A MATHEMATICAL PROGRAMMING APPROACH TO ANALYZE THE ACTIVITY-BASED COSTING PRODUCT-MIX DECISION WITH CAPACITY EXPANSIONS

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1 A MATHEMATICAL PROGRAMMING APPROACH TO ANALYZE THE ACTIVITY-BASED COSTING PRODUCT-MIX DECISION WITH CAPACITY EXPANSIONS Wen-Hsien Tsai and Thomas W. Lin ABSTRACT In recent years, Activity-Based Costing (ABC) has become a popular cost management technique in both accounting academics and business practice. It uses a two-stage procedure to assign resource costs to products: first from resources to activities, then from activities to products. It improves the accuracy of product cost data derived from traditional direct labor-based costing systems. Product-mix decision analysis is an important part of ABC information. The purpose of this paper is to incorporate the capacity expansion features into an ABC product-mix decision model by using a mathematical programming approach. The current traditional ABC product-mix decision models do not explicitly consider capacity expansions. We developed a new mixed integer programming product-mix model that maximizes a firm s profit with five major types of ABC constraints: () unit-level direct material constraints; () unit-level piecewise direct labor constraints; () batch-level activity constraints (e.g. scheduling and setup activities); () product-level Mathematical Programming Applications of Management Science, Volume, Copyright 0 by Elsevier Ltd. All rights of reproduction in any form reserved ISSN: 0-/doi:./S0-(0)-

2 WEN-HSIEN TSAI AND THOMAS W. LIN activity constraints (e.g. product design); and () stepwise facility-level activity cost with machine hour constraints (e.g. plant guard and management). With the model presented in this paper, we can evaluate the benefits of simultaneously expanding the various kinds of capacity.. INTRODUCTION In recent years, Activity-Based Costing (ABC) has become a popular cost management technique in both accounting academics and business practice. ABC has been gradually adopted to overcome the shortcomings of traditional cost systems. Through the interactive developments between practical and academic circles, ABC has been applied to various business functions and different industries. The latest ABC model is composed of both the cost assignment view and the process view with activities as the intersection of these two views (Turney, a, pp. 0, b). The cost assignment view provides information about resources, activities, and cost objects. This information can be used to analyze critical decisions such as pricing, product mix, sourcing, and product design. The process view provides financial and non-financial information about cost drivers and performance measures for each activity or process. This information can be used for activity/process improvements to reduce costs and/or enhance value to customers. Thus, product-mix design analysis is an important part of the cost assignment view of ABC. In the early ABC literature, authors often used realistic numerical examples to show that the products ABC costs will be significantly different from the ones derived from traditional direct labor-based costing systems. However, they seldom demonstrated how to select the optimal product-mix. ABC had also been criticized for its failure to incorporate constraints into production-related decisions (Kee & Schmidt, 00; Spoede et al., ). In light of this, some authors (Kee, ; Kee & Schmidt, 00; Malik & Sullivan, ; Tsai, ; Yahya-Zadel, ) proposed various mathematical programming models to conduct the product-mix decision analysis under ABC. All of these product-mix models can be used to select the optimal product-mix that maximizes a firm s profit with various constraints. Kee () integrated ABC with the Theory of Constraints (TOC) in the product-mix decision analysis. As quoted by Kee and Schmidt (00) from Goldratt (): The TOC consists of a set of focusing procedures for identifying a bottleneck managing the production system with respect to this constraint, while resources are expended to relieve this limitation on the system. When a bottleneck is relieved, the firm moves to a higher level of goal attainment and

3 A Mathematical Programming Approach to Analyze the Activity-Based Costing one or more new bottlenecks will be encountered. The cycle of managing the firm with respect to the new bottleneck(s) is repeated, leading to successive improvements in the firm s operations and performance. This means that TOC successively relieves the bottlenecks and the associated constraints by expanding the obtainable resources (capacities) or by improving the firm s operations. The current traditional ABC product-mix decision models do not explicitly consider capacity expansions. The feasibility or benefits of capacity expansions has usually been evaluated by using post-optimal (sensitivity) analysis. It is a tedious job to successively expand various resources. Sometimes, a manager may end up with a sub-optimal solution because he/she determines the combination of the expanded levels of various resources by a trial-and-error method, instead of a systematic method. Furthermore, it is difficult to simultaneously consider two or more kinds of capacity expansions by using the current ABC product-mix decision models. In light of this, the purpose of this paper is to incorporate the capacity expansion features into an ABC product-mix decision model by using a mathematical programming approach. The detailed cost assignment view of ABC is described in Section. Section presents an ABC product-mix decision model with capacity expansion features. In Section, a numerical example is used to demonstrate how to apply the model. Finally, the discussion and conclusions are presented in Sections and, respectively.. COST ASSIGNMENT VIEW OF ABC The detailed cost assignment view of ABC is shown in Fig. (Tsai, a; Turney, a, b). ABC assumes that cost objects (e.g. products, product lines, processes, Fig.. The Detailed Cost Assignment View of ABC.

