# Fuzzy Representation of Principal’s Preferences in Inspire Negotiation Support System

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## Abstract

**:**

## 1. Introduction

- (i)
- we discuss the problem of interpretation by an agent of the principal preferences visualised imprecise by circles;
- (ii)
- we design a new procedure for building a fuzzy scoring system by an agent using simultaneous recommendations provided by many independent interpreters;
- (iii)
- we identify some problems with an evaluation of preferential information by such interpreters linked with the normalisation procedures.

## 2. Fuzzy Numbers—Selected Facts

**Definition**

**1.**

## 3. Negotiation Template and Scoring Function

**Example**

**1.**

- where the symbol $V\left(\mathcal{C}\left(\varphi \right)\right)$ denotes the size of the circle $\mathcal{C}\left(\varphi \right)$.

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

- comparison between radii causes overestimation of the relative utility of the worse object;
- comparison between areas causes underestimation of the relative utility of the worse object.

## 4. The Prenegotiation Experiment

## 5. Fuzzy Scoring System

- the least underestimated utility rating$${\stackrel{\u02c7}{u}}_{i,j}^{\left(Q\right)}=\mathrm{min}\left\{y:y\in {\mathcal{U}}_{i,j}^{\left(Q\right)}\right\},$$
- the greatest underestimated utility rating$${\overline{u}}_{i,j}^{\left(Q\right)}=\mathrm{min}\left\{y:\text{}\frac{\mathrm{card}\{z:z\le y,\text{}z\in {\mathcal{U}}_{i,j}^{\left(Q\right)}\}\text{}}{l}\ge \frac{1}{3},\text{}y\in {\mathcal{U}}_{i,j}^{\left(Q\right)}\right\},$$
- the least overestimated utility rating$${\stackrel{=}{u}}_{i,j}^{\left(Q\right)}=\mathrm{max}\left\{y:\text{}\frac{\mathrm{card}\{z:z\ge y,\text{}z\in {\mathcal{U}}_{i,j}^{\left(Q\right)}\}\text{}}{l}\ge \frac{1}{3},\text{}y\in {\mathcal{U}}_{i,j}^{\left(Q\right)}\right\},$$
- the greatest overestimated utility rating$${\widehat{u}}_{i,j}^{\left(Q\right)}=\mathrm{max}\left\{y:y\in {\mathcal{U}}_{i,j}^{\left(Q\right)}\right\}.$$

