Advances in Fuzzy Systems

Volume 2018, Article ID 1569860, 12 pages

https://doi.org/10.1155/2018/1569860

## On Application of Ordered Fuzzy Numbers in Ranking Linguistically Evaluated Negotiation Offers

^{1}Department of Investment and Real Estate, Poznań University of Economics and Business, 61-875 Poznań, Poland^{2}Faculty of Economics and Management, University of Bialystok, 15-062 Bialystok, Poland

Correspondence should be addressed to Krzysztof Piasecki; lp.nanzop.eu@ikcesaip.fotzsyzrk

Received 22 May 2018; Accepted 26 July 2018; Published 1 November 2018

Academic Editor: Zeki Ayag

Copyright © 2018 Krzysztof Piasecki and Ewa Roszkowska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The main purpose of this paper is to investigate the application potential of ordered fuzzy numbers (OFN) to support evaluation of negotiation offers. The Simple Additive Weighting (SAW) and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) methods are extended to the case when linguistic evaluations are represented by OFN. We study the applicability of OFN for linguistic evaluation negotiation options and also provide the theoretical foundations of SAW and TOPSIS for constructing a scoring function for negotiation offers. We show that the proposed framework allows us to represent the negotiation information in a more direct and adequate way, especially in ill-structured negotiation problems, allows for holistic evaluation of negotiation offers, and produces consistent rankings, even though new packages are added or removed. An example is presented in order to demonstrate the usefulness of presented fuzzy numerical approach in evaluation of negotiation offers.

#### 1. Introduction

The process of defining, evaluating, and building a negotiation template is an important part of negotiation analysis, as well as constructing a scoring function, which is realized in the prenegotiation phase [1–3]. The negotiation template specifies a negotiation space by defining negotiation issues and acceptable resolution levels (options). The scoring function helps evaluating negotiation packages which take into account negotiator’s preferences with respect to all given issues, as well as their relative importance. Because negotiation packages are often characterized by several contradictory criteria, the multicriteria techniques are useful for evaluating negotiation offers and building negotiation-scoring functions [4]. The most popular techniques used for supporting a negotiation process are(i)Simple Additive Weighting method (SAW)/The Simple Multi Attribute Rating Technique [5, 6].(ii)The Analytic Hierarchy Process [7].(iii)Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [8, 9].

There are also studies showing applicability of other techniques for building scoring functions such as(i)Measuring Attractiveness by a Categorical Based Evaluation Technique [10].(ii)Utility Additive Method [11].(iii)Measuring Attractiveness near Reference Situations [10, 11].(iv)Generalized Regression with Intensities of Preference [11].

Each of those methods has its advantaged and disadvantages; thus selecting the “best” method for a particular problem is a really difficult task. The choice between mentioned techniques depends on the negotiation problem, types of criteria, available information, decision maker’s cognitive abilities, and properties of the multicriteria technique. In real negotiation situations, the options cannot be assessed precisely in a quantitative form, but still they may be in a qualitative one. This implies the usability of the linguistic approach for evaluating negotiation offers. The approximate technique may represent qualitative/quantitative options verbally by means of linguistic variable, i.e., variable whose values are not numbers but words or sentences in a natural or artificial language. One possibility of modelling linguistic values is the application of fuzzy sets and fuzzy numbers. The application of fuzzy sets in negotiations was competently discussed in [12].

The linguistic approach in negotiation support was considered in [10]. In this paper, we focus on the problem of extending the scale of values used in evaluating the negotiation options to include the linguistic scale that is, for example, expressions: very bad, bad, average, good, and very good together with intermediate values such as “at least good" or “at most good”*. *To deal with a problem defined in this way and involving uncertain information given in linguistic terms* at least* or* at most* we used ordered fuzzy numbers (OFN). The main contribution of the paper is the discussion about applicability of Oriented Fuzzy SAW (OF-SAW) and Oriented Fuzzy TOPSIS (OF-TOPSIS) procedure based on OFNs in scoring negotiation offers.

The main goal is to investigate the effectivity of ordered fuzzy numbers (OFN) application to support evaluation of negotiation offers. The paper is organized as follows. Section 2 outlines the mathematical formulation OFNs. The general linguistic approach based on OFNs is outlined in Section 3. The OF-SAW and OF-TOPSIS are presented in Section 4. The example of application of the OF-SAW and OF-TOPSIS methods for negotiation-scoring system is presented in the Section 5. The numerical example of using OF-SAW and OF-TOPSIS is discussed in Section 6. Finally, Section 7 concludes the article, summarizes the main findings of this research, and proposes some future research directions.

#### 2. Ordered Fuzzy Number Concept

The ordered fuzzy numbers (OFNs) were intuitively introduced by Kosiński and his cowriters [13] as an extension of the concept of fuzzy numbers introduced by Dubois and Prade [14]. OFNs are also called Kosiński’s numbers [15]. The Kosiński’s theory was revised in [16]. In this paper, we restrict our considerations to the case of trapezoidal OFN (TrOFN) defined in the following way.

*Definition 1. *For any monotonic sequence , the trapezoidal ordered fuzzy number (TrOFN) is determined explicitly by its membership functions as follows:Let us note that this identity describes additionally extended notation of numerical intervals which are used in this work.

