Combining the Cost Drivers Based on the Singular Value Decomposition of Matrix

Abstract—Reducing the complexity of activity-based costing system is one of the key factor when successfully implement the activity-based costing system. Based on dimensionality reduction of the singular value decomposition (SVD) of the matrix, this paper proposes and researches the method of SVD-based combination of cost drivers under activity-based costing. Through singular value’s decomposition for coefficient matrix in the costing model under activity-based costing, this paper classifies the cost drivers based on singular values and eigenvectors in the right singular matrix. On this basis, using integer programming, representative cost drivers are selected for combination to simplify the activity-based costing model. Numerical examples indicate that the method of SVD-based combination of cost drivers reduces the complexity of activity-based costing system and significantly improves the accuracy of the combined product cost. The result shows, whether reducing model complexity or ensuring product cost accuracy, the method of SVD-based combination of cost drivers is superior to the existing other methods, such as integer programming method, clustering method, etc.


I. INTRODUCTION
Since the generation of Activity-Based Costing (ABC) in the 1980s, it has received widespread attention because of its ability to more accurately allocate overhead to costing objects such as products, special orders, and customers (Cooper, Kaplan, 1988;Dopuch, 1993;Gupta and Galloway, 2003;Velmurugan, 2010;Tsar et al., 2014;Almeida & Cunha, 2017) [1]- [7]. ABC is also gradually applied from the manufacturing industry to service industries such as medical and logistics, and non-profit organizations such as government and universities. ABC's research has also shifted from cost accounting to cost control, production management and so on. When using ABC to calculate costs of product, the costing procedures divided into two stages according to production process activities. The first stage is from resource to activities. In this stage, according to the causal relationship of resource costs incurring, the resource costs incurred in the production process are collected into the activity cost pools, and the activity costs of the activities are formed. The second stage is from activities to cost objects such as product. In this stage, according to the causal relationship of the activity costs Manuscript  incurring, the activity costs are allocated to product, etc., the product cost and other costs are formed. Whether it is the causal relationship of resource costs incurring or activities costs incurring, they are collectively referred to cost drivers. With the diversification of customer needs and increased complexity of production processes, the number of activities involved in the activity-based costing is increasing, resulting to the greatly increased implementing cost when collecting, processing and handling the activity information, so the costing system are becoming more and more complex. This not only adds the burden of activity-based costing system, but also leads to excessive implementation costs (Kaplan & Anderson, 2004) [8], which hinder the application of ABC. Therefore, how to reduce the number of activities, complexity of the activity-based costing system and implementation costs have become the topics of academic research.

II. LITERATURE REVIEW
In order to reduce the complexity of activity-based costing system, many scholars have carried out some useful explorations. Babad & Balachandran (1993) firstly studies the effect of cost drivers' combination on product cost accuracy [9]. Their research shows that the combination of completely correlation cost drivers doesn't compromise the accuracy of product costs. At the same time, they further study the cost drivers' combination by constructing a planning model that balances the information cost savings with the accuracy loss. The results show that the cost drivers' combination will lose the accuracy of product costs. However, their researches are only applicable to the combination of two cost drivers and consider the importance of product in the form of weights in the planning model. Based on the cost drivers and their causal relationships, Schierderjans & Garvin (1997) uses the Analytic Hierarchy Process (AHP) and Zero One Goal Programming (ZOGP) to study the selection of representative cost driver from a set of candidate cost drivers. Their results show that the AHP method is conform to the realistic selection process of cost drivers [10]. Based on Babad & Balachandran's research, Homburg (2001) constructs a combined alternative model where a cost driver is replaced by a set of cost drivers, proving the rationality of multi-cost drivers' combination [11]. Zhao & Ouyang (2009) improves Homburg's model by quantifying the importance of cost drivers and integrating them into the model, therefore achieving the balance of cost drivers' importance and relevance [12]. Li and Wang (2009) note that, although the above optimization model minimizes the loss of accuracy directly or indirectly, the minimum accuracy loss may still be very large. They establish an optimization model that maximizes the number of combined cost drivers and has limited error between product costs before and after combination of cost drivers at the same time, thus ensuring the controllability of the product cost costs [13].
Based on the fact that the combination of cost drivers with perfect positive correlation does not change the accuracy of product cost, Wang et al. (1999) propose the homogeneity concept of amount of cost drivers, and conclude that "satisfactory accuracy of product cost can be achieved by combining cost drivers with positive and negative correlation" [14]. Li et al. (2005) use matrix theory to study the theory of combining cost drivers, and extend the homogeneity concept of the number of cost drivers to the situations that the cost driver ratio of an activity is equal or proportional to linear combination of the cost driver ratio of several activities [15]. Afterwards, Li and Wang (2007) research the theory of the multiple cost drivers' combination, and further explore the mechanism of "satisfactory accuracy of product cost can be achieved by combining cost drivers with positive and negative correlation", therefore proposing the concept of the weighted average cost driver However, whether it is the clustering method or the combination of clustering analysis and principal component analysis, it is difficult to overcome the problem of uncontrollable cost error by using these methods to combine cost drivers.
In summary, the existing cost driver selection and combination research has proposed various methods to reduce the complexity of activity-based cost system, also consider the accuracy of product costs, but it is difficult to reducing the complexity and ensuring the accuracy of product cost at the same time. In this paper, based on the dimension reduction and denoising roles of singular value decomposition (SVD) [19], it further researches the combination of cost drivers under activity-based costing.

