The evaluation and selection process can be regarded as a complex multiple criteria decision analysis (MCDA) problem which involves various interaction relationships among criteria under high uncertain environment. In addition, the decision-makers are always bounded rational in the risk decision-making process. However, the current robot evaluation and selection approach seldom considers the decision-maker’s risk preference and interactive criteria under high uncertain environment. Thus, the purpose of this paper is to develop a hybrid MCDA approach for solving the robot evaluation and selection problem. In the proposed framework, the interval type-2 fuzzy set is used to express the uncertain evaluation information provided by decision-makers. Next, the distance measure of interval type-2 fuzzy numbers is developed to determine the fuzzy measure of each criterion. Then, the extended prospect theory based on developed Choquet integral is proposed to evaluate and prioritize the robot by considering the decision-maker’s risk preference and interactive criteria. Finally, a case study of robot evaluation and selection in the auto industry is selected to exemplify the application of the proposed framework. After that, comparison and sensitivity studies are conducted to further demonstrate the robustness, effectiveness, and reasonableness of the developed approach.
In recent years, Industry 4.0 has been used to describe the features of the digitized and automated manufacturing industry [
In the literature, in the robot evaluation and selection process, the application of expert elicitation is one of the most important approaches to depict the uncertainties of robot evaluation and selection. Consequently, some important techniques are used to depict the uncertain evaluation information from various experts. These techniques include triangular fuzzy numbers (TFNs) [
In general, evaluation and selection of optimal robot for an industrial application is completed by a group of decision-makers from various fields, in which different kinds of factors, such as cost, quality, and functions man-machine interface, should be taken into account [
In addition, the robot evaluation and selection process in the practice includes many objective and subjective factors, some of which are affected by each other [
As mentioned above, little attention has been paid to addressing robot selection problem by considering the risk preference of decision-makers and the interactive criteria within high type uncertain environment. Compared with the type-1 fuzzy set, the type-2 fuzzy set is an effective technique to depict high type uncertainty [
As the discussion mentioned above, the summaries of the main contributions of this study to the relevant literature are provided as follows: A developed hybrid decision-making framework for robot evaluation and selection is constructed. The results of comparison analysis and sensitivity analysis indicate that the proposed approach embodies a few desirable features to address the robot evaluation and selection problem compared with the extant approaches. The distance measure of interval type-2 fuzzy numbers is introduced to determine the fuzzy measure of each criterion, which not only can provide an objective weight vector for robot evaluation and selection but also can provide a more reasonable result than the weight vector assigned beforehand. A developed Choquet integral is incorporated into the traditional prospect theory for solving the robot evaluation and selection problem, in which the decision-maker’s risk preference and interactive criteria are considered. In addition, the positive and negative reference points are introduced into the traditional prospect theory to determine the optimal robot, which can express the complex risk preference of decision-makers more effectively than the existing methods for robot selection.
The remainder of this paper is organized as follows. In Section
In this section, the basic concepts related to interval type-2 fuzzy set, prospect theory, and Choquet integral are briefly reviewed, which shall be used in the subsequent sections.
A type-2 fuzzy set in the universe of discourse can be expressed by a type-2 membership function, which can be defined as follows:
Let
In general, the interval type-2 trapezoidal fuzzy number (IT2TrFN) is selected to express the IT2FS. Let
Visual representation of an IT2TrFN.
Assume that Addition operation Subtraction operation Multiplication operation Multiplication by crisp number operation Power operation
Let
The prospect theory was introduced by Kahneman D [
A value function of prospect theory.
Based on the value function defined in Figure
The theory of fuzzy measure, primely introduced by [
Let
If the set
According to equation (
Assume that
In this section, we incorporate the extended prospect theory into the performance evaluation and prioritization determination procedures to address the robot evaluation and selection problems with interactive criteria and interval type-2 fuzzy numbers. According to the literature review of current robot evaluation and selection methods mentioned in Section
The flowchart of the proposed method.
