5. Procedure of Scoring Negotiation Space Using the Fuzzy Clustering Model
5.1. Fuzzy Numbers in Scoring the Limiting Profiles
The preliminary definitions of fuzzy sets, fuzzy numbers, fuzzy operations and fuzzy preferences are presented below.
Fuzzy number. A fuzzy number is defined as a fuzzy subset of the universe of discourse ℜ that is both convex and normal. The most commonly used form of fuzzy numbers is Triangular Fuzzy Numbers (TFNs).
Triangular fuzzy number. Triangular fuzzy number
is defined on ℜ as a fuzzy subset with the membership function
:
Then, TFN can be represented by , where: is the left threshold value, —the midpoint, and —the right threshold value.
Operations on TFNs. For any given two TFNs
and
and a positive real number
, the main operations of fuzzy numbers
and
can be expressed as follows:
The fuzzy weights and normalization formula. We adopt the notion of fuzzy weights proposed by Wang and Elhag [
78]. Let
be the set of criteria. Set
constitutes the set of fuzzy criteria weights, where:
) expresses the importance degrees of
, if all
are positive fuzzy numbers and
for every
.
Let us assume that the decision maker (DM) assigns to each criterion
individual positive fuzzy weights
representing the importance of the criteria
). To normalize the positive fuzzy numbers
, where:
we apply the following formula [
78]:
Fuzzy approximation procedure. Let , be fuzzy representations of negotiation options described by real numbers, respectively. Then:
- (a)
for
, we have
where:
- (b)
for
, we have
, where:
Comparison and rank ordering fuzzy numbers. To compare two fuzzy numbers, we used the fuzzy preference relation with the membership function representing the preference degree proposed by Wang [
79]. This approach seems more reasonable because defuzzification on ranking fuzzy numbers does not take into account the preference degree between two fuzzy numbers; therefore, some information may be lost.
Let
and
be two TFNs. A fuzzy preference relation
is a fuzzy subset of ℜxℜ with membership function
representing the preference degree of
over
defined by [
79] as follows:
where:
,
.
We say that is preferred to if . On the other hand, A is equal to B if .
The notion of fuzzy scoring system represented by TFN. Let
be a negotiation template (see Formula (1)). Then, the negotiation fuzzy offer scoring system can be represented by the following n + 1 tuple
where:
is a fuzzy set of weights of negotiation issues and
is a set of representation options from
by TFNs.
In this way, the fuzzy representation of package
from
has the form
, where
(
) and the final fuzzy score of package
is calculated as follows:
where:
is the fuzzy value of the
th package with respect to the
th criterion and
is the weight of the
th criterion.
5.2. Algorithm of Scoring Negotiation Space Using the Fuzzy Clustering Model
The procedure of scoring negotiation space in bilateral negotiation using the fuzzy clustering model is presented in the
Figure 4.
The steps of algorithm are as following:
Step 1. Defining the negotiation template, i.e., set of negotiation issues and the negotiation space.
Let us assume that and denote the negotiator one and two, respectively. The negotiation template for both parties has the form as described in Formula (1).
For each party, we have , where: is the set of benefit criteria, is the set of cost criteria for , where: . In the case of quantitative issues, we assume that they are monotonic.
Let as was noted in Formula (2), where: denotes an option of issue in th package, .
Step 2. Determining the fuzzy vector of the importance of issues for negotiators .
Let be the set of fuzzy weights for issues from the set for (see Formula (15)).
The weights can be determined using a linguistic evaluation scale. Let
LT denote the set of linguistic terms,
—the set of linguistic labels. An example of such linguistic scale for
is presented in
Table 1. Note, however, that some other alternative methods may be used here to elicit the issue importance, for instance, when the cognitive limitations of negotiators require deeper or more transparent facilitation of the process of preference impartation.
Step 3. Defining the —point linguistic scale for the evaluation of negotiation options represented by TFNs .
An example of a 7-point linguistic scale is presented in
Table 2.
Step 4. Defining the aspiration and reservation package for negotiators .
Let represent the aspiration package and reservation package for .
Determining the levels of aspiration and reservation is consistent with the assumptions of the negotiation analysis [
1,
5]. If the quantitative criterion is of the profit type, the aspiration level can be designated as the maximum option value, and the reservation level as the minimum option value of the given criterion. In the case of the quantitative cost type criterion, we proceed the other way round. If the criterion is qualitative, then each option is assigned a linguistic term according to the adopted scale in Step 3. Then, the level of aspiration and reservation is assigned with the highest and the lowest linguistic label, respectively, used to order the options for this criterion.
Step 5. Defining the set of limiting profiles.