**Example**

**5.**

## 6. Discussion

## 7. Final Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Raiffa, H.; Richardson, J.; Metcalfe, D. Negotiation Analysis: The Science and Art of Collaborative Decision Making; Harvard University Press: Cambridge, MA, USA, 2002; ISBN 978-0-674-00890-8. [Google Scholar]
- Kersten, G.E.; Noronha, S.J. WWW-based negotiation support: Design, implementation, and use. Decis. Support Syst.
**1999**, 25, 135–154. [Google Scholar] [CrossRef] - Kersten, G.E.; Lai, H. Negotiation Support and E-negotiation Systems: An Overview. Group Decis. Negot.
**2007**, 16, 553–586. [Google Scholar] [CrossRef] - Brams, S.J. Negotiation Games: Applying Game Theory to Bargaining and Arbitration; Psychology Press: Hove, UK, 2003; Volume 2, ISBN 0-415-30894-1. [Google Scholar]
- Vetschera, R.; Kersten, G.; Köszegi, S. User assessment of internet-based negotiation support systems: An exploratory study. J. Organ. Comput. Electron. Commer.
**2006**, 16, 123–148. [Google Scholar] [CrossRef] - Wu, S. Design Science Research Approach in Studying Enegotiations: Models, Systems, Experiments. Control Cybern.
**2021**, 50. in print. [Google Scholar] - Kersten, G.E.; Roszkowska, E.; Wachowicz, T. Representative Decision-Making and the Propensity to Use Round and Sharp Numbers in Preference Specification. In Group Decision and Negotiation in an Uncertain World. GDN 2018; Chen, Y., Kersten, G.E., Vetschera, R., Xu, H., Eds.; Lecture Notes in Business Information Processing; Springer: Cham, Switzerland, 2018; Volume 315, pp. 43–55. [Google Scholar]
- Vetschera, R.; Koeszegi, S.T.; Schoop, M. Electronic negotiation systems. Wiley Encycl. Oper. Res. Manag. Sci.
**2011**, 1–8. [Google Scholar] [CrossRef] - Kersten, G.; Roszkowska, E.; Wachowicz, T. An Impact of Negotiation Profiles on the Accuracy of Negotiation Offer Scoring System? Experimental Study. Mult. Criteria Decis. Mak.
**2016**, 11, 77–103. [Google Scholar] [CrossRef] - Kersten, G.E.; Chen, E.; Rios, J.; Strecker, S. A study on preference impartation and decision support in e-negotiation. In Proceedings of the 2010 43rd Hawaii International Conference on System Sciences, Honolulu, HI, USA, 5–8 January 2010; IEEE: New York, NY, USA, 2010; pp. 1–10. [Google Scholar]
- Kersten, G.; Roszkowska, E.; Wachowicz, T. The Heuristics and Biases in Using the Negotiation Support Systems. In Proceedings of the Group Decision and Negotiation. A Socio-Technical Perspective; Schoop, M., Kilgour, D.M., Eds.; Springer International Publishing: Cham, Switzerland, 2017; pp. 215–228. [Google Scholar]
- Wachowicz, T.; Kersten, G.E.; Roszkowska, E. How do I tell you what I want? Agent’s interpretation of principal’s preferences and its impact on understanding the negotiation process and outcomes. Oper. Res.
**2019**, 19, 993–1032. [Google Scholar] [CrossRef] [Green Version] - Kersten, G.; Noronha, S. Negotiation via the World Wide Web: A cross-cultural study of decision making. Group Decis. Negot.
**1999**, 8, 251–279. [Google Scholar] [CrossRef] - Bottom, W.P.; Holloway, J.; Miller, G.J.; Mislin, A.; Whitford, A. Building a pathway to cooperation: Negotiation and social exchange between principal and agent. Adm. Sci. Q.
**2006**, 51, 29–58. [Google Scholar] [CrossRef] [Green Version] - Pepper, A.; Gore, J. Behavioral agency theory: New foundations for theorizing about executive compensation. J. Manag.
**2015**, 41, 1045–1068. [Google Scholar] [CrossRef] [Green Version] - Bazerman, M.H.; Neale, M.A.; Valley, K.L.; Zajac, E.J.; Kim, Y.M. The effect of agents and mediators on negotiation outcomes. Organ. Behav. Hum. Decis. Process.
**1992**, 53, 55–73. [Google Scholar] [CrossRef] - Rubin, J.Z.; Sander, F.E. When should we use agents? Direct vs. representative negotiation. Negot. J.
**1988**, 4, 395–401. [Google Scholar] [CrossRef] - Miettinen, K. Survey of methods to visualize alternatives in multiple criteria decision making problems. Spectrum
**2014**, 36, 3–37. [Google Scholar] [CrossRef] [Green Version] - Korhonen, P.; Wallenius, J. Behavioral Issues in MCDM: Neglected Research Questions. In Multicriteria Analysis; Springer: Berlin/Heidelberg, Germany, 1997; pp. 412–422. [Google Scholar]
- Liu, S.; Cui, W.; Wu, Y.; Liu, M. A survey on information visualization: Recent advances and challenges. Vis. Comput.
**2014**, 30, 1373–1393. [Google Scholar] [CrossRef] - Roselli, L.R.P.; Frej, E.A.; de Almeida, A.T. Neuroscience Experiment for Graphical Visualization in the FITradeoff Decision Support System. In Proceedings of the International Conference on Group Decision and Negotiation; Springer: Berlin/Heidelberg, Germany, 2018; pp. 56–69. [Google Scholar]
- Macdonald-Ross, M. How numbers are shown. AV Commun. Rev.
