3.1. ER approach to modeling MADA problems

Since attribute weights are intended for MADA issues under ER scenario, the ways of modeling and assessing MADA issues under ER scenario are presented as shown below.

Assuming M alternatives a_l (l = 1, …, M) along with L attributes e_i (i = 1, …, L) are present in a MADA issue. For L attributes, their relative weights are signified by w = (w₁, w₂, …, w_L), enabling 0 ≤ w_i ≤ 1 and = 1.

A decision maker can determine a grade set Ω = {H₁, H₂, …, H_N}, which is sorted in an ascending order of quality, so that every alternative on every standard is assessable in the ER context. Semantics of grade H_n are described through utilities settings for assessment grades u(H_n) (n = 1, …, N), which fulfill constraint 0 = u(H₁) < u(H₂) < … < u(H_N) = 1. Suppose that the evaluation of alternative a_l on standard e_i is profiled using a belief distribution B(e_i(a_l)) = {(H_n, β_n,i(a_l)), n = 1, …, N; (Ω, β_Ω,i(a_l))}, with β_n,i(a_l) ≥ 0, ≤ 1, and β_Ω,i(a_l) = 1 − . In the distribution, β_n,i(a_l) refers to the belief degree allocated to grade H_n; β_Ω,i(a_l) stands for the global ignorance degree. The evaluation is complete in case β_Ω,i(a_l) = 0, while is incomplete in other cases. Where B(e_i(a_l)) (i = 1, …, L, l = 1, …, M) is designated, a belief decision matrix S_L×M can be acquired.

Through using attribute weights along with ER rule, the total assessment B(y(a_l)) = {(H_n, β_n(a_l)), n = 1, …, N; (Ω, β_Ω(a_l))} is derived by integrating individual assessments B(e_i(a_l)). Here, β_Ω(a_l) symbolizes the total global ignorance degree. Next, by integrating total assessment and grade utilities, the anticipated minimum and maximum utilities were derived for alternative a_l as

In the light of decision rules like the minimax regret rule or Hurwicz rule, the M alternatives can be compared from [u⁻(a_l), u⁺(a_l)], thereby deriving a solution to the MADA issue. Especially, when the individual ignorance on each criterion is assigned by following some way, the expected utility of each alternative reduces to a precise number and a solution can be directly made from the expected utilities of alternatives.

3.2. Measurement of reliability of aggregated solution

Following the obtainment of minimal satisfaction of alternative V(a_l), the aggregated solution reliability is presented as follows by refer to relevant concepts of Fu and Xu [41].

Definition 1. [41] Assuming that V(b_l) (l = 1, …, M) stands for the ordered least satisfaction of alternatives enabling V(b₁) ≥ … ≥ V(b_m). In other words, V(b_l) indicates the lth greatest of V(a₁), …, V(a_m). Accordingly, for alternative b_l (l = 1, …, M – 1), ∆V(b_l), its superior intensity is expressed as ∆V(b_l) = (l = 1, …, M − 1), while the formula for aggregated solution reliability is Q = , with ϕ_l (l = 1, …, M − 1) denoting the weights of ∆V(b_l) (l = 1, …, M − 1) for Q that enables 0 ≤ ϕ_l ≤ 1 (l = 1, …, M − 1) and .

Definition 1 unveils the establishment of aggregated solution reliability as the weighted mean for the alternative's superior intensity. Explanations on properties associated with ∆V(b_l) (l = 1, …, M − 1) and Q are shown below.

Property 1. [41] Given ∆V(b_l) (l = 1, …, M − 1) and Q in Definition 1, the following relationships hold:

(1)

(2)

(3)

(4)

Assumption 1 [41] Assuming ϕ_l (l = 1, …, M − 1) denotes the weight of ∆V(b_l) (l = 1, …, M − 1) for Q in Definition 1. In order to underline the declining contribution of ∆V(b_l) (l = 1, …, M − 1) to Q, the following is subsequently required

(5)

(6)

(7)

Based on Assumption 1, ϕ_l (l = 1, …, M − 1) is estimable in case the greatest weight ϕ₁ is designated.

Theorem 1. [41] Assuming ϕ_l (l = 1, …, M − 1) denotes the weight of ∆V(b_l) (l = 1, …, M − 1) for Q and ∆d in Definition 1. Where the greatest weight ϕ₁ is designated, ϕ_l (l = 2, …, M − 1) is calculated using ϕ₁ − (l − 1)d₁ + ∆d by the minimax disparity approach exploiting Assumption 1. That is, the greatest disparity of ϕ_l from ϕ_l+1 (l = 1, …, M − 2) is minimized, with d₁ = + ε, ∆d = + , and ε being a tiny positive number that approximates 0.

With the purpose of ensuring the validity of Theorem 1, the following theorem was exploited to ascertain the permissible ranges of ϕ₁.

Theorem 2. [41] Suppose that ϕ_l (l = 1, …, M − 1) denotes the weight of ∆V(b_l) (l = 1, …, M − 1) for Q in Theorem 1, the calculation of ϕ_l (l = 2, …, M − 1) in Theorem 1 following Assumption 1 requires that < ϕ₁ < , with ε being a tiny positive number that approximates 0.