The set of limiting profiles is defined in the form of complete packages and their fuzzy representation based on the linguistic scale (chosen in Step 3). The sets of limiting profiles are built by negotiators separately out of the options evaluated according to the subset of linguistic terms (SLT) from LT, . Moreover, let SL be the set of linguistic labels describing linguistic terms from SLT. We assumed that the linguistic terms representing reservation and aspiration packages are in SLT.
Let:
where:
is an option with the
—linguistic label from
for
issue (
), and
.
Let us note that
. Moreover,
is less preferred than
(
if
Now, define the set of fuzzy limiting profiles as follows:
where:
is the fuzzy profile represented by
label options and
—fuzzy representation of the
label options for the
criterion.
Step 6. Define the set of limiting fuzzy sub-profiles.
Let
be the number of divisions. The fuzzy sub-profiles
are determined in the following way:
In this way, we have ) fuzzy sub-profiles. The fuzzy sub-profiles are determined under a subjective negotiator evaluation, while for are technically calculated.
Step 7. Determining the fuzzy value of packages from the set .
Let , for The fuzzy representation of option is determined according to Formula (16) if and Formula (17) if , for some . Then, the fuzzy score of package is calculated by using Formula (21).
Step 8. Determining the set of categories based on the set of limiting sub-profiles and classifying packages to categories.
Let where: .
Moreover, note that the set of pairs
is an ordered finite set, so each pair can be assigned a natural number
in the following way (see
Figure 5).
Then, , where: denotes the number of categories and and if and only if , .
Step 9. Presenting the obtained classification of all packages for both parties of the negotiations in two-dimensional space as the points , where: , denoted the number of the cluster obtained for the same package in the case of the negotiators and .
Step 10. Determining the set of the most favorable packages for both parties of the negotiations.
Packages, which are on the efficient frontier and which are fair, must fulfill the following two conditions:
The most desirable situation is when , therefore, in a case when the given package was assigned to the same category by both negotiation parties.
6. Numerical Example
To verify the theoretical approach proposed in
Section 5.2, the numerical example, based on the negotiation case found in the eNego system [
15], will be presented. In Step 1 of the algorithm the negotiation template is defined. We consider bilateral negotiation. One party is the bicycle producer (
N1), and the second one is the parts supplier (
N2). Their aim is to negotiate a new contract for the delivery of rear-wheel gears. Four issues that are taken into account (price (
), delivery time (
), payment (
), and returns conditions (
)) and for each of them, the sets of defined options (
) made the following negotiation template (see
Table 3).
Considering all combinations within the issues, we obtained the set of all packages
where
. Examples of packages
where
are presented in
Table 4.
The levels of realizations of issues are described by means of numerical values. For the recipient (N1), the issues , are the cost issues, while is the profit. From the supplier’s (N2) point of view, the criteria , are the profit issues, while is the cost issue.
For the returns conditions (
), the negotiators provide the linguistic evaluations represented by TFNs according to
Table 2. The results are presented in
Table 5.
In Step 2, the negotiators
N1 and
N2 determine the fuzzy weights of the importance of issues using the 9-point scale linguistic evaluation from
Table 1.
Table 6 and
Table 7 present the fuzzy weights as well as the fuzzy normalized weights for both negotiators using Formula (15).
In Step 3, the 7-point linguistic scale was chosen (see
Table 2) represented by TFNs for evaluation options.
In Step 4, the negotiators define the aspiration packages as and the reservation packages as =, = for N1, N2, respectively. The reservation packages are evaluated as very poor, and the aspiration ones as very good.
Moreover, in Step 5, the negotiators are asked to choose these options for every criterion which they evaluate as poor, fair, medium good, and good. The distinguished set of linguistic terms SLT = {very poor, poor, fair, medium good, good, very good} is represented by the set of linguistic labels SL = {1, 2, 4, 5, 6, 7}, respectively. Then, we obtained the following packages with the th linguistic labels ( in order from the least preferred to the most preferred for recipient (N1): , and for supplier (N2): These packages defined the sets of limiting profiles and where for two negotiators separately. Every option in the package of limiting profiles has its fuzzy representation: , , , , and , where: and .
