## 2 Literature Review on the HV Model and the WPM  
Hadi-Vencheh (2018) proposed a multi-criteria weighted nonlinear model for ABC inventory classification. The proposed model (hereafter the HV model) is an extension of the Ng model ([PERSON] and others, 2018). The Ng model transforms the inventory object to a scalar value. The grouping, according to the measured values, is then applied according to the ABC theory. [PERSON] also extended the Ng model to resolve the condition in which the score is independent of the weights from the model for each item. Despite the improvement in maintaining the influences of weights in the final score, one notable problem remains: the HV model calculates the scores assigned to each item using the weighted sum method (WSM) for criteria with different measurement units, which therefore requires converting the performance of individual inventory items into a normalized attribute value, expressed in terms of every criterion. [PERSON] (2018) contended that if the problem involved criteria with different measurement units, the weighted product method (WPM) would be a more suitable tool to calculate the scores given to each item. To avoid an erroneous extreme value leading to inventory item misclassification, we propose the following nonlinear WPM to model the classification problem involving criteria with different measurement units. This study presents a broader version of the HVmodel, taking weight values into account in the ABC inventory classification for multiple criteria by using the WPM, which applies multiplication weights and forms a nonlinear optimization problem. To solve the nonlinear optimization problem efficiently, the canonical coordinates method (CCM) algorithm is used to calculate the weights of the criteria for each inventory item.  
Suppose that \(I\) inventory items are present, and that the items must be graded as A, B, or C based on their results, according to \(J\) criteria. In particular, let the output of the \(i\)th inventory item be referred to as \(y_{i,j}\) with respect to each criterion. For simplicity, all parameters are beneficial; in other words, they are positively connected with the degree of value of an item. The goal is to combine many performance scores in the subsequent ABC inventory classification, in regard to different parameters, into a single score. In both the Ng and HV models, a nonnegative weight \(w_{i,j}\) is the weight of performance contribution of the \(i\)th item under the \(j\)th criteria to the score of the item. The parameters are supposed to be listed in descending order such that \(w_{i,1}\geq w_{i,2}\geq\cdots\geq w_{i,J}\) for all items \(i\). The proposed model by [PERSON] [13] is as follows:  
\[\begin{array}{ccc}\max\ S_{i}=&&\sum\limits_{j=1}^{J}y_{i,j}w_{i,j}\\ \text{s.t.}&&\sum\limits_{j=1}^{J}w_{i,j}{}^{2}=1\\ &&w_{i,j}-w_{i,j+1}\geq 0,\ j=1,2,\ldots,J-1\\ &&w_{i,j}\geq 0,\ j=1,2,\ldots,J\end{array} \tag{1}\]  
In the HV model, the performance in each criterion of the \(i\)th inventory item \(y_{i,j}\) is further normalized to \(s_{i,j}\), and the objective function of the nonlinear programming (NLP) model from Equation (1) is found to be  
\[\max\ \ \ S_{i}=\sum\limits_{j=1}^{J}s_{i,j}w_{i,j}\]  
[PERSON] [1] indicated that normalization scaling involves extreme measurement values and would thus have an effect on all normalized measurements if the extremes changed. To avoid an invalid extreme value leading to inventory item misclassification, we propose the following nonlinear WPM to model the classification problem involving criteria with different measure units:  
\[\begin{array}{ccc}\max\ S_{i}=&&\sum\limits_{j=1}^{J}y_{i,j}w_{i,j}\\ \text{s.t.}&&\sum\limits_{j=1}^{J}w_{i,j}{}^{2}=1\\ &&w_{i,j}-w_{i,j+1}\geq 0,\ j=1,2,\ldots,J-1\\ &&w_{i,j}\geq 0,\ j=1,2,\ldots,J\end{array} \tag{2}\]