The scope of this stage is to obtain the evaluation information by the application of expert elicitation. Various types of uncertainties may exist in the robot evaluation and selection problems because of the intra- and interuncertain evaluation information produced by decision-makers [ In this study, we assume that there are In this study, the trapezoid interval type-2 fuzzy numbers are utilized to express the linguistic decision preference information. The transformation from the linguistic information into trapezoid interval type-2 fuzzy numbers can be represented in Table
In the course of robot evaluation and selection, the decision-makers may not be equal in the realistic process because of their knowledge, experience, and different work department. These different characteristics may lead to different preference and evaluation information under the same alternative and criteria [ First, according to the knowledge, experience, and work department of each decision-maker, we assign the importance weight Let
Linguistic terms and their corresponding IT2TrFNs [
Linguistic terms | Interval type-2 trapezoid fuzzy number |
---|---|
Very high (VH) | [(0.9, 1, 1, 1; 1); (0.95, 1, 1, 1; 0.9)] |
High (H) | [(0.7, 0.9, 0.9, 1; 1); (0.8, 0.9, 0.9, 0.95; 0.9)] |
Slightly high (SH) | [(0.5, 0.7, 0.7, 0.9; 1),(0.6, 0.7, 0.7, 0.8; 0.9)] |
Medium (M) | [(0.3, 0.5, 0.5, 0.7; 1),(0.4, 0.5, 0.5, 0.6; 0.9)] |
Slight low (SL) | [(0.1, 0.3, 0.3, 0.5; 1),(0.2, 0.3, 0.3, 0.4; 0.9)] |
Low (L) | [(0, 0.1, 0.1, 0.3; 1),(0.05, 0.1, 0.1, 0.2; 0.9)] |
Very low (VL) | [(0, 0, 0, 0.1; 1),(0, 0, 0, 0.05; 0.9)] |
In Stage 3, the priority of each alternative robot is calculated by constructing an extended prospect theory-based prioritization approach. In this prioritization approach, the risk preference of decision-makers and interaction relationships among criteria are taken into account. First, the distance measure of IT2TrFNs is introduced to determine the fuzzy measure. Then, the extended prospect theory based on Choquet integral is proposed to determine the ranking order of each alternative robot. The main steps of the extended prospect theory-based prioritization approach are described as follows: In general, the fuzzy measure of each criterion is always assigned beforehand, which depends on the judgment of decision-makers. The fuzzy measure obtained in this way is a subjective result that may cause a misleading decision-making result. Consequently, an objective method for determining fuzzy measure is necessary for the priority calculating process. In such case, drawing the experience of [ The first step of this method is to identify the positive and negative evaluation values, which can be completed as the following form: where
The comparison between each two IT2TrFNs can be performed by using the method proposed by Qin et al. [ Next, we can obtain the distance between each evaluation rating and the positive and negative evaluation values as follows: Then, the dispersion of each criterion can be expressed as follows: Finally, according to the experience in literature [ According to the discussion mentioned above, current robot selection approaches are insufficient to determine optimal robot by considering the risk preference of decision-makers and interactive relationships among criteria. Although the prospect theory can depict the risk preference of decision-makers, it is unable to model the interactive relationships among criteria. Thus, we extend the prospect theory with Choquet integral into priority calculation problem by considering decision-makers’ risk preference and interactive criteria. The extended prospect theory-based prioritization approach is described as follows. First, in order to subjectively determine the reference point, we use equations ( Next, the value functions under positive and negative reference point can be defined as follows: Then, the prospect value of each alternative robot under positive and negative reference point can be obtained by the following form: Motivated by the mechanism of the TOPSIS method, the priority of each alternative robot can be derived as follows:
According to the substeps of the above three stages, we can make a summary of the solving procedures for robot selection problem, which is shown as follows:
In this section, an illustrative example of robot evaluation and selection problem in the auto industry [
Linguistic evaluation information from decision-maker
P | VG | SP | G | SG | VG | G | |
G | SP | P | M | VG | VG | SG | |
VG | SP | P | M | M | SP | SG | |
SP | M | G | G | M | SG | M | |
VG | P | SP | P | SP | M | SP | |
P | SG | SG | SP | VG | G | VG | |
G | SP | VP | SP | SP | VP | SP | |
VG | P | P | VP | P | SP | M |
After the preliminary simulation, there are eight alternative robots
The criteria for alternative robot selection.