4 WEN-HSIEN TSAI AND THOMAS W. LIN customers, channels, markets, etc.) create the need for activities, and activities create the need for resources. Accordingly, ABC uses a two-stage procedure to assign resource costs to cost objects. In the first stage, resource costs are assigned to various activities by resource drivers. Resource drivers are the factors chosen to approximate the consumption of resources by the activities. Each type of resource traced to an activity becomes a cost element of an activity cost pool. Thus, an activity cost pool is the total costs associated with an activity. An activity center is composed of related activities, usually clustered by function or process. In the second stage, costs in each activity cost pool are distributed to cost objects by an adequate activity driver which is used to measure the consumption of activities by the cost objects. In this paper, we regard products as the cost objects. The total costs of a specific product can be calculated by adding the costs of various activities assigned to that product. The unit cost of the product is achieved by dividing the total costs by the quantity of the product. The resources used in manufacturing companies may include people, machines, facilities, and utilities, while the corresponding resource costs could be assigned to activities in the first stage of cost assignment view by using the resource drivers: time, machine hours, square footage, and kilowatt hours, respectively (Brimson,, p. ). The following are the categories for manufacturing activities: () unit-level activities (performed one time for one unit of product, e.g. machining, finishing); () batch-level activities (performed one time for a batch of products, e.g. setup, scheduling); () product-level activities (performed to benefit all units of a particular product, e.g. product design); and () facility-level activities (performed to sustain the manufacturing facility, e.g. plant guard and management) (Cooper, ). The costs of different levels of activities can be traced to products by using the different kinds of activity drivers in the second stage of cost assignment view. For example, number of machine hours is used for the activity machining, setup hours for machine setup, and number of drawings for product design. Usually, the costs of facility-level activities cannot be traced to products with definite causal relationships and should be allocated to products with the appropriate allocation bases (Metzger, ; Tsai, b). For the purpose of product-mix decisions, we regard the costs of facility-level activities as the common fixed cost in this paper.. ABC PRODUCT-MIX DECISION MODEL WITH CAPACITY EXPANSIONS Jadicke () applied a Linear Programming (LP) technique to a Cost-Volume- Profit (CVP) model, called Product Mix model in many management accounting or LP texts, which could aid management in determining the optimal product mix,

5 A Mathematical Programming Approach to Analyze the Activity-Based Costing maximizing total profit under some limits (constraints) to production or sales in the case of multi-product companies. In recent years, some authors (Kee, ; Kee & Schmidt, 00; Malik & Sullivan, ; Tsai, ; Yahya-Zadel, ) utilized various mathematical programming approaches to conduct the productmix decision analysis under ABC. In this paper, we will extend their research to incorporating capacity expansion features into the product-mix decision model under ABC... Assumptions The ABC product-mix decision model presented in this paper has several assumptions. First, the activities in a multi-product company have been classified as unit-level, batch-level, product-level, and facility-level activities, and the related resource drivers and activity drivers have been chosen by the company s ABC team through an ABC study. Second, the data on actual running activity cost per activity driver for each activity (Tyson et al., ) has been collected and used in the model. Third, the unit selling prices and the unit direct material costs are constant within the relevant range. Fourth, only the facility-level activity cost is regarded as the common fixed cost, and its cost function is a stepwise function that varies with machine hour. Fifth, renting additional machines can expand machine hour resources. Sixth, using overtime work or additional night shifts with higher wage rates can expand direct labor resources... Capacity Expansion Features... Stepwise Facility-Level Activity Cost Because the total cost of facility-level activities (e.g. plant guard and management) cannot be traced to products with definite causal relationships, we regard it as the common fixed cost and assume that its cost function is a stepwise function (as shown in Fig. ) which varies with machine hours, observed from a prior cost behavior analysis. The total facility-level activity cost is F 0 under the current capacity H 0 machine hours. If the capacity is successively expanded to H, H,... H t machine hours, the total facility-level activity cost increases to F, F,...F t, respectively. Let X i be the production quantity of product i and ih the requirement of machine hours for one unit of product i. As a result, the total facility-level activity cost and the associated machine hour constraints are (Tsai & Lin, ): t Total facility-level activity cost = F k k () k=0

