#
Reference-Dependent Aggregation in Multi-AttributeGroup Decision-Making

^{*}

## Abstract

**:**

## 1. Introduction

**Methods developed for aggregating information can be classified into three types**. The first is the weighted-average method, which aggregates the information by using different importance degrees of the arguments [3,4]. The second is the probabilistic aggregation method, which unifies the ordered weighted averaging (OWA) operator and the corresponding probabilities by incorporating the importance degree of each case in the aggregation process [5,6]. The third is the deviation aggregation method, which minimizes the deviation between the aggregation result and evaluation information characterized by distance metrics or penalty functions [7,8].

**Reference-dependent utility function to characterize psychological factors.**There is a consensus that the psychological factors of the DM, such as reference wealth [9], cognitive elements [10] and the behavior towards risk [11], play important roles in decision analysis. Nevertheless, the aforementioned aggregation methods fail to capture the psychological character of DMs in the aggregation process. In this paper, we attempt to partially fill this gap by modeling psychological factors via reference-dependent utility functions (RUs). The most famous RU is the value function of Prospect theory [12,13], which involves a basic utility function, loss aversion coefficient and a reference outcome. This is the fundamental framework of RUs.

**Weight models of attributes and decision makers.**Another crucial step in the application of aggregation operators to MAGDM is to determine the associated weights for both attributes and DMs (see, Figure 1). Relevant methods include the minimum variance method [19], minimum dispersion method [20], minimum chi-square method [21], minimum disparity method [22], and maximum Bayesian entropy method [23]. An unresolved issue in the aforementioned methods is how to factor the influence of input arguments in the process of determining weights. In practice, an attribute with similar attribute values across most alternatives is deemed less important, so it should be assigned a smaller weight; on the other hand, an attribute with values fluctuating across alternatives is considered more important, it then should be assigned a larger weight.

**Our contributions.**We summarize our contributions in three directions.

- (1)
- To investigate the impact of DMs psychological factors on the decision-making result, we propose for the first time two new operators based on RUs: the S-shaped and non-S-shaped operators. The DM can choose different RU operators to get the result according to his/her attitude toward the relative losses. To be specific, if the attitude of the DM is risk-seeking for relative losses, he/she can use the S-shaped operators (see Equation (11)) to select the optimal alternative. If the attitude of the DM is risk-averse for relative losses, he/she can apply the non-S-shaped operators (see, Equation (16)) in the decision-making process. If the attitude of a DM is risk-neutral, he/she can make decisions via the generalized ordered weighted multiple averaging (GOWMA) operator (see, Equation (18)) which is degenerated by the non-S-shaped operator. The main advantage of the RU operators is that they not only reflect the psychological character of the DM while the aforementioned aggregation methods fail to capture in the aggregation process, but also generate a family of aggregation operators by taking different parameters. Specifically, the RU operators can degenerate to the existing aggregation operators (see Table A1, Table A2 and Table A3 in Appendix A.3), which can be seen as the particular case of the RU operators.
- (2)
- To determine the associated weights for the multiple attributes and DMs, we propose an attribute- deviation weight model and a DMs-deviation weight model (see, models (19) and (20)). Going beyond the framework of existing weight models which ignored the dependence on the attribute variation (deviation), our new weight models consider the variations impacts of the attribute values on the determination of the weight in aggregation process. In addition, the attributes weights and the DMs weights are calculated by using attribute-deviation and DM-deviation models respectively, while the most research uses the same model to determine the associated weights for both attributes and DMs, sometimes leading to biased decision results.
- (3)
- We develop a new approach for MAGDM based on the RU operators and the weight models. In addition, the numerical examples are given to illustrate the application of the approach. Two novel findings emerge from the numerical analysis in Section 6. First, the optimal alternative will change to a relatively prudent alternative with the absolute risk aversion coefficient increasing. Second, the optimal alternative changes to a relatively risky one with the reference point (or, loss aversion coefficient) increasing.

## 2. Aggregation Operators for Risk-Seeking DMs Regarding Relative Losses

#### 2.1. General Framework

- (1)
- v is strictly increasing;
- (2)
- v is convex for $x<b$ and concave for $x>b$;
- (3)
- v is asymmetry for $x>b$: $v(x-b)<-v(b-x)$;
- (4)
- ${v}^{\prime}(x-b)<{v}^{\prime}(b-x)$ for all $x>b$, and $v\left(b\right)=0$.

