A Proposed Extended Version of the Hadi-Vencheh Model to Improve Multiple-Criteria ABC Inventory Classification

: The ABC classification problem is approached as a ranking problem by the most current classification models; that is, a group of inventory items is expressed according to its overall weighted score of criteria in descending order. In this paper, we present an extended version of the Hadi-Vencheh model for multiple-criteria ABC inventory classification. The proposed model is one based on the nonlinear weighted product method (WPM), which determines a common set of weights for all items. Our proposed nonlinear WPM incorporates multiple criteria with different measured units without converting the performance of each inventory item, in terms of converting each criterion into a normalized attribute value, thereby providing an improvement over the model proposed by Hadi-Vencheh. Our study mainly includes various criteria for ABC classification and demonstrates an efficient algorithm for solving nonlinear programming problems, in which the feasible solution set does not have to be convex. The algorithm presented in this study substantially improves the solution efficiency of the canonical coordinates method (CCM) algorithm when applied to large-scale, nonlinear programming problems. The modified algorithm was tested to compare our proposed model results to the results derived using the Hadi-Vencheh model and demonstrate the algorithm’s efficacy. The practical objectives of the study were to develop an efficient nonlinear optimization solver by optimizing the quality of existing solutions, thus improving time and space efficiency.


Introduction
To facilitate the successful management of a growing number of stock-keeping units (SKUs), inventory managers have found that inventory classification systems provide essential context for evaluating inventory management. ABC analysis is one of the most frequently used inventory classification techniques. Raw materials, subassemblies, intermediate products, parts, and end products can be divided into three classes: A (very important items), B (moderately important items), and C (relatively unimportant items). The ABC classification problem is presented as a ranking problem by the most current classification models [1][2][3]; that is, a group of inventory items is represented according to its overall weighted score of criteria in descending order. The idea of ABC analysis was applied to the inventory management at General Electric during the 1950s. This approach is based on Pareto's famous theory of inequality in the distribution of incomes. A conventional ABC study is conducted on the basis of one criterion: the annual dollar usage (value of an item times its annual usage) of SKUs.
Under Pareto's theory, all items are ranked based on a single criterion; within inventory management, dollar usage is the only criterion for managers to classify items into the A, B, and C categories. However, managers sometimes want to consider more attributes of an item when classifying goods. Many item characteristics could influence inventory control policy and must be considered. Flores [2] noted that other vital criteria can be adopted in addition to dollar usage, such as commonality, reparability, substitutability, lead time, and commonality. For instance, an enterprise must pursue efficient operations that can both minimize total costs and maximize satisfaction brought to their internal or external customers. If SKUs are only classified based on the single criterion of dollar usage, an item with a lower dollar usage, but a long lead time and high criticality, would be misclassified into the C category, resulting in serious damage to the company if the item were to run out of stock.
Detailed literary research has been carried out on multi-choice programming (MCP) theories and applications. MCP is a branch of multi-objective programming that stems from multiple-criteria decision-making (MCDM). MCDM tests several overlapping criteria of decision-making in various areas [4,5]. Multi-criteria inventory classification (MCIC) can be viewed as an application of multicriteria decision analysis [6,7]. To solve the MCIC problem, the joint criteria matrix [8] is a simple and easy-to-understand tool, but it is not practical for more than two criteria and involves too much subjectivity. The analytic hierarchy process (AHP) is a popular methodology, but it involves subjectivity as well. Methods for solving the ABC inventory classification problem have been systematically and thoroughly reviewed and discussed in the relevant literature [9][10][11][12]. A number of methods were suggested in order to achieve multi-criteria classification of SKUs. These methods contribute much to the classification of items and help improve the efficiency and performance of a firm through better inventory management. However, these approaches contain some shortcomings, such as involving too much subjectivity or being overly complicated.
To facilitate better allocation of the priorities of items and further classification, it is worth developing a model that can accommodate multiple criteria to create guidelines for inventory control. This study builds a proper model for categorizing SKUs and demonstrates an efficient algorithm for solving the nonlinear programming model, in which the feasible solution set does not have to be convex. The rest of this paper is structured as follows. Section 2 provides the details of the model's development. The solution algorithm and its improvement are presented in Section 3. Section 4 details the results of the model constructed herein, with comparisons to previous studies using a benchmark data set. Conclusions and recommendations for future research are offered in the final section.

