# Comprehensive Interval-Induced Weights Allocation with Bipolar Preference in Multi-Criteria Evaluation

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## Abstract

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## 1. Introduction

## 2. Bipolar Preferences-Involved Aggregations and Related Weight Allocations

**w**) $W{A}_{w}:{[0,1]}^{n}\to [0,1]$ is defined by

**d**, the IOWA operator with weight vector

**w**$IOW{A}_{w}:{[0,1]}^{n}\to [0,1]$ is defined by

**w**will mainly decide the represented bipolar preference, Yager defined the orness [20] of any weight vector

**w**by $orness(w)\triangleq {\displaystyle \sum _{i=1}^{n}\frac{n-i}{n-1}}{w}_{i}$ and dually the andness of it is defined by $andness(w)\triangleq 1-orness(w)$. Cognitively, a weight vector with larger orness will be considered to embody larger optimism in an OWA-based evaluation (or larger preference extent in an IOWA-based evaluation). For some new development and strict analysis in relation to orness/andness, we recommend the Reference [16].

**w**) (IvWA) $IvW{A}_{w}:{\mathcal{I}}^{n}\to \mathcal{I}$ is defined by

**w**) $IvW{A}_{w}:{\mathcal{I}}^{n}\to \mathcal{I}$,

**w**is defined in the following steps:

**v**, we obtain the normalized weight vector $w={({w}_{i})}_{i=1}^{n}$ by

## 3. Some Analysis for Bipolar Preferences-Involved Weighting and Comprehensive Evaluation

#### 3.1. Comprehensive Weights Determination for Criteria with Relative Importance Information

**c**,

**d**, and

**z**by ${s}^{<c>}={({s}_{i}^{<c>})}_{i=1}^{n}$, ${s}^{<d>}={({s}_{i}^{<d>})}_{i=1}^{n}$, and ${s}^{<z>}={({s}_{i}^{<z>})}_{i=1}^{n}$, respectively.

**z**to obtain the weight vector ${s}^{*}=({s}_{i}^{*})$ for criteria. When $z=0$ (i.e., $[c,d]=[0,0]$), we may directly set the corresponding weight vectors obtained from them using the Laplace principle with ${s}^{*}=({s}_{i}^{*})=(1/n,\dots ,1/n)$.

#### 3.2. Comprehensive Weights Determination for Criteria with Optimism–Pessimism Preference

**q**to obtain the weight vector ${r}^{*}=({r}_{i}^{*})$ from inputs. When $q=0$ (i.e., $[a,b]=[0,0]$), we may directly set the corresponding weight vectors obtained from them using the Laplace principle with ${r}^{*}=({r}_{i}^{*})=(1/n,\dots ,1/n)$. Similarly, we may also take a combinational form to obtain a comprehensive weighting result embodying the optimism–pessimism preference, i.e., $r=0.3{r}^{<I>}+0.1{r}^{<a>}+0.1{r}^{<b>}+0.1{r}^{<q>}+0.4{r}^{*}$.

#### 3.3. Adjusted Weights with the Optimism–Pessimism Preference under Known Weights

**s**and

**r**are reasonable from different types of inducing information, it is necessary to consider the weights allocation with the optimism–pessimism preference under the situation in which an original weight vector for criteria has already been known and will matter.

**v**, we obtain the normalized weight vector $w={({w}_{i})}_{i=1}^{n}$ by

**s**,

**r**, and

**w**from different perspectives, we may take a weighted average of them to yield a final resulting weight vector for criteria $u=0.3s+0.3r+0.4w$ (note that the involved weight vector $(0.3,0.3,0.4)$ can be changed by any other weight vector according to different situations in practice) and finally perform the interval-valued weighted arithmetic mean (with

**u**) (IvWA) $IvW{A}_{u}:{\mathcal{I}}^{n}\to \mathcal{I}$ with

## 4. Detailed Comprehensive Weighting and Evaluation Model with Bipolar Preferences

**Stage 1.**Evaluation background determination and evaluation information collection

**Stage 2.**Comprehensive weights determination for criteria with relative importance information

**v**with

**v**, we obtain the weight vector ${s}^{<I>}=\frac{1}{24}(2,5,2,15)\doteq (0.0833,0.2084,0.0833,0.625)$.

**v**with

**v**is already normalized, we have ${s}^{<c>}=(\frac{1}{4},\frac{1}{4},\frac{1}{16},\frac{7}{16})=(0.25,0.25,0.0625,0.4375)$.