**Definition**

**1.**

**Proposition**

**1**(Properties of SOMR)

**.**

- (1)
- (Monotonicity) For two vectors $\mathbf{x}$ and $\overline{\mathbf{x}}$ with ${x}_{i}\ge {\overline{x}}_{i}$ and the same reference points, then $\mathrm{SOMR}\left(\mathbf{x}\right)\ge \mathrm{SOMR}\left(\overline{\mathbf{x}}\right)$.
- (2)
- (Boundedness) If ${b}_{1}\le {y}_{1}=\underset{i}{max}\left\{{x}_{i}\right\}$ and ${b}_{n}>{y}_{n}=\underset{i}{min}\left\{{x}_{i}\right\}$, then ${v}_{1}^{-1}\left({v}_{2}\left({y}_{n}\right)\right)\le \mathrm{SOMR}\left(\mathbf{x}\right)\le {y}_{1}$.
- (3)
- (Commutativity) If $\widehat{\mathbf{x}}$ is a permutation of $\mathbf{x}$, then $\mathrm{SOMR}\left(\mathbf{x}\right)=\mathrm{SOMR}\left(\widehat{\mathbf{x}}\right)$.
- (4)
- (Idempotency) If ${x}_{i}=x\ge {x}_{0}$ for all $1\le i\le n$, then $\mathrm{SOMR}\left(\mathbf{x}\right)=x$.

#### 2.2. Prospect Reference-Dependent Aggregation Operator

**Definition**

**2.**

#### 2.3. S-Shaped Hyperbolic Absolute Risk Aversion (HARA) Reference-Dependent Aggregation Operator

**Definition**

**3.**

## 3. Aggregation Operators for Risk-Averse DMs Regarding Relative Losses

#### 3.1. General Framework

**Definition**

**4.**

#### 3.2. Non-S-Shaped HARA Reference-Dependent Aggregation Operator

**Definition**

**5.**

**Remark**

**1**(special cases of NHOMR operator)

**.**

## 4. New Weight Models for Reference-Dependent Aggregation Operators

#### 4.1. Weight Model for Attributes

- A weak ranking: $\{{w}_{i}^{\left(k\right)}\ge {w}_{j}^{\left(k\right)}\}$;
- A strict ranking: $\{{w}_{i}^{\left(k\right)}-{w}_{j}^{\left(k\right)}\ge {\alpha}_{i}\}$;
- A ranking with multiples: $\{{w}_{i}^{\left(k\right)}\ge {\alpha}_{i}{w}_{j}^{\left(k\right)}\}$;
- An interval form: $\{{\alpha}_{i}\le {w}_{i}^{\left(k\right)}\le {\alpha}_{i}+{\epsilon}_{i}\}$;
- A ranking of differences: $\{{w}_{i}^{\left(k\right)}-{w}_{j}^{\left(k\right)}\ge {w}_{h}^{\left(k\right)}-{w}_{l}^{\left(k\right)}\}$, for $j\ne h\ne l$, ${\alpha}_{i}$, ${\epsilon}_{i}>$ 0.

**Remark**

**2**(Generality of the attribute-weight model (19))

**.**

#### 4.2. Weight Model for Decision Makers

## 5. A New Reference-Dependent Aggregation Approach for MAGDM

**Input data of our new MAGDM algorithm.**Let $\mathcal{A}=\left\{\left.{A}_{1},{A}_{2},\cdots ,{A}_{m}\right\}\right.$ be the set of m alternatives, $\mathcal{G}=\left\{\left.{G}_{1},{G}_{2},\cdots ,{G}_{n}\right\}\right.$ be the set of n attributes, and $\mathcal{D}=\left\{\left.{D}_{1},{D}_{2},\cdots ,{D}_{l}\right\}\right.$ be the set of l DMs. Assume that ${\mathbf{w}}^{\left(DM\right)}=({w}_{1}^{\left(DM\right)},{w}_{2}^{\left(DM\right)},\cdots ,{w}_{l}^{\left(DM\right)})$ is the weight vector for the DMs and ${\mathbf{w}}^{\left(k\right)}=({w}_{1}^{\left(k\right)},{w}_{2}^{\left(k\right)},\cdots ,{w}_{n}^{\left(k\right)})$ is the attribute weight vector for the kth DM such that ${w}_{j}^{\left(k\right)}\ge 0$, $\sum _{j=1}^{n}}{w}_{j}^{\left(k\right)}=1$, ${w}_{k}^{\left(DM\right)}\ge 0$ and $\sum _{k=1}^{l}}{w}_{k}^{\left(DM\right)}=1$. Opinions of DM ${D}_{k}$, $1\le k\le l$, are characterized by the decision matrix ${\mathbf{S}}^{\left(k\right)}={({s}_{ij}^{\left(k\right)})}_{m\times n}$ and reference point vector ${\mathbf{B}}^{\left(k\right)}=({b}_{1}^{\left(k\right)},{b}_{2}^{\left(k\right)},\cdots ,{b}_{n}^{\left(k\right)})$, where ${s}_{ij}^{\left(k\right)}$ is the input argument of ${D}_{k}\in D$ for alternative ${A}_{i}\in \mathcal{A}$ and attribute ${G}_{j}\in \mathcal{G}$, and ${b}_{j}^{\left(k\right)}$ is the reference point of ${D}_{k}\in D$ for attribute ${G}_{j}\in \mathcal{G}$. We summarize all input data below,