Literature Review on the HV Model and the WPM
Hadi-Vencheh [13] 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 [1]. 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. Hadi-Vencheh 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. Triantaphyllou [14] 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 HV model, 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 , i j y 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 , i j w 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 ,1 for all items i . The proposed model by Hadi-Vencheh [13] is as follows: In the HV model, the performance in each criterion of the i th inventory item , Ng [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:

The CCM Algorithm
This section presents the canonical coordinates method (CCM) algorithm [15,16], which is applied to solve nonlinear programming problems in which the feasible solution set does not have to be convex. Convexity is a strong property that often replaces differentiability as a desirable property in most constrained optimization problems. However, the CCM efficiently addresses continuous search spaces and benefits from the low computational cost for solving constrained optimization. A set containing nonlinear constraints may or may not be convex. This study mainly demonstrates an efficient algorithm for solving nonlinear programming problems in which the feasible solution set does not have to be convex. The main difference between linear and nonlinear programming is that linear programming helps find the best solution from a set of parameters or requirements that have a linear relationship, whereas nonlinear programming helps find the best solution from a set of parameters or requirements that have a nonlinear relationship. The prerequisite for applying the CCM algorithm is that the implicit function theorem can be used in any feasible set. That is, at any point in the feasible set, one can find m variables, such as ( ) , in such a way that the Jacobian matrix of the constraint functions ( ) , with respect to z , are nonsingular. In the implicit function theorem, there exist functions j g such that ( ) We describe the CCM algorithm below: Input: The nonlinear program: Steps: 2. For , 1, , i j m =  and 1, , k n =  , we calculate the following partial derivatives at point 3. We then calculate the m n × matrix for the implicit function g (i.e., 4. We perform a line search along the ray through To do so, we need to solve for ( ) z t , which is done by solving the following system of ordinary differential equations: 1, , j m =  using Taylor polynomial approximation and then apply Newton's method to solve the system of ordinary differential equations at x z and repeat the procedure.
The CCM algorithm helped us identify the local optimal points of NLP so that the feasible set fulfilled the requirements of the implicit function theorem. The problem could then be turned into an NLP problem on a subspace n ℜ of the original space m n + ℜ .

Improvement of the Algorithm Using Efficient Selection of Bases
Step 1 in the CCM algorithm is to find a subset of m variables among the ( ) m n + variables so that a resulting Jacobian matrix is non-singular [8]. This is equivalent to finding a subset of column In order to find a subset of m columns of A that is linearly independent, the original method by Chang and Prabhu [15] was to choose any m subset of the ( ) matrix, which has the same complexity as a Gaussian elimination process. We will show that, by using Gaussian elimination on A to reach its reduced row echelon form, we can find one subset of columns of A that is linearly independent. The process of Gaussian elimination involves performing a sequence of row operations to a given matrix to reach its reduced row echelon form. Each type of row operation corresponds to a type of elementary matrix, all of which are nonsingular. Each time a row operation is performed, it is equivalent to multiplying the original matrix by an elementary matrix on the left. We can also see that in an ( ) m m n × + matrix with rank m , there is a subset of column vectors that is linearly independent. Now, let us state the proposition that yields the discovery of the desired linearly independent subset of columns of A.
Proof: Matrix B must also have a rank of m because it is the reduced row echelon form of A, whose rank is m . Thus, there are m columns of B that form the m m × identity matrix. That is, During the process of Gaussian elimination, performed to obtain the reduced row echelon form of A, we can find elementary matrices 1 2 , , , Note that the k th column of B is also obtained from performing the same row operations on the k th column of A. Thus, we can say Additionally, we can say Since all the elementary matrices 1 2 , , , p E E E  and the identity matrix I are nonsingular,   must also be nonsingular. Now, we can simply apply Gaussian elimination to the matrix and find a linearly independent subset of column vectors that allows the implicit function theorem and the CCM algorithm to be applied. □