For the monotonicity assessment of Q relative to ϕ₁ during Q maximization, the relationship between Q and ϕ₁ are investigated using the theorem below.

Theorem 3. [41] Let aggregated solution reliability be Q = , with 0 ≤ ϕ_l ≤ 1 (l = 1, …, M − 1) and , ϕ₁ fulfills requisites provided in Theorem 2, and ∆V(b_l) = (l = 1, …, M − 1), Q shows monotonous elevation respecting ϕ₁.

4.1. Identification of potentially optimal alternatives

As demonstrate in Section 3.1, with the known w and the specified u(H_n) (n = 1, …, N), the least satisfaction of alternatives V(a_l) (l = 1, …, M) is obtainable from the evaluations B(e_i(a_l))_L×M, yielding a sole solution for a MADA issue. On the downside, under some circumstances, w and u(H_n) (n = 1, …, N) maybe completely unknown or partially available. Multiple alternatives may have the minimal satisfaction and be identified as the optimal alternative. To cope with the situation where a possible set of optimal alternatives may exist due to the fully unspecified or partially available of attribute weights and evaluation grade utilities, the potentially best alternatives are determined by creating an optimization model.

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

In the above model, the parameters , , , δ_u, u⁻(H_n) and u⁺(H_n) are designated by the decision maker, with being the constraints on , which used to control each attribute to contribute to a solution. and denote inferior and superior limits of w_i(a_l) for a potentially best alternative a_l. u⁻(H_n) and u⁺(H_n) refer to inferior and superior limits of u(H_n)(a_l). δ_u represents the constraint on least disparity of u(H_n)(a_l) from u(H_n−1)(a_l). w_i(a_l) (i = 1, …, L; l = 1, …, M) and u(H_n)(a_l) (n = 1, …, N; l = 1, …, M) stand for decision variables in the intervals.

According to the minimax regret approach (MRA) [38], the smaller R(a_l), the better an alternative a_l compared with the others. Thus, if the optimized objective of an alternative a_l in Eq. (8) is equal to 0, R(a_l) is the smallest; that is, the alternative al can become the optimal. In Eqs. (10) and (11), u⁻(a_l) and u⁺(a_l) integrate V(a_l) plus β_Ω(a_l) with u(H_n)(a_l) (n = 1, …, N, l = 1, …, M) to form V(a_l) in Eq. (9), which can produce a ranking order of alternative. In Eq. (17), u(H_n)(a_l) is limited to [u⁻(H_n), u⁺(H_n)]. The intrinsic constraint on u(H_n)(a_l) is expressed as 0 = u(H₁)(a_l) < u(H₂)(a_l) < … < u(H_N)(a_l) = 1 in Eq. (15). The minimal difference between u(H_n)(a_l) and u(H_n−1)(a_l) is limited by δ_u in Eq. (16). In Eq. (13), w_i(a_l) is limited to [, ] and ≤ is needed. Eq. (14) shows the constraint on w_i(a_l) (i = 1, …, L). The alternatives with the zero−valued optimized objectives in the above model are called potentially optimal alternatives, which form M_b.

In Eq. (9), exploiting nonlinear analytical algorithm, the evaluations B(y(a_l)) (l = 1, …, M) are generated through summation of evaluations B(e_i(a_l))_L×M weighted by w_i(a_l), and through their further summation, the anticipated minimal and maximal utilities u⁻(a_l) and u⁺(a_l) (l = 1, …, M) with u(H_n)(a_l) are generated, as displayed in Section 3.1. With u⁻(a_l) and u⁺(a_l), computation of least satisfaction is accomplished for alternative V(a_l) (l = 1, …, M). The normalized constraint on w is expressed in Eq. (10). In Eq. (12), the trait of partially available attribute weights information is a range of constraints on w, which are represented by G_w. Table 1 lists the six possible constraint types in G_w. The inherent constraint on u(H_n)(a_l) is denoted in Eq. (13). Similarly to Eq. (12), in Eq. (16), the partially available information on utilities of assessments grades are typified by a range of constraints on u(H_n)(a_l) and represented by G_u. Table 2 lists the five possible constraint types in G_u. Hereinto, p₂₃ and r₂₃ refer separately to the reciprocal fuzzy and multiplicative preference associations, whose value intervals are [0, 1] and [1/9, 9].

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Table 1. The different types of constraints on attribute weights.

https://doi.org/10.1371/journal.pone.0317438.t001

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Table 2. The different types of constraints on utilities of assessments grades.

https://doi.org/10.1371/journal.pone.0317438.t002

According to the definition of minimal satisfaction in Section 3.1, greater preference is given to the alternative a_l with higher minimal satisfaction. Hence, the alternative a_l is identified as the potentially optimal alternative if and only if the optimized purpose of alternative a_l in Eq. (8) is equivalent to 0. In other words, a_l is identified as the potentially best alternative if the minimal satisfaction of alternative al is largest in comparison with those of other alternatives. Suppose that M_b denotes the set of potentially best alternatives. The alternatives whose optimized objective values are 0 in the foregoing model forms M_b.

