Interval Generalized Ordered Weighted Utility Multiple Averaging Operators and Their Applications to Group Decision-Making

Multiple attributes decision-making (MADM) refers to considering the problem of evaluating or selecting alternatives that are associated with incommensurate and conflicting criteria. However, the increasing complexity of the socioeconomic environment makes it less possible for a single decision-maker (DM) to consider all relevant aspects of a problem as many decision-making processes take place in group settings. As a result, multiple attributes group decision-making (MAGDM) has been receiving more and more attentions from researchers and has been extensively applied to our daily life [1,2,3,4,5,6,7]. Nevertheless, due to the complexities of objects and the vagueness of human mind, it is more appropriate for DMs to use interval/uncertain information than other descriptors to express their assessments in the actual process of MAGDM.

The objective of interval MAGDM (IMAGDM) is to find the optimal solution(s) from a set of feasible alternatives by means of interval information provided by DMs. To realize this objective, the construction of aggregating operators and the determination of aggregation operators’ weights are the two key issues.

A variety of operators have been developed in the past few decades, among them, ordered weighted averaging (OWA) operators introduced by Yager [8] are the most widely used ones. Motivated by these, an extension of the OWA operator is the uncertain OWA (UOWA) operator [9], which aggregates the input arguments taking the form of intervals rather than exact numbers. Some new developments about it have mostly appeared from 2008 to 2013. For example, Xu [10] proposed dependent uncertain ordered weighted averaging (DUOWA) operators and dependent uncertain ordered weighted geometric (DUOWG) operators. Xu and Yager [11] introduced the uncertain power ordered weighted geometric (UPOWG) operator on the basis of the power average operator and the geometric mean. Merigó and Casanovas [12] proposed uncertain induced quasi-arithmetic OWA (Quasi-UIOWA) operator which uses the main characteristics of the induced OWA, the quasi-arithmetic OWA and the UOWA operator. Merigó and Casanovas [13] presented the uncertain heavy OWA (UHOWA) operator and uncertain induced heavy OWA (UIHOWA) operator. Xu and Cai [14] developed the uncertain power ordered weighted average (UPOWA) operator on the basis of the power averaging (PA) operator and the UOWA operator. To extend the generalized ordered weighted averaging operator (GOWA), Zhou et al. [15] developed the uncertain generalized ordered weighted averaging (UGOWA) operator. Zeng et al. [16] present the uncertain probabilistic ordered weighted averaging distance (UPOWAD) operator which uses distance measures in a unified framework between the probability and the OWA operator that considers the degree of importance of each concept in the aggregation. However, the above uncertain aggregation operators only focus on using the mean to eliminate the difference of variable values but fail to consider DMs’ risk attitude in the aggregation process. In actual decision making, different DMs may show different risk attitudes, and the risk attitudes of the same DM may also vary at different times. Obviously, ignoring the risk attitude of DMs will lead to a wrong decision-making.

Another important issue of applying the aggregation operator for IMAGDM is how to determine the associated weights. Many researchers have focused on this issue and developed some useful approaches for determining weights [9,10,15,17,18,19]. For instance, Xu and Da [9] established a linear objective-programming model to generate the UOWA weights by utilizing the given partial weight information, the input interval arguments, and the associated aggregated values. Ahn [18] presented several weighting functions having constant values of orness irrespective of the number of objectives aggregated. Considering the impact of optimism degree of DMs on the decision-making results and by applying the absolute deviation measure, Xu [10] derived the weights of DUOWA operator which only depend on the aggregated interval arguments. To determine UGOWA weights, Zhou et al. [15] used the relative deviation measure to construct a nonlinear model. However, we find that they did not provide a uniform weighting vector based on the weighting models in [10,15], which can affect the accuracy of the decision results and lead to unreasonable results.