III. PRODUCT COSTING MODEL UNDER ABC AND COMBINATION OF COST DRIVERS
For convenient analysis, product costing model under ABC is established firstly.
Assume there are n kinds of products and m activities. Each activity corresponds to a cost driver. R ij represents the cost driver ratio of activity j consumed by product i to activity j consumed by all products, and is known as coefficient on cost driver j of product i. Obviously, cost driver coefficient R ij satisfy: And given S j ( =1, 2, , jm ) represents resource costs or total activity costs corresponded to cost driver j, C i ( 1, 2, , in  ) represents total activity cost of product i. The product's indirect costing model (hereinafter referred as the costing model) under the activity-based costing is： Expressed in matrix form as： Matrix R is referred as the coefficient matrix of product cost drivers，it reflects the structural proportion of the cost driver amount consumed by each product in the total amount of different cost drivers. The number of rows of the matrix reflects the number of products, and the number of columns of the matrix reflects the number of activities in the production process. In the coefficient matrix R of product activity driver, the column vector represents the cost driver ratio of activity j consumed by various products, the row vector 12 R reflects the cost driver ratios of various activities consumed by product i . According to the ratios, the total cost of the product i is calculated. Therefore, the product costing model clearly reflects the forming process of cost from activities to product cost.
When the column number in coefficient matrix (the number of cost drivers) is large, the product costing model becomes very complicated. At the same time, in order to calculate the cost of all products, it is necessary to collect a large amount of relevant information, and the cost of implementing the activity-based costing system is high (Kaplan & Anderson, 2004). Therefore, simplifying the activity-based costing system has become the key to the successful implementation of the activity-based costing method.
How to simplify the activity-based costing model? The common method is to classify and combine the activities or cost drivers represented by the column vector of the coefficient matrix in the costing model (1), which is usually known as cost drivers' combination. It means the activity cost (or resource cost) which is collected by one or more activities will be combined into another activity cost (or resource cost), and then aggregated cost will be assigned to all products in corresponding cost driver proportion of this another activity, thereby the products cost are calculated. First, before combining the activity cost (or resource cost) of multiple activities, the cost drivers are classified according to certain characteristics. Only the activities that are classified into the same category can be combined. Second, in each category of the cost driver, a cost driver needs to be selected as a representative cost driver. If the representative cost drivers chosen are different, the final calculated cost of each product may different. The above classifying cost drivers and selecting representative cost driver will affect the accuracy of the final calculated product cost.