4.2. Determination of the intervals of reliability of the aggregated solution for the identified optimal alternatives

After all the potentially optimal alternatives are identified and included in M_b, the issue left is further comparison of alternatives in M_b if more than one alternative is identified as potentially optimal alternative. In particular, if only one alternative in M_b, the one is identified as the optimal alternative. In theory, the alternatives in M_b correspond to different solutions to MADA problems. In terms of the decision quality of different solutions, more credible solution is often favored by the decision maker when making decisions [41]. To guarantee the solution in line with the preference of decision maker, it may be feasible to measure the aggregated solution reliability for the alternatives in M_b, thus achieving reliability comparison among alternatives. On the other hand, an alternative in M_b may correspond to multiple sets of attribute weights along with assessments grade utilities. Aggregated solution reliability for the alternatives in M_b may vary along with the fluctuations of attribute weights, as well as assessments grade utilities. To reflect the solution reliability robustness of alternatives in M_b to the fluctuations of attribute weights and assessments grade utilities, the possible intervals of aggregated solution reliability for the alternatives in M_b is measured, which is elaborated as follows.

Suppose that Q(a_l) denotes the aggregated solution reliability for the alternative a_l when a_l is identified as the potentially optimal alternative. Then, Q⁻(a_l) and Q⁺(a_l) respectively stand for the lowest and highest reliabilities of aggregated solution for the alternative a_l, which are estimated by addressing a pair of nonlinear optimization issues shown below:

(19)

(20)

To evade repetition, we elide the constraints in Eqs. (9)–(18) in the foregoing optimization models in section 4.1. The calculation of Q(a_l) is demonstrated in Definition 1. In theory, Q(a_l) is considered as a continuous function associated with the variables of w_i(a_l) (i = 1, …, L; l = 1, …, M), as well as u(H_n)(a_l) (n = 1, …, N; l = 1, …, M). Besides, maximization of Q(a_l) is closely associated with the ϕ_l (l = 1, …, M − 1). As is shown in Theorem 3, Q(a_l) monotonously increasing with respect to ϕ₁. In other words, Q(a_l) value is greatest if and only if maximal ϕ₁ is achieved, while minimum Q(a_l) is reached if and only if ϕ₁ is minimized. In order to obtain the maximum Q(a_l), the maximal ϕ₁ should be guaranteed. On the contrary, to obtain the minimum Q(a_l), the minimal ϕ₁ should be guaranteed. On the assumption that the maximum ϕ₁ and minimum ϕ₁ are determined by using Theorem 2, the intervals of reliability of the aggregated solution for the identified optimal alternatives are assessable by addressing the foregoing pair of optimization issues. In the ER approach, an alternative with high reliability is better than others. To facilitate the comparison of alternatives, we introduce the reliability into the foregoing pair of optimization issues, which is opposed to the maximal regret. According, an alternative with larger reliability is better than the others. From Property 1, we can infer that [Q⁻(a_l), Q⁺(a_l)] belongs to [0, M − 1]. For the alternative in M_b, an aggregated solution with larger reliability intervals is more favored by decision maker.

4.3. Determination of the intervals of attribute weights and utilities of assessment grades for the identified optimal alternatives

In the present section, attribute weights and evaluation grader utilities for identified optimal alternatives will be determined under the condition of making the aggregation solution most reliable. Interval attribute weights are determined for each alternative by creating three optimization problem pairs, such that al Mb and the intervals for utilities of assessments grades, as described below.

To determine [, ] enabling al Mb in case one, an optimization problem pair is established as shown below:

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

The constraint in Eq. (23) is computed by an identical procedure to that followed for objective function computation in Eq. (8), which is therefore elided from the foregoing optimization problem. After the model shown in Eqs. (19)–(20) is solved, its optimal objectives denoted by Q⁻(a_l) and Q⁺(a_l) can be obtained. The optimal objective is definitely accompanied with a set of criterion weights and utilities of grades. The set of criterion weights and utilities of grades may not be unique. There may be other sets of criterion weights and utilities of grades leading to the same objective values Q⁻(a_l) and Q⁺(a_l). To cover all possible sets of criterion weights and utilities of grades resulting in Q⁻(a_l) and Q⁺(a_l) , a slightly enlarged post−optimal solution space is defined as Q⁺(a_l) − φ ≤ Q(a_l) for further analysis, where φ is a very small positive number such as 0.001 to compensate computational error incurred in the process of generating Q(a_l). In case objective function in Eq. (21) becomes u(H_n)(a_l), we can get [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] by seeking best solution of the foregoing optimization problem. In general, the larger range of w_i(a_l) and u(H_n)(a_l) of the alternative a_l is with the others, the more robust a solution is.

Under scenario two, another optimization problems pair is created for evaluating [, ], which enables al M_b.

(31)

(32)

(33)

(34)

(35)

(36)

The calculation of the constraint in Eq. (33) is the same as the objective function in Eq. (8). It is thus omitted in the above optimization model. After the model shown in Eqs. (19)–(20) is solved, its optimal objectives denoted by Q⁻(a_l) and Q⁺(a_l) can be obtained. The optimal objective is definitely accompanied with a set of criterion weights. The set of criterion weights may not be unique. There may be other sets of criterion weights leading to the same objective values Q⁻(a_l) and Q⁺(a_l). To cover all possible sets of criterion weights resulting in Q⁻(a_l) and Q⁺(a_l) , a slightly enlarged post−optimal solution space is defined as Q⁺(a_l) − φ ≤ Q(a_l) for further analysis, where φ is a very small positive number such as 0.001 to compensate computational error incurred in the process of generating Q(a_l).

