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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: (1) unit-level direct material
constraints; (2) unit-level piecewise direct labor constraints; (3) batch-level
activity constraints (e.g. scheduling and setup activities); (4) product-level
Mathematical Programming
Applications of Management Science, Volume 11, 163–178
Copyright © 2004 by Elsevier Ltd.
All rights of reproduction in any form reserved
ISSN: 0276-8976/doi:10.1016/S0276-8976(04)11012-2
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activity constraints (e.g. product design); and (5) 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.
1. 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,
1992a, pp. 80–89, 1992b). 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, 2000; Spoede et al., 1994). In light of this, some
authors (Kee, 1995; Kee & Schmidt, 2000; Malik & Sullivan, 1995; Tsai, 1994;
Yahya-Zadel, 1998) 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 (1995) integrated ABC with the Theory of Constraints (TOC) in the
product-mix decision analysis. As quoted by Kee and Schmidt (2000) from
Goldratt (1990): “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
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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 2. Section 3 presents an ABC product-mix decision model with capacity
expansion features. In Section 4, a numerical example is used to demonstrate
how to apply the model. Finally, the discussion and conclusions are presented in
Sections 5 and 6, respectively.
2. COST ASSIGNMENT VIEW OF ABC
The detailed cost assignment view of ABC is shown in Fig. 1 (Tsai, 1996a; Turney,
1992a, b). ABC assumes that cost objects (e.g. products, product lines, processes,
Fig. 1. The Detailed Cost Assignment View of ABC.
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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, 1991, p. 135). The following are the categories for
manufacturing activities: (1) unit-level activities (performed one time for one unit
of product, e.g. machining, finishing); (2) batch-level activities (performed one
time for a batch of products, e.g. setup, scheduling); (3) product-level activities
(performed to benefit all units of a particular product, e.g. product design); and
(4) facility-level activities (performed to sustain the manufacturing facility, e.g.
plant guard and management) (Cooper, 1990). 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,
1993; Tsai, 1996b). For the purpose of product-mix decisions, we regard the costs
of facility-level activities as the common fixed cost in this paper.
3. ABC PRODUCT-MIX DECISION MODEL
WITH CAPACITY EXPANSIONS
Jadicke (1961) applied a Linear Programming (LP) technique to a Cost-VolumeProfit (CVP) model, called “Product Mix” model in many management accounting
or LP texts, which could aid management in determining the optimal product mix,
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maximizing total profit under some limits (constraints) to production or sales in
the case of multi-product companies. In recent years, some authors (Kee, 1995;
Kee & Schmidt, 2000; Malik & Sullivan, 1995; Tsai, 1994; Yahya-Zadel, 1998)
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.
3.1. 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., 1989) 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.
3.2. Capacity Expansion Features
3.2.1. 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. 2) which varies with machine hours, observed from a prior cost
behavior analysis. The total facility-level activity cost is F0 under the current
capacity H0 machine hours. If the capacity is successively expanded to H1 , H2 , . . .
Ht machine hours, the total facility-level activity cost increases to F1 , F2 , . . . Ft ,
respectively. Let Xi 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, 1990):
Total facility-level activity cost =
t
k=0
F k ␪k
(1)
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Fig. 2. Stepwise Facility-Level Activity Cost.
Constraints:
n
␭ih X i ≤
i=1
t
t
H k ␪k
(2)
k=0
␪k = 1
(3)
k=0
where (␪0 , ␪1 , . . . ␪t ) is an SOS1 set of 0–1 variables within which exactly one
variable must be non-zero (Beale & Tomlin, 1970; Williams, 1985). When ␪q = 1
(q = 0), we know that the capacity needs to be expanded to the qth level, i.e. Hq
machine hours.
3.2.2. 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. 3. The available
Fig. 3. Piecewise Direct Labor Cost.
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normal direct labor hour is G1 and the direct labor hour can be expanded to G2 ;
the total direct labor cost is L1 and L2 at G1 and G2 , respectively. As a result, the
total direct labor cost and the associated constraints are (Tsai & Lin, 1990):
Total Direct Labor Cost = L 1 ␮1 + L 2 ␮2
(4)
Constraints:
TL = G 1 ␮1 + G 2 ␮2
(5)
␮0 − ␩1 ≤ 0
(6)
␮1 − ␩1 − ␩2 ≤ 0
(7)
␮2 − ␩2 ≤ 0
(8)
␮0 + ␮1 + ␮2 = 1
(9)
␩1 + ␩2 = 1
(10)
where (␩1 , ␩1 ) is an SOS1 set of 0–1 variables within which exactly one variable
must be non-zero; (␮0 , ␮1 , ␮2 ) is an SOS2 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, 1970; Williams, 1985); TL is the total direct labor hour we need
and its function depends on the case under study.
If ␩1 = 1, then ␩2 = 0 [from Eq. (10)], ␮2 = 0 [from Eq. (8)], ␮0 , ␮1 ≤ 1
[from Eqs (6) and (7)], and ␮0 + ␮1 = 1 [from Eq. (9)]. Thus, from Eqs (4) and
(5) we know that total direct labor cost and total labor hour needed are L1 ␮1 and
G1 ␮1 , respectively; this means that we will not need the overtime work.