In what follows, we adopt the proposed hybrid robot evaluation and selection method to evaluate and prioritize the performance of all of the eight alternative robots, which includes the following specific steps: According to the linguistic term sets shown in Table According to the linguistic evaluation information provided in Tables According to the importance weight of each expert, the final group decision matrix is constructed via equation ( First, we calculate the distances Based on the value functions obtained by Step 4, the prospect value of each robot is derived by using equations ( In order to clearly illustrate the calculation process of prospect value, the alternative robot According to the prospect values of each alternative robot under positive and negative reference points, the final priorities of alternative robots are determined by using equation (
According to the result provided in Table
Linguistic evaluation information from decision-maker
P | VG | G | SG | G | G | G | |
VG | SP | VP | SP | G | M | SP | |
G | VP | M | SP | M | P | G | |
P | SP | SG | M | SP | SG | M | |
G | SP | SP | VP | P | SP | M | |
VP | VG | G | SP | VG | G | VG | |
VG | M | P | M | SP | P | P | |
G | VP | P | P | VP | M | M |
Linguistic evaluation information from decision-maker
P | VG | M | G | G | VG | VG | |
G | SP | SP | M | SG | VG | M | |
SG | VP | P | M | M | SP | SG | |
SP | M | G | VG | SP | SG | M | |
VG | VP | SP | P | M | SP | SP | |
P | SG | SG | P | G | SG | VG | |
VG | G | SP | M | SP | VP | SP | |
VG | VP | SP | VP | P | SP | VP |
Interval type-2 fuzzy evaluation matrix from decision-maker
[(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | |
[(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | |
[(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | |
[(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | |
[(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] |
Interval type-2 fuzzy evaluation matrix from decision-maker
[(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | |
[(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | |
[(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | |
[(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | |
[(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | |
[(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] |
Interval type-2 fuzzy evaluation matrix from decision-maker
[(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | |
[(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | |
[(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | |
[(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | |
[(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | [(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.7, 0.9, 0.9, 1; 1), (0.8, 0.9, 0.9, 0.95; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] | [(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0, 0, 0.1; 1), (0, 0, 0, 0.05; 0.9)] |
The group evaluation matrix.
[(0, 0.1, 0.1, 0.3; 1), (0.05, 0.1, 0.1, 0.2; 0.9)] | [(0.8, 0.95, 0.95, 1; 1), (0.88, 0.95, 0.95, 0.98; 0.9)] | [(0.72, 0.89, 0.89, 0.98; 1), (0.81, 0.89, 0.89, 0.94; 0.9)] | [(0.05, 0.2, 0.2, 0.4; 1), (0.13, 0.2, 0.2, 0.3; 0.9)] | |
[(0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.03, 0.09, 0.09, 0.22; 1), (0.06, 0.09, 0.09, 0.16; 0.9)] | [(0.2, 0.4, 0.4, 0.6; 1), (0.3, 0.4, 0.4, 0.5; 0.9)] | |
[(0.44, 0.64, 0.64, 0.79; 1), (0.54, 0.64, 0.64, 0.72; 0.9)] | [(0.02, 0.09, 0.09, 0.24; 1), (0.06, 0.09, 0.09, 0.17; 0.9)] | [(0.15, 0.3, 0.3, 0.5; 1), (0.23, 0.3, 0.3, 0.4; 0.9)] | [(0.6, 0.8, 0.8, 0.95; 1), (0.7, 0.8, 0.8, 0.88; 0.9)] | |
[(0.6, 0.8, 0.8, 0.95; 1), (0.7, 0.8, 0.8, 0.88; 0.9)] | [(0.2, 0.4, 0.4, 0.6; 1), (0.3, 0.4, 0.4, 0.5; 0.9)] | [(0.2, 0.4, 0.4, 0.6; 1), (0.3, 0.4, 0.4, 0.5; 0.9)] | [(0.54, 0.72, 0.72, 0.85; 1), (0.63, 0.72, 0.72, 0.79; 0.9)] | |
[(0.64, 0.84, 0.84, 0.97; 1), (0.74, 0.84, 0.84, 0.91; 0.9)] | [(0.72, 0.89, 0.89, 0.98; 1), (0.81, 0.89, 0.89, 0.94; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] | [(0.16, 0.36, 0.36, 0.56; 1), (0.26, 0.36, 0.36, 0.46; 0.9)] | |
[(0.8, 0.95, 0.95, 1; 1), (0.88, 0.95, 0.95, 0.98; 0.9)] | [(0.6, 0.75, 0.75, 0.85; 1), (0.68, 0.75, 0.75, 0.8; 0.9)] | [(0.05, 0.2, 0.2, 0.4; 1), (0.13, 0.2, 0.2, 0.3; 0.9)] | [(0.5, 0.7, 0.7, 0.9; 1), (0.6, 0.7, 0.7, 0.8; 0.9)] | |
[(0.74, 0.92, 0.92, 1; 1), (0.83, 0.92, 0.92, 0.96; 0.9)] | [0.26, 0.46, 0.46, 0.66; 1], (0.36, 0.46, 0.46, 0.56; 0.9)] | [(0.6, 0.8, 0.8, 0.95; 1), (0.7, 0.8, 0.8, 0.88; 0.9)] | [(0.3, 0.5, 0.5, 0.7; 1), (0.4, 0.5, 0.5, 0.6; 0.9)] |
The group evaluation matrix.