Under scenario three, another optimization problems pair is created for evaluation of [u⁻(H_n)(a_l), u⁺(H_n)(a_l)].

(37)

(38)

(39)

(40)

(41)

(42)

(43)

In the foregoing three optimization issue pairs, subjective restrictions on u(H_n) (n = 1, ..., N) and w are manageable as well. An example of this is presented in Section 6 below, where restrictions on u(H_n) (n = 1, ..., N) are dealt with.

4.4. Generation of a ranking order of potentially optimal alternatives

For above mentioned three scenarios in Section 4.3, the interval difference enabling a_l M_b for an alternative a_l is defined by utilizing [, ] (i = 1,…, L) and [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] (n = 1, ..., N), which are applied further for the comparison of potential best alternatives in M_b.

Definition 1. Assuming [, ] and [, ] enabling l ≠ m and a_l, a_mM_b are derived by resolving optimization issues stated in Section 4.3, the definition of interval difference of [, ] to [, ] is given as =.

Here, d([, ], [0,0]) and d([, ], [1,1]) represent the distance between [, ] and [0,0] and that between [, ] and [1,1], respectively. d([, ], [0,0]) and d([, ], [1,1]) stand severally for the distance from [, ] to [0,0] and that from [, ] to [1,1]. Their computation is accomplished using the distance measure between interval numbers by Li et al. (2008) [43], as presented in Definition A.2 in Appendix A.

When the interval difference of [, ] to [, ] is obtained as , definition of interval difference for an alternative a_l in M_b under scenario two can be given by (i = 1, ..., L).

Definition 2. Assuming the interval difference of [, ] to [, ] is obtained as , which enables l ≠ m and a_l, a_mM_b, the differences in interval of alternatives a_l to a_m and of alternative a_l are formulated as

(44)

and

(45)

respectively.

Similarly, for an alternative al in Mb under scenario three, the interval difference is expressible with the utilization of [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] (n = 1, ..., N).

Definition 3. Assuming [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] and [u⁻(H_n)(a_m), u⁺(H_n)(a_m)] that enable l ≠ m and a_l, a_mM_b are derivable by solving optimization issues described in Section 4.3, the definition of interval difference of [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] to [u⁻(H_n)(a_m), u⁺(H_n)(a_m)] is given as = .

Because [u⁻(H₁)(a_l), u⁺(H₁)(a_l)] =[0, 0] and [u⁻(H_N)(a_l), u⁺(H_N)(a_l)] =[1, 1], only [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] (n = 2,..., N ˗ 1) are utilized in Definition 3.

When the interval difference of [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] from [u⁻(H_n)(a_m), u⁺(H_n)(a_m)] is derived as , for an alternative a_l in M_b under scenario two, definition of its interval difference can be given by (n = 2, ..., N ˗ 1).

Definition 4. Suppose that the interval difference of [u⁻(H_n)(a_l), u⁺(H_n)(a_l)] from [u⁻(H_n)(a_m), u⁺(H_n)(a_m)] is derived as , which enables l ≠ m and a_l, a_mM_b, the interval differences of alternatives a_l to a_m and of alternative a_l are expressable as

(46)

and

(47)

respectively.

The , , , , and have the properties as follows.

Property 1. For , , , , and defined in Definitions 1−4, it is satisfied that

1. (1) and ,
2. (2) and ,
3. (3) and , and
4. (4) and .

On the bases of Definitions 1−4, the difference in interval for the optional al in Mb under three scenarios is formulated as

(48)

Because and , the scope ofis within [0,1]. In M_b, an alternative a_l with greater difference in interval is favored over other alternatives. Thus, generation of alternative ranking order is achievable with the exploitation of interval difference between potentially best alternatives in M_b.

For MADA problems, we usually only need to deal with one of three situations. Through several iterations, we can identify and compare the potential optimal solutions, and then produce a complete solution ranking, as shown below.

4.5. Procedure of the proposed method

Taking into account what has been introduced in Sections 4.1–4.4, the solution seeking process in which the aggregated solution reliability is incorporated is depicted in Fig 2.