- Step 1
- Transform the decision matrixes ${\mathbf{S}}^{\left(k\right)}$ to the corresponding normalized version ${\mathbf{R}}^{\left(k\right)}={({r}_{ij}^{\left(k\right)})}_{m\times n}$ [8]:$$\begin{array}{c}\hfill {r}_{ij}^{\left(k\right)}={s}_{ij}^{\left(k\right)}/\phantom{\rule{0.0pt}{0ex}}\underset{i}{max}{s}_{ij}^{\left(k\right)}\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}j\in {I}_{1}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{r}_{ij}^{\left(k\right)}=\underset{i}{min}{s}_{ij}^{\left(k\right)}/\phantom{\rule{0.0pt}{0ex}}{s}_{ij}^{\left(k\right)}\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}j\in {I}_{2}\end{array}$$
- Step 2
- Calculate the attribute weight vector of the $k\mathrm{th}$ decision matrix ${\mathbf{w}}^{\left(k\right)}$ by solving the optimization problem in model (19) for $k=1,2,\cdots ,l$.
- Step 3
- Aggregate all individual decision matrixes ${\mathbf{R}}^{\left(k\right)}$ to obtain a collective decision matrix $\mathbf{R}={\left({r}_{ik}\right)}_{m\times l}$ by using the attribute weight vector and the aggregation operator, and following Equations (11) and (16) for cases of risk-seeking and risk-averse attitude.
- Step 4
- Calculate the DMs weights ${\mathbf{w}}^{\left(DM\right)}$ by solving the optimization problem in model (20).
- Step 5
- Aggregate all attribute values ${r}_{ik}$ to obtain an overall preference value ${t}_{i}$ of the alternative ${A}_{i}$ by using the DM weight vector ${\mathbf{w}}^{\left(DM\right)}=({w}_{1}^{\left(DM\right)},{w}_{2}^{\left(DM\right)},\cdots ,{w}_{l}^{\left(DM\right)})$ and the HOM operator given in Equation (17).
- Step 6
- Rank the collective overall preference values ${t}_{1},{t}_{2},\cdots ,{t}_{m}$ in the descending order and consequently, select the optimal alternative(s) (e.g., the one(s) with the greatest value ${t}_{i}$).

## 6. Numerical Examples

#### 6.1. An Investment Selection Problem

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

**Remark**

**3**(Characteristics of the new approach)

**.**

- (1)
- The SHOMR operators (see, Equation (11)) and NHOMR operators (see, Equation (16)) can capture the psychological preference of the DM with regard to the input argument information, while the aggregation operators in Merigó & Casanovas [7] fail to consider in the decision-making process. Specifically, the above three cases clearly show that the optimal alternative highly depends on the reference point $\mathbf{B}$ and the loss aversion coefficient θ; this confirms the significance of capturing psychological factors of DMs in the aggregation process. The DMs can choose different $\mathbf{B}$ and θ based on their risk preference to select the optimal alternative.
- (2)
- The attribute-deviation weight model (see, model (19)) and DM-deviation weight model (see, model (20)) are constructed to determine the associated weights of the attributes and DMs, while the weights of the attributes and DMs are completely known in Merigó & Casanovas [7]. In fact, owing to the complexity and uncertainty of things in reality, the weights of the attributes and DMs are generally incomplete known.
- (3)
- The new aggregation operators can degenerate to the traditional aggregation operators including the OWGA operator [4], OWMA operator [8], CC-OWGA operator [32] and GOWMA operator [8], etc. (see, Table A1, Table A2 and Table A3 in Appendix A.3). In this way, the new aggregation approach can consider a wide range of scenarios according to the interests of the DM and select the alternative which is closest to his/her real interests.

#### 6.2. Sensitive Analysis of Reference-Dependent Aggregation Operators

#### 6.2.1. Sensitive Analysis of Parameters in the Basic Utility Function

#### 6.2.2. Sensitive Analysis of Reference Points

#### 6.2.3. Sensitive Analysis of the Loss Aversion Coefficient

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof for Properties of SOMR Operator

**Proposition**

**A1.**

- (1)
- (Monotonicity) For two vectors $\mathbf{x}$ and $\overline{\mathbf{x}}$ with ${x}_{i}\ge {\overline{x}}_{i}$ and the same reference points, then $\mathrm{SOMR}\left(\mathbf{x}\right)\ge \mathrm{SOMR}\left(\overline{\mathbf{x}}\right)$.
- (2)
- (Boundedness) If ${b}_{1}\le {y}_{1}=\underset{i}{max}\left\{{x}_{i}\right\}$ and ${b}_{n}>{y}_{n}=\underset{i}{min}\left\{{x}_{i}\right\}$, then ${v}_{1}^{-1}\left({v}_{2}\left({y}_{n}\right)\right)\le $ $\mathrm{SOMR}\left(\mathbf{x}\right)\le {y}_{1}$.
- (3)
- (Commutativity) If $\widehat{\mathbf{x}}$ is a permutation of $\mathbf{x}$, then $\mathrm{SOMR}\left(\mathbf{x}\right)=\mathrm{SOMR}\left(\widehat{\mathbf{x}}\right)$.
- (4)
- (Idempotency) If ${x}_{i}=x\ge {x}_{0}$ for all $1\le i\le n$, then $\mathrm{SOMR}\left(\mathbf{x}\right)=x$.