[(0.8, 0.95, 0.95, 1; 1), (0.88, 0.95, 0.95, 0.98; 0.9)] | [(0, 0.05, 0.05, 0.2; 1), (0.03, 0.05, 0.05, 0.13; 0.9)] | [(0.84, 0.97, 0.97, 1; 1), (0.91, 0.97, 0.97, 0.99; 0.9)] | [(0.8, 0.95, 0.95, 1; 1), (0.88, 0.95, 0.95, 0.98; 0.9)] | |
[(0.05, 0.18, 0.18, 0.36; 1), (0.12, 0.18, 0.18, 0.27; 0.9)] | [(0.7, 0.85, 0.85, 0.95; 1), (0.978, 0.85, 0.85, 0.9; 0.9)] | [(0.32, 0.52, 0.52, 0.7; 1), (0.42, 0.52, 0.52, 0.61; 0.9)] | [(0, 0.03, 0.03, 0.16; 1), (0.02, 0.03, 0.03, 0.1; 0.9)] | |
[(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0.6, 0.8, 0.8, 0.95; 1), (0.7, 0.8, 0.8, 0.88; 0.9)] | [(0.02, 0.11, 0.11, 0.28; 1), (0.07, 0.11, 0.11, 0.2; 0.9)] | [(0.02, 0.14, 0.14, 0.34; 1), (0.08, 0.14, 0.14, 0.24; 0.9)] | |
[(0, 0.05, 0.05, 0.2; 1), (0.03, 0.05, 0.05, 0.13; 0.9)] | [(0.08, 0.26, 0.26, 0.46; 1), (0.17, 0.26, 0.26, 0.36; 0.9)] | [(0.24, 0.44, 0.44, 0.64; 1), (0.3, 0.44, 0.44, 0.54; 0.9)] | [(0, 0.05, 0.05, 0.2; 1), (0.03, 0.05, 0.05, 0.13; 0.9)] | |
[(0.09, 0.24, 0.24, 0.44; 1), (0.17, 0.24, 0.24, 0.34; 0.9)] | [(0.86, 0.98, 0.98, 1; 1), (0.92, 0.98, 0.98, 0.99; 0.9)] | [(0.1, 0.3, 0.3, 0.5; 1), (0.2, 0.3, 0.3, 0.4; 0.9)] | [(0, 0.05, 0.05, 0.2; 1), (0.03, 0.05, 0.05, 0.13; 0.9)] | |
[(0.16, 0.36, 0.36, 0.56; 1), (0.26, 0.36, 0.36, 0.46; 0.9)] | [(0.66, 0.86, 0.86, 0.98; 1), (0.76, 0.86, 0.86, 0.92; 0.9)] | [(0, 0.05, 0.05, 0.2; 0.9), (0.03, 0.05, 0.05, 0.13; 0.9)] | [(0.2, 0.4, 0.4, 0.6; 1), (0.3, 0.4, 0.4, 0.58; 0.9)] | |
[(0.05, 0.15, 0.15, 0.3; 0.9), (0.1, 0.15, 0.15, 0.23; 0.9)] | [((0.9, 1, 1, 1; 1), (0.95, 1, 1, 1; 0.9)] | [(0.05, 0.2, 0.2, 0.4; 1), (0.13, 0.2, 0.2, 0.3; 0.9)] | [(0.24, 0.4, 0.4, 0.58; 1), (0.32, 0.4, 0.4, 0.49; 0.9)] |
The result of distance
0.845 | 0.969 | 0.601 | 0.809 | 0.844 | 0.917 | 0.786 | |
0.008 | 0.352 | 0.000 | 0.419 | 0.874 | 0.728 | 0.360 | |
0.051 | 0.072 | 0.252 | 0.419 | 0.519 | 0.200 | 0.689 | |
0.725 | 0.452 | 0.761 | 0.716 | 0.379 | 0.719 | 0.400 | |
0.008 | 0.200 | 0.272 | 0.000 | 0.240 | 0.379 | 0.000 | |
0.925 | 0.860 | 0.761 | 0.272 | 0.929 | 0.862 | 0.817 | |
0.000 | 0.567 | 0.032 | 0.459 | 0.319 | 0.000 | 0.080 | |
0.008 | 0.000 | 0.080 | 0.000 | 0.000 | 0.419 | 0.280 |
The result of distance
0.080 | 0.000 | 0.160 | 0.000 | 0.085 | 0.000 | 0.031 | |
0.917 | 0.617 | 0.761 | 0.389 | 0.055 | 0.189 | 0.457 | |
0.874 | 0.897 | 0.509 | 0.389 | 0.410 | 0.717 | 0.128 | |
0.200 | 0.517 | 0.000 | 0.092 | 0.550 | 0.198 | 0.417 | |
0.917 | 0.769 | 0.489 | 0.809 | 0.689 | 0.538 | 0.817 | |
0.000 | 0.109 | 0.000 | 0.537 | 0.000 | 0.055 | 0.000 | |
0.925 | 0.402 | 0.729 | 0.349 | 0.610 | 0.917 | 0.737 | |
0.917 | 0.969 | 0.680 | 0.809 | 0.929 | 0.498 | 0.537 |
The value function under positive reference point.
