1. Introduction
Negotiation is the process of exchanging offers, concessions, and argumentation, where conflicting issues need to be evaluated [
1]. In the pre-negotiation phase, negotiators must structure the negotiation problem, build the negotiation template, elicit their preferences and build negotiation scoring systems. Such scoring systems help in the further negotiation process in evaluating offers, measuring the scale of concessions, and estimating negotiation progress. Finally, in the post-negotiation phase, they can be used to search for fair and satisfying final solutions or improvements to the negotiated agreement [
2].
The evaluation of negotiation offers is possible if each party thoroughly analyzes their preferences during the pre-negotiation phase. Such an analysis requires mapping the preference information expressed by the decision maker, usually verbally, in the direct discussion with an analyst or facilitator, into the system of corresponding quantitative meanings. Then, this quantitative preference information needs to be organized and processed according to the previously recognized DM’s individual preference model. The latter should be done particularly diligently to ensure that the scoring system and scoring formulas reliably copy the DM’s intrinsic behavior in using their preferences for evaluating the negotiation offers. Therefore, such a preference analysis is usually conducted with the support of various multiple criteria decision-making (MCDM) or fuzzy multiple criteria decision-making (FMCDM) methods, as from the viewpoint of individual negotiator analyzing preferences for multi-issue negotiation resembles analyzing the preferences in any single-DM MCDM problem [
2,
3,
4]. The choice between multiple criteria techniques depends on the negotiation problem, types of issues, available information, and properties of the multiple criteria technique, among others. MCDM methods are particularly useful where the negotiation problem is well structured, i.e., the issues and options can be precisely specified while defining the negotiation problems, and their evaluations (e.g., weights) can be measured using crisp numbers. The MCDM methods used to determine the scoring systems in such situations are DR (Direct Rating) [
5], AHP (Analytic Hierarchy Process) [
6], TOPSIS (Technique for Order Preferences by Similarity to Ideal Solution) [
7], MARS (Measuring Attractiveness near Reference Situations) [
8,
9], UTA (UTilités Additives) [
10], ELECTRE (ÉLimination Et Choix Traduisant la Realité) [
11], among others. Within this group of MCDM methods, especially vital are those based on a reference point (ideal solution) or two reference points (ideal and anti-ideal solutions) such as TOPSIS, VIKOR (Serb. Vlse Kriterijumska Optimizacija i Kompromisno Resenje) [
12], and BIPOLAR [
13]. The ideal solution can correspond with the aspiration level defined in pre-negotiations, while the anti-ideal solution does with the reservation level [
14]. On the other hand, FMCDM can be applied in the ill-structured negotiation problem where the negotiators express ratings and criteria weights imprecisely, subjectively, or vaguely. Such imprecise evaluations can result from the lack of information, measurement error, cognitive limitations, or subjective evaluation of the options, which are often observed in real-life negotiation [
15]. Applications of fuzzy multiple criteria decision-making methods to determine the scoring systems can be found in many papers, e.g., [
7,
15,
16,
17].
In some situations, it may be helpful and more flexible to operate with the linguistic evaluation of options and offers and describe preferences naturally and intuitively, e.g., when qualitative negotiation issues need to be considered. An example of such a variable in typical business negotiations may be the returns policy, terms of warranty, or quality. The numerical values may also be evaluated through linguistic variables if the granularity of such an evaluation is sufficient for the supported negotiator. This granularity, i.e., the cardinality of the linguistic term set used in preference declarations, should be small enough not to provide the negotiator with too many evaluation options to declare (a useless precision). On the other hand, it should be rich enough to allow the discrimination of the assessments in a limited number of degrees.