**Proof**

**of**

**Proposition**

**A1.**

- For $i\in {Y}_{1}$, by taking the first-order condition of S with respect to ${y}_{i}$, we have that,$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial S}{\partial {y}_{i}}=}& {\displaystyle \frac{1}{2{v}_{1}^{\prime}\left(S\right)}\left[{\left(\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{\lambda}\left({y}_{i}\right)\right)/\phantom{\rule{0.0pt}{0ex}}\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{-\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{-\lambda}\left({y}_{i}\right)\right)\right)}^{1/2\lambda -1}\right]\times}\hfill \\ & {\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{-\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{-\lambda}\left({y}_{i}\right)\right)}^{-2}\left[\left({w}_{i}{v}_{1}^{\lambda -1}\left({y}_{i}\right){v}_{1}^{\prime}\left({y}_{i}\right)\right)\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{-\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{-\lambda}\left({y}_{i}\right)\right)+\right.\hfill \\ & \left.\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{\lambda}\left({y}_{i}\right)\right)\left({w}_{i}{v}_{1}^{-\lambda -1}\left({y}_{i}\right){v}_{1}^{\prime}\left({y}_{i}\right)\right)\right].\hfill \end{array}\end{array}$$
- For $i\in {Y}_{2}$, by taking the first-order condition of S with respect to ${y}_{i}$, we have that,$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial S}{\partial {y}_{i}}=}& {\displaystyle \frac{1}{2{v}_{1}^{\prime}\left(S\right)}\left[{\left(\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{\lambda}\left({y}_{i}\right)\right)/\phantom{\rule{0.0pt}{0ex}}\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{-\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{-\lambda}\left({y}_{i}\right)\right)\right)}^{1/2\lambda -1}\right]\times}\hfill \\ & {\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{-\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{-\lambda}\left({y}_{i}\right)\right)}^{-2}\left[\left(-{w}_{i}{v}_{2}^{\lambda -1}\left({y}_{i}\right)({v}_{2}\left({y}_{i}\right){)}^{\prime}\right)\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{-\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{-\lambda}\left({y}_{i}\right)\right)+\right.\hfill \\ & \left.\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{v}_{1}^{\lambda}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{v}_{2}^{\lambda}\left({y}_{i}\right)\right)\left(-{w}_{i}{v}_{2}^{-\lambda -1}\left({y}_{i}\right)({v}_{2}\left({y}_{i}\right){)}^{\prime}\right)\right].\hfill \end{array}\end{array}$$

#### Appendix A.2. Proof for Properties of NOMR Operator

**Proposition**

**A2.**

- (1)
- (Monotonicity) For two vectors $\mathbf{x}$ and $\overline{\mathbf{x}}$ with ${x}_{i}\ge {\overline{x}}_{i}$ and the same reference points, then $\mathrm{NOMR}\left(\mathbf{x}\right)\ge \mathrm{NOMR}\left(\overline{\mathbf{x}}\right)$.
- (2)
- (Boundedness) If ${b}_{1}\le {y}_{1}=\underset{i}{max}\left\{{x}_{i}\right\}$ and ${b}_{n}>{y}_{n}=\underset{i}{min}\left\{{x}_{i}\right\}$, then ${u}^{-1}\left({u}_{1}\left({y}_{n}\right)\right)\le $$\mathrm{NOMR}\left(\mathbf{x}\right)\le {y}_{1}$. Especially, ${y}_{n}\le \mathrm{NOMR}\left(\mathbf{x}\right)\le {y}_{1}$ while ${y}_{n}={b}_{n}$.
- (3)
- (Commutativity) If $\widehat{\mathbf{x}}$ is a permutation of $\mathbf{x}$, then $\mathrm{NOMR}\left(\mathbf{x}\right)=\mathrm{NOMR}\left(\widehat{\mathbf{x}}\right)$.
- (4)
- (Idempotency). If ${x}_{i}=x\ge {x}_{0}$ for all $1\le i\le n$, then $\mathrm{NOMR}\left(\mathbf{x}\right)=x$.