**1977**, 25, 359–409. [Google Scholar] [CrossRef] - Wachowicz, T.; Roszkowska, E. Holistic Preferences and Prenegotiation Preparation. In Handbook of Group Decision and Negotiation; Kilgour, D.M., Eden, C., Eds.; Springer International Publishing: Cham, Switzerland, 2021; pp. 255–289. ISBN 978-3-030-49629-6. [Google Scholar]
- Roszkowska, E.; Wachowicz, T. Inaccuracy in Defining Preferences by the Electronic Negotiation System Users. In Outlooks and Insights on Group Decision and Negotiation; Lecture Notes in Business Information Processing; Springer International Publishing: Cham, Switzerland, 2015; pp. 131–143. [Google Scholar]
- Roszkowska, E.; Wachowicz, T. The Application of Item Response Theory for Analyzing the Negotiators’ Accuracy in Defining Their Preferences. In Group Decision and Negotiation. Theory, Empirical Evidence, and Application; Springer: Cham, Switzerland, 2016; pp. 3–15. [Google Scholar]
- Dubois, D.; Prade, H. Operations on fuzzy numbers. Int. J. Syst. Sci.
**1978**, 9, 613–626. [Google Scholar] [CrossRef] - Matos, N.; Sierra, C. Evolutionary Computing and Negotiating Agents. In Proceedings of the International Workshop on Agent-Mediated Electronic Trading; Springer: Berlin/Heidelberg, Germany, 1998; pp. 126–150. [Google Scholar]
- Zuo, B.; Sun, Y. Fuzzy Logic to Support Bilateral Agent Negotiation in E-commerce. In Proceedings of the Proceedings of the 2009 International Conference on Artificial Intelligence and Computational Intelligence; IEEE Computer Society: Washington, DC, USA, 2009; Volume 4, pp. 179–183. [Google Scholar]
- Zhan, J.; Luo, X.; Feng, C.; He, M. A multi-demand negotiation model based on fuzzy rules elicited via psychological experiments. Appl. Soft Comput.
**2018**, 67, 840–864. [Google Scholar] [CrossRef] [Green Version] - Roszkowska, E.; Wachowicz, T. Application of Fuzzy TOPSIS to scoring the negotiation offers in ill-structured negotiation problems. Eur. J. Oper. Res.
**2015**, 242, 920–932. [Google Scholar] [CrossRef] - Piasecki, K.; Roszkowska, E. On application of ordered fuzzy numbers in ranking linguistically evaluated negotiation offers. Adv. Fuzzy Syst.
**2018**, 2018, 1569860. [Google Scholar] [CrossRef] - Kowalczyk, R. Fuzzy e-negotiation agents. Soft Comput.
**2002**, 6, 337–347. [Google Scholar] [CrossRef] - Kim, J.S. Negotiation Support in Electronic Commerce Using Fuzzy Membership Functions and AHP. In Proceedings of the 6th Pacific Rim International Workshop on Multi-Agents (PRIMA), Seoul, Korea, 7–8 November 2003; Citeseer: Princeton, NJ, USA, 2003; pp. 93–104. [Google Scholar]
- Masero, E.; Francisco, M.; Maestre, J.M.; Revollar, S.; Vega, P. Hierarchical distributed model predictive control based on fuzzy negotiation. Expert Syst. Appl.
**2021**, 176, 114836. [Google Scholar] [CrossRef] - Yang, Y.; Luo, X. A multi-Demand Negotiation Model with Fuzzy Concession Strategies. In Proceedings of the International Conference on Artificial Intelligence and Soft Computing; Springer: Berlin/Heidelberg, Germany, 2019; pp. 689–707. [Google Scholar]
- Chou, S.-Y.; Chang, Y.-H. A decision support system for supplier selection based on a strategy-aligned fuzzy SMART approach. Expert Syst. Appl.
**2008**, 34, 2241–2253. [Google Scholar] [CrossRef] - Filho, J.L.S.; Morais, D.C. Negotiation protocol based on ordered weighted averaging and Fuzzy metrics. J. Organ. Comput. Electron. Commer.
**2019**, 29, 190–208. [Google Scholar] [CrossRef] - Francisco, M.; Mezquita, Y.; Revollar, S.; Vega, P.; De Paz, J.F. Multi-agent distributed model predictive control with fuzzy negotiation. Expert Syst. Appl.
**2019**, 129, 68–83. [Google Scholar] [CrossRef] - Piasecki, K.; Roszkowska, E.; Łyczkowska-Hanćkowiak, A. Simple additive weighting method equipped with fuzzy ranking of evaluated alternatives. Symmetry
**2019**, 11, 482. [Google Scholar] [CrossRef] [Green Version] - Piasecki, K.; Roszkowska, E.; łyczkowska-Hanćkowiak, A. Impact of the Orientation of the Ordered Fuzzy Assessment on the Simple Additive Weighted Method. Symmetry
**2019**, 11, 1104. [Google Scholar] [CrossRef] [Green Version] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning—II. Inf. Sci.
**1975**, 8, 301–357. [Google Scholar] [CrossRef] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning-III. Inf. Sci.
**1975**, 9, 43–80. [Google Scholar] [CrossRef] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning—I. Inf. Sci.
**1975**, 8, 199–249. [Google Scholar] [CrossRef] - Liu, Y.; Eckert, C.M.; Earl, C. A review of fuzzy AHP methods for decision-making with subjective judgements. Expert Syst. Appl.
**2020**, 161, 113738. [Google Scholar] [CrossRef] - Kersten, G.E. E-negotiation systems: Interaction of people and technologies to resolve conflicts. In Proceedings of the UNESCAP Third Annual Forum on Online Dispute Resolution, Melbourne, Australia, 5–6 July 2004; pp. 5–6. [Google Scholar]
- Chen, S.-M. Fuzzy system reliability analysis using fuzzy number arithmetic operations. Fuzzy Sets Syst.