Accuracy Improvement
A line search in the fourth step, whereby a system of nonlinear ordinary differential equations with initial values is resolved [15], must be carried out in the implementation of the CCM algorithm. The desired unidimensional direction can be approximated numerically from any line search, but its explicit functional expression cannot be calculated. This drawback impedes the output of the points found in any line search system. We present a modification of the CCM algorithm used by Chang and Prabhu [15], which adopted the gradient method to determine the next point without any line searching.
will stay in U if the distance is small enough.
One common issue with solving an NLP model using the gradient method is that it is likely to leave S by traveling along the gradient direction of a feasible point. This causes a big problem in keeping the NLP model's feasibility. Applying the CCM algorithm does not present such a problem, as every iteration remains within the feasible region. This is because the gradient of the inducted objective function F , with regard to the selected value 1 , , n x x  , will locally move inside the feasible set U if selected carefully. When using the CCM algorithm in a small-scale NLP model, one can conduct a line search along the gradient direction of the induced objective function. The relation between the induced line search and the movement along the original feasible set is illustrated in Figure 1. Because there is one-to-one mapping between U and a neighborhood of ( ) The problem can be viewed as lifting a straight line in n R to a curve in m n + R . In performing a line search, one has to find a one-dimensional, local, optimal point on such a curve with only the knowledge of the projection of the curve, while the other coordinates are unknown. Fortunately, we also know the explicit objective and constraint functions, so we can approximate the change of the unknown coordinates z Δ with the derivatives dz dt . That is, we can approximate z Δ by ( ) In addition, We can now move the previous point along its gradient direction ( ) First, it requires the knowledge of the lifted curve for any 0 z > . When an NLP model is a smallscale problem and the corresponding system of ordinary differential equations is possible to solve, the CCM algorithm can locate or approximate the exact lifted curve C at any 0 t > . To do so, we may need to apply some Ordinary Differential Equation methods, such as the Euler and Runge-Kutta methods. As t gets larger, the feasibility of the point Δ . This way, we have control over staying as close to the feasible set as we need. Not only is the feasibility better kept, but the calculation is also reduced, since we are not solving for the system of ordinary differential equations globally.

Illustrative Example
We applied the WPM to the same problem of the multi-criteria inventory classification problem as reported in the referenced literature [1,6,7,13,17]. Following Ng [1] and Hadi-Vencheh [13], we considered three criteria for inventory classification: annual dollar usage (ADU), average unit cost (AUC), and lead time (LT). We also assumed the importance of the criteria to be ADU, AUC, and LT, in descending order. All of the criteria held positive inventory item scores.

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 ,1 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 ( 8,1 2640 y = > 29,1 268.68 y = ). Although item 29 outperformed item 8 in AUC ( 8,2 55 y = < 29,2 134.34 y = ) and LT ( 8,3 4 y = < 29,3 7 y = ), 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.

