## 4 Illustrative Example
### Quality of Solutions  
The 47 inventory items' optimal scores and weights are shown in Table 1. As demonstrated, the optimal scores derived by using the CCM algorithm to solve the WPM of multicriteria ABC classification were as good as those derived using LINGO [18], the off-the-shelf optimization software. If we look carefully at the weights derived using LINGO, it is obvious that most item weights for ADU and AUC were identical, and that some item weights (4, 25, 27, and 30) for LT were zero. This is because the primary underlying technique used by LINGO's nonlinear solver is to reach a feasible solution for nonlinear models quickly. The weight values derived by using the CCM better fit the assumption that the criteria were graded in descending order, such that \(w_{i,1}\geq w_{i,2}\geq\cdots\geq w_{i,l}\) for all items \(i\). Therefore, the CCM is superior to LINGO in terms of solution quality in this illustrative example. Next, the maximal overall scores were sorted in descending order, and inventory classification was conducted based on the WPM (shown in Table 2). For comparison purposes, we maintained the same distribution of items in the A, B, and C classes as in studies within the cited literature [1; 13; 17]; that is, there were 10 class A items, 14 class B items, and 23 class C items. The ABC analysis using the Ng [1], Hadi-Vencheh [13], and Zhou and Fan (ZF) models [17] are also shown in Table 2. There were ten items (8, 29, 15, 16, 27, 33, 39, 40, 34, and 45) that did not have the same classifications in the WPM model as in the Ng, HV, and ZF models. The difference in classification was due to the difference in score computation schemes. Of the 10 class A items identified in the WPM, only item 8 was recognized as a class B item in the Ng, ZF, and HV models. Moreover, in these models, item 29 was classified as a class A item, while the WPM reclassified it as a class B item. A comparison of items 8 and 29 revealed that item 8 was superior to item 29 in terms of ADU value (\(y_{8,1}=2640>y_{29,1}=268.68\)). Although item 29 outperformed item 8 in AUC (\(y_{8,2}=55<y_{29,2}=134.34\)) and LT (\(y_{8,3}=4<y_{29,3}=7\)), the differences were not significant. Therefore, based on the most important consideration, the value of the annual consumption of inventory items (ADU) in a year, the WPM provided a more reasonable classification. Regarding the 14 class B items in the HV model, eight items (6, 7, 23, 18, 19, 28, 12, and 31) were retained in class B when the WPM was adopted, five of the class B items (33, 39, 40, 34, and 45) were reclassified as C, and the remaining one (item 8) was moved up to class A. Out of the 23 class C items, 18 items were retained as such, whereas the remaining five (15, 16, 22, 20, and 27) were moved up to class B. Items 33, 39, 40, 34, and 45, classified as class B items in the Ng, HV, and ZF models, were reclassified as class C items when using the WPM (see Table 3) and had relatively higher LT measurements (more than 4), but lower performance in terms of AUC and ADU, which were the two more important criteria. However, the maximum ADU value (197.92) of these five items was much less than the minimum ADU value (336.12) of items 15, 16 and 27, which were reclassified as class C items by the WPM, while their AUC values were about even. Therefore, the WPM provided a more reasonable ranking of items.