**1994**, 64, 31–38. [Google Scholar] [CrossRef] - Orlovsky, S.A. Decision-making with a fuzzy preference relation. Fuzzy Sets Syst.
**1978**, 1, 155–167. [Google Scholar] [CrossRef] - Chen, S.-M.; Munif, A.; Chen, G.-S.; Liu, H.-C.; Kuo, B.-C. Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights. Expert Syst. Appl.
**2012**, 39, 6320–6334. [Google Scholar] [CrossRef] - Chang, D.-Y. Applications of the extent analysis method on fuzzy AHP. Eur. J. Oper. Res.
**1996**, 95, 649–655. [Google Scholar] [CrossRef] - Junior, F.R.L.; Osiro, L.; Carpinetti, L.C.R. A comparison between Fuzzy AHP and Fuzzy TOPSIS methods to supplier selection. Appl. Soft Comput.
**2014**, 21, 194–209. [Google Scholar] [CrossRef] - Piasecki, K. On imprecise investment recommendations. Stud. Log. Gramm. Rhetor.
**2014**, 37, 179–194. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.-J. Ranking triangle and trapezoidal fuzzy numbers based on the relative preference relation. Appl. Math. Model.
**2015**, 39, 586–599. [Google Scholar] [CrossRef] - Wei, S.-H.; Chen, S.-M. A new approach for fuzzy risk analysis based on similarity measures of generalized fuzzy numbers. Expert Syst. Appl.
**2009**, 36, 589–598. [Google Scholar] [CrossRef] - Rao, P.P.B.; Shankar, N.R. Ranking fuzzy numbers with an area method using circumcenter of centroids. Fuzzy Inf. Eng.
**2013**, 5, 3–18. [Google Scholar] [CrossRef] [Green Version] - Liou, T.-S.; Wang, M.-J.J. Ranking fuzzy numbers with integral value. Fuzzy Sets Syst.
**1992**, 50, 247–255. [Google Scholar] [CrossRef] - Cheng, C.-H. A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets Syst.
**1998**, 95, 307–317. [Google Scholar] [CrossRef] - Brunelli, M.; Mezei, J. How different are ranking methods for fuzzy numbers? A numerical study. Int. J. Approx. Reason.
**2013**, 54, 627–639. [Google Scholar] [CrossRef] - Lee-Kwang, H.; Lee, J.-H. A method for ranking fuzzy numbers and its application to decision-making. IEEE Trans. Fuzzy Syst.
**1999**, 7, 677–685. [Google Scholar] [CrossRef] - Leekwijck, W.V.; Kerre, E.E. Defuzzification: Criteria and classification. Fuzzy Sets Syst.
**1999**, 108, 159–178. [Google Scholar] [CrossRef] - Łyczkowska-Hanćkowiak, A. On Application Oriented Fuzzy Numbers for Imprecise Investment Recommendations. Symmetry
**2020**, 12, 1672. [Google Scholar] [CrossRef] - Thompson, L. The Mind and Heart of the Negotiator, 6th ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2015. [Google Scholar]
- Peterson, R.M.; Lucas, G.H. Expanding the antecedent component of the traditional business negotiation model: Pre-negotiation literature review and planning-preparation propositions. J. Mark. Theory Pract.
**2001**, 9, 37–49. [Google Scholar] [CrossRef] - Young, H.P. Negotiation Analysis; University of Michigan Press: Ann Arbor, MI, USA, 1991; ISBN 978-0-472-08157-8. [Google Scholar]
- Lee, S.; Thompson, L. Do agents negotiate for the best (or worst) interest of principals? Secure, anxious and avoidant principal–agent attachment. J. Exp. Soc. Psychol.
**2011**, 47, 681–684. [Google Scholar] [CrossRef] - Angur, M.G.; Lotfi, V.; Sarkis, J. A hybrid conjoint measurement and bi-criteria model for a two group negotiation problem. Socioecon. Plann. Sci.
**1996**, 30, 195–206. [Google Scholar] [CrossRef] - Brinton, W.C. Graphic Methods for Presenting Facts; The Engineering Magazine Company: New York, NY, USA, 1914. [Google Scholar]
- Chang, K. Circle size judgment and map design. Am. Cartogr.
**1980**, 7, 155–162. [Google Scholar] [CrossRef] - Ekman, G.; Junge, K. Psychophysical relations in visual perception of length, area and volume. Scand. J. Psychol.
**1961**, 2, 1–10. [Google Scholar] [CrossRef] - Hollands, J.G.; Spence, I. Judgments of change and proportion in graphical perception. Hum. Factors
**1992**, 34, 313–334. [Google Scholar] [CrossRef] [PubMed] - Davis, F.D. Perceived usefulness, perceived ease of use, and user acceptance of information technology. MIS Q.
**1989**, 13, 319–340. [Google Scholar] [CrossRef] [Green Version]

