Scenario and the moderation process for Case III

In Case III, the Pseudo-Order Preference Model (POPM) is introduced to collect consumers’preferences in the situation which does not require the users to express their preference for alternatives in complete order. The scenario of Case III is also based on

searching an appropriate flight booking service. Four consumers and ten airline services are included in Case III. Consumers have their different subjective opinions on the ingredients of the composite term Satisfaction. In Case III, six primitive terms comprise the composite term, Satisfaction, to represent the overall quality of a flight ticket. These primitive terms, or alternatives, are Cheap(a ), MultimediaEquipment(₁ a ), Food(₂ a ), Airtime(₃ a ), Seatsize(₄ a ),₅

and FlightServiceOfCrew(a ).₆ It is assumed that consumers in Case III cannot easily distinguish the relative importance of some of the alternatives and only the top-3 alternatives are required, hence gaining better performance by reducing the system complexity.

First, POPM can be adopted to prioritize the order of various alternatives by identifying the relatively most important criteria (top-3) accepted by the four consumers in order to filter out the remaining three less significant criteria. For each consumer k, his / her preference over these six alternatives is not collected (transformed) from a complete order of the sorted alternatives. Instead, it is gathered pair by pair by using preference relations ( p_ij^k ) [66],[68],[75], as follows:



In fuzzy preference relations, the importance of alternatives is collected pair by pair.

For example, p₁₆² p₁₆³ 1 means that both User₂ and User₃ completely prefer a to₁

a .6 p₅₁¹ 0.6 and p₁₅¹ 0.4 indicate that User₁ slightly prefer a to₅ a .₁

Each consumer possesses a fuzzy preference relation and all fuzzy preference relations can be aggregated to calculate the collective preference relation ( p_ij^c) by Equation (8) or (17) (Section 3.3.2, Section 3.4). In this experiment, the linguistic quantifier ‘most’with pair [0.3, 0.8] is applied to conclude that“mostofthe consumers agree to six alternatives / criteria for the flight booking service discovery”. The corresponding OWA operator with the weighting vector for ‘most’will be w(0.00,0.40,0.50,0.10) and the p_ij^c is as follows:

By applying various thresholds (q=0.1~0.9) and Equations (18~19), the distinct preference order for six alternatives can be derived, as shown in Table 5-29. Finally, the grouped preference order for the six primitive terms is determined when a common indifference threshold is applied [72]. To identify a distinct top-3 it can be seen from the following table that q = 0.3, 0.4 or 0.5 will provide such a division. In these cases {Cheap( a ), Airtime(₁ a ), Seatsize(₄ a )} > {MultimediaEquipment(₅ a ), Food(₂ a ),₃ FlightServiceOfCrew(a )}.₆

Table 5-29 Derived distinct preference orders for six alternatives with various thresholds Indifference

Threshold Derived Preference Order q=0.1 {a₁,a₄}{a₅}{a₂}{a₃,a₆} q=0.2 {a₁,a₄}{a₅}{a₂,a₃,a₆} q=0.3 {a₁,a₄,a₅}{a₂,a₃,a₆} q=0.4 {a₁,a₄,a₅}{a₂,a₃,a₆} q=0.5 {a₁,a₄,a₅}{a₂,a₃,a₆} q=0.6 {a₁,a₄,a₅,a₂}{a₃,a₆} q=0.7 {a₁,a₄,a₅,a₂}{a₃,a₆} q=0.8 {a₁,a₄,a₅,a₂,a₃,a₆} q=0.9 {a₁,a₄,a₅,a₂,a₃,a₆}

According to Table 5-29, the top-3 alternatives (a ,₁ a ,₄ a ) are selected to be the most₅ important primitive terms which are used in Fuzzy Moderator for calculating the consensus weightings. These three primitive terms will be denoted as Cheap(C~

), Airtime(T~ ), and Seatsize(S~

). The four consumers should bring themselves to an agreement over the definition of these three primitive terms (C~

,T~ ,S~

). For instance, Cheap was initially defined as C~_init (0,0,14500,16500). However, the four consumers have different views on these definitions which are formulated as the following fuzzy sets:

In the following, Similarity Aggregation Method (SAM) is applied to calculate the consensus value for Cheap(C~

), Airtime(T~

), and Seatsize(S~

). After the application of

SAM, the initial subjective value, C~_init (0,0,14500,16500), which was given for the Fuzzy Classifier to carry out reasoning has been modified to the new derived consensus value:

14925.007) 13314.333,

0,

~ (0,

C . The same principle is applicable to the other two

primitive fuzzy terms Airtime(T~

) and Seatsize(S~

), and therefore their consensus values are )

9379 . 2 , 0666 . 2 , 0 , 0

~ (

T and S~ (0.8911,1.2008,2.0105,2.5181) respectively. For more detailed procedure, please refer to Section 5.1.1.

When the preference order and consensus value of the three primitive terms have been resolved, it is able to adopt the RMGDP process (Section 3.3) to carry out transformation, aggregation, and exploitation processes in order to reach a consensus on the weightings of criteria which comprise the composite terms Satisfaction.

Assume that each consumer provides his / her preferences on a list of alternatives, }

, , {a₁ a₅ a₄

A where a is Cheap,₁ a₄ is Airtime, and a is Seatsize, using a preference₅

ordering O^k {o₁^k,o₂^k,..,o_m^k} (m is the number of alternatives). For instance, each consumer k, denoted as User_k(k1,2,3,4), provides his / her preferences on alternatives by the following preference ordering O¹ {3,12} , O² {1,3,2} , O³ {1,2,3} and

} 1 , 3 , 2

4 {

O . For any two ordering preference values, o ,_i^k o^k_j , assessed by User_k , a

difference-scale transformation function, p_ij^k in Equation (7), shows that User_k has a subjective ordering preference of the alternative a_i over alternative a_j . For each

consumer, the preference ordering O^k {o₁^k,o₂^k,..,o_m^k} can be transformed into fuzzy

preference relation (p_ij^k) as follows:

After transforming preference orderings into fuzzy preference relations, the collective preference relation ( p_ij^c) can be calculated by Equation (8). In this case, consumers’ opinions are treated on an equal basis so that the corresponding OWA operator with the weighting vector will be w = (1/4, 1/4, 1/4, 1/4) and p_ij^c is as follows:

Moreover, Quantifier Guided Non-Dominance Degree (QGNDD) and Quantifier Guided Dominance Degree (QGDD) could be obtained by using Equation (10, 11), and the consensus of four consumers is reached as shown in Table 5-30. In the exploitation process, the derived values of QGDD and QGNDD are used to determine the complete order and weightings for each alternative. The complete order of primitive terms is the same as the preference order of the service criteria. The consensus weightings, as shown in Table 5-30, for alternatives derived from QGDD and QGNDD are formulated as

)

weighting for Cheap is 0.3959 when consensus weighting from QGDD is adopted. Thus, the initial composite term, Satisfaction, with new consensus weightings can be moderated as:

Q~

= 0.3959 × C~

+ 0.3333 × T~

+ 0.2708 × S~

Table 5-30 QGDD, QGNDD and the consensus weightings for alternatives

0.5938 0.5000 0.4063

Consensus Weightings for

Alternatives

from QGDD 0.3959 0.3333 0.2708 a1

Alternatives 1.000 0.9375 0.8125

Consensus Weightings for

Alternatives

from QGNDD 0.3636 0.3409 0.2955