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Fig 2. A flow diagram for our approach.

https://doi.org/10.1371/journal.pone.0317438.g002

1. Step 1: Construct a multi−attribute decision problem
2. L attributes along with respective classes (benefits or costs) are identified by decision maker, as well as N evaluation levels. Further, a multi−attribute decision problem is established by listing M alternatives.
3. Step 2: To tackle the established decision problem, an ER method exploiting high solution reliability is put forward.
4. One scenario is specified by decision maker among three scenarios. Under scenarios one and three, the decision maker provides δ_u. Under scenarios two and three, u(H_n) (n = 1, ..., N) are provided, which enable 0 = u(H₁) < u(H₂) < < u(H_N) = 1 and the attribute weight set w. In addition, assuming M_b = {Ø} and M_r = {a₁, ..., a_M}.
5. Step 3: Gather evaluation values for the alternatives.
6. The decision maker provides an estimate of the solution on every attribute.
7. Step 4: Gather the interval values for attribute weights and/or evaluate the assessment grades utility.
8. In a particular case, an interval of attribute weights and/or the assessment grade utility is/are provided by decision maker.
9. Step 5: Determine the minimum and maximum reliabilities for aggregated solution.
10. Minimum and maximum aggregated solution reliabilities for the alternative a_l, i.e. Q⁻(a_l) and Q⁺(a_l), are determined by tackling nonlinear optimization problems expressed in Eqs. (19)–(20).
11. Step 6: Exclude M_b from M_r.
12. In the present iteration, collection M_b is excluded from M_r.
13. Step 7: Determine if |M_r| falls below or is equivalent to 1.
14. In case |M_r| falls below or is equivalent to 1, the complete order is derived for M alternatives, followed by proceeding to step 13; Otherwise, enter step 8.
15. Step 8: Identify M_b from M_r.
16. In a specific case, the set M_b under the present iteration is identified based on alternatives set M_r, according to the optimal solution of the optimization model in Eqs. (8)–(18).
17. Step 9: Determine if |M_b| is equivalent to 1.
18. In case |M_b| is equivalent to 1, proceed to step 13. Otherwise, enter step 10.
19. Step 10: Determine optimal attribute weight interval or evaluate the utility of the level.
20. In the current iteration, for the scheme from the set M_b, the utility of the optimal interval or evaluation level of its attribute weights is derived by addressing the pairwise optimization model under a particular circumstance.
21. Step 11: Calculate interval deviation for every scheme in M_b.
22. To tackle the multi−attribute decision problem in a specific case, computation of interval deviation is accomplished for every alternative in M_b under the present iteration.
23. Step 12: Derive a rank−order for alternatives in the collection M_b + {a₁,..., a_M}− M_r.
24. In case |Mb| is equivalent to 1, the current iteration directly produces alternatives order in the set M_b + {a₁,..., a_M} − M_r. Otherwise, inter−alternative interval differences in the present iteration Mb are used to produce alternative order, which in addition can be blended with the order obtained in the prior iteration, and then produced alternative sorting in M_b + {a₁,..., a_M} − M_r. Thereafter, return to step 5.
25. Step 13: Complete the process.
26. Finally, the order of M alternative can be obtained, i.e., the solution of multi−attribute decision problem.

5.1. Description of the CPA problem

CPA issue assesses six commercial vehicles c_l (l =1, …, 6) regarding seven performance attributes, namely acceleration, handling, braking, power transmission, fuel economy, horsepower and ride mass. Assuming the assessment grades are denoted as Ω = {H_n, n=1, …, 6} = {Worst, Poor, Average, Good, Excellent, Top} = {W, P, A, G, E, T}. With the utilization of assessment grades, the six vehicles are evaluated by decision maker, and Table 3 details relevant results, which comprise complete ignorance assessments (β_Ω,i(a_l) = 1) and incomplete assessments (β_Ω,i(a_l) > 0). The description for the CPA dataset is given in Appendix B. Decision maker wants to get their ranking order, a CPA issue solution, through performance contrast among six cars. CPA issues in these three cases are addressed to illustrate the flexibility of our proposed method. Decision maker provides relevant parameters for these three scenarios below.

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Table 3. Assessments for six executive vehicles in CPA issue.

https://doi.org/10.1371/journal.pone.0317438.t003

Scenario one: Assuming that (i = 1,. . .,7) = (0.01, 0.01,0.01,0.01,0.01,0.01,0.01), [, ](i = 1,. . .,7) = {[0.05,0.45], [0.05,0.45], [0.05,0.45], [0.05,0.45], [0.05,0.45], [0.05,0.45], [0.05,0.45]}, and [u⁻(H_n), u⁺(H_n)] (n = 1,. . .,6) = {[0,0], [0.1,0.3], [0.3,0.5], [0.5,0.7], [0.7,0.9], [1,1]}.

Scenario two: Assuming that u(H_n) (n = 1,. . .,6) = (0, 0.2, 0.4, 0.6, 0.8, 1), (i = 1,. . .,7) = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01), and [, ](i = 1,. . .,7)= {[0.1,0.35], [0.1,0.35], [0.1, 0.35], [0.1,0.35], [0.1,0.35], [0.1,0.35], [0.1,0.35]}.

Scenario three: Assuming w = (1/7,1/7,1/7,1/7,1/7,1/7,1/7) and [u⁻(H_n), u⁺(H_n)] (n = 1, ..., 6) = {[0,0], [0.1,0.3], [0.3,0.5], [0.5,0.7], [0.7,0.9], [1,1]}. Under scenarios one and three, δ_u = 0.05 is specified by decision maker. The first step is designation of M_b = {Ø} and M_r = {c₁, ..., c₆}. Null collection is symbolized by ‘Ø’.