Negotiations Issues | Lists of Predefined Options |
---|---|

Number of promotional concerts (per year) | 5; 6; 7 or 8 concerts |

Number of new songs introduced and performed each year | 11; 12; 13; 14 or 15 songs |

Royalties for CDs (in per cent) | 1.5; 2; 2.5 or 3% |

Contract signing bonus (in dollars) | $125,000; $150,000 or $200,000 |

Issue | Issue Importance $\mathbf{\left(}{\mathit{R}}_{\mathit{i},0}\mathbf{\right)}$ | Preferences between Options | ||||
---|---|---|---|---|---|---|

${\mathit{R}}_{\mathit{i},\mathbf{1}}$ | ${\mathit{R}}_{\mathit{i},\mathbf{2}}$ | ${\mathit{R}}_{\mathit{i},\mathbf{3}}$ | ${\mathit{R}}_{\mathit{i},\mathbf{4}}$ | ${\mathit{R}}_{\mathit{i},\mathbf{5}}$ | ||

Concerts | 5.59 | 4.30 | 3.85 | 3.45 | 1.85 | |

Songs | 4,74 | 2.00 | 2.70 | 3.70 | 4.90 | 4.20 |

Royalties | 3.54 | 3.80 | 4.50 | 4.00 | 2.90 | |

Bonus | 2.89 | 4.00 | 3.40 | 2.50 |

Issue | Standardised Radii for: | |||||
---|---|---|---|---|---|---|

$\mathbf{Issue}\text{}\mathbf{Importance}\text{}\left({\mathit{r}}_{\mathit{i},0}\right)$ | Option Importance | |||||

${\mathit{r}}_{\mathit{i},1}^{\left(1\right)}$ | ${\mathit{r}}_{\mathit{i},2}^{\left(1\right)}$ | ${\mathit{r}}_{\mathit{i},3}^{\left(1\right)}$ | ${\mathit{r}}_{\mathit{i},4}^{\left(1\right)}$ | ${\mathit{r}}_{\mathit{i},5}^{\left(1\right)}$ | ||

Concerts | 0.3335 | 1 | 0.8162 | 0.6531 | 0 | |

Songs | 0.2828 | 0 | 0.2414 | 0.5862 | 1 | 0.7582 |

Royalties | 0.2112 | 0.5625 | 1 | 0.6875 | 0 | |

Bonus | 0.1724 | 1 | 0.6000 | 0 |

Issue | Standardised Radii for: | |||||
---|---|---|---|---|---|---|

$\mathbf{Issue}\text{}\mathbf{Importance}\text{}\left({\mathit{r}}_{\mathit{i},0}\right)$ | Option Importance | |||||

${\mathit{r}}_{\mathit{i},1}^{\left(2\right)}$ | ${\mathit{r}}_{\mathit{i},2}^{\left(2\right)}$ | ${\mathit{r}}_{\mathit{i},3}^{\left(2\right)}$ | ${\mathit{r}}_{\mathit{i},4}^{\left(2\right)}$ | ${\mathit{r}}_{\mathit{i},5}^{\left(2\right)}$ | ||

Concerts | 0.3335 | 1 | 0.8953 | 0.8023 | 0.4302 | |

Songs | 0.2828 | 0.4081 | 0.5510 | 0.7551 | 1 | 0.8571 |

Royalties | 0.2112 | 0.8444 | 1 | 0.8888 | 0.6444 | |

Bonus | 0.1724 | 1 | 0.8500 | 0.6250 |

Options | INSPIRE 1 (l = 38) | INSPIRE 2 (l = 103) | |
---|---|---|---|

${x}_{1,1}$ | 8 promotional concerts | 0.592 | 0.586 |

${x}_{1,2}$ | 7 promotional concerts | 0.892 | 0.601 |

${x}_{1,3}$ | 6 promotional concerts | 1.426 | 1.122 |

${x}_{1,4}$ | 5 promotional concerts | - | 2.699 |

${x}_{2,1}$ | 11 new songs | - | 2.255 |

${x}_{2,2}$ | 12 new songs | 0.953 | 1.534 |

${x}_{2,3}$ | 13 new songs | 0.864 | 1.173 |

${x}_{2,4}$ | 14 new songs | 0.333 | 0.599 |

${x}_{2,5}$ | 15 new songs | 0.424 | 1.035 |

${x}_{3,1}$ | 1.5% royalties for CDs | 1.540 | 1.685 |

${x}_{3,2}$ | 2% royalties for CDs | 0.763 | 1.021 |

${x}_{3,3}$ | 2.5% royalties for CDs | 1.081 | 0.926 |

${x}_{3,4}$ | 3% royalties for CDs | - | 2.715 |

${x}_{4,1}$ | $125,000 contract signing bonus | 0.991 | 1.943 |

${x}_{4,2}$ | $150,000 contract signing bonus | 1.090 | 2.615 |

${x}_{4,3}$ | $200,000 contract signing bonus | - | 4.240 |

Options | INSPIRE 1 (l = 38) | INSPIRE 2 (l = 103) |
---|---|---|

${x}_{1,1}$ | $Tr\left(0.35,0.40,\text{}0.43,\text{}060\right)$ | $Tr\left(0.29,0.39,\text{}0.40,\text{}0.53\right)$ |

${x}_{12}$ | $Tr\left(0.20,0.30,\text{}0.32,\text{}0.48\right)$ | $Tr\left(0.23,0.30,\text{}0.32,\text{}0.42\right)$ |

${x}_{1,3}$ | $Tr\left(0.10,0.20,\text{}0.22,\text{}0.40\right)$ | $Tr\left(0.13,0.20,\text{}0.24,\text{}0.39\right)$ |

${x}_{1,4}$ | $Tr\left(0.00,0.00,\text{}0.00,\text{}0.00\right)$ | $Tr\left(0.00,0.07,\text{}0.10,\text{}0.24\right)$ |