0.862 | 0.000 | −0.449 | 0.000 | −0.257 | 0.000 | −0.107 | |
0.014 | −1.472 | −1.769 | −0.980 | −0.176 | −0.520 | −1.130 | |
0.073 | −2.045 | −1.241 | −0.980 | −1.026 | −1.680 | −0.369 | |
0.754 | −1.260 | 0.000 | −0.276 | −1.329 | −0.541 | −1.043 | |
0.014 | −1.786 | −1.199 | −1.866 | −1.621 | −1.304 | −1.884 | |
0.934 | −0.319 | 0.000 | −1.301 | 0.000 | −0.176 | 0.000 | |
0.000 | −1.008 | −1.703 | −0.891 | −1.455 | −2.085 | −1.720 | |
0.014 | −2.188 | −1.603 | −1.866 | −2.109 | −1.281 | −1.303 |
The value function under negative reference point.
−0.245 | 0.973 | 0639 | 0.829 | 0.861 | 0.927 | 0.809 | |
−2.085 | 0.399 | 0.000 | 0.465 | 0.888 | 0.756 | 0.407 | |
−1.998 | 0.098 | 0.297 | 0.465 | 0.562 | 0.243 | 0.720 | |
−0.546 | 0.467 | 0.786 | 0.746 | 0.426 | 0.748 | 0.446 | |
−2.085 | 0.243 | 0.318 | 0.000 | 0.285 | 0.426 | 0.000 | |
0.000 | 0.876 | 0.786 | 0.318 | 0.937 | 0.877 | 0.837 | |
−2.101 | 0.607 | 0.049 | 0.504 | 0.366 | 0.000 | 0.109 | |
−2.085 | 0.000 | 0.109 | 0.000 | 0.000 | 0.465 | 0.326 |
The prospect value of each robot.
V+ ( | 0.131 | −0.938 | −1.099 | −0.496 | −1.182 | −0.107 | −1.244 | −1.307 |
V− ( | 0.764 | 0.241 | 0.213 | 0.472 | 0.058 | 0.585 | 0.064 | 0.019 |
The final priority of each robot.
Priority | 0.895 | −0.696 | −0.885 | −0.025 | −1.124 | 0.478 | −1.181 | −1.288 |
Rank | 1 | 4 | 5 | 3 | 6 | 2 | 7 | 8 |
The radar plot showing the ranking result of robots.
As mentioned in Section
The result of sensitivity analysis.