6 WEN-HSIEN TSAI AND THOMAS W. LIN Constraints: Fig.. Stepwise Facility-Level Activity Cost. n ih X i i= t H k k () k=0 t k = () k=0 where ( 0,,... t ) is an SOS set of 0 variables within which exactly one variable must be non-zero (Beale & Tomlin, ; Williams, ). When q = (q 0), we know that the capacity needs to be expanded to the qth level, i.e. H q machine hours.... Piecewise Direct Labor Cost In this paper, we assume that using overtime work or additional night shifts with higher wage rates can expand direct labor resources. Thus, the total direct labor cost function will be a piecewise linear function as shown in Fig.. The available Fig.. Piecewise Direct Labor Cost.

7 A Mathematical Programming Approach to Analyze the Activity-Based Costing normal direct labor hour is G and the direct labor hour can be expanded to G ; the total direct labor cost is L and L at G and G, respectively. As a result, the total direct labor cost and the associated constraints are (Tsai & Lin, ): Constraints: Total Direct Labor Cost = L + L () TL = G + G () 0 0 () 0 () 0 () = () + = () where (, ) is an SOS set of 0 variables within which exactly one variable must be non-zero; ( 0,, ) is an SOS set of non-negative variables within which at most two adjacent variables, in the order given to the set, can be non-zero (Beale & Tomlin, ; Williams, ); TL is the total direct labor hour we need and its function depends on the case under study. If =, then = 0 [from Eq. ()], = 0 [from Eq. ()], 0, [from Eqs () and ()], and 0 + = [from Eq. ()]. Thus, from Eqs () and () we know that total direct labor cost and total labor hour needed are L and G, respectively; this means that we will not need the overtime work. If =, then = 0 [from Eq. ()], 0 = 0 [from Eq. ()],, [from Eqs () and ()], and + = [from Eq. ()]. Thus, from Eqs () and () we know that total direct labor cost and total labor hour needed are L + L and G + G, respectively; this means that we will need the overtime work... Description of the Model The ABC product-mix decision model with capacity expansions is as follows: Maximize = Total Revenue Total Direct Material Cost Total Direct Labor Cost Total Unit-, Batch-, Product- & Facility-Level Activity Costs

8 WEN-HSIEN TSAI AND THOMAS W. LIN n n s n = p i X i c m a im X i (L + L ) d j ij X i i= i= m= i= j U n n t d j ij B ij d j ij R i F k k () i= j B i= j P k=0 Subject to: Unit-Level Direct Material Constraints: n a im X i Q m, m =,,... s () i= Piecewise Unit-Level Direct Labor Constraints: TL = G + G () 0 0 () 0 () 0 () = () + = () Stepwise Facility-Level Machine Hour Constraints: n t ih X i H k k 0, () i= k=0 t k = () k=0 Batch-Level Activity Constraints: X i ij B ij, i =,,...n; j B () n ij B ij T j, j B () i= Product-Level Activity Constraints: X i D i R i, i =,,... n ()

9 A Mathematical Programming Approach to Analyze the Activity-Based Costing n ij R i V j, j P () i= X i 0, i =,,... n () ( 0,, ) : An SOS set of non-negative variables. () (, ) : An SOS set of 0 variables () ( 0,,... t ) : An SOS set of 0 variables () R i : 0 variables, i =,,... n () B ij : Non-negative integer variables, i =,,... n, j B () where X i = The production quantity of product i; p i = The unit selling price of product i; C m = The unit cost of the mth material; a im = The requirement of the mth material for one unit of product i; Q m = The available quantity of the mth material; d j = The actual running activity cost per activity driver for activity j; ij = The requirement of the activity driver of unit-level activity j ( j U) for one unit of product i; ij = The requirement of the activity driver of batch-level activity j ( j B) for product i; B ij = The number of batches of batch-level activity j ( j B) for product i; ij = The number of units per batch of batch-level activity j ( j B) for product i; T j = The capacity limit of the activity driver of batch-level activity j ( j B); ij = The requirement of the activity driver of product-level activity j ( j P) for product i; R i = The indicator for producing product i (R i = ) or not producing product i (R i = 0); V j = The capacity limit of the activity driver of product-level activity j ( j P); D i = The maximum demand of product i; Other variables and parameters are as mentioned before. Equation () represents the total profit function, and Eqs () () are the constraints associated with various resources and activities. Equation () is the direct material constraint. Equations () () are the direct labor constraints described in Section... Equations () and () are the machine hour constraints described in Section...