${x}_{2,1}$ | $Tr\left(0.00,0.00,\text{}0.00,\text{}0.00\right)$ | $Tr\left(0.00,0.06,\text{}0.06,\text{}0.14\right)$ |

${x}_{2,2}$ | $Tr\left(0.05,0.10,\text{}0.12,\text{}0.15\right)$ | $Tr\left(0.05,0.12,\text{}0.14,\text{}0.25\right)$ |

${x}_{2,3}$ | $Tr\left(0.10,0.18,\text{}0.21,\text{}0.27\right)$ | $Tr\left(0.08,0.18,\text{}0.21,\text{}030\right)$ |

${x}_{2,4}$ | $Tr\left(0.25,0.30,\text{}0.30,\text{}0.35\right)$ | $Tr\left(0.20,0.30,\text{}0.30,\text{}0.38\right)$ |

${x}_{2,5}$ | $Tr\left(0.20,0.21,\text{}0.25,\text{}0.30\right)$ | $Tr\left(0.10,0.23,\text{}0.25,\text{}0.34\right)$ |

${x}_{3,1}$ | $Tr\left(0.02,0.07,\text{}0.09,\text{}0.15\right)$ | $Tr\left(0.03,0.10,\text{}0.13,\text{}0.23\right)$ |

${x}_{3,2}$ | $Tr\left(0.10,0.15,\text{}0.20,\text{}0.24\right)$ | $Tr\left(0.10,0.20,\text{}0.20,\text{}0.30\right)$ |

${x}_{3,3}$ | $Tr\left(0.05,0.12,\text{}0.15,\text{}0.19\right)$ | $Tr\left(0.08,0.15,\text{}0.16,\text{}0.21\right)$ |

${x}_{3,4}$ | $Tr\left(0.00,0.00,\text{}0.00,\text{}0.00\right)$ | $Tr\left(0.00,0.05,\text{}0.06,\text{}0.14\right)$ |

${x}_{4,1}$ | $Tr\left(0.05,0.10,\text{}0.10,\text{}0.14\right)$ | $Tr\left(0.00,0.10,\text{}0.10,\text{}0.21\right)$ |

${x}_{4,2}$ | $Tr\left(0.02,0.05,\text{}0.05,\text{}0.08\right)$ | $Tr\left(0.00,0.06,\text{}0.07,\text{}0.19\right)$ |

${x}_{4,3}$ | $Tr\left(0.00,0.00,\text{}0.00,\text{}0.00\right)$ | $Tr\left(0.00,0.03,\text{}0.04,\text{}0.16\right)$ |

**Table 7.**Global utility values for negotiation packages from the set ${\mathbb{P}}^{\nabla}$ (INSPIRE 1).

No | Negotiation Packages | INSPIRE 1 |
---|---|---|

${\overline{P}}_{1}$ | 5 concerts, 11 songs, 1.5% royalties, $125 000 contract | $Tr\left(0.07,0.17,\text{}0.19,\text{}0.29\right)$ |

${\overline{P}}_{2}$ | 7 concerts, 11 songs, 1.5% royalties, $125 000 contract | $Tr\left(0.27,0.47,\text{}0.51,\text{}0.77\right)$ |

${\overline{P}}_{3}$ | 6 concerts, 12 songs, 1.5% royalties, $150 000 contract | $Tr\left(0.20,0.42,\text{}0.48,\text{}0.78\right)$ |

${\overline{P}}_{4}$ | 5 concerts, 11 songs, 2.5% royalties, $200 000 contract | $Tr\left(0.05,0.12,\text{}015,\text{}0.19\right)$ |

${\overline{P}}_{5}$ | 5 concerts, 13 songs, 3.0% royalties, $125 000 contract | $Tr\left(0.15,0.28,\text{}0.31,\text{}0.41\right)$ |

${\overline{P}}_{6}$ | 8 concerts, 15 songs, 2.5% royalties, $200 000 contract | $Tr\left(0.60,0.73,\text{}0.83,\text{}1.09\right)$ |

**Table 8.**Global utility values for negotiation packages from the set ${\mathbb{P}}^{\nabla}$ (INSPIRE 2).

No | Negotiation Packages | INSPIRE 2 |
---|---|---|

${\overline{P}}_{1}$ | 5 concerts, 11 songs, 1.5% royalties, $125 000 contract | $Tr\left(0.03,0.33,\text{}0.39,\text{}0.81\right)$ |

${\overline{P}}_{2}$ | 7 concerts, 11 songs, 1.5% royalties, $125 000 contract | $Tr\left(0.26,0.56,\text{}0.61,\text{}0.99\right)$ |

${\overline{P}}_{3}$ | 6 concerts, 12 songs, 1.5% royalties, $150 000 contract | $Tr\left(0.21,0.48,\text{}0.58,\text{}1.05\right)$ |