The linguistic values can be represented in various ways, e.g., through fuzzy sets [
18,
19], intuitionistic fuzzy sets [
20], or ordered fuzzy sets [
17]. The linguistic approach in negotiation preference elicitation and support has been considered so far in a few papers (see, e.g., [
17,
21]). In paper [
17], the scale of values used in evaluating the negotiation options included the following expressions: very bad, bad, average, good, and very good together with intermediate values such as “at least good” or “at most good” and was represented by oriented fuzzy numbers (OFNs). This scale was used to verify the applicability of the Oriented Fuzzy SAW (OF-SAW) and Oriented Fuzzy TOPSIS (OF-TOPSIS) methods based on oriented fuzzy numbers in scoring negotiation offers.
The motivation for this study is the following. We want to take advantage of oriented fuzzy numbers represented in linguistic terms for dealing with unprecise information in evaluating negotiation offers presented in [
17]. The second motivation is applying Hellwig’s framework, a progenitor to TOPSIS and VIKOR, for building a negotiation scoring system. Two variants of the extended linguistic Hellwig’s method, i.e., Oriented Fuzzy Hellwig’s methods (OF-Hellwig’s), are presented and compared. The first one, named
, uses one reference point, and the other,
, operates with two reference points. Although various researchers have earlier proposed many modifications of both variants of Hellwig’s method (see Tables 2 and 3), this paper’s novelty is the application of oriented fuzzy numbers in the modified Hellwig’s measure and their use to rank-ordering negotiation offers. We also compare these methods with OF-TOPSIS and OF-SAW in an illustrative example. The TOPSIS method is based on the concept that the chosen alternative should be the closest to the positive ideal solution and the farthest from the negative ideal solution [
22]. On the contrary, the first variant of classical Hellwig’s method takes into account the distance to the positive ideal solution only [
23], while in the second variant, the distances between the best (positive ideal) and the worst (negative ideal solution) solutions are used in the normalizing measure.
The advantages of this new approach are the following:
It allows for linguistic evaluation negotiation offers;
The scoring procedure implemented does not need the normalization procedure for options;
It allows using either one or two reference points;
In , procedure rank reversal can be avoided when new offers are added to the evaluation process; additionally, also avoids changing scores points;
Both Hellwig’s methods based on oriented fuzzy numbers are intuitive and easy tools for rank ordering negotiation offers and can be alternatives to the methods presented in [
17], i.e., OF-TOPSIS and OF-SAW.
The remainder of this paper is structured as follows. In
Section 2, reviews of oriented fuzzy sets and linguistic approach are briefly outlined. The classical Hellwig’s methods (
and
) and their extended variants based on a linguistic approach represented by oriented fuzzy numbers (
and
) are presented in
Section 3. The case study and implementation of the proposed method for evaluating negotiation offers are then presented in
Section 4. It additionally provides a discussion and comparative analyses of the proposed method to the current approaches showing the advantages and disadvantages of the former in
Section 5. In
Section 6, concluding remarks and future research directions are presented.
5. Discussion
Let us first note that all techniques considered in
Section 4, i.e., OF-SAW, OF-TOPSIS, and both OF-Hellwig’s methods, allow for assigning a quantitative score to each offer and ordering them from best to worst. It is a desired property of the negotiation offer scoring system because we can estimate the value of each offer and counter offer and evaluate the value of the concessions made. What is also important is that all considered methods are based on a verbal evaluation of options using the linguistic scale. It allows us to implement them in the same negotiation situation, in which parties cannot (e.g., due to some cognitive limitations) classically evaluate the template employing crisp values assigned, for instance, through the direct rating approach. Consequently, Steps 1–3, which are used in both OF-Hellwig’s methods to score the negotiation template (and described in
Section 3.3), are also performed in the same way in OF-SAW and OF-TOPSIS.
Similarly, all the methods require defining the issue importance in the form of a vector of weights (as shown in Step 4 for the OF-Hellwig’s procedure). As a result, they all use the same formally defined scoring system. What makes the methods different is the aggregation procedure used to produce the global scores of alternatives. Below, we discuss the main methodological differences in algorithms. In particular, we pay attention to similarities and dissimilarities among methods and their advantages and disadvantages when applied to support building a negotiation scoring system.