The condition fulfilment determines the positive orientation of TrOFN . Any positively oriented TrOFN is interpreted as such imprecise number, which may increase. The condition fulfilment determines the negative orientation of TrOFN . Negatively oriented TrOFN is interpreted as such imprecise number, which may decrease. For the case , TrOFN represents crisp number , which is not oriented. The space of all TrOFNs we will denote by the symbol . The following definition fits well with the colloquial understanding of the concept of fuzziness.

*Definition 2. *A functional is called significantly fuzzy iff it is dependent on all parameters of its argument.

Kosiński has introduced the arithmetic operators of dot product of K-sum for TrOFN in following way:Kosiński has shown that there exists TrOFNs that their K-sum is not TrOFN [17]. For this reason, in [16] K-sum is replaced by the sum determined as follows:The distance between TrOFNs is calculated in the following way:Ranking OFN plays a very important role in fuzzy decision-making. Despite many ranking methods proposed in literature, there is no universal technique. Decision makers have to rank fuzzy numbers with their own intuition, preferences, and consistently with considered problem. In this paper we use concept of defuzzification technique extended for TrOFN.

*Definition 3. *Defuzzification functional is the map satisfying for any monotonic sequence the following conditions:In [18] we find following defuzzification methods:(i)the weighted maximum functional(ii)the first maximum functional(iii)the last maximum functional(iv)the middle maximum functional(v)the gravity center functional(vi)the geometrical mean functionalWe see that only the gravity center and the geometrical mean functionals are significantly fuzzy ones. Only these functionals will be applied for determining benchmarks for research described in the Section 5. Let us note that all defuzzification formulas do not imply much computational effort and does not require a priori knowledge of the set of all alternatives.

#### 3. The Linguistic Approach in Decision-Making

Information in a quantitative setting is usually expressed by means of numerical values. However, there are situations dealing with uncertainty or vague information in which the use of linguistic assessments instead of numerical values may be more useful. Linguistic decision analysis is based on the use of a linguistic approach and it is applied for solving decision-making problems under linguistic information [19]. Herrera-Viedma pointed out also that "*its application in the development of the theory and methods in decision analysis is very beneficial because it introduces a more flexible framework which allows us to represent the information in a more direct and adequate way when we are unable to express it precisely. In this way, the burden of quantifying a qualitative concept is eliminated*” [19].

In information sciences, natural language word is considered as linguistic variable defined as fuzzy subset in the predefined space . Then these linguistic variables may be transformed with the use of fuzzy set theory [20–22]. From decision-making point view*, *the linguistic variable transformation methodologies are reviewed in [1, 13, 19]. In the literature, there are many applications of linguistic decision analysis to solve real-world problems such as, e.g., group decision-making, multicriteria decision-making, consensus, marketing, software development, education, material selection, and personnel management as well as negotiation support. For review variety of application linguistic models in decision-making, see, for example, [19].

In [19] the steps of fuzzy linguistic analysis are systematically described. In general, any linguistic value is characterized by means of a label with semantic value. The label is an expression belonging to given linguistic term set. Finally, a mechanism for generating the linguistic descriptors is provided.

The semantic value meaning may be imprecise. Thus each label from applied linguistic term set is represented by fuzzy subset in the real line .

An important parameter to be determined at the first step is the granularity of uncertainty, i.e., the cardinality of the linguistic term set used for showing the information. The uncertainty granularity indicates the capacity of distinction that may be expressed. The knowledge value is increasing with the increase in granularity. The typical values of cardinality used in the linguistic models are odd ones, usually between 5 and 13. It is worth noting that the idea of granular computing goes from Zadeh [23] who wrote “*fuzzy information granulation underlies the remarkable human ability to make rational decisions in an environment of imprecision, partial knowledge, partial certainty and partial truth*.” Also Yao [24] pointed up that “*the consideration of granularity is motivated by the practical needs for simplification, clarity, low cost, approximation, and the tolerance of uncertainty”*.

In our model all linguistic terms are linked with Tentative Order Scalewhich is previously determined as discrete set of linear ordered linguistic values. Each element of Tentative Order Scale is called reference point. Reference points are ordered from the worst one to the best one. In our paper, we shall use following Tentative Order Scale:

It is obvious that each reference point is equivalent to the numerical diagnosis .

Now, we focus on the problem of Tentative Order Scale enlargement of intermediate values. For this purpose we use the phrases “at least” described by the symbol and “at most” described by the symbol .

The expression means “no worse than and worse that ”. The expression means “no better than and better that ”. Moreover, we assume that the “is better than” .

In this way we obtain extended Order Scale

Let us note that expressions and do not belong to above extended order scale.

The numerical representation of the phrases “at least” is the relation “great or equal” denoted by the GE. Thus, expression is equivalent to numerical diagnosis .

The numerical representation of the phrases “at most” is the relation “less or equal” denoted by the LE. Thus, expression is equivalent to numerical diagnosis .

In this way we obtain the Numerical Diagnosis Set

All numerical diagnoses and are imprecise ones. Therefore according to suggestion given in [25], numerical diagnosis will be represented for any by TrONF in following way:

Let us note that the sum and dot product of the numbers , , and are also TrOFN. Taking into account (19) together with (20) and (21), we determine the converting system, which transforms numerical diagnoses into performance ratings described by TrOFN in the following way:It is worth noting that the orientation OFN is used for representing relation LE (negative orientation) and GE (positive orientation).

In this way, we obtain following numerical order scale:determined by trapezoidal OFN.

The consolidated mechanism of transformation of considered linguistic values into performance ratings is presented in Table 1.