A novel multi-attribute decision-making framework based on Z-RIM: an illustrative example of cloud service selection

Abstract

In the multi-attribute decision-making (MADM) problems, decision makers refer to the extreme attribute values in general when evaluating the alternatives. However, in the real world, the ideal solution may lie in somewhere between the extreme values. The recently proposed reference ideal method (RIM) is able to solve the problem rightly. This study aims at developing a novel MADM framework combining best–worst method (BWM), maximizing deviation method (MDM), and RIM under Z-number environment. In this framework, Z-number is used to depict the inherent uncertainty and reliability of information in the decision makers’ judgments. And BWM and MDM are combined to determine the comprehensive attribute weights, in which BWM is utilized to obtain the subjective weights, while MDM is utilized to obtain the objective weights. In addition, Z-RIM is proposed by extending the traditional RIM under Z-number environment, which is employed for ranking the alternatives. An illustrative example of cloud service selection problem is implemented to illustrate the proposed framework. By comparison analysis, we demonstrate that Z-RIM can not only avoid rank reversal problem, but also generate reasonable results during the MADM processes.

Appendix

Parameter	Explanation
$\mu _{\tilde{A}}(x)$	The membership function of a fuzzy number $\tilde{A}$
$\tilde{A}$	A fuzzy number which is expressed as $(a_1,a_2,a_3,a_4)$, and $a_1<a_2<a_3<a_4$
$\tilde{B}$	A fuzzy number which is expressed as $(b_1,b_2,b_3,b_4)$, and $b_1<b_2<b_3<b_4$
r	A positive real number
$R(\tilde{A})$	$R(\tilde{A})$ is a ranking value calculated by the graded mean integration representation method (GMIR)
$\text {dist}(\tilde{A},\tilde{B})$	The distance between two fuzzy numbers $\tilde{A}$ and $\tilde{B}$
$\alpha$	A real number which represents the reliability part of a Z-number
$\tilde{W}_B^s$	The subjective weight of the best attribute $C_B$
$\tilde{W}_W^s$	The subjective weight of the worst attribute $C_W$
$\tilde{A}_{Bj}$	The fuzzy preference of the attribute $C_B$ over the attribute $C_j$
$\tilde{A}_{jW}$	The fuzzy preference of the attribute $C_j$ over the worst attribute $C_W$
$\tilde{W}^s=(\tilde{W}_j^s)_n$	$\tilde{W}^s$ is the optimal subjective fuzzy weight vector, and $\tilde{W}_j^s$ is the subjective weight of the attribute $C_j$
$W^s=(W_j^s)_n$	$W^s$ is the optimal subjective weight vector, and $W_j^s$ is the defuzzified subjective weight of the attribute $C_j$
$\tilde{D}=[\tilde{x}_{ij}]_{m\times n}$	$\tilde{D}$ is the decision matrix over the alternatives established by decision makers, and $\tilde{x}_{ij}$ is the performance rating of the alternative $A_i$ regarding the attribute $C_j$
$\overline{W}^o=(\overline{W}^o_j)_n$	$\overline{W}^o$ is the objective weight vector, and $\overline{W}^o_j$ is the objective weight of the attribute $C_j$
$W^o=(W^o_j)_n$	$W^o$ is the normalized objective weight vector, and $W^o_j$ is the normalized objective weight of the attribute $C_j [\tilde{A}_j,\tilde{B}_j]$
	$[\tilde{A}_j,\tilde{B}_j]$ is the value of alternatives regarding the attribute $C_j$ can be selected
$[\tilde{C}_j,\tilde{D}_j]$	$[\tilde{C}_j,\tilde{D}_j]$ is the ideal reference range, which counts most in the given value range
$\tilde{D}'=[\tilde{x}'_{ij}]_{m\times n}$	$\tilde{D}'$ is the regular fuzzy decision matrix, and $\tilde{x}'_{ij}$ is calculated by Eq. (12)
$D=[x_{ij}]_{m\times n}$	D is the normalized decision matrix, and $x_{ij}$ is calculated by Eq. (27)
$\text {dist}_{\min }(\tilde{x}'_{ij},[\tilde{C}_j,\tilde{D}_j])$	$\text {dist}_{\min }(\tilde{x}'_{ij},[\tilde{C}_j,\tilde{D}_j])$ is the smaller one between $dist(\tilde{x}'_{ij},\tilde{C}_j)$ and $dist(\tilde{x}'_{ij},\tilde{D}_j)$
$W=(W_j)_n$	W is the comprehensive weight vector, and $W_j$ is the comprehensive weight of the attribute $C_j$
$I_i^+$	$I_i^+$ represents the extent to which the evaluation of the alternative $A_i$ deviates from the positive ideal point
$I_i^-$	$I_i^-$ represents the extent to which the evaluation of the alternative $A_i$ deviates from the negative ideal point
$R_i$	$R_i$ represents the ideal reference index of the alternative $A_i$