Let us observe that linguistic evaluation of negotiation options eliminates the problem of differentiation between benefit and cost criteria and the necessity of criteria normalization (see Step 4, Formulas (18) and (19)). The and measures are normalized in the range [0, 1], but is not. The latter allows some offers to be evaluated below 0. These are the worst offers in the negotiation space, the performance of which is so bad that they occur more distant from PIS than those with a distance equal to average plus two values of standard deviation.
The and measure require building both an ideal solution (PIS) and an anti-ideal (NIS) solution, while requires only an ideal solution. The measure does not require such reference points. Those reference points can be considered aspiration (PIS) and reservation (NIS) levels in the negotiation analysis. We identify the PIS as a solution consisting of the maximum values from the scale. Simultaneously, we identify the NIS as a solution consisting of the minimum values from the scale. Let us recall that the OF-TOPSIS method is based on the idea that the chosen alternative should be the closest to the positive ideal solution and the farthest from the negative ideal solution. Yet, in OF-Hellwig-based techniques, the chosen alternative should be the closest to the positive ideal solution, although in the distances between the best (positive ideal) and the worst (negative ideal) solutions were used to normalize the measure. It is challenging to univocally confirm which use of reference points results in the definition of the global scores more precisely and adequately reflecting the negotiator’s preferences. It seems it should be a DM’s individual decision based on prior training in using all these methods that would show the hands-on results of each approach using some numerical examples.
The advantage of constructing
PIS and
NIS using TrOFNs corresponding to the minimum and maximum levels from the linguistic scale is that if a new offer is included or one of the existing ones is removed from
, there is no need to reevaluate the offers previously scored through
and
techniques. In addition, scores of all alternatives remain stable. Thus, those techniques avoid rank reversals. Unfortunately, it does not hold for the
technique, which is normalized by aggregated scores of alternatives. Therefore, adding or removing alternatives provides a stable ranking, but the scores of alternatives can be changed. A good example of how significant these changes can be is shown in
Table 10 below, where the global scores of 15 alternatives obtained through
are shown, first determined based on the limited negotiation space
(consisting of 15 alternatives) and then based on the entire space of feasible alternatives
(consisting of 7176) built out of the template presented in
Table 5.
Such differences are the result of the peculiarity of the scoring formula, which is sensitive to the number and performance of all alternatives m that are processed to determine the average distance, standard deviation across the negotiation space and value according to Formula (28). Even if differences may not be significant for some offers, as for P15, where the discrepancies reach only rating points, for others, they may reach as much as 0.1 rating points, e.g., for P1 and P9. It means nearly 10% of the entire rating space. Such a situation does not seem comfortable from the perspective of negotiation support, as the negotiator receives an ambiguous recommendation on the potential good quality of the same offers and is left confused about which scores should be considered a reliable basis when analyzing the potential concessions.
It is also worth noting that even though does not use the second reference point (NIS) explicitly in preference declarations, it is implicitly used to determine the global score. An equivalent of NIS is considered an offer, the distance of which to PIS equals an average value of all distances in negotiation space plus two values of their standard deviation. It is used to set a reference threshold of a global score equal to 0. Hence, despite the global values in not being limited to [0; 1]-range as they are in and , those that fit this range may be considered as varying between two reference points: the best and the “very bad” one. additionally allows considering some offers to be “worse than very bad” and scoring them with negative values if they are worse than the one implicitly defined as NIS (with a score equal to 0).
The discussion above makes us propose an additional minor modification to the
-based Hellwig’s method to determine the global scores that would better fit the negotiator’s individual problem understanding. Instead of using an implicit reference point that refers to what statistics consider an outlier but which may be too abstract for any negotiation party, the negotiator may clearly define what their BATNA is pointing out to one (or more) borderline offers, i.e., the one (ones) that embody maximum concessions they are going to offer to their counterpart. Then, in Formula (28), the denominator in the subtracting fraction
should be determined as
where:
is the set of alternatives defined by the negotiator as their BATNA offers, and
.