Rank | Rank | Rank | Rank | Rank | Rank | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
0.989 | 1 | 0.951 | 1 | 0.914 | 1 | 0.895 | 1 | 0.876 | 1 | 0.838 | 1 | |
−0.167 | 4 | −0.379 | 4 | −0.591 | 4 | −0.696 | 4 | −0.892 | 4 | −1.014 | 4 | |
−0.255 | 5 | −0.507 | 5 | −0.759 | 5 | −0.885 | 5 | −1.011 | 5 | −1.263 | 5 | |
0.398 | 3 | 0.229 | 3 | 0.060 | 3 | −0.025 | 3 | −0.109 | 3 | −0.278 | 3 | |
−0.459 | 6 | −0.725 | 6 | −0.991 | 6 | −1.124 | 6 | −1.258 | 6 | −1.524 | 6 | |
0.718 | 2 | 0.622 | 2 | 0.526 | 2 | 0.478 | 2 | 0.430 | 2 | 0.334 | 2 | |
−0.483 | 7 | −0.762 | 7 | −1.041 | 7 | −1.181 | 7 | −1.320 | 7 | −1.599 | 7 | |
−0.553 | 8 | −0.847 | 8 | −1.141 | 8 | −1.288 | 8 | −1.435 | 8 | −1.729 | 8 |
The priorities of robots with different values of parameter
As shown in Table
In addition, Table
In order to further illustrate the rationality and applicability of the proposed robot evaluation and selection approach, a comparison study is conducted with a number of previous alternative robot evaluation and selection methods, which includes hesitant fuzzy linguistic MULTIMOORA (Method 1) [
The result of comparison analysis.
Method 1 | Method 2 | Method 3 | Method 4 | The proposed method | |
---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | |
4 | 5 | 5 | 5 | 4 | |
5 | 4 | 4 | 4 | 5 | |
3 | 3 | 3 | 3 | 3 | |
7 | 7 | 8 | 8 | 6 | |
2 | 2 | 2 | 2 | 2 | |
6 | 6 | 6 | 6 | 7 | |
8 | 8 | 7 | 7 | 8 |
The result of comparative analysis.
Kendall’s coefficient of concordance in Table
The priority ranking orders of alternative robots
The priority ranking orders of alternative robots
The priority ranking orders of the eight alternative robots obtained by Method 3 and Method 4 are consistent. However, the worst alternative robot obtained by the two methods is
According to the discussion mentioned above, the proposed robot evaluation and selection approach can obtain a more reasonable and reliable priority ranking result than the existing approaches. The advantages of the proposed approach can be summarized as follows: In the extended prospect theory, the positive and negative reference points are used as a substitute for single reference point. The using two reference points provides a more flexible and effective way to represent the complex risk preference of decision-maker in the robot evaluation and selection process. The distance measure of IT2TrFNs is incorporated into the developed Choquet integral for determining the fuzzy measure of each criterion. This cannot only provide a more objective means for modeling the interactive relationships among criteria but also provide a more reliable and reasonable result of robot evaluation and selection. The proposed hybrid framework is constructed based on the prospect theory and Choquet integral, which considers the decision-maker’s risk preference and interactive criteria in the robot evaluation and selection process.
In this paper, a developed hybrid MCDA framework is proposed for addressing the robot evaluation and selection problem by considering the decision-maker’s risk preference and interactive criteria under high uncertain environment. In the hybrid framework, the interval type-2 fuzzy set is adopted to express the uncertain evaluation information from various decision-makers. In addition, the distance measure of IT2TrFNs is incorporated into the Choquet integral to depict the interactive relationships among criteria. Furthermore, motived by the TOPSIS method, the extended prospect theory with two reference points is proposed to determine the optimal alternative robot by considering the risk preference of decision-maker and interactive relationships among criteria. Finally, the proposed approach is applied to evaluate and select an optimal robot in the auto industry. The illustrative example demonstrates the proposed hybrid framework in detail. Moreover, the sensitivity and comparison studies are conducted in the illustrative example. The results show the robustness, effectiveness, and advantages of the proposed framework.
We also point out some limitations of the developed robot evaluation and selection framework and directions of the future research. First, in the course of evaluation information aggregation, the importance weight of each decision-maker is determined by adopting a subjective weighting method, which may be highly dependent on the expert’s personal judgments. For future research, the objective weight determination method is suggested to be used for obtaining the objective importance weight of each decision-maker. Second, only seven criteria are utilized to evaluate and select the optimal alternative robot. In the future, more related criteria should be considered to make more thorough and reasonable evaluation criteria. Third, the influence of decision-maker’s capability on robot evaluation and selection is not considered in this research. Thus, the consensus-based decision-making approaches can be extended to this problem for overcoming this gap.
The research data are all provided in the manuscript.
The authors declare that they have no conflicts of interest.
Guobao Zhang wrote the original manuscript. Shuping Cheng contributed reviewed and edited the manuscript. Guobao Zhang contributed to project administration and is responsible for funding acquisition.
This work was supported by the Youth Project of Anhui Province Philosophy and Social Sciences Planning (no. AHSKQ2016D25).