5.2. Generation of a solution to the CPA problem in the first scenario

Under scenario one, optimization model in Eqs. (8)–(18) is addressed with specified , [, ](i = 1,. . .,7), [u⁻(H_n), u⁺(H_n)], and δ_u, thereby obtaining the set M_b = {c₁, c₂, c₃, c₅, c₆} from optimal solutions. It be can then derived that M_r = {c₄} during iteration one. Since |M_b| > 1, computation of interval difference should be accomplished for five vehicles in M_b for their comparison purpose. Then, five sets of [(c_l), (c_l)] (i = 1, ..., 7) and [u⁻(H_n)(c_l), u⁺(H_n)(c_l)] are calculated through resolution of the optimization problem pair in Eqs. (21)–(30). Table 4 details relevant results, based on which we calculate that = {0.5453,0.5906,0.5602,0.2134,0.5894} and = {0.5341, 0.5341,0.5341,0.3421,0.5341} using Definitions 1-4.

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Table 4. Attribute weights of five vehicles in Mb during iteration one under scenario one, and the optimization interval for evaluating the rating utility.

https://doi.org/10.1371/journal.pone.0317438.t004

Base on and , we have = (+)/2 = {0.5397, 0.5624, 0.5472, 0.2778, 0.5617}. We then obtain a rank order for five vehicles in M_b by exploiting as c₂ > c₆ > c₃ c₁ > c₅. Since |M_r| = 1 during iteration one, a complete ranking order for six vehicles is infer that c₂ > c₆ > c₃ > c₁ > c₅ > c₄. We deem it as the CPA issue solution under scenario one. As is clear from the ranking order for six vehicles, c₂ is the best-performing vehicle.

For a more universal situation, assuming that (i = 1, ..., 7) = (1,1,1,1,1,1,1) and u⁺(H_n) (n = 2, ..., 5) = (1, 1, 1, 1). The set M_b and M_r are respectively obtained as {c₁, c₂, c₃, c₄, c₅, c₆} and {Ø} during iteration one. We then obtain that = {0.5399, 0.5487, 0.5346, 0.448, 0.3597, 0.5609}, = {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, and = {0.52, 0.5244, 0.5173, 0.474, 0.4299, 0.5305} for six vehicles in M_b. Base on , a thorough ranking order for six vehicles is infer that c₆ > c₂ > c₁ > c₃ > c₄ > c₅, which is considered as the CPA issue solution. The solution and highest vehicle performance have changed significantly.

5.3. Generation of a solution to the CPA problem in the second scenario

Under scenario two, optimization model in Eqs. (8)–(18) is addressed using exact u(H_n) (n = 1,. . .,6), (i = 1,. . .,7), and [, ] (i = 1,. . .,7), thereby obtaining a set M_b = {c₁, c₂, c₃, c₆}. It is clear from set M_b that M_r = {c₄, c₅} during iteration one. Since |M_b| > 1, computation of interval difference should be accomplished for four vehicles in M_b for comparison among these cars. To achieve this, the optimization problem pair in Eqs. (31)–(36) for four vehicles in M_b during iteration one are solved to derive four sets of [(c_l), (c_l)] (i = 1,. . . ,7), which are shown in Table 5. Then, based on Definitions 1 and 2, enabling c_l M_b is computed in terms of Table 5 outcomes as {0.4651,0.5237,0.502,0.507} during such iteration. A rank-order of four vehicles in M_b is produced as c₂ > c₆ > c₃ > c₁. Since |M_r| > 1, iteration should be carried out. The optimization model in Eqs. (8)–(18) is addressed, thereby getting M_b = {c₄, c₅} during iteration two. It can be obtained the M_r = {Ø} during such iteration. By tackling the optimization problems pair in Eqs. (31)–(36), the [(c_l), (c_l)] (l = 4,5) are calculated as {[0.1,0.35], [0.1,0.35], [0.1,0.35], [0.1,0.35], [0.1,0.35], [0.1,0.35], [0.1,0.35]} and {[0.1,0.35], [0.1,0.15], [0.1,0.35], [0.1,0.12], [0.1,0.35], [0.1,0.35], [0.1,0.135]}, respectively. It has > in terms of Definitions 1 and 2. Therefore, an intact ranking order for six vehicles is inferred as c₂ > c₆ > c₃ > c₁ > c₄ > c₅.

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Table 5. In the second case, the optimization interval of the attribute weights for four vehicles in Mb during iteration one.

https://doi.org/10.1371/journal.pone.0317438.t005

Resembling scenario one, assuming (i = 1, ..., 7) = (1,1,1,1,1,1,1), M_b and M_r during iteration one are then calculated as {c₁, c₂, c₃, c₅, c₆} and {c₄}, respectively. Then, = {0.4875, 0.6231,0.5231,0.3647,0.5501} is obtained for five vehicles in set M_b. Since |M_r| = 1 during iteration one, an intact rank order for six vehicles is derivable as c₂ > c₆ > c₃ > c₁ > c₅ > c₄, representing the CPA issue solution. Based on the result, c₂ remains to be the best-performing vehicle although the solution is changed.

