*7.2. Illustrative Example*

An illustrative example (based on consumer's buying behavior) for eliciting the numerical applicability of our proposed approach is given below:

In a company's production oriented decision-making processes, consumers or buyers play a vital role. In order to increase sales and to be in good books of every customer, every production company pays a grea<sup>t</sup> attention to customer's buying behavior. This consumer behavior is the main driving force behind the change of trends, need of updation in the products etc., to which the production company must remain in contact to have a grea<sup>t</sup> mutual relationship with the customers and to maintain a strong position in the competitive market environment.

Suppose a multi-national company wants to launch the new products on the basis of different consumers in different countries. For that, they have delegated works to the company heads of three different countries viz. India, Canada, and Australia. The company heads of these countries have to

analyze the customer's buying behavior and for that, they have information available in the form of PDHFEs. Each expert (*d* = 1, 2, 3) from the three different countries accessed the available information oriented to four company products *Ai*'s where (*i* = 1, 2, 3, 4) classified under four criteria determining the customer's buying behavior namely *C*1 : 'Suitability to cultural environment'; *C*2 : 'Global trend accordance'; *C*3 : 'Suitability to weather conditions' ; *C*4 : 'Good quality after-sale services'. The aim of the company is to access the main criteria which affect the customer's buying behavior so as to figure out which product among *Ai*'s (*i* = 1, 2, 3, 4) has to be launched first. Following steps are adopted to find the most suitable product for the first launch.

Step 1: The preference information corresponding to three decision-makers (d = 1; 2; 3) is given in Tables 1–3.


**Table 1.** Preference values provided by decision-maker 1.


**Table2.**Preferencevaluesprovidedbydecision-maker2.

**Table 3.** Preference values provided by decision-maker 3.


Step 2: Since number of decision makers i.e., *d* ≥ 2, therefore, using Algorithm 1, the comprehensive matrix obtained after integrating all the preferences given by the panel of experts is given in Table 4.



*Q*1 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.52130.0056, 0.54390.0006, 0.55460.0154, 0.57600.0017, . . . . . . . . . . . . , 0.63470.0037 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.26170.0444, 0.25310.0222, 0.19090.0222, 0.18440.0111, . . . . . . . . . . . . , 0.31200.0074 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠ *Q*2 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.60800.0469, 0.62010.0614, 0.48380.0253, 0.49850.0331, . . . . . . . . . . . . , 0.42400.0157 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.25310.0123, 0.23590.0198, 0.22660.0049, 0.53720.0062, . . . . . . . . . . . . , 0.64270.0025 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠ *Q*3 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.33840.0173, 0.33520.0173, 0.35150.0173, 0.39630.0074, . . . . . . . . . . . . , 0.73790.0123 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.43910.0444, 0.42510.0346, 0.42560.0691, 0.22260.0222, . . . . . . . . . . . . , 0.15400.0026 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠ *Q*4 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.42250.0474, 0.40360.0474, 0.39470.0474, 0.46670.0435, . . . . . . . . . . . . , 0.31100.0078 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.30160.0017, 0.34130.0039, 0.36980.0111, 0.25330.0006, . . . . . . . . . . . . , 0.52590.0197 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠


Thus, it is clear that according to the experts product *A*1 should be launched first.

However, on the other hand, if we utilize the PDHFWEG operator instead of PDHFWEA operator to aggregate the different preferences, then the following steps of the proposed approach are executed to reach the optimal alternative(s) as.


*Q*1 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.39590.0056, 0.40920.0006, 0.46420.0154, 0.47920.0017, . . . . . . . . . . . . , 0.59080.0037 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.29170.0444, 0.28270.0222, 0.20080.0222, 0.19130.0111, . . . . . . . . . . . . , 0.35410.0074 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠ *Q*2 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.50900.0469, 0.54150.0614, 0.43910.0253, 0.46850.0331, . . . . . . . . . . . . , 0.29590.0157 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.39500.0123, 0.33120.0198, 0.30780.0049, 0.63760.0062, . . . . . . . . . . . . , 0.65160.0025 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠ *Q*3 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.16670.0173, 0.16150.0173, 0.18280.0173, 0.29500.0074, . . . . . . . . . . . . , 0.61640.0123 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.48900.0444, 0.46460.0346, 0.52030.0691, 0.32560.0222, . . . . . . . . . . . . , 0.27420.0026 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠ *Q*4 = ⎛⎜⎜⎝⎧⎪⎪⎨⎪⎪⎩0.41500.0474, 0.39810.0474, 0.38860.0474, 0.45800.0435, . . . . . . . . . . . . , 0.27740.0078 ⎫⎪⎪⎬⎪⎪⎭ , ⎧⎪⎪⎨⎪⎪⎩0.33950.0017, 0.37440.0039, 0.41570.0111, 0.29740.0006, . . . . . . . . . . . . , 0.56560.0197 ⎫⎪⎪⎬⎪⎪⎭⎞⎟⎟⎠

Step 5: The score values are obtained as *<sup>S</sup>*(*Q*1) = 0.0937, *<sup>S</sup>*(*Q*2) = −0.0073, *<sup>S</sup>*(*Q*3) = −0.0202 and *<sup>S</sup>*(*Q*4) = −0.0545

Step 6: Since, the ranking order is *<sup>S</sup>*(*Q*1) > *<sup>S</sup>*(*Q*3) > *<sup>S</sup>*(*Q*2) > *<sup>S</sup>*(*Q*4), thus the ranking is obtained as *A*1 *A*2 *A*3 *A*4.

The most desirable alternative is *A*1.

If we analyze the impact of the all the proposed operators along with the distance *d*1 and *d*2 onto the final ranking order of the alternative, we perform an experiment where the steps of the proposed algorithms are executed. The final score values of each alternative *Ai* (*i* = 1, 2, 3, <sup>4</sup>), are obtained and are summarized in Table 5. It is seen that utilizing different distance measures i.e., *d*1 and *d*2 do not affect the best alternative *A*1 in most of the cases. Moreover, the score values obtained by the proposed operators namely: PDHFWEA, PDHFWEG, and PDHFOWEG represent the same alternative *A*1 as the best alternative which is to be launched first while the operator PDHOWEA represents the alternative *A*3 as the best one. However, it can be seen that corresponding average PDHFWEA, PDHFOWEA score values are greater than that of PDHFWEG, PDHFOWEG aggregation operators showing that the average aggregation operators offer the decision maker more optimistic score-values as compared to the geometric ones. Also, it can be seen that both the distances, despite providing, a huge variation in numerical evaluation and data processing flexibility lead to the same result as *A*1 as the best choice in most of the cases among the alternatives to be launched first.


**Table 5.** Score values of proposed approach.
