-th alternative regarding j-th criterion, h = 1, 2, …, H ; i = 1, 2, …, n; j = 1, 2, …, m (see Remark 2).

• •

  X˜jh=(x˜ijh)n×m: the unified information matrix with respect to Xth, where x˜ijh denotes the unified fuzzy information corresponding to xijh, h = 1, 2, …, H ; i = 1, 2, …, n; j = 1, 2, …, m.

• •

  X_t = (x_ij)_n×m: denotes the aggregated information matrix regarding X˜th, at decision moment, t, i = 1, 2, …, n; j = 1, 2, …, m.

• •

  Wth=(wc1h,wc2h,…,wcmh): the assessments vector regarding the criteria importance provided by the h-th expert at decision moment, t, where wcjh denotes the h-th expert’s assessments on the criterion c_j, h = 1, 2, …, H , j = 1, 2, …, m (see Remark 3).

• •

  W˜th=(w˜c1h,w˜c2h,…,w˜cmh): the unified fuzzy information vector with respect to Wth, where w˜cjh denotes the unified fuzzy information corresponding to wcjh, h = 1, 2, …, H , j = 1, 2, …, m.

• •

  W¯t=(w¯c1,w¯c2,…,w¯cm): the aggregated information vector regarding W˜th, at decision moment, t, j = 1, 2, …, m.

3.2. Gathering information

The opinions/assessments, xijh, over the alternatives, p_i, regarding criteria, c_j, and the assessments over the criteria importance, wcjh , provided by the expert, e_h, at decision moment, t, are gathered below.

where xijh∈{I∈[ηL,ηU]S={s0,s1,…,sg}Sll, i = 1, 2, …, n; j = 1, 2, …, m; h = 1, 2, …, H . where wcjh∈{S={s0,s1,…,sg}Sll, j = 1, 2, …, m; h = 1, 2, …, H .

3.3. Unification process

In this proposal, the expression domains used by experts can be interval values (I), linguistic terms (s_i), or complex linguistic expressions (S_ll).

• •

  Interval values. Assessments represented by interval values, I , belong to a special domain, [η^L, η^U], that is, I ∈ [η^L, η^U].

• •

  Linguistic terms. Assessments represented by linguistic terms s_i, belong to a linguistic term set S = {s₀, s₁, …, s_g}, that is, s_i ∈ S , where g + 1 is the granularity of S .

• •

  Complex linguistic expressions. Assessments represented by S_ll, generated by G_H (see Definition 4).

As mentioned in section 2.1, to deal with quantitative and qualitative information, a unification process is needed to facilitate the computations.

In order to retain uncertain information, including experts’ hesitation, and obtain more reliable results, the assessments, Xth=(xijh)n×m, and criteria importance, Wth=(wc1h,wc2h,…,wcmh), are transformed into its corresponding fuzzy domains, X˜th=(x˜ijh)n×m and W˜th=(w˜c1h,w˜c2h,…,w˜cmh), by using transformation functions (see Fig. 4).

Fig. 4.

Transformation functions

The transformation functions are detailed below:

1) Interval values, I , are first normalized into [0,1], and then, transformed into trapezoidal fuzzy numbers by using a transformation function, T_I . Let [η^L,η^U] be the domain of the interval values for quantitative contexts; let xijh=[dL,dU] be the information provided by the expert, e_h, over the i-th alternative concerning the j-th criterion at decision moment, t, where xijh=[dL,dU]∈[ηL,ηU]. The interval values, [d^L, d^U], are normalized into [ξ_,ξ¯]∈[0,1] as follows:

The transformation function, T_I , is defined as follows.

According to Eqs. (24)–(26), the gathered information, Xth=(xijh)n×m and Wth=(wc1h,wc2h,…,wcmh), can be transformed into its corresponding fuzzy domain X˜th=(x˜ijh)n×m and W˜th=(w˜c1h,w˜c2h,…,w˜cmh), respectively.

3.4. Aggregation process

The aggregation process is the process wherein experts’ opinions are aggregated to obtain collective values for each alternative and criteria weights.