IV. SVD OF COEFFICIENT MATRIX AND COMBINATION OF COST DRIVERS
Essentially, the simplification of the costing model under activity-based cost is to simplify the column vector of the coefficient matrix R in the model. The combination of cost drivers is the process of classifying the column vectors of the coefficient matrix R and selecting a column vector in each category to reconstruct the coefficient matrix, which is also the dimension reduction process of the matrix. Therefore, it is necessary to research the characteristics of column vectors in the coefficient matrix. The significance of singular value decomposition of a matrix is that any form of matrix can be decomposed into the product of several simpler matrices. These simple matrices reflect the characteristics of different aspects of the original matrix.
According to the principle of the SVD of the matrix, the where, U is a nn  matrix and referred as left singular matrix, it reflects the important characteristics of the product.
V is a mm  matrix and usually known as right singular matrix, it reflects the characteristics of cost drivers.
U and V are orthogonal matrices satisfy: In formula, i u and i v is the i-th column vector of matrix U and V respectively. In singular value decomposition of matrix, singular values often correspond to important information in the matrix, and the importance of information is positively correlated with the size of singular values. However, those small singular values are usually considered to be caused by random interferences (noise). When these small singular values are set to 0, we can eliminate the interference of random factors. Suppose that the first s larger singular values are retained and the other smaller singular values are discarded. It is obtained that It can be seen that order of coefficient matrix R change from nm  to ns  because of matrix ms V  , that is, the number of columns of the coefficient matrix is compressed. This can be understood as combination of the cost drivers corresponded to the column vectors. Therefore, the matrix Based on the above analysis, the cost drivers' classification and combination steps are as follows: The step 1: perform singular value decomposition for the coefficient matrix R , obtain a left singular matrix U, a right singular matrix V and a singular value , 1, 2, minus cost of product k before combination, i C indicates the cost of product k before combination. The step 6: in the same category, incorporate the activity costs which corresponded to other cost drivers into activity cost which corresponded to representative cost driver. Then allocate aggregated costs to products with the ratio of a representative cost driver and calculate the cost of each product. Finally, complete combination.

V. NUMERICAL EXAMPLES
This paper uses the data from the XAMC case used by Jin et al. (2005): 6 products are denoted as P 1 , P 2 , P 3 , P 4 , P 5 , P 6 [20]. The corresponding 14 cost drivers are denoted as j 1 , j 2 , … , j 14 . Through calculating the original data, the cost driver ratio of different activities consumed by various products (cost driver coefficient) and the activity cost of each activity are shown in Table I as follow.

A. Singular Value Decomposition for Coefficient Matrix
Using above data, the non-zero singular value, left singular matrix and the right singular matrix are calculated in Table II,  Table III and Table IV as follows:

B. Cost Driver Classification
Based on SVD principle, the column vector 1 v of the right singular matrix corresponded to the largest singular value 1  reflects the most important feature of the column vector (cost drivers) in the coefficient matrix. Therefore, this paper regards 1 v as 14 points in the one-dimensional space (cost drivers) and uses cluster analysis method to classify, thus obtaining categories of all 14 cost driver as Fig. 1 as following. It can be seen that all 14 cost drivers are divided into 5 categories at most, and these 5 categories can be further clustered into 2 categories at least. In the 5 categories, only the cost driver 13 j is clustered as a separate category, and the other 4 categories contain at least two cost drivers, and the representative cost drivers are selected in each category, as shown in Table V. The bold words in the Table IV indicate selected representative cost drivers.

C. Error between Product Costs before and after Combination of Cost Drivers
According to the different number of categories of 14 cost drivers, the different categories and the representative cost drivers in each category, the calculated pre-combination product costs, and the combined product costs, the errors between them, the error degree and accuracy of the total product cost are shown in Table VI. In Table VI, l denotes number of categories, e and a respectively represents the error degree and accuracy of the total product costs before and after the combination.
It can be seen that the eigenvector corresponded to the maximum singular value 1.90497 reflects the most important feature of cost driver (column vector) in the coefficient matrix. According to this eigenvector, the all cost drivers are clustered into five categories. After selecting a representative cost driver for each category of cost drivers that contain multiple cost drivers and combining other activity costs into activity cost corresponded to a representative cost driver in this category, the product costing model is reduced from the original 14 cost drivers to 5 cost drivers, and the complexity of the costing model is greatly reduced. Further clustering, the number of categories of cost drivers can be clustered into 2 categories, and the complexity of the costing model will be further reduced. At the same time, it can be seen from Table VI that when the number of categories is 5, the maximum one of the error between products costs before and after the combination is 7.43%, the error degree of all product costs is 4.4%, and the accuracy is 95.6%. When further clustered into 4 categories, the error degree of all products cost reaches a minimum of 3.99%, and the accuracy reaches the highest, which is 96.01%. After that, with the number of categories of cost drivers decreasing, the error degree of all product costs gradually increases, and the cost accuracy decreases. When the number of categories is 2, the error degree of all products costs reaches the maximum which is 22.21%, and the accuracy reaches the lowest which is 77.79%.
The above results show that the product costing model is simplified from 14 cost drivers to 4 cost drivers, which is the best simplification result. It is superior to the methods of Wang et al. (2009) and Liu et al. (2014). Therefore, the method of SVD-based combination of cost drivers is better.