Therefore, the aim of this paper is to develop some interval utility aggregation operators to deal with interval preference information, which incorporates the risk attitude of DMs in the aggregation process. Under the general framework of utility function, a new operator called the interval generalized ordered weighted utility multiple averaging (IGOWUMA) operator by applying the penalty theory is derived firstly, and then by studying its properties we find that it is commutative, idempotent, monotone increasing and bounded. Furthermore, we focus on a concrete utility function-hyperbolic absolute risk aversion (HARA), which is rather rich, e.g., by suitable adjustments of the parameters one can respectively obtain power utility, exponential utility and logarithm utility. Under the HARA utility, we propose another new operator named as the interval generalized ordered weighted hyperbolic absolute risk aversion utility multiple averaging-HARA (IGOWUMA-HARA) operator, and investigate its families which include a wide range of aggregation operators such as interval generalized ordered weighted multiple averaging (IGOWMA) operator, interval ordered weighted multiple averaging (IOWMA) operator and interval ordered weighted geometric averaging (IOWGA) operator. The merits of the IGOWUMA-HARA operator are that it not only reflects the DMs’ risk attitude towards the aggregation information, but also provides a very general formula including a wide range of aggregation operators. To determine the IGOWUMA-HARA operator’ weights, we propose a preemptive nonlinear objective programming model, which can provide a uniform weighting vector to guarantee the uniform standard comparison between the alternatives and reduce the influence of unfair arguments on the aggregated results. After that, based on the IGOWUMA-HARA operator and weight determination model, a new approach for IMAGDM is developed. This approach is also effectively applicable to different group decision-making problems such as engineering management and financial management, etc. Finally, an example of the investment selection is provided to examine our approach.

The rest of the paper is organized as follows. Section 2 reviews some basic concepts. Section 3 presents the IGOWUMA operator and analyzes its properties as well as provides an IGOWUMA-HARA operator and identifies its families. We then present a preemptive nonlinear objective programming model for determining IGOWUMA-HARA weights in Section 4. Section 5 develops an approach for IMAGDM under the IGOWUMA-HARA operator. An illustrative example is provided in Section 6 and the conclusions are drawn in Section 7.

This section briefly reviews definitions and operations of interval values, UGOWA operator and HARA utility function.

Following [26,27,28,29], in this section, we will introduce a new aggregation operator by minimizing a new penalty function and obtain the IGOWUMA operator for the general utility function. Then, under the HARA utility function, we propose another new operator called the IGOWUMA-HARA operator.

In order to determine the weights of the IGOWUMA-HARA operator, we propose the concept of similarity degree of interval numbers and analyze its properties. We further provide a new optimization model, which can obtain an optimal weight vector for the IGOWUMA-HARA operator.

In the literature [10], Xu presented a weighting model of DUOWA operator by applying the absolute deviation measure and the optimism degree. Actually, as far as measuring deviation, the relative deviation measure is more effective than the absolute deviation measure. Thus, Zhou et al. [15] developed a new model to determine the weights of UGOWA operator via the relative deviation measure. However, the weighted models do not provide a uniform weighting vector in the literature [10,15], which will affect the accuracy of the decision results. Therefore, we will present a new preemptive nonlinear objective programming model to determining the IGOWUMA-HARA operator weights, which can provide a uniform weighting vector to guarantee the uniform standard comparison between the alternatives and reduce the influence of unfair arguments on the aggregated results.

Let $X=\{x_1,x_2,\cdots ,x_m\}$ be a finite set of alternatives, $C=\{c_1,c_2,\cdots ,c_n\}$ a finite set of attributes, $\tilde{B}={({\tilde{b}}_{ij})}_{m\times n}$ is the interval decision matrix, where ${\tilde{b}}_{ij}=\left[b_{ij}^L,b_{ij}^U\right]\in \mathsf{\Omega }$ is a preference value, which takes the form of interval argument, given by the decision maker for the alternative $x_i\in X$ with regard to the attribute $c_j\in C$. Assume that the partial weight information of attributes is:

Standardize matrix $\tilde{B}={({\tilde{b}}_{ij})}_{m\times n}$ into matrix $\tilde{R}={({\tilde{r}}_{ij})}_{m\times n}={\left(\left[r_{ij}^L,r_{ij}^U\right]\right)}_{m\times n}$, then the integrated attribute value, which is obtained by using the aggregation operator IGOWUMA-HARA to gather the attribute preference information value of each alternative, is still the interval number ${\tilde{z}}_i(W)=[{z_i}^L(W),{z_i}^U(W)](i=1,2,\cdots ,m)$, where:

In order to calculate a uniform weighting vector to guarantee the uniform standard comparison between the alternatives and measure their fair competition under the condition of valid comparison between various alternatives. Following [31], we can construct the nonlinear programming model as follows: where $H$ is partial weight information.