Another issue should also be considered when comparing OF-Hellwig’s approaches to the OF-SAW one. The latter is based on a weighted average of performance ratings and results in the global scores represented by oriented fuzzy numbers. To compare the offers and determine the scale of concessions between them, the negotiator has to use one of the methods of defuzzification (see
Table 1). However, various defuzzification approaches process imprecise information differently, resulting in different evaluations of alternatives and offer rankings. Again, the selection of an “appropriate” defuzzification approach requires additional cognitive effort from the decision maker to understand how these approaches work and what are the numerical consequences of selecting each of them.
Analyzing the differences between the methods using the example shown in
Section 4.2 also provides some interesting insights from the viewpoint of negotiation analysis and support. The results obtained from two OF-Hellwig’s methods and OF-TOPSIS and OF-SAW may, at first glance, look similar. However, more detailed analyses reveal the differences that may have a crucial impact on further use of the global ratings obtained from each method in the forthcoming actual negotiations phase. First, if both Hellwig’s methods and OF-TOPSIS are compared, one may easily notice that all three produce the same offers rankings. Apart from the Kendall correlation coefficient, which is naturally equal to 1 for all pairs of ranking considered, the Person correlations among the series of offers ratings are also exceptionally high (>0.999). The data series visualizing the differences in rating for all three methods and the 15 alternatives evaluated with the negotiation space
are shown in
Figure 4.
From the viewpoint of developing the concession strategy in prenegotiation [
1], all three methods indicate the same sequence of offers to make the subsequent minimal concessions. However, the conclusions vary if we analyze how Itex may interpret the relative value of subsequent offers and the scale of concessions related to submitting the successive packages from the list.
and
measures generate nearly the same offer ratings and assure that Itex will interpret the quality of offers likewise, no matter which scoring aggregation mechanism is used. For instance, P15 is evaluated equally as fairly good by these methods, i.e.,
=
= 0.73. The next offer is also evaluated the same by both methods, i.e.,
=
= 0.63. Consequently, the concession is considered worth 0.1 rating points, no matter which method is used. However, if
is applied, the relative interpretation of offers value and concessions differ, i.e.,
= 0.68, which is 0.05 rating points worse than when evaluated by
or
. One may consider such difference to be minor, but note that the rating space in the OF-TOPSIS and OF-Hellwig’s
H2 methods is scaled to the [0; 1]-range (and for
H1, this range still allows for considering the distance between the very best and the significantly bad offer); thus, the difference of 0.05 means (at least) five percentage points. The difference in interpreting offers quality increases more and more for subsequent offers in the order depending on whether
-based ratings or those resulting from
or
are used (the gap between blue/orange line and the grey one increases while moving to the right). The least attractive offer P1 is scored 0.2 by
and
-based rating formulas, but only 0.04 according to
. The discrepancy is significant. Similarly, the interpretation of the scale of concessions may differ heavily for those methods. For instance, the first concession, made when Itex resigns from P15 and submits to the negotiation table P13 instead, is worth
=
=
but as much as
points when
-based scoring system is used. Even minor differences in measuring the amounts of concessions may impact negotiators’ attitudes toward the counterpart and perception of reciprocity. Therefore, it is crucial which of these methods will be implemented to support negotiators, and the meanings of the scores and scale of scoring space should be thoroughly explained to them to avoid future misinterpretations of negotiation moves. All these discrepancies in the evaluation of the offers discussed above have a systemic character and influence all the offers within the entire negotiation space defined for this negotiation problem by the template
, i.e., when the negotiation space
consisting of 7176 offers is evaluated (see
Figure 5).
What also should be noticed is that
and
generate ratings that produce more similar ranks of offers than those obtained from
. It can be easily seen from
Figure 5 that
and
consequently decrease for subsequent negotiation offers (when moving from left to right of the chart) in regular and symmetric deviations. This cannot be observed for
, which, despite generating ratings similar in values to those from
, results in ranks that frequently differ. It can be observed through the regular picks of the blue data series when confronted with the smooth decrease of the grey one.