If ␩2 = 1, then ␩1 = 0 [from Eq. (10)], ␮0 = 0 [from Eq. (6)], ␮1 , ␮2 ≤ 1
[from Eqs (7) and (8)], and ␮1 + ␮2 = 1 [from Eq. (9)]. Thus, from Eqs (4)
and (5) we know that total direct labor cost and total labor hour needed are
L 1 ␮1 + L 2 ␮2 and G 1 ␮1 + G 2 ␮2 , respectively; this means that we will need the
overtime work.
3.3. 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
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=
n
pi Xi −
i=1
−
s
n c m a im X i − (L 1 ␮1 + L 2 ␮2 ) −
d j ␦ij B ij −
i=1 j∈B
d j ␭ij X i
i=1 j∈U
i=1 m=1
n n n d j ␳ij R i −
i=1 j∈P
t
F k ␪k
(11)
k=0
Subject to:
Unit-Level Direct Material Constraints:
n
a im X i ≤ Q m ,
m = 1, 2, . . . s
(12)
i=1
Piecewise Unit-Level Direct Labor Constraints:
TL = G 1 ␮1 + G 2 ␮2
(13)
␮0 − ␩1 ≤ 0
(14)
␮1 − ␩1 − ␩2 ≤ 0
(15)
␮2 − ␩2 ≤ 0
(16)
␮0 + ␮1 + ␮2 = 1
(17)
␩1 + ␩2 = 1
(18)
Stepwise Facility-Level Machine Hour Constraints:
n
␭ih X i −
i=1
t
t
H k ␪k ≤ 0,
(19)
k=0
␪k = 1
(20)
k=0
Batch-Level Activity Constraints:
X i ≤ ␴ij B ij ,
n
i = 1, 2, . . . n;
␦ij B ij ≤ T j ,
j∈B
j∈B
(21)
(22)
i=1
Product-Level Activity Constraints:
Xi ≤ Di Ri ,
i = 1, 2, . . . n
(23)
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␳ij R i ≤ V j ,
j∈P
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(24)
i=1
X i ≥ 0,
i = 1, 2, . . . n
(25)
(␮0 , ␮1 , ␮2 ) : An SOS2 set of non-negative variables.
(26)
(␩1 , ␩2 ) : An SOS1 set of 0–1 variables
(27)
(␪0 , ␪1 , . . . ␪t ) : An SOS1 set of 0–1 variables
(28)
R i : 0–1 variables,
i = 1, 2, . . . n
B ij : Non-negative integer variables,
(29)
i = 1, 2, . . . n,
j∈B
(30)
where
Xi =
pi =
Cm =
aim =
Qm =
dj =
␭ij =
␦ij =
B ij =
␴ij =
Tj =
␳ij =
Ri =
Vj =
Di =
The production quantity of product i;
The unit selling price of product i;
The unit cost of the mth material;
The requirement of the mth material for one unit of product i;
The available quantity of the mth material;
The actual running activity cost per activity driver for activity j;
The requirement of the activity driver of unit-level activity j ( j ∈ U) for
one unit of product i;
The requirement of the activity driver of batch-level activity j ( j ∈ B) for
product i;
The number of batches of batch-level activity j ( j ∈ B) for product i;
The number of units per batch of batch-level activity j ( j ∈ B) for product
i;
The capacity limit of the activity driver of batch-level activity j ( j ∈ B);
The requirement of the activity driver of product-level activity j ( j ∈ P) for
product i;
The indicator for producing product i (Ri = 1) or not producing product i
(Ri = 0);
The capacity limit of the activity driver of product-level activity j ( j ∈ P);
The maximum demand of product i;
Other variables and parameters are as mentioned before.
Equation (11) represents the total profit function ␲, and Eqs (12)–(24) are the
constraints associated with various resources and activities. Equation (12) is the
direct material constraint. Equations (13)–(18) are the direct labor constraints
described in Section 3.2.2. Equations (19) and (20) are the machine hour
constraints described in Section 3.2.1.
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Equations (21) and (22) are the constraints associated with batch-level activities,
where Eq. (22) 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, Tj 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 1 and Tj is the available
number of setup.
Equations (23) and (24) are the constraints associated with product-level
activities. Equation (23) is the market demand constraint and Eq. (24) 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, Vj is the available number of drawings for the
firm’s capacity, and ␳ij is the number of drawings needed for product i.
4. A NUMERICAL EXAMPLE
In this section, we present a numerical example. Assume that a manufacturing
company is considering producing product 1, product 2, and product 3 (i = 1, 2,
3) and that these products need two kinds of the same direct material (m = 1, 2).
We also assume that it needs five main activities in producing these three products:
two unit-level activities, Machining and Finishing (U = {1, 2}), two batch-level
activities, Scheduling and Setup (B = {3, 4}), and one product-level activity,
Product Design (P = {5}).
The related data for this example are shown in Table 1.
From Table 1, we know that the total facility-level activity cost is $40,000
under the current capacity H 0 = 3,000 machine hours and that the capacity can
be expanded to H 1 = 4,000 or H 2 = 5,000 machine hours by renting additional
machines with the total facility-level activity cost increasing to F 1 = $55,000
or F 2 = $70,000, respectively. Besides, the available normal direct labor hour is
G 1 = 7,000 hours with the normal wage rate of $1.2/hr and the direct labor hour
can be expanded to G 2 = 9,000 hours with the overtime wage rate of $1.8/hr.
Further, assume that two unit-level activities, Machining and