10 WEN-HSIEN TSAI AND THOMAS W. LIN Equations () and () are the constraints associated with batch-level activities, where Eq. () is the capacity constraint for batch-level activity j ( j B). For example, we use setup hours as the activity driver of the batch-level activity setup because each product needs different setup hours. In this case, T j are the available setup hours, ij are the needed setup hours for product i, B ij is the number of setup needed for product i, and ij is the average number of units in each setup batch. In fact, there may be different number of units in each setup batch for a specific product. However, we can use the average number of units for the purpose of planning. On the other hand, we may use number of batches as the activity driver of the batch-level activity setup because the setup hours needed is the same for each product. In this situation, ij can be set to and T j is the available number of setup. Equations () and () are the constraints associated with product-level activities. Equation () is the market demand constraint and Eq. () is the capacity constraint for product-level activity j ( j P). For example, we may use number of drawings as the activity driver of the product-level activity product design. In this case, V j is the available number of drawings for the firm s capacity, and ij is the number of drawings needed for product i.. A NUMERICAL EXAMPLE In this section, we present a numerical example. Assume that a manufacturing company is considering producing product, product, and product (i =,, ) and that these products need two kinds of the same direct material (m =, ). We also assume that it needs five main activities in producing these three products: two unit-level activities, Machining and Finishing (U ={, }), two batch-level activities, Scheduling and Setup (B ={, }), and one product-level activity, Product Design (P ={}). The related data for this example are shown in Table. From Table, we know that the total facility-level activity cost is $,000 under the current capacity H 0 =,000 machine hours and that the capacity can be expanded to H =,000 or H =,000 machine hours by renting additional machines with the total facility-level activity cost increasing to F = $,000 or F = $0,000, respectively. Besides, the available normal direct labor hour is G =,000 hours with the normal wage rate of $./hr and the direct labor hour can be expanded to G =,000 hours with the overtime wage rate of $./hr. Further, assume that two unit-level activities, Machining and Finishing, need direct labor, and Machining activity needs / i labor hour for one unit of product i;

11 Table. Example Data. Product (i) Available Capacity Maximum demand D i,000,000,000 Selling price p i 0 Direct material m = c = a i Q =,000 m = c = a i Q =,000 Unit-level activity j Activity driver Cost/driver d j Machining Machine hours $ i Finishing Labor hours $ i Batch-level activity Scheduling Production $0 i T = 0 Orders i 0 Setup Setup $0 i T = 0 Hours i Product-level activity Design Drawings $0 i V = Facility-level cost Cost F 0 = $,000 F = $,000 F = $0,000 Machine hrs H 0 =,000 H =,000 H =,000 Direct labor constraint Cost L = $,0 L = $,000 Labor hrs G =,000 G =,000 Wage rate r = $./hr r = $./hr A Mathematical Programming Approach to Analyze the Activity-Based Costing

12 WEN-HSIEN TSAI AND THOMAS W. LIN these two activities utilize the same group of multi-function workers. Thus, Eq. () will be: n ( ) i + i X i = G + G () i= By using Eqs () (), the product-mix decision model for the example is formulated as follows: Maximize = X + X + X,0,000 0B 0B 0B 0B 0B 00B,000R,000R,000R,000 0,000 0,000 Subject to: Unit-Level Direct Material Constraints: X + X + X,000 X + X + X,000 Piecewise Unit-Level Direct Labor Constraints: X + X +.X,000,000 = = + = Stepwise Facility-Level Machine Hour Constraints: X + X + X,000 0,000, = Batch-Level Activity Constraints: Scheduling X B 0 X B 0

13 A Mathematical Programming Approach to Analyze the Activity-Based Costing Setup Product-Level Activity Constraints: Design X 0B 0 B + B + B 0 X B 0 X B 0 X B 0 B + B + B 00 X,000R 0 X,000R 0 X,000R 0 R + R + R where X i, B ij 0, i =,,, j B; 0,, 0; R i,,, k = 0,, i =,,, k = 0,,. This is a Mixed-Integer Programming (MIP) model. We solve this problem by utilizing the software, LINDO (Schrage, ), and obtain the following optimal solution: X = X = 0 X =,000 0 = 0 = 0. = 0. = 0 = B = B = 0 B = B = B = 0 B = R = R = 0 R = 0 = 0 = = 0 Accordingly, the optimal product-mix is (X, X, X ) = (, 0, 00), which requires,000 units (= + 0 +,000) of the first kind of material,, units (= + 0 +,000) of the second kind of material,,0 (= + 0 +,000) machine hours, and,00 (= ,000) direct labor hours. The total profit is $,. This means that the machine capacity is expanded to,000 hours (i.e. there are 0 excess machine hours) by renting machines, and that the direct labor capacity is expanded to,00 labor hours by adding,00 overtime hours.