Given the above, the comparison of results obtained from OFN-based reference-points-based approaches (i.e., TOPSIS and Hellwig) with those coming from OF-SAW and various defuzzification techniques seems interesting. To show the similarities in the evaluations on a common graph, we normalized the OF-SAW ratings from
Table 9 using max–min normalization with scores 1 and 5 as min and max, respectively. The global scores of 15 offers are shown in
Figure 6.
Generally, we may observe that OF-SAW-based global ratings indicate that offers are more attractive to Itex. Their relative scores are higher (after scaling them) than the corresponding ones obtained from OF-TOPSIS or OF-Hellwig approaches. The differences in offers evaluations, important from the viewpoint of measuring the scale of the concessions, are even smaller for measure than for or - (the data series are far more flat). As a result, interpreting the value of the worst offer considered by Itex may bring different conclusions, depending on the approach used. Offer P1 evaluated by any OF-SAW techniques will be considered somewhat weak (with a rating of about 0.36 points), while from the viewpoint of -based rating it would be totally unacceptable (with a rating of 0.04). Again, the discrepancy in evaluating the best offer (P15) will not be so evident.
6. Conclusions
In this paper, we proposed two variants of the OF-Hellwig’s approaches as extensions of the classical Hellwig’s method and analyzed the usability of those techniques in negotiation support.
The contribution of this paper is as follows. Firstly, we proposed two variants of Hellwig-based methods based on oriented fuzzy numbers and a linguistic approach. The oriented fuzzy numbers are used to deal with imprecise and incomplete information in the negotiation process. In the first classical variant (, we used Hellwig’s proposition of normalization measure based on the average and the standard deviation of distances between alternatives and the ideal solution. In the second variant (), we used a normalized measure based on the distance between ideal and anti-ideal solutions represented by a linguistic scale which makes the OF-Hellwig’s algorithm more straightforward and intuitive than the classical approach.
Secondly, we showed how these fuzzy Hellwig-based techniques might be applied for building a negotiation scoring system in multi-issue negotiations and analyzed their usability in the illustrative example. Finally, the proposed methods were compared with other techniques proposed earlier in the literature, i.e., SAW and TOPSIS, based on the oriented fuzzy numbers. In this comparison, we considered both the technical properties of algorithms and the results from the illustrative example. The advantages and disadvantages of all techniques seen from the viewpoint of the negotiation support focused on building a scoring system were also outlined.
The comparative analysis showed that the measures might be an alternative for other techniques based on oriented fuzzy numbers such as or . The main advantage of OF-Hellwig’s techniques compared to OF-SAW is that it does not use defuzzification formulas. In contrast, compared to OF-TOPSIS, it may be based on one reference point (ideal solution, as in ) or adopt the second reference point to the individual needs of the negotiators (additional definition of BATNA used to normalize the measure). Finally, comparing and , the latter seems simpler and more intuitive and additionally avoids changing scores when a set of offers is modified. However, we should note that DM has to decide which of the methods will be the most useful in a specific negotiation situation and which of them best reflects preferences through the global score points. Let us recall that in our example, and produced highly similar scores of offers, though they may result in different rankings. At the same time, and algorithms produce highly similar rankings but with quite high differences in scoring points. Consequently, using different methods will impact the interpretation of not only the offers’ scores, but also the entire negotiation process, i.e., the concession made by both parties, the negotiation progress and the agreement. Additionally, not discussed broader in this paper, it will have a significant impact on the recommendation of compromise improvement in the post-negotiation phase, as well as the results of the potential negotiation support offered by the third party (e.g., the suggestions of fair agreements made by the arbitrator).
Our future research will be focused on empirically verifying the negotiators’ acceptability of the OF-Hellwig’s methods. The negotiation experiments conducted with the support of and measures could allow us to verify if the reception of these negotiation support mechanisms is positive and whether it may depend on the cognitive profiles of the parties. Additionally, the applicability of these approaches may also be verified in decision-making contexts other than negotiations.