D. Discussion: The Number of Non-zero Singular Values Considered for Cost Driver Classification
The above-mentioned cost driver classification is mainly based on the eigenvector in the right singular matrix corresponded to the largest singular value, and discards the eigenvectors corresponded to other non-zero singular values. One question is: When classify the cost drivers, should we discard non-zero singular values except the largest singular values? Or, how many larger singular values should we used to classify the cost drivers? In order to answer this question, we use dividing the cost drivers into four categories as an example because the error degree of all product costs is the lowest when the number of categories is 4. On the basis of classification of eigenvector in the he right singular matrix corresponded to the largest singular value, we increase the eigenvectors in the right singular matrix corresponded to the singular values one by one from large to small, and then cluster the cost drivers.
Journal of Economics, Business and Management, Vol. 8, No. 1, February 2020   It can be seen from Table VIII and Table VI that when the number of non-zero singular on which the cost driver's classification depends is increased from 1 to 2 up to 5, the error degree of all product costs is increased from 3.99% to 8.44%, and finally increased to 18.25%; while the accuracy of all product costs decreases from 96.01% to 91.56%, and finally decreases to 81.72%.When the classification considers all six non-zero singular values, the error degree of all products reaches 17.43% ,the accuracy of all product costs reaches 82.57%.
We also consider the case where all 14 cost drivers are divided into five categories. That is, when the costing model is reduced to five cost drivers, as the number of singular values considered increases from 1 to 2 and gradually increases to 6, the error degree of all product costs increases from 4.4% to 6.68% and gradually increases to 19.6%. The accuracy of all product costs decreased from 95.6% to 93.32%, and gradually decreased to 80.4%. This conclusion is basically the same as when all 14 cost drivers are divided into four categories.
The above discussion shows that, based on the principle of SVD of the matrix, when classifying and combining the cost drivers represented by the column vectors of the coefficient matrix, we only need to consider the eigenvector corresponded to the largest singular values in the right singular matrix. The greater the number of singular values considered, the greater the error degree of all product costs calculated by classifying and combining the cost drivers, and the lower the accuracy of product cost.

VI. CONCLUSION
Reducing the complexity of the activity-based costing system is one of the key factors in the successful implementation of the activity-based costing system. Based on the dimension reduction effect of singular value decomposition of a matrix, this paper proposes and studies the SVD-based combination method of cost driver under activity-based costing. By establishing a product costing model under activity-based costing, the singular value decomposition of the coefficient matrix in the model is carried out. The singular value and the eigenvectors in and the right singular matrix are used to classify the cost drivers. Then the representative cost drivers are selected to perform the combination of cost drivers and simplify the costing model. Numerical examples show that the method of SVD-based combination of cost drivers significantly reduces the error between product costs before and after combination and improves the accuracy of the combined product cost while reducing the complexity of the activity-based costing system. Compared with existing methods of cost driver combination such as integral value planning and clustering methods, the method of SVD-based combination of cost drivers is superior to other methods in reducing model complexity and ensuring accuracy of product cost. j is clustered as a separate category, and the other 4 categories contain at least two cost driveIt can be seen that all 14 cost drivers are divided into 5 categories at most, and these 5 categories can be further clustered into 2 categories at least. In the 5 categories, only the cost driver 13 j is clustered as a separate category, and the other 4 categories contain at least two cost drivers, and the representative cost drivers are selected in each category, as shown in Table V. The bold words in the Table IV