5.4. Generation of a solution to the CPA problem in the third scenario

Under scenario three, optimization model in Eqs. (8)–(18) is addressed using exact w, [u⁻(H_n), u⁺(H_n)] (n =  1,...,6), and δ_u, thereby calculating set M_b as {c₁, c₂, c₃, c₆} during iteration one. The set M_r is inferred as {c₄, c₅}. Since | M_b | >  1, interval difference computation should be accomplished for four vehicles in M_b to allow their comparison. By tackling the optimization problems pair in Eqs. (37)–(43), four sets of [u⁻(H_n)(c_l), u⁺(H_n)(c_l)] (n =  1,...,6) are obtained during iteration one, as listed in Table 6. Based on the table outcomes, Definitions 3 and 4 are exploited to derive = {0.1674, 0.6772, 0.5766, 0.6145} for four vehicles in M_b during iteration one. Subsequent step is creation of a rank-order for these four vehicles, that is, c₂ > c₆ > c₃ > c₁. Since | M_r | >  1, iteration should be carried out. Optimization model in Eqs. (8)–(18) is addressed for c₄ and c₅, thereby determining M_b =  {c₄} during iteration two from best solutions, which means that the corresponding M_r =  {c₅}. Therefore, the complete ranking order is generated as c₂ > c₆ > c₃ > c₁ > c₄ > c₅ for the six vehicles. Resembling the first scenario, assuming u⁺(H_n) (n =  2,..., 5) =  (1,1,1,1), M_b and M_r during iteration one are calculated separately as {c₁, c₂, c₃, c₆} and {c₄, c₅}. For the four cars in the set M_b, the is recalculated as {0.3763,0.6456,0.499,0.5218}, leading to creation of their ranking order, i.e., c₂ > c₆ > c₃ > c₁. This finishes iteration one. Since | M_r | >  1, it is necessary to carry out iteration two, during which M_b =  {c₄, c₅} is determined by seeking best solutions for the optimization model in Eqs. (8)–(18). Next step is determination of corresponding M_r as {Ø}. Then, through resolution of optimization problem pair in Eqs. (37)–(43) for c₄ and c₅, it can be generated that [u⁻(H_n)(c_l), u⁺(H_n)(c_l)] (n =  1,..., 6, l =  4,5) =  {[0], [0.04,0.7], [0.15,0.75], [0.25,0.85], [0.25,0.9], [1]}, {[0], [0.05,0.2784], [0.15,0.334], [0.45,0.498], [0.453,0.95], [1]}. This infers that c₄ > c₅ by Definitions 3 and 4. Hence, an intact ranking order is derivable as c₂ > c₆ > c₃ > c₁ > c₄ > c₅ for all six vehicles, without changing the solution.

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Table 6. In the third case, optimization intervals of evaluation level for four vehicles in M_b during iteration one.

https://doi.org/10.1371/journal.pone.0317438.t006

In conclusion, under the parameters provided in Section 5.1, the solution to the CPA problem in these three cases is very similar apart from the varying order of c₄ and c₅. In the solution, c₂ consistently has the best performance. Nonetheless, solution for scenario one is found by single iteration, while that for scenarios two and three are found by double iterations. Specifically, during the second iteration, c₄ is contrasted against c₅ by the interval difference in the second situations, while in the third situation, we distinguish them directly due to M_b =  {c₄}. Additionally, when (i =  1,..., 7) along with u⁺(H_n) (n =  2,..., 5) are given, the best−performing car is probably replaced, as under the first scenario.

5.5. Comparison analysis

To compare the objective derivation of attribute weights herein with that in representative references, we describe the way of objectively identifying attribute weights based on a belief decision matrix by following original processes of entropy [32], CRITIC [33], SD [33], as well as CCSD [27], the detail processes of the four typical methods are presented in Appendix C.

To contrast our proposed method against these four methods, estimation of attribute weights and production of CPA issue solutions are accomplished with four extended models under requisites prescribed in Section 5.1. Get the attribute weight [u_min(c_l), u_max(c_l)] (l =  1,..., 6), as well as rank-order of c_l (l =  1,..., 6). Table 7 details the computational outcomes.

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Table 7. Attribute weights, [u_min(c_l), u_max(c_l))] (l =  1,..., 6), Reliability along with rank-orders of c_l (l =  1,..., 6) derived using four approaches in CPA issue.

https://doi.org/10.1371/journal.pone.0317438.t007

As can be seen from Table 7, unlike c₃ which is derived from proposed method, c₁ is the optimal choice derived based on the four approaches. This is primarily attributable to failure to address the alternative insensitivities to the attribute weight alterations with these four approaches. The optimization model generalized by the four approaches involves L equation. Hence, an exclusive w_i (i =  1,..., L) set can be decided by every model, thus generating an exclusive solution. Contrastively, our proposed method addresses the solution insensitivity to the attribute weight alterations during generation of a high reliability solution for the CPA issue.

From the results in Table 7, the different reliability of the aggregated solutions is derived from the four representative methods and the proposed method. Compared with the proposed method, the low recognizable expected utilities are created using the four representative methods, which further lead to the low reliability of the aggregated solutions. Different from the four methods, the high recognizable interval-valued expected utilities are generated by using the proposed method, which contribute to the generation of highest reliability of the aggregated solutions. Due to the high solution reliability created by the proposed method, the preferential order of six cars seems more reliable and convincing than the other four methods.

