# Reducing Cognitive Effort in Scoring Negotiation Space Using the Fuzzy Clustering Model

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## Abstract

**:**

## 1. Introduction

## 2. Decision Support in Negotiation—Selected Facts

#### 2.1. Negotiation Template, Negotiation Space, and Scoring Systems

#### 2.2. Mutual Evaluation of the Negotiation Space

## 3. Methods for Scoring the Negotiation Template—Literature Review

#### 3.1. Classic Multiple Criteria Decision Aiding Approaches

#### 3.2. Fuzzy Approaches to Negotiation Support

## 4. Using Sorting Approach and Limiting Profiles to Evaluate the Negotiation Space

## 5. Procedure of Scoring Negotiation Space Using the Fuzzy Clustering Model

#### 5.1. Fuzzy Numbers in Scoring the Limiting Profiles

- (a)
- for $x<z<y$, we have $\tilde{z}=\left({z}_{l},{z}_{m},{z}_{r}\right),$ where:$${z}_{\alpha}=\left({y}_{\alpha}-{x}_{\alpha}\right)\frac{y-z}{y-x}+{x}_{\alpha}(\alpha =\left\{l,m,r\right\}),$$
- (b)
- for $y<z<x$, we have $\tilde{z}=\left({z}_{l},{z}_{m},{z}_{r}\right)$, where:$${z}_{\alpha}=-\left({y}_{\alpha}-{x}_{\alpha}\right)\frac{y-z}{y-x}+{y}_{\alpha}(\alpha =\left\{l,m,r\right\}).$$

#### 5.2. Algorithm of Scoring Negotiation Space Using the Fuzzy Clustering Model

**Step 1.**Defining the negotiation template, i.e., set of negotiation issues and the negotiation space.

**Step 2.**Determining the fuzzy vector of the importance of issues for negotiators $N1,N2$.

**Step 3.**Defining the $s$—point linguistic scale for the evaluation of negotiation options represented by TFNs $\left(s\ge 3\right)$.

**Step 4.**Defining the aspiration ${P}_{as}^{Q}$ and reservation package ${P}_{res}^{Q}$ for negotiators $N1,N2$.

**Step 5.**Defining the set of limiting profiles.

**Step 6.**Define the set of limiting fuzzy sub-profiles.

**Step 7.**Determining the fuzzy value of packages from the set $\mathbb{P}\backslash {B}^{Q}\left(Q=N1,N2\right)$.

**Step 8.**Determining the set of categories ${C}^{Q}$ based on the set of limiting sub-profiles ${B}^{Q}$ and classifying packages to categories.

**Step 9**. Presenting the obtained classification of all packages for both parties of the negotiations in two-dimensional space as the points $\left(t\left(N1\right),t\left(N2\right)\right)$, where: $t\left(N1\right)$, $t\left(N2\right)$ denoted the number of the cluster obtained for the same package in the case of the negotiators $N1$ and $N2$.

**Step 10.**Determining the set of the most favorable packages for both parties of the negotiations.

- $t\left(N1\right),t\left(N2\right)\to \mathrm{max}$
- $\left|t\left(N1\right)-t\left(N2\right)\right|\to \mathrm{min}$

## 6. Numerical Example

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Representation of the negotiation space in the two-dimensional scoring space of both negotiators.

**Figure 3.**Sub-categories and their limiting profiles defined additionally in negotiation space $\mathbb{T}$.

**Figure 5.**Assigning numbers z,r consecutive natural numbers. (Note: i is the next label after 1 from SL).

**Figure 7.**Negotiations space for various numbers of sub-profiles: (

**a**) 31 sub-profiles; (

**b**) 36 sub-profiles; (

**c**) 56 sub-profiles; (

**d**) 506 sub-profiles.

L | LT | FTN |
---|---|---|

1 | Absolutely low important | (0.0, 0.1, 0.2) |

2 | Very low important | (0.1, 0.2, 0.3) |

3 | Low important | (0.2, 0.3, 0.4) |

4 | Medium low important | (0.3, 0.4, 0.5) |

5 | Medium important | (0.4, 0.5, 0.6) |

6 | Medium high important | (0.5, 0.6, 0.7) |

7 | Hight important | (0.6, 0.7, 0.8) |

8 | Very high important | (0.7, 0.8, 0.9) |

9 | Absolutely high important | (0.8, 0.9, 1.0) |

L | LT | FTN |
---|---|---|

1 | Very poor | (0, 0, 1) |

2 | Poor | (0, 1, 3) |

3 | Medium poor | (1, 3, 5) |

4 | Fair | (3, 5, 7) |

5 | Medium good | (5, 7, 9) |

6 | Good | (7, 9, 10) |

7 | Very good | (9, 10, 10) |

$\mathbf{Issues}\mathbf{to}\mathbf{Negotiate}\left({\mathit{g}}_{\mathit{i}}\right)$ | $\mathbf{Options}({\mathit{O}}_{\mathit{i}})$ |
---|---|

Price (in US$) (${g}_{1}$) | 10; 10.5; 11; 11.5, …, 24.5; 25 |

Delivery time (in days) (${g}_{2}$) | 14; 21; 30; 45; 75; 90 |

$\mathrm{Payment}(\mathrm{in}\mathrm{days})\left({g}_{3}\right)$ | 1; 7; 14; 30; 45; 60 |

Returns conditions (${g}_{4}$) | A; B; C; D; E ^{1} |

^{1}where: A—3% defects; no penalty, B—any defects; no penalty, C—7% defects; 4% penalty, D—5% defects; 2% penalty, E—5% defects; 4% penalty.