Model (19) can not only determine the uniform weight vector, making the alternative is comparable, but also can ensure that the alternatives can compete in a fair environment. As can be seen from the model (19), the expected value of the objective function $z_1(W)$ is ${\displaystyle \sum _{i=1}^m\left\{\frac{1-\gamma }{\beta }\left[{\left({\displaystyle \sum _{j=1}^n{\rho }_j^L{(\frac{\beta }{1-\gamma }{y}_{ij}^L+\eta )}^{\lambda \gamma }}/\left({\displaystyle \sum _{j=1}^n{\rho }_j^L/{(\frac{\beta }{1-\gamma }{y}_{ij}^L+\eta )}^{\lambda \gamma }}\right)\right)}^{1/2\lambda \gamma }-\eta \right]\right\}}$ while expected value of the objective function $z_2(W)$ is ${\displaystyle \sum _{i=1}^m\left\{\frac{1-\gamma }{\beta }\left[{\left({\displaystyle \sum _{j=1}^n{\rho }_j^U{(\frac{\beta }{1-\gamma }{y}_{ij}^U+\eta )}^{\lambda \gamma }}/\left({\displaystyle \sum _{j=1}^n{\rho }_j^U/{(\frac{\beta }{1-\gamma }{y}_{ij}^U+\eta )}^{\lambda \gamma }}\right)\right)}^{1/2\lambda \gamma }-\eta \right]\right\}}$.

In order to solve the multi-objective programming model, the model (19) can be transformed into the following model: where $P_1,P_2$ are the priority factor which indicates the relative importance of each target; $d^{-},d^{+}$ are the lower and upper deviation variables of the objective function ${\displaystyle \sum _{i=1}^m{z}_i^L(W)}$ which is lower or higher than the expected value ${\displaystyle \sum _{i=1}^m\left\{\frac{1-\gamma }{\beta }\left[{\left({\displaystyle \sum _{j=1}^n{\rho }_j^L{(\frac{\beta }{1-\gamma }{y}_{ij}^L+\eta )}^{\lambda \gamma }}/\left({\displaystyle \sum _{j=1}^n{\rho }_j^L/{(\frac{\beta }{1-\gamma }{y}_{ij}^L+\eta )}^{\lambda \gamma }}\right)\right)}^{1/2\lambda \gamma }-\eta \right]\right\}}$, respectively; $e^{-},e^{+}$ are the lower and upper deviation variables of the objective function ${\displaystyle \sum _{i=1}^m{z}_i^U(W)}$ which is lower or higher than the expected value ${\displaystyle \sum _{i=1}^m\left\{\frac{1-\gamma }{\beta }\left[{\left({\displaystyle \sum _{j=1}^n{\rho }_j^U{(\frac{\beta }{1-\gamma }{y}_{ij}^U+\eta )}^{\lambda \gamma }}/\left({\displaystyle \sum _{j=1}^n{\rho }_j^U/{(\frac{\beta }{1-\gamma }{y}_{ij}^U+\eta )}^{\lambda \gamma }}\right)\right)}^{1/2\lambda \gamma }-\eta \right]\right\}}$, respectively.

This section develops a new approach for MAGDM based on the IGOWUMA-HARA operator.

For a MAGDM problem, let $X=\{x_1,x_2,\cdots ,x_m\}$ be a finite set of alternatives, $C=\{c_1,c_2,\cdots ,c_n\}$ be a finite set of attributes and $V^{(k)}={\left(v_1^{(k)},v_2^{(k)},\cdots ,v_n^{(k)}\right)}^T$ be the vector of attributes such that $v_j^{(k)}\ge 0$ and ${\displaystyle \sum _{j=1}^n{v}_j^{(k)}}=1$. $D=\{d_1,d_2,\cdots ,d_l\}$ is a finite set of DMs and $W={\left(w_1,w_2,\cdots ,w_l\right)}^T$ is the vector of DMs satisfying $w_k\ge 0$ and ${\displaystyle \sum _{k=1}^l{w}_k}=1.$ In addition, assume that ${\tilde{B}}^{(k)}={\left({\tilde{b}}_{ij}^{(k)}\right)}_{m\times n}$ is the interval decision matrix, where ${\tilde{b}}_{ij}^{(k)}=\left[b_{ij}^{(k)L},b_{ij}^{(k)U}\right]\in \mathsf{\Omega }$ is a preference value, which takes the form of interval argument, given by the decision maker $d_k\in D$ for the alternative $x_i\in X$ with regard to the criterion $c_j\in C$.

Since different attributes have different measurement scales in MAGDM problem, it is necessary for the standardization of attributes to avoid the variance among different attributes. In this paper, we consider two attributes, i.e., profit type and cost type. Let $I_1$ be a set of benefit attributes and $I_2$ be a set of cost attribute. The decision matrix ${\tilde{B}}^{(k)}$ can be transformed into a corresponding decision matrix ${\tilde{R}}^{(k)}$ via the following formulas [31]:

Based on the above explanation, we develop an approach to MAGDM problem. The concrete steps are shown as follows.

Step 1. Standardize the decision matrixes.