When the methods of entropy, SD, CRITIC, and CCSD are used to evaluate the performance of the six vehicles, the most best-performing vehicle is c₁. For the CPA problem, the decision maker attempts to take into account the worst scenarios of each vehicle. In accordance with the willingness of the decision maker, the reliability of each vehicle is considered. Under the conditions, the solution generated by the proposed method, i.e., c₁,..., c₆ with the preferential order of c₆ > c₂ > c₁ > c₃ > c₄ > c₅, is clearly different from those associated with the four methods. When the solution reliability is considered, the solution generated by the proposed method is also different from those associated with the four methods. The reason why the solution generated by the proposed method is considered as what the decision maker anticipates is that the DM is involved in the process of obtaining all possible sets of attribute weights when generating the expected utilities of the six cars. The proposed method to determine order reflects the interaction between the experts and decision maker making the obtained solution consistent with the real preferences of the decision maker. Relevant theoretical analysis can be found in Section 4.2. The reliability of the ranking order, however, is not considered in the obtainment of solution by using the four methods.

In brief, a method for objectively determining attribute weights is proposed with the consideration of solution reliability. The proposed method can help guarantee the generation of a solution with high reliability.

6.1. Numerical analysis

In this section, we conduct a numerical experiment to analyze the superiority of the proposed method. This numerical experiment is based on the practical example presented in Fu et al (2014) [42]. This numerical experiment investigates the selection of leading industries in the industry-cluster region in the north of the Yangtze River. The data related with the leading industries selection problem is presented in Table D.1 of Appendix D.

In this numerical experiment, following the steps outlined in Sect. 4.5, the reliability of the aggregated solutions and the rank-order of the twelve industries is calculated firstly using the assessment data in Table D.1, as presented in Table 8. The four methods in Appendix C are used to generate four solutions to the problem of selecting leading industries. The resulting reliability of the aggregated solutions are compared with those using the proposed method, as presented in Table 8.

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Table 8. Comparison of rank-order and reliability of the aggregated solutions and generated by the five methods and proposed method.

https://doi.org/10.1371/journal.pone.0317438.t008

It is shown from Table 4 that the rank-order of the industries generated by the five methods is correspondingly different although the top five industries obtained by the five methods are the same. In particular, in terms of the minimal satisfaction produced, the second and the third industries are very close if using SD, CRITIC, CCSD, and the proposed methods, while using the entropy method, it is the second and the sixth that are very close. This results in a clear difference between the rank-order of the second, the third, and the sixth industries generated by the entropy method and those by the other four methods. In brief, the proposed method can generate a reasonable set of attribute weights by using solution reliability and uses the attribute weights to guarantee high reliability of the aggregated solution.

6.2. Simulation experiments analysis

In the simulation experiment, six possible alternative and attribute combinations as shown in Table 9 were selected. In order to ensure the statistical significance of the simulation experiment, 100 random belief decision matrices were generated for each combination to conduct the simulation experiment. Then, with the changes of alternatives and attributes, simulation experiments are designed, as shown in Appendix D of Table D.2.

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Table 9. The random decision matrices with different combinations of alternatives and attributes.

https://doi.org/10.1371/journal.pone.0317438.t009

Based on the simulation process in Table D.1, 600 groups of randomized experiments were conducted to verify the superiority of the proposed method. From the results of simulation experiments, five different curves of the reliability of the aggregated solutions generated by the four representative methods and the proposed methods are shown in Fig 3.

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Fig 3. Variation of the reliability of the aggregated solution for the four representative methods and proposed method.

https://doi.org/10.1371/journal.pone.0317438.g003

As shown in Fig 3, with the increase of L, the reliability of the aggregated solution generated by the proposed method is always greater than that of the four representative methods. In addition, the reliability trend of the aggregated solutions generated by this method is consistent with that of the four representative methods. These results show that the proposed method is the most effective way to identify and measure the performance differences of alternative solutions on various attributes. In other words, the attribute weights generated using the methods presented in this paper may be more reasonable than the weights of the four representative methods. The above results in Fig 3 show that compared with the four representative methods, the proposed method is the best method to meet the requirements of solving high reliability.

On the other hand, in order to further prove the superiority of the proposed method, the significance of testing the superiority of the proposed method is investigated by the following statistical methods. From a statistical point of view, since the overall distribution of the results of 600 randomized experiments was unknown, this paper adopted non-parametric Wilcoxon rank sum test. Table 10 shows the basic content of Wilcoxon rank sum test. As shown in Table 10, when the significance level is 5%, it can be considered that the reliability of the polymerization solution generated by the method in this paper is significantly higher than that of other methods. The test results are consistent with the simulation results shown in Fig 3.

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Table 10. Wilcoxon signed-rank test results at the significance level α =  0.05.

https://doi.org/10.1371/journal.pone.0317438.t010

Based on the above simulation and test results, it can be concluded that the proposed method is the most suitable method to achieve high reliability solutions. In other words, compared to other methods, the proposed method can reveal the preferences of decision makers well and identify the solutions that decision makers prefer the most.