- For $i\in {Y}_{1}$, the first-order condition of N with respect to ${y}_{i}$ implies that,$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial N}{\partial {y}_{i}}=}& {\displaystyle \frac{1}{2{u}^{\prime}\left(N\right)}\left[{\left(\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{\lambda}\left({y}_{i}\right)\right)/\phantom{\rule{0.0pt}{0ex}}\left(\sum _{i\in {Y}_{1}}{w}_{i}{u}^{-\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{-\lambda}\left({y}_{i}\right)\right)\right)}^{1/\phantom{\rule{0.0pt}{0ex}}2\lambda -1}\right]\times}\hfill \\ & \left[\left({w}_{i}{u}^{\lambda -1}\left({y}_{i}\right){u}^{\prime}\left({y}_{i}\right)\right)\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{-\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{-\lambda}\left({y}_{i}\right)\right)+\left({w}_{i}{u}^{-\lambda -1}\left({y}_{i}\right){u}^{\prime}\left({y}_{i}\right)\right)\times \right.\hfill \\ & \left.\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{\lambda}\left({y}_{i}\right)\right)\right]{\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{-\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{-\lambda}\left({y}_{i}\right)\right)}^{-2}.\hfill \end{array}\end{array}$$
- For $i\in {Y}_{2}$, the first-order condition of N with respect to ${y}_{i}$ implies that,$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\partial N}{\partial {y}_{i}}=}& {\displaystyle \frac{1}{2{u}^{\prime}\left(N\right)}\left[{\left(\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{\lambda}\left({y}_{i}\right)\right)/\phantom{\rule{0.0pt}{0ex}}\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{-\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{-\lambda}\left({y}_{i}\right)\right)\right)}^{1/\phantom{\rule{0.0pt}{0ex}}2\lambda -1}\right]\times}\hfill \\ & \left[\left({w}_{i}{u}_{1}^{\lambda -1}\left({y}_{i}\right){u}_{1}^{\prime}\left({y}_{i}\right)\right)\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{-\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{-\lambda}\left({y}_{i}\right)\right)+\left({w}_{i}{u}_{1}^{-\lambda -1}\left({y}_{i}\right){u}_{1}^{\prime}\left({y}_{i}\right)\right)\right.\times \hfill \\ & \left.\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{\lambda}\left({y}_{i}\right)\right)\right]{\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{u}^{-\lambda}\left({y}_{i}\right)+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{u}_{1}^{-\lambda}\left({y}_{i}\right)\right)}^{-2}.\hfill \end{array}\end{array}$$

#### Appendix A.3. Families of the Reference-Dependent Aggregation Operators

λ | ${\mathit{\alpha}}_{0},{\mathit{\beta}}_{0},{\mathit{\theta}}_{0}$ | ${\mathit{b}}_{\mathit{i}}$ | Formulation | The Name of Aggregation Operator |
---|---|---|---|---|

λ is odd and $\lambda >0$ | $\begin{array}{c}0<{\alpha}_{0}<1,\\ 0<{\beta}_{0}<1,\\ {\theta}_{0}>1\end{array}$ | $\begin{array}{c}{y}_{i}\ge {b}_{i},\\ {b}_{i}\ne 0\end{array}$ | ${\left(\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{({y}_{i}-{b}_{i})}^{{\alpha}_{0}\lambda}\right)/\phantom{\rule{0.0pt}{0ex}}\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{({y}_{i}-{b}_{i})}^{-{\alpha}_{0}\lambda}\right)\right)}^{1/\phantom{\rule{0.0pt}{0ex}}2\lambda {\alpha}_{0}}$ | Prospect gain ordered multiple reference-dependent operator (PGOMR) |

$\begin{array}{c}{y}_{i}\ge {b}_{i},\\ {b}_{i}=0\end{array}$ | ${\left(\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{y}_{i}^{{\alpha}_{0}\lambda}\right)/\phantom{\rule{0.0pt}{0ex}}\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{y}_{i}^{-{\alpha}_{0}\lambda}\right)\right)}^{1/\phantom{\rule{0.0pt}{0ex}}2\lambda {\alpha}_{0}}$ | Prospect gain ordered multiple operator (PGOM) | ||

$\begin{array}{c}{y}_{i}<{b}_{i},\\ {b}_{i}\ne 0\end{array}$ | ${\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}})}^{\lambda}/\phantom{{\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}})}^{\lambda}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}})}^{-\lambda}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}})}^{-\lambda}\right)}^{1/\phantom{12\lambda {\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}2\lambda {\alpha}_{0}}$ | Prospect loss ordered multiple reference-dependent operator (PLOMR) | ||

$\begin{array}{c}{y}_{i}<{b}_{i},\\ {b}_{i}=0\end{array}$ | ${\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}})}^{\lambda}/\phantom{{\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}})}^{\lambda}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}})}^{-\lambda}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{(-{\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}})}^{-\lambda}\right)}^{1/\phantom{12\lambda {\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}2\lambda {\alpha}_{0}}$ | Prospect loss ordered multiple operator (PLOM) | ||