14 WEN-HSIEN TSAI AND THOMAS W. LIN. DISCUSSION In the model shown in this paper, we only considered the capacity expansions for machine hours and direct labor hours. In future research, we also can consider the capacity expansions for various levels of activities by using a similar process of model formulation. In addition to capacity expansions, we can further improve this product-mix model by relaxing the strong assumption: the unit selling prices and the unit direct material costs are constant within the relevant range. This may be achieved by using piecewise linear functions to approximate the non-linear revenue or the non-linear direct material costs (Tsai & Lin, ). The future research can also build a product-mix model to evaluate the impact on profits or product mixes of decreasing unit activity costs through ABC activity improvements.. CONCLUSIONS In recent years, activity-based costing has become a popular cost management technique in both accounting academics and business practice. ABC uses a two-stage procedure to assign resource costs to products: first from resources to activities, then from activities to cost objects (products). It improves the accuracy of product cost data derived from the traditional volume or unit-based (e.g. direct labor hours) costing systems. One of the special features of ABC is that it uses both volume-based (i.e. unit-level) and non-volume-based (i.e. batch-level, product-level, or facility-level) drivers to assign activity costs to products according to the nature of activities. Product-mix decision analysis is an important part of the ABC information. To extend the existing research literature, this paper incorporates capacity expansion features into a ABC product-mix decision model by using a mathematical programming approach. The current traditional ABC product-mix decision models do not explicitly consider capacity expansions. This paper contributes to the management sciences and accounting literature by developing a new mixed integer programming product-mix model that maximizes a firm s profit with five major types of ABC constraints: () unit-level direct material constraints; () unit-level piecewise direct labor constraints; () batch-level activity constraints (e.g. scheduling and setup activities); () product-level activity constraints (e.g. product design); and () stepwise facility-level activity cost with machine hour constraints (e.g. plant guard and management). With the model presented in this paper, we could evaluate the benefit of simultaneously expanding the various kinds of capacity.

15 A Mathematical Programming Approach to Analyze the Activity-Based Costing ACKNOWLEDGMENT This research was supported by the National Science Council of the Republic of China under grant NSC0 -H REFERENCES Beale, E. M. L., & Tomlin, J. A. (). Special facilities in a general mathematical programming system for nonconvex problems using ordered sets of variables. In: J. Lawrence (Ed.), Proceedings of the th International Conference on Operational Research (pp. ). London: Tavistock. Brimson, J. A. (). Activity accounting An activity-based costing approach. New York, NY: Wiley. Cooper, R. (). Cost classification in unit-based and activity-based manufacturing cost systems. Journal of Cost Management, (),. Goldratt, E. (). What is this thing called theory of constraints and how should it be implemented? Croton-on-Hudson, NY: North River Press. Jadicke, R. K. (). Improving breakeven analysis by linear programming techniques. NAA Bulletin,. Kee, R. (, December). Integrating activity-based costing with the theory of constraints to enhance production-related decision making. Accounting Horizons, (),. Kee, R., & Schmidt, C. (00). A comparative analysis of utilizing activity-based costing and the theory of constraints for making product-mix decisions. International Journal of Production Economics, (),. Malik, S. A., & Sullivan, W. G. (). Impact of ABC information on product mix and costing decisions. IEEE Transactions on Engineering Management, (),. Metzger, L. M. (). The power to compete: The new math of precision management. National Public Accountant, (),,. Schrage, L. (). Linear, integer, and quadratic programming with LINDO-user s manual (rd ed.). Redwood City, CA: Scientific Press. Spoede, C., Henke, E., & Umble, M. (). Using activity analysis to locate profitability drivers: ABC can support a theory of constraints management process. Management Accounting, (),. Tsai, W.-H. (). Product-mix decision model under activity-based costing. Proceedings of Japan U.S.A. Symposium on Flexible Automation, Vol. I, Institute of Systems, Control and Information Engineers, Kobe, Japan, July, pp. 0. Tsai, W.-H. (a). A technical note on using work sampling to estimate the effort on activities under activity-based costing. International Journal of Production Economics, (),. Tsai, W.-H. (b). Activity-based costing model for joint products. Computers & Industrial Engineering, (/),. Tsai, W.-H., & Lin, T.-M. (, July). Nonlinear multiproduct CVP analysis with 0 mixed integer programming. Engineering Costs and Production Economics, (),. Turney, P. B. B. (a). COMMON CENTS-The ABC performance breakthrough, how to succeed with activity-based costing. Hillsboro, OR: Cost Technology.

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