${\overline{P}}_{4}$ | 5 concerts, 11 songs, 2.5% royalties, $200 000 contract | $Tr\left(0.08,0.31,\text{}0.36,\text{}0.76\right)$ |

${\overline{P}}_{5}$ | 5 concerts, 13 songs, 3.0% royalties, $125 000 contract | $Tr\left(0.08,0.40,\text{}047,\text{}0.89\right)$ |

${\overline{P}}_{6}$ | 8 concerts, 15 songs, 2.5% royalties, $200 000 contract | $Tr\left(0.47,0.80,\text{}0.85,\text{}1.25\right)$ |

No | ${\overline{\mathit{P}}}_{1}$ | ${\overline{\mathit{P}}}_{2}$ | ${\overline{\mathit{P}}}_{3}$ | ${\overline{\mathit{P}}}_{4}$ | ${\overline{\mathit{P}}}_{5}$ | ${\overline{\mathit{P}}}_{6}$ |
---|---|---|---|---|---|---|

${\overline{P}}_{1}$ | 1 | 0.07 | 0.29 | 1 | 0.62 | 0 |

${\overline{P}}_{2}$ | 1 | 1 | 1 | 1 | 1 | 0.44 |

${\overline{P}}_{3}$ | 1 | 1 | 1 | 1 | 1 | 0.42 |

${\overline{P}}_{4}$ | 0.86 | 0 | 0 | 1 | 0.25 | 0 |

${\overline{P}}_{5}$ | 1 | 0.46 | 0.66 | 1 | 1 | 0 |

${\overline{P}}_{6}$ | 1 | 1 | 1 | 1 | 1 | 1 |

No | ${\overline{\mathit{P}}}_{1}$ | ${\overline{\mathit{P}}}_{2}$ | ${\overline{\mathit{P}}}_{3}$ | ${\overline{\mathit{P}}}_{4}$ | ${\overline{\mathit{P}}}_{5}$ | ${\overline{\mathit{P}}}_{6}$ |
---|---|---|---|---|---|---|

${\overline{P}}_{1}$ | 1 | 0.76 | 0.87 | 1 | 0.99 | 0.45 |

${\overline{P}}_{2}$ | 1 | 1 | 1 | 1 | 1 | 0.73 |

${\overline{P}}_{3}$ | 1 | 1 | 1 | 1 | 1 | 0.73 |

${\overline{P}}_{4}$ | 1 | 0.72 | 0.82 | 1 | 0.95 | 0.40 |

${\overline{P}}_{5}$ | 1 | 0.87 | 0.98 | 1 | 1 | 0.56 |

${\overline{P}}_{6}$ | 1 | 1 | 1 | 1 | 1 | 1 |

**Table 11.**Membership functions indicating non-dominated negotiation packages within ${\mathbb{P}}^{\nabla}$.

Negotiation Packages | ${\mathit{\psi}}_{\mathbf{max}\mathit{\mathbb{P}}}\left({\overline{\mathit{P}}}_{\mathit{i}}\right)$ | |
---|---|---|

INSPIRE 1 | INSPIRE 2 | |

${\overline{P}}_{1}$ | 0.00 | 0.45 |

${\overline{P}}_{2}$ | 0.44 | 0.73 |

${\overline{P}}_{3}$ | 0.42 | 0.73 |

${\overline{P}}_{4}$ | 0.00 | 0.40 |

${\overline{P}}_{5}$ | 0.00 | 0.56 |

${\overline{P}}_{6}$ | 1.00 | 1.00 |

**Table 12.**Absolute utilities assigned to the negotiation template elements obtained by the INSPIRE 1 method.

${\mathit{u}}_{1,1}^{\left(1\right)}$ | ${\mathit{u}}_{1,2}^{\left(1\right)}$ | ${\mathit{u}}_{1,3}^{\left(1\right)}$ | ${\mathit{u}}_{1,4}^{\left(1\right)}$ | ${\mathit{u}}_{2,1}^{\left(1\right)}$ | ${\mathit{u}}_{2,2}^{\left(1\right)}$ | ${\mathit{u}}_{2,3}^{\left(1\right)}$ | ${\mathit{u}}_{2,4}^{\left(1\right)}$ | ${\mathit{u}}_{2,5}^{\left(1\right)}$ | ${\mathit{u}}_{3,1}^{\left(1\right)}$ | ${\mathit{u}}_{3,2}^{\left(1\right)}$ | ${\mathit{u}}_{3,3}^{\left(1\right)}$ | ${\mathit{u}}_{3,4}^{\left(1\right)}$ | ${\mathit{u}}_{4,1}^{\left(1\right)}$ | ${\mathit{u}}_{4,2}^{\left(1\right)}$ | ${\mathit{u}}_{4,3}^{\left(1\right)}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Principal | 0.33 | 0.27 | 0.22 | 0.00 | 0.00 | 0.07 | 0.17 | 0.28 | 0.21 | 0.12 | 0.21 | 0.15 | 0.00 | 0.17 | 0.10 | 0.00 |