$\begin{array}{c}{\alpha}_{0}\to 1,\\ {\beta}_{0}\to 1,\\ {\theta}_{0}\to 1\end{array}$ | ${b}_{i}\ne 0$ | ${\left({\displaystyle \sum _{i=1}^{n}{w}_{i}{({y}_{i}-{b}_{i})}^{\lambda}/\sum _{i=1}^{n}{w}_{i}{({y}_{i}-{b}_{i})}^{-\lambda}}\right)}^{1/2\lambda}$ | Ordered multiple reference-dependent operator (OMR) | |

${b}_{i}=0$ | ${\left({\displaystyle {\displaystyle \sum _{i=1}^{n}}{w}_{i}{{y}_{i}}^{\lambda}/{\displaystyle \sum _{i=1}^{n}}{w}_{i}{{y}_{i}}^{-\lambda}}\right)}^{1/2\lambda}$ | GOWMA operator [8] | ||

$\lambda \to 0$ | $\begin{array}{c}0<{\alpha}_{0}<1,\\ 0<{\beta}_{0}<1,\\ {\theta}_{0}>1\end{array}$ | ${b}_{i}\ne 0$ | $\prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{\left({\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}}\right)}^{{w}_{i}/\phantom{{w}_{i}{\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}{\alpha}_{0}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{\left({\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}}\right)}^{{w}_{i}/\phantom{{w}_{i}{\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}{\alpha}_{0}$ | Prospect ordered multiple geometric reference-dependent operator (POMGR) |

${b}_{i}=0$ | $\prod _{i\in {Y}_{1}}}{y}_{i}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{y}_{i}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{\left({\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}}\right)}^{{w}_{i}/\phantom{{w}_{i}{\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}{\alpha}_{0}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{\left({\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}}\right)}^{{w}_{i}/\phantom{{w}_{i}{\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}{\alpha}_{0}$ | Prospect ordered multiple geometric operator (POMG) | ||

$\begin{array}{c}{\alpha}_{0}\to 1,\\ {\beta}_{0}\to 1,\\ {\theta}_{0}\to 1\end{array}$ | ${b}_{i}\ne 0$ | $\prod _{i\in {Y}_{1}}{({y}_{i}-{b}_{i})}^{{w}_{i}}/\prod _{i\in {Y}_{2}}{({b}_{i}-{y}_{i})}^{{w}_{i}}$ | Ordered multiple geometric reference-dependent aggregation operator (OMGR) | |

${b}_{i}=0$ | $\prod _{i\in {Y}_{1}}{{y}_{i}}^{{w}_{i}}/\prod _{i\in {Y}_{2}}{(-{y}_{i})}^{{w}_{i}}$ | Ordered multiple geometric aggregation operator (OMG) | ||

$\lambda =1$ | $\begin{array}{c}0<{\alpha}_{0}<1,\\ 0<{\beta}_{0}<1,\\ {\theta}_{0}>1\end{array}$ | ${b}_{i}\ne 0$ | ${\left({\displaystyle \frac{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{({y}_{i}-{b}_{i})}^{{\alpha}_{0}}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}}}{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{({y}_{i}-{b}_{i})}^{-{\alpha}_{0}}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\left({\theta}_{0}{({b}_{i}-{y}_{i})}^{{\beta}_{0}}\right)}^{-1}}}\right)}^{1/\phantom{12{\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}2{\alpha}_{0}}$ | Constant prospect ordered multiple reference-dependent operator (CPOMR) |

${b}_{i}=0$ | ${\left({\displaystyle \frac{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{y}_{i}^{{\alpha}_{0}}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}}}{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{y}_{i}^{-{\alpha}_{0}}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\left({\theta}_{0}{(-{y}_{i})}^{{\beta}_{0}}\right)}^{-1}}}\right)}^{1/\phantom{12{\alpha}_{0}}\phantom{\rule{0.0pt}{0ex}}2{\alpha}_{0}}$ | Constant prospect ordered multiple dependent operator (CPOM) | ||

$\begin{array}{c}{\alpha}_{0}\to 1,\\ {\beta}_{0}\to 1,\\ {\theta}_{0}\to 1\end{array}$ | ${b}_{i}\ne 0$ | $\sqrt{{\displaystyle {\displaystyle \sum _{i=1}^{n}}{w}_{i}({y}_{i}-{b}_{i})}/{\displaystyle {\displaystyle \sum _{i=1}^{n}}{{w}_{i}({y}_{i}-{b}_{i})}^{-1}}}$ | Constant ordered multiple reference-dependent operator (COMR) | |

${b}_{i}=0$ | $\sqrt{{\displaystyle \sum _{i=1}^{n}{w}_{i}{y}_{i}}/{\displaystyle \sum _{i=1}^{n}{w}_{i}{{y}_{i}}^{-1}}}$ | OWMA operator [8] | ||

$\begin{array}{c}{\alpha}_{0}\to 0,\\ {\beta}_{0}\to 0,\\ {\theta}_{0}\to 1\end{array}$ | ${b}_{i}\ne 0$ | ${\left({\displaystyle \prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}\right)}^{\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}\right)}.$ | Constant prospect ordered multiple geometric reference-dependent operator (CPOMGR) | |