Based on Equations (21) and (22), the matrixes ${\tilde{B}}^{(k)}={\left({\tilde{b}}_{ij}^{(k)}\right)}_{m\times n}(k=1,2,\cdots ,l)$ can be transformed into standardization matrixes ${\tilde{R}}^{(k)}={\left({\tilde{r}}_{ij}^{(k)}\right)}_{m\times n}(k=1,2,\cdots ,l)$.

Step 2. Calculate the weight vector of attributes.

Based on model (19), the IGOWUMA-HARA weight vector can be derived: $V^{(k)}={\left(v_1^{(k)},v_2^{(k)},\cdots ,v_n^{(k)}\right)}^T$ is the vector of attributes such that $v_j^{(k)}\ge 0$ and ${\displaystyle \sum _{j=1}^n{v}_j^{(k)}}=1$.

Step 3. Aggregate the decision matrixes into a collective decision matrix.

According to IGOWUMA-HARA operator ${\tilde{\tilde{r}}}_i^{(k)}=IGOWUMA-HARA({\tilde{r}}_{i1}^{(k)},{\tilde{r}}_{i2}^{(k)},\cdots ,{\tilde{r}}_{in}^{(k)}),i=1,2,\cdots ,m,k=1,2,\cdots ,l,$ we can aggregate all the decision matrices ${\tilde{R}}^{(k)}(k=1,2,\cdots ,l)$ into a collective decision matrix $\tilde{\tilde{R}}={({\tilde{\tilde{r}}}_{ij})}_{m\times l}$.

Step 4. Calculate the weight vector of the DMs.

Based on model (19), the weight vector of decision makers can be calculated $W={\left(w_1,w_2,\cdots ,w_l\right)}^T$, where $w_k\ge 0(k=1,2,\cdots ,l)$ and ${\displaystyle \sum _{k=1}^l{w}_k}=1$.

Step 5. Aggregate the collective overall the interval preference value.

According to IGOWUMA-HARA operator ${\tilde{t}}_i=IGOWUMA-HARA({\tilde{\tilde{r}}}_{i1},{\tilde{\tilde{r}}}_{i2},\cdots ,{\tilde{\tilde{r}}}_{il}),i=1,2,\cdots ,m,$ we can obtain the collective overall preference value ${\tilde{t}}_i$ of the alternative $x_i$.

Step 6. Compare each interval argument ${\tilde{t}}_i$ with all ${\tilde{t}}_i$ by Equation (1) and get the complementary matrix $P={(p_{ij})}_{m\times m}.$

Step 7. Synthesize all elements of matrix $P={(p_{ij})}_{m\times m}$ by Equation (3) and get $p_i(i=1,2,\cdots ,m).$

Step 8. Rank the collective overall preference values ${\tilde{t}}_j$ in descending order in accordance with the values of $p_i\left(i=1,2,\cdots ,m\right).$

Step 9. Select the best alternative.

Rank all the alternatives $x_i(i=1,2,\cdots ,m)$ in descending order and consequently select the best one in the light of the collective overall preference value ${\tilde{t}}_i(i=1,2,\cdots ,m)$.

In this section, a practical group decision making problem involving the prioritization of a set of information technology improvement projects adapted from Xu [10] is used to illustrate the proposed uncertain aggregation operator.

The steering committee of information management department of Midwest American’s Manufacturing Corp must consider expanding and implementing a set of six projects preferentially. The projects, $x_1$ quality management information, $x_2$ inventory control, $x_3$ customer order tracking, $x_4$ materials purchasing management, $x_5$ fleet management and $x_6$ design change management, are proposed by different regional managers, aiming at improving information and technology. What the committee is concerned about is the projects’ prioritized potential contribution, accordingly, from the highest level to the lowest. At a strategic level, company acquires competitive advantages over its rivals in the industry. In evaluating the potential contribution of each project, three factors are taken into consideration, namely, $c_1$-productivity, $c_2$-differentiation, and $c_3$-management.

Suppose that the decision experts of the investment company are four persons $d_1$, $d_2$, $d_3$ and $d_4,$ they offer their own opinions with scores ranging from 0 to 100, over the projects $x_i(i=1,2,\cdots ,6)$ with respect to the factors $c_j(j=1,2,3)$, and the results are shown as in Table A1, Table A2, Table A3 and Table A4 (See Appendix B).

In addition, for simplicity, without loss of generality, we assume that the parameters of the IGOWUMA-HARA operator are respectively: $\lambda =1$, $\beta =3$, $\eta =1$ and $\gamma =0.1$.