Elapsed Runtime and Iterations
The efficiency of solving a problem is also an important criterion for algorithm comparisons; a good algorithm should solve problems within an acceptable time. This section compares the number of iterations of implementations of the CCM and LINGO to solve the WPM of multi-criteria ABC classification. The main difference between implementing the CCM algorithm and the LINGO solver was that we did not need to specify a starting point or moving step size. When the solution region is a polyhedron, determining the first basic solution (the starting point) would be vital, as the local optimal solution is usually located near the basic feasible solution (BFS); the quality of the solution is highly related to the location of the BFS. An unsuitable starting point could lead to a worse local optimal solution. The search region of an algorithm is related to the step size of the line search, and the search region decides whether a feasible solution can be found. The step size also determines the quality of the final solution. A large step size within a line search could jump over the optimal solution, whereas smaller search distances could trap in and require a significant amount of time to reach the local optimal solution. A good starting point and step size for a search can help researchers find acceptable solutions in less time. Table 4 illustrates the process of tuning the step size in order to reach a feasible solution. The CCM algorithm provides more flexibility than other solvers from commercial package software, which presents a higher probability of finding better solutions. In this study, using CCM to solve the problem required more time to achieve the local optimal solution than using LINGO because the step size of the line search determined whether the CCM could find feasible solutions. In this study, LINGO took mere seconds to find a feasible solution, whereas CCM took more time-sometimes nearly a minute-to find a solution. Table 4. Tuning the step size to reach a feasible solution.

Item 4
Step size Step size  When looking at the number of iterations each algorithm needs, CCM requires more iterations to solve the problem, even with a larger step size. As presented in Table 5, LINGO can solve most problems in only 60 iterations, whereas CCM might need approximately one thousand iterations. In conclusion, LINGO is more efficient than CCM in this study, which contradicts the results of the previous study, which stated that CCM can solve nonlinear problems more quickly than other

Conclusions
In this paper, we presented an extended version of the HV model to improve multi-criteria ABC inventory classification. Our proposed nonlinear weighted product model (WPM) incorporates multiple criteria with different measurement units without converting the performance of each inventory item in terms of each criterion into a normalized attribute value. This represents an improvement over the model proposed by Hadi-Vencheh. The WPM could also be viewed as providing a more reasonable classification for inventory items from the illustrated example, presented and used to compare our model with the HV model. In this paper, we also presented the improved CCM algorithm for solving the WPM, in which nonconvex nonlinearity was present in both the objective function and the constraints. The strategy presented here involved greatly reducing the steps in choosing m variables among ( ) m n + variables, such that the corresponding m m × Jacobian matrix was nonsingular. Using the improved algorithm, we applied Gaussian elimination to the original matrix to determine which m variables to choose. Our second improvement was to remove solving the nonlinear differential equations system, which occurs in the line search method of the CCM algorithm. This paper demonstrates an efficient algorithm for solving nonlinear programming problems, in which the feasible solution set does not have to be convex. The practical implication of this study is to further improve the efficient nonlinear optimization solver based on the CCM by optimizing the quality of existing solutions, thus improving time and space efficiency. Future research should continue to investigate the feasibility of implementing this proposed CCM algorithm in discrete domain issues for engineering applications in order to decide if the algorithm could be superior to off-the-shelf software. Future studies could apply the CCM to other nonlinear programs that arise in practice. For instance, autonomous vehicles represent one of many developments that will influence future mobility needs and planning needs. Traffic assignment models seek the same objective as route guidance strategies and provide turning points with information for implementing route guidance control strategies. Faster algorithms developed specifically for traffic assignment can be adapted and used in vehicle route guidance systems. The minimization of total travel time is a common goal, both globally and from a traffic administration perspective. The current road network manages more traffic by achieving system optimization. Some researchers have focused their efforts on dynamic traffic assignment because of the unrealistic assumptions of static traffic assignment. Difficulties encountered by the dynamic model result from route calculation being related to travel time on an arc, which is also dependent on the traffic along the route. It is difficult to solve such relationships analytically under a dynamic circumstance. In response to the difficulties of dynamic traffic modeling, Jahn et al. [19] therefore developed a model in which flow represents the traffic patterns in a steady state, and the results are the boundary for the total travel time. However, the algorithm by Jahn et al. [19] only solves problems with convex, nonlinear objective functions and linear constraints. To avoid this restriction, future studies could adopt the CCM to solve nonlinear optimization models and provide strategies for route guidance.