${\mathit{g}}_{\mathit{i}}\backslash {\mathit{P}}_{\mathit{k}}$ | ${\mathit{P}}_{1}$ | ${\mathit{P}}_{2}$ | ${\mathit{P}}_{3}$ | ${\mathit{P}}_{4}$ | … | ${\mathit{P}}_{5577}$ | ${\mathit{P}}_{5578}$ | ${\mathit{P}}_{5579}$ | ${\mathit{P}}_{5580}$ |
---|---|---|---|---|---|---|---|---|---|

g_{1} | 10 | 10 | 10 | 10 | … | 25 | 25 | 25 | 25 |

g_{2} | 90 | 75 | 45 | 30 | … | 45 | 30 | 21 | 14 |

g_{3} | 1 | 1 | 1 | 1 | … | 60 | 60 | 60 | 60 |

g_{4} | A | A | A | A | … | E | E | E | E |

${\mathit{g}}_{4}$ | N1 | N2 |
---|---|---|

A | (0, 0, 1) | (8, 9, 10) |

B | (0, 1, 3) | (5, 7, 9) |

C | (5, 7, 9) | (3, 5, 7) |

D | (3, 5, 7) | (0, 1, 3) |

E | (8, 9, 10) | (0, 0, 1) |

${\mathit{g}}_{\mathit{i}}$ | ${\mathit{w}}_{\mathit{i}}^{\mathit{N}1}$ | ${\mathit{w}}_{\mathit{i}}^{\mathit{N}2}$ |
---|---|---|

${g}_{1}$ | (0.8, 0.9, 1.0) | (0.8, 0.9, 1.0) |

${g}_{2}$ | (0.5, 0.6, 0.7) | (0.7, 0.8, 0.9) |

${g}_{3}$ | (0.1, 0.2, 0.3) | (0.5, 0.6, 0.7) |

${g}_{4}$ | (0.8, 0.9, 1.0) | (0.1, 0.2, 0.3) |

${\mathit{g}}_{\mathit{i}}$ | ${\tilde{\mathit{w}}}_{\mathit{i}}^{\mathit{N}1}$ | ${\tilde{\mathit{w}}}_{\mathit{i}}^{\mathit{N}2}$ |
---|---|---|

${g}_{1}$ | (0.29, 0.35, 0.42) | (0.30, 0.36, 0.43) |

${g}_{2}$ | (0.18, 0.23, 0.29) | (0.26, 0.32, 0.39) |

${g}_{3}$ | (0.04, 0.08, 0.12) | (0.18, 0.24, 0.30) |

${g}_{4}$ | (0.29, 0.35, 0.42) | (0.04, 0.08, 0.13) |

${\mathit{g}}_{\mathit{i}}$ | ${\tilde{\mathit{p}}}_{\mathit{i}}^{\mathit{N}1}$ | ${\tilde{\mathit{p}}}_{\mathit{i}}^{\mathit{N}2}$ |
---|---|---|

${g}_{1}$ | (8.87, 9.94, 10.00) | (0.00, 0.25, 1.50) |

${g}_{2}$ | (0.00, 0.00, 1.00) | (9.00, 10.00, 10.00) |

${g}_{3}$ | (0.00, 0.00, 1.00) | (9.00, 10.00, 10.00) |

${g}_{4}$ | (0.00, 1.00, 3.00) | (7.00, 9.00, 10.00) |

Package | ${\mathit{g}}_{1}$ | ${\mathit{g}}_{2}$ | ${\mathit{g}}_{3}$ | ${\mathit{g}}_{4}$ | Category |
---|---|---|---|---|---|

1 | 17.5 | 45 | 1 | E | 16 |

2 | 18 | 45 | 1 | E | 16 |

3 | 18.5 | 45 | 1 | E | 16 |

4 | 18.5 | 45 | 7 | E | 16 |

5 | 20 | 30 | 1 | E | 16 |

Package | ${\mathit{g}}_{1}$ | ${\mathit{g}}_{2}$ | ${\mathit{g}}_{3}$ | ${\mathit{g}}_{4}$ | Category (a) | Category (b) | Category (c) | Category (d) |
---|---|---|---|---|---|---|---|---|

1 | 17.5 | 45 | 1 | E | 19 | 22 | 34 | 306 |

2 | 18 | 45 | 1 | E | 19 | 22 | 35 | 313 |

3 | 18.5 | 45 | 1 | E | 19 | 22 | 34 | 304 |

4 | 18.5 | 45 | 7 | E | 19 | 22 | 34 | 309 |

5 | 20 | 30 | 1 | E | 19 | 22 | 34 | 304 |

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**MDPI and ACS Style**

Filipowicz-Chomko, M.; Mierzwiak, R.; Nowak, M.; Roszkowska, E.; Wachowicz, T.
Reducing Cognitive Effort in Scoring Negotiation Space Using the Fuzzy Clustering Model. *Entropy* **2021**, *23*, 752.
https://doi.org/10.3390/e23060752

**AMA Style**

Filipowicz-Chomko M, Mierzwiak R, Nowak M, Roszkowska E, Wachowicz T.
Reducing Cognitive Effort in Scoring Negotiation Space Using the Fuzzy Clustering Model. *Entropy*. 2021; 23(6):752.
https://doi.org/10.3390/e23060752

**Chicago/Turabian Style**

Filipowicz-Chomko, Marzena, Rafał Mierzwiak, Marcin Nowak, Ewa Roszkowska, and Tomasz Wachowicz.
2021. "Reducing Cognitive Effort in Scoring Negotiation Space Using the Fuzzy Clustering Model" *Entropy* 23, no. 6: 752.
https://doi.org/10.3390/e23060752