The unified information, X˜th and W˜th, are aggregated to calculate the static rating of alternatives at each decision moment, t. This phase consists of two sub-aggregation processes (see Fig. 5): 1) aggregation of unified fuzzy information and 2) aggregation of unified criteria importance, which are explained as:

• 1)

  Aggregation of unified fuzzy information.

  The aggregated fuzzy information matrix at decision moment, t, X_t = (x_ij)_n×m, where xij=(xij1,xij2,xij3,xij4) is obtained by means of the unified fuzzy information, X˜th=(x˜ijh)n×m , where x˜ijh=(x˜ijh1,x˜ijh2,x˜ijh3,x˜ijh4) is given by:

  (27)xij1=minh{x˜ijh1},     xij2=1H∑h=1Hx˜ijh2xij3=1H∑h=1Hx˜ijh3,    xij4=maxh{x˜ijh4}

  where h = 1, 2, … , H ; i = 1, 2, …, n, j = 1, 2, … , m.

• 2)

  Aggregation of unified criteria importance.

  The aggregated fuzzy criteria weights at decision moment, t, W¯t={w¯c1,w¯c2,…,w¯cm}, where w¯cj=(w¯cj1,w¯cj2,w¯cj3,w¯cj4) can be obtained according to W˜th={w˜c1h,w˜c2h,…,w˜cmh}, where w˜cjh=(w˜cjh1,w˜cjh2,w˜cjh3,w˜cjh4), utilizing similar equations to Eq. (27):

  (28)w¯cj1=minh{w˜cjh1},     w¯cj2=1H∑h=1Hw˜cjh2w¯cj3=1H∑h=1Hw¯cjh3,   w¯cj4=maxh{w˜cjh4}

  where h = 1, 2, … , H , j = 1, 2, … , m.

Fig. 5.

Aggregation process

The advantages of the aggregation equations above are not only to retain uncertain information as much as possible and take into account all involved experts’ opinions in the dynamic MAGEDM process, but also to ease computation.

3.5. Computing static rating

As noted earlier, the fuzzy TOPSIS method based on alpha-level sets is utilized as the static MADM method to obtain the static ratings of alternatives at each decision moment, t, in our proposal. Since the aggregated fuzzy information matrix, X_t = (x_ij)_n×m and W¯t, have been already normalized in the unification process, it is not necessary to normalize X_t = (x_ij)_n×m and W¯t again. Thus, the positive and negative ideal solutions are P^* = {1, …, 1}, and P ⁻ = {0, …, 0}, respectively.

Let (yij)α=[(yij)αL,(yij)αU] and (wcj)α=[(wcj)αL,(wcj)αU] be the alpha-level sets of x_ij and w¯cj , respectively, at decision moment, t. The RC of the alternative, p_i, based on different alpha levels, (RCit)αU, and (RCit)αL can be obtained by using Eqs. (29) and (30), respectively.

Similar to Eqs. (20) and (21), the fuzzy RC of the alternative, p_i, with different alpha-level sets at decision moment, t, with our notation can be expressed as follows: According to Eq. (21), the static ratings of alternatives in this study can be obtained as follows:

3.6. Computing dynamic rating

Since EE information changes along with time (alternatives, criteria, and experts), leading to dynamic evolution, it seems necessary to consider the dynamic rating of each alternative. This is a comprehensive factor that indicates the performance of the alternative not only in its current stage, but also in the previous one. In this proposal, the dynamic rating of alternatives based on Eq. (6) is as follows:

where Φ is an associative aggregation operator that can apply different types of reinforcements (such as, downward reinforcement, upward reinforcement, and full reinforcement) for enhancing different performances in the dynamic context. (see Ref. 4 for details).

The operator selection and reinforcement depend on the characteristics of the problem.

The ranking of the alternatives is obtained according to the dynamic ratings, E_t(p_i); the higher dynamic rating the better alternative.

4.1. Case study

A big explosion took place at a container storage station at the Port of Tianjin, which contained hazardous and flammable chemicals, including sodium nitrate, calcium carbide, and ammonium nitrate, among others. The local government organized relevant departments (fire department, traffic management department, hygiene department, etc.) to collaborate in order to address the emergency situation. Short messages were sent to inform citizens within one kilometer to evacuate to safe areas. In this example, when the explosion occurred, the decision moment, t = 1.