R1 | 0.35 | 0.30 | 0.25 | 0.00 | 0.00 | 0.15 | 0.22 | 0.30 | 0.25 | 0.12 | 0.22 | 0.18 | 0.00 | 0.13 | 0.08 | 0.00 |

R2 | 0.48 | 0.33 | 0.19 | 0.00 | 0.00 | 0.12 | 0.24 | 0.33 | 0.26 | 0.02 | 0.14 | 0.10 | 0.00 | 0.05 | 0.03 | 0.00 |

R3 | 0.40 | 0.30 | 0.20 | 0.00 | 0.00 | 0.05 | 0.10 | 0.30 | 0.20 | 0.15 | 0.20 | 0.05 | 0.00 | 0.10 | 0.05 | 0.00 |

**Table 13.**Rating and ranks (in brackets) of selected offers determined for the respondents’ individual and principal’s scoring systems in INSPIRE 1.

No | Principal | R1 | R2 | R3 |
---|---|---|---|---|

${\overline{P}}_{1}$ | 0.29 (5) | 0.25 (5) | 0.07 (6) | 0.25 (4) |

${\overline{P}}_{2}$ | 0.56 (2) | 0.55 (3) | 0.40 (2) | 0.55 (2) |

${\overline{P}}_{3}$ | 0.51 (3) | 0.60 (2) | 0.36 (3) | 0.45 (3) |

${\overline{P}}_{4}$ | 0.15 (6) | 0.18 (6) | 0.10 (5) | 0.05 (6) |

${\overline{P}}_{5}$ | 0.34 (4) | 0.35 (4) | 0.29 (4) | 0.20 (5) |

${\overline{P}}_{6}$ | 0.69 (1) | 0.78 (1) | 0.83 (1) | 0.65 (1) |

**Table 14.**Rating and ranks (in brackets) of selected offers determined for the respondents’ individual and principal’s scoring systems in INSPIRE 2.

No | Principal | R’1 | R’2 | R’3 |
---|---|---|---|---|

${\overline{P}}_{1}$ | 0.61 (5) | 0.13 (6) | 0.20 (6) | 0.40 (4) |

${\overline{P}}_{2}$ | 0.76 (2) | 0.44 (2) | 0.44 (3) | 0.63 (2) |

${\overline{P}}_{3}$ | 0.75 (3) | 0.33 (3) | 0.46 (2) | 0.58 (3) |

${\overline{P}}_{4}$ | 0.55 (6) | 0.15 (5) | 0.22 (5) | 0.34 (6) |

${\overline{P}}_{5}$ | 0.67 (4) | 0.24 (4) | 0.36 (4) | 0.38 (5) |

${\overline{P}}_{6}$ | 0.87 (1) | 0.80 (1) | 0.87 (1) | 0.83 (1) |

**Table 15.**Membership functions indicating non-dominated negotiation packages within subsequent ${\mathbb{P}}_{\mathrm{t}}^{\nabla}$ in the tth negotiation round for the INSPIRE 1 group.

Negotiation Packages | ${\mathit{\psi}}_{\mathbf{max}\mathbf{\mathbb{P}}}\mathbf{\left(}{\overline{\mathit{P}}}_{\mathit{i}}\mathbf{\right)}$ | |||
---|---|---|---|---|

Round 1 | Round 2 | Round 3 | Round 4 | |

${\overline{P}}_{1}$ | 0.00 | 0.07 | 0.62 | 1.00 |

${\overline{P}}_{2}$ | 0.44 | 1.00 | X | X |

${\overline{P}}_{3}$ | 0.42 | 1.00 | X | X |

${\overline{P}}_{4}$ | 0.00 | 0.00 | 0.25 | 0.86 |

${\overline{P}}_{5}$ | 0.00 | 0.46 | 1.00 | X |

${\overline{P}}_{6}$ | 1.00 | X | X | X |

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**MDPI and ACS Style**

Piasecki, K.; Roszkowska, E.; Wachowicz, T.; Filipowicz-Chomko, M.; Łyczkowska-Hanćkowiak, A.
Fuzzy Representation of Principal’s Preferences in Inspire Negotiation Support System. *Entropy* **2021**, *23*, 981.
https://doi.org/10.3390/e23080981

**AMA Style**

Piasecki K, Roszkowska E, Wachowicz T, Filipowicz-Chomko M, Łyczkowska-Hanćkowiak A.
Fuzzy Representation of Principal’s Preferences in Inspire Negotiation Support System. *Entropy*. 2021; 23(8):981.
https://doi.org/10.3390/e23080981

**Chicago/Turabian Style**

Piasecki, Krzysztof, Ewa Roszkowska, Tomasz Wachowicz, Marzena Filipowicz-Chomko, and Anna Łyczkowska-Hanćkowiak.
2021. "Fuzzy Representation of Principal’s Preferences in Inspire Negotiation Support System" *Entropy* 23, no. 8: 981.
https://doi.org/10.3390/e23080981