${b}_{i}=0$ | ${\left({\displaystyle \prod _{i\in {Y}_{1}}}{{y}_{i}}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{{y}_{i}}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{{y}_{i}}^{{w}_{i}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{{y}_{i}}^{{w}_{i}}\right)}^{\left({\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}\right)}.$ | Constant prospect ordered multiple geometric operator (CPOMG) |

**Table A2.**Families of the SHOMR operator $(\beta ={\beta}_{1},\gamma ={\gamma}_{1},\eta ={\eta}_{1})$.

λ | $\mathit{\beta},\mathit{\eta},\mathit{\gamma},{\mathit{\theta}}_{1}$ | ${\mathit{b}}_{\mathit{i}}$ | Formulation | The Name of Aggregation Operator |
---|---|---|---|---|

λ is odd and $\lambda >0$ | $\begin{array}{c}\beta =1-\gamma ,\\ \eta \to 0,\\ \gamma \to 1\\ {\theta}_{1}>1\end{array}$ | ${b}_{i}\ne 0$ | ${\left({\displaystyle \frac{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{({y}_{i}-{b}_{i})}^{\lambda}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\theta}_{1}^{\lambda}{({b}_{i}-{y}_{i})}^{\lambda}}{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{({y}_{i}-{b}_{i})}^{-\lambda}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\theta}_{1}^{-\lambda}{({b}_{i}-{y}_{i})}^{-\lambda}}}\right)}^{1/\phantom{12\lambda}\phantom{\rule{0.0pt}{0ex}}2\lambda}$ | S-shaped ordered multiple reference-dependent operator (SOMR) |

${b}_{i}=0$ | ${\left({\displaystyle \frac{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{y}_{i}^{\lambda}+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\theta}_{1}^{\lambda}{y}_{i}^{\lambda}}{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{y}_{i}^{-\lambda}+{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\theta}_{1}^{-\lambda}{y}_{i}^{-\lambda}}}\right)}^{1/\phantom{12\lambda}\phantom{\rule{0.0pt}{0ex}}2\lambda}$ | S-shaped ordered multiple operator (SOM) | ||

$\begin{array}{c}{y}_{i}\ge {b}_{i}or\\ {y}_{i}<{b}_{i},\\ {b}_{i}\ne 0\end{array}$ | ${\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{({y}_{i}-{b}_{i})}^{\lambda}/\phantom{{\displaystyle \sum _{i=1}^{n}}{w}_{i}{({y}_{i}-{b}_{i})}^{\lambda}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{({y}_{i}-{b}_{i})}^{-\lambda}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{({y}_{i}-{b}_{i})}^{-\lambda}\right)}^{1/\phantom{12\lambda}\phantom{\rule{0.0pt}{0ex}}2\lambda}$ | Ordered multiple reference-dependent operator (OMR) | ||

$\begin{array}{c}{y}_{i}\ge {b}_{i}or\\ {y}_{i}<{b}_{i},\\ {b}_{i}=0\end{array}$ | ${\left({\displaystyle \sum _{i=1}^{n}}{w}_{i}{y}_{i}^{\lambda}/\phantom{{\displaystyle \sum _{i=1}^{n}}{w}_{i}{y}_{i}^{\lambda}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{y}_{i}^{-\lambda}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \sum _{i=1}^{n}}{w}_{i}{y}_{i}^{-\lambda}\right)}^{1/\phantom{12\lambda}\phantom{\rule{0.0pt}{0ex}}2\lambda}$ | GOWMA operator [8] | ||

$\lambda \to 0$ | $\begin{array}{c}\beta ,\eta >0,\\ \gamma \in {R}^{-}\cup (0,1)\\ {\theta}_{1}>1\end{array}$ | ${b}_{i}\ne 0$ | $\frac{1-\gamma}{\beta}\left({\left({\displaystyle \frac{{\displaystyle \prod _{i\in {Y}_{1}}}{\left({\left(\frac{\beta}{1-\gamma}({y}_{i}-{b}_{i})+\eta \right)}^{\gamma}-{\eta}^{\gamma}\right)}^{{w}_{i}}}{{\displaystyle \prod _{i\in {Y}_{2}}}{\theta}_{1}^{{w}_{i}}{\left({\left(\frac{\beta}{1-\gamma}({b}_{i}-{y}_{i})+\eta \right)}^{\gamma}-{\eta}^{\gamma}\right)}^{{w}_{i}}}+{\eta}^{\gamma}}\right)}^{1/\phantom{1\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}-\eta \right)$ | S-shaped HARA ordered geometric reference-dependent operator (SHOGR) |