In Step 6, by using Formula (25) for arbitrarily taken , twenty-six limiting fuzzy sub-profiles ( where ) for each negotiator are determined. The set of all fuzzy sub-profiles for N1 is as follows:
{(0.00, 0.00, 1.25), (0.00, 0.20, 1.75), (0.00, 0.40, 2.25), (0.00, 0.60, 2.75), (0.00, 0.80, 3.25), (0.00, 1.00, 3.75), (0.47, 1.80, 4.75), (0.94, 2.60, 5.75), (1.41, 3.48, 6.75), (1.89, 4.20, 7.75), (2.36, 5.00, 8.75), (2.67, 5.40, 9.25), (2.99, 5.80, 9.75), (3.30, 6.20, 10.25), (3.61, 6.60, 10.75), (3.93, 7.00, 11.25), (4.24, 7.40, 11.50), (4.56, 7.80, 11.75), (4.87, 8.20, 12.00), (5.19, 8.60, 12.25), (5.50, 9.00, 12.50), (5.81, 9.20, 12.50), (6.13, 9.40, 12.50), (6.44, 9.60, 12.50), (6.76, 9.80, 12.50), (7.07, 10.00, 12.50)}, while for N2 it is as follows:
{(0.00, 0.00, 1.26), (0.00, 0.20, 1.76), (0.00, 0.40, 2.27), (0.00, 0.60, 2.77), (0.00, 0.80, 3.28), (0.00, 1.00, 3.78), (0.47, 1.80, 4.79), (0.93, 2.60, 5.80), (1.40, 3.40, 6.81), (1.87, 4.20, 7.82), (2.33, 5.00, 8.83), (2.64, 5.40, 9.33), (2.95, 5.80, 9.83), (3.27, 6.20, 10.34), (3.58, 6.60, 10.84), (3.89, 7.00, 11.35), (4.20, 7.40, 11.60), (4.51, 7.80, 11.85), (4.82, 8.20, 12.10), (5.13, 8.60, 12.36), (5.44, 9.00, 12.61), (5.76, 9.20, 12.61), (6.07, 9.40, 12.61), (6.38, 9.60, 12.61), (6.69, 9.80, 12.61), (7.00, 10.00, 12.61)}.
In Step 7, the global fuzzy values of all packages which are between consecutive two limiting profiles are calculated. In the further part of this example, let us consider the package
. In
Table 8, one can find the fuzzy representations
,
of options from package
for both negotiators,
N1,
N2, obtained by using Formulas (16) and (17).
Next, by using Formula (21) for fuzzy values from
Table 7 and
Table 8, we obtained the following fuzzy score of package
:
for negotiator ,
for negotiator .
In Step 8, each of the fuzzy representations of 5580 packages is compared with all 26 fuzzy limiting sub-profiles by using Formulas (18) and (19), for both negotiators. The values of for are the following:
[1.00, 1.00, 1.00, 0.97, 0.90, 0.84, 0.72, 0.62, 0.53, 0.44, 0.36, 0.33, 0.29, 0.25, 0.22, 0.19, 0.15, 0.12, 0.09, 0.06, 0.03, 0.00, 0.00, 0.00, 0.00, 0.00] for recipient (N1), and [1.00, 1.00, 1.00, 1.00, 1.00, 1.00, 0.94, 0.85, 0.76, 0.68, 0.60, 0.57, 0.54, 0.50, 0.46, 0.43, 0.40, 0.37, 0.34, 0.32, 0.29, 0.27, 0.25, 0.23, 0.20, 0.18] for supplier (N2).
According to the classification described in Step 8, the values of allow the package to be classified into the appropriate category. An exemplary package was assigned by the recipient to cluster no. 9, and by the supplier to cluster no. 13.
In Step 9, the obtained classifications for both parties are presented as the points of two-dimensional space (see
Figure 6). The horizontal axis shows the cluster number assigned to the supplier, and the vertical axis shows the cluster number assigned to the recipient.
The first step in determining the best packages for both parties is to indicate the efficient frontier. Next, among the offers on this frontier, the best ones are those which are Pareto-efficient. The best packages for both
N1 and
N2 turn out to be the following ones presented in
Table 9.
In the example shown, it is assumed arbitrarily that twenty-six limiting sub-profiles will be created in Step 6. It was also analyzed, as part of the model sensitivity analysis, whether a greater level of granularity of the division, i.e., the adoption of a greater number of limiting sub-profiles, will cause changes in the best packages for both
N1 and
N2.
Figure 7 presents four cases in which the number of sub-categories was 5, 6, 10, and 100, respectively (in the above example, 4 sub-categories were used). This number of sub-categories generates 31, 36, 56, and 506 sub-profiles, respectively (see
Figure 7).
In each of the analyzed cases, it turned out that, regardless of the increase in the number of limiting sub-profiles, exactly the same five packages turned out to be the best packages for both
N1 and
N2.
Table 10 shows the results of the category determination for each of the five analyzed packages, assuming 31, 36, 56, and 506 sub-profiles, respectively.
When negotiators have the goal of choosing not a set but one absolutely best offer, then the number of fuzzy sub-profiles should be increased. In the analyzed empirical example, using ten sub-categories (56 sub-profiles) it was possible to choose the best package.
Table 10 also presents the result of categorization of offers, assuming the presence of 506 sub-profiles. The differences between the individual packages became clearer. It was possible to develop a precise ranking of the packages. Apart from the second package, the fourth and the first package turned out to be the best.