**v**with

**v**is already normalized, we have ${s}^{<d>}=(\frac{1}{16},\frac{5}{16},\frac{3}{16},\frac{7}{16})=(0.0625,0.3125,0.1875,0.4375)$.

**v**with

**v**is already normalized, we have ${s}^{<z>}=(\frac{2}{16},\frac{5}{16},\frac{2}{16},\frac{7}{16})=(0.125,0.3125,0.125,0.4375)$.

**z**into the weight vector ${s}^{*}={({s}_{i}^{*})}_{i=1}^{4}=\frac{1}{17}(3,4,3,7)\doteq (0.1765,0.2352,0.1765,0.4118)$.

**Stage 3.**Comprehensive weights determination for criteria with the optimism–pessimism preference

**v**with

**v**is already normalized, we have ${r}^{<I>}=(\frac{3}{8},\frac{1}{8},\frac{1}{4},\frac{1}{4})=(0.375,0.125,0.25,0.25)$.

**v**with

**v**is already normalized, we have ${r}^{<a>}=(\frac{5}{16},\frac{1}{16},\frac{7}{16},\frac{3}{16})=(0.3125,0.0625,0.4375,0.1875)$.

**v**with

**v**is already normalized, we have ${r}^{<b>}=(\frac{7}{16},\frac{4}{16},\frac{1}{16},\frac{4}{16})=(0.4375,0.25,0.0625,0.25)$.

**v**with

**v**is already normalized, we have ${r}^{<q>}=(\frac{6}{16},\frac{1}{16},\frac{6}{16},\frac{3}{16})=(0.375,0.0625,0.375,0.1875)$.

**q**into the weight vector ${r}^{*}={({r}_{i}^{*})}_{i=1}^{4}=\frac{1}{2.25}(0.5,0.65,0.5,0.6)\doteq (0.2222,0.2889,0.2222,0.2667)$.

**Stage 4.**Determine an adjusted weight vector with the optimism–pessimism preference with the known weight vector $s=(0.13934,0.2441,0.13309,0.48347)$.

**v**, we obtain the normalized weight vector $w={({w}_{i})}_{i=1}^{4}=(0.2407,0.0921,0.1331,0.5341)\doteq v$.

**Stage 5.**Obtain the final resulting weight vector for criteria and perform the interval-valued weighted arithmetic mean (with

**u**)

**u**) (IvWA) $IvW{A}_{u}:{\mathcal{I}}^{4}\to \mathcal{I}$ with

## 5. Conclusions

**s**for criteria is comprehensively generated from five aspects with a given concave BUM function, four of which are concerned with the absolute importance being inducing information and the fifth with direct normalization of the absolute importance. In a similar way, we comprehensively generate another weight vector, namely

**r**, for criteria with any BUM function using interval inputs as inducing information without the intervention of

**s**. As the third suggested method and in a tangled way, we successfully applied the weighted OWA allocation on a convex poset in an interval environment and obtained the weight vector

**w**for criteria with the intervention of

**s**. Finally, the resulting weight vector

**u**for criteria is obtained by comprehensively considering all the three types using a weighted form.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Jin, X.; Yager, R.R.; Mesiar, R.; Borkotokey, S.; Jin, L.
Comprehensive Interval-Induced Weights Allocation with Bipolar Preference in Multi-Criteria Evaluation. *Mathematics* **2021**, *9*, 2002.
https://doi.org/10.3390/math9162002

**AMA Style**

Jin X, Yager RR, Mesiar R, Borkotokey S, Jin L.
Comprehensive Interval-Induced Weights Allocation with Bipolar Preference in Multi-Criteria Evaluation. *Mathematics*. 2021; 9(16):2002.
https://doi.org/10.3390/math9162002

**Chicago/Turabian Style**

Jin, Xu, Ronald R. Yager, Radko Mesiar, Surajit Borkotokey, and Lesheng Jin.
2021. "Comprehensive Interval-Induced Weights Allocation with Bipolar Preference in Multi-Criteria Evaluation" *Mathematics* 9, no. 16: 2002.
https://doi.org/10.3390/math9162002