${b}_{i}=0$ | $\frac{1-\gamma}{\beta}\left({\left({\displaystyle \frac{{\displaystyle \prod _{i\in {Y}_{1}}}{\left({\left(\frac{\beta}{1-\gamma}{y}_{i}+\eta \right)}^{\gamma}-{\eta}^{\gamma}\right)}^{{w}_{i}}}{{\displaystyle \prod _{i\in {Y}_{2}}}{\theta}_{1}^{{w}_{i}}{\left({\left(\frac{\beta}{1-\gamma}(-{y}_{i})+\eta \right)}^{\gamma}-{\eta}^{\gamma}\right)}^{{w}_{i}}}+{\eta}^{\gamma}}\right)}^{1/\phantom{1\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}-\eta \right)$ | S-shaped HARA ordered geometric operator (SHOG) | ||

$\begin{array}{c}\beta =1-\gamma ,\\ \eta \to 0,\\ {\theta}_{1}>1\end{array}$ | ${b}_{i}\ne 0$ | $\prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{\theta}_{1}^{{w}_{i}/\phantom{{w}_{i}\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}{({b}_{i}-{y}_{i})}^{{w}_{i}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{\theta}_{1}^{{w}_{i}/\phantom{{w}_{i}\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}{({b}_{i}-{y}_{i})}^{{w}_{i}$ | S-shaped ordered geometric reference-dependent operator (SOGR) | |

${b}_{i}=0$ | $\prod _{i\in {Y}_{1}}}{y}_{i}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{y}_{i}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{\theta}_{1}^{{w}_{i}/\phantom{{w}_{i}\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}{(-{y}_{i})}^{{w}_{i}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{\theta}_{1}^{{w}_{i}/\phantom{{w}_{i}\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}{(-{y}_{i})}^{{w}_{i}$ | S-shaped ordered geometric operator (SOG) | ||

$\begin{array}{c}\beta =1-\gamma ,\\ \eta \to 0,\gamma \to 1\\ {\theta}_{1}\to 1\end{array}$ | ${b}_{i}\ne 0$ | $\prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{({y}_{i}-{b}_{i})}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{({b}_{i}-{y}_{i})}^{{w}_{i}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{({b}_{i}-{y}_{i})}^{{w}_{i}$ | Ordered multiple geometric reference-dependent operator (OMGR) | |

${b}_{i}=0$ | $\prod _{i\in {Y}_{1}}}{y}_{i}^{{w}_{i}}/\phantom{{\displaystyle \prod _{i\in {Y}_{1}}}{y}_{i}^{{w}_{i}}{\displaystyle \prod _{i\in {Y}_{2}}}{(-{y}_{i})}^{{w}_{i}}}\phantom{\rule{0.0pt}{0ex}}{\displaystyle \prod _{i\in {Y}_{2}}}{(-{y}_{i})}^{{w}_{i}$ | Ordered multiple geometric operator (OMG) | ||

$\lambda =1$ | $\begin{array}{c}\beta ,\eta >0,\\ \gamma \in {R}^{-}\cup (0,1)\\ {\theta}_{1}>1\end{array}$ | ${b}_{i}\ne 0$ | $\frac{1-\gamma}{\beta}\left({\left(\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{\mu}_{1}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\mu}_{2}\left({y}_{i}\right)}{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{\mu}_{1}^{-1}\left({y}_{i}\right)-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{\mu}_{2}^{\phantom{\rule{0.277778em}{0ex}}-1}\left({y}_{i}\right)}}}+{\eta}^{\gamma}\right)}^{1/\phantom{1\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}-\eta \right)$ | Constant HARA ordered multiple reference-dependent operator(CHOMR) |

${b}_{i}=0$ | $\begin{array}{c}{\displaystyle \frac{1-\gamma}{\beta}\left({\left(\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}A-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}B}{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{A}^{-1}-{\displaystyle \sum _{i\in {Y}_{2}}}{w}_{i}{B}^{-1}}}}+{\eta}^{\gamma}\right)}^{1/\phantom{1\gamma}\phantom{\rule{0.0pt}{0ex}}\gamma}-\eta \right),}\\ A={\left({\displaystyle \frac{\beta {y}_{i}}{1-\gamma}+\eta}\right)}^{\gamma}-{\eta}^{\gamma},B={\theta}_{1}\left({\left({\displaystyle -\frac{\beta {y}_{i}}{1-\gamma}+\eta}\right)}^{\gamma}-{\eta}^{\gamma}\right)\end{array}$ | Constant HARA ordered multiple operator (CHOM) | ||

$\begin{array}{c}\beta >0,\eta \to 0,\\ \gamma \in {R}^{-}\cup (0,1)\\ {\theta}_{1}>1\end{array}$ | ${b}_{i}\ne 0$ | ${\left({\displaystyle \frac{{\displaystyle \sum _{i\in {Y}_{1}}}{w}_{i}{({y}_{i}-{b}_{i})}^{\gamma}-{\displaystyle \sum _{i\in {Y}_{2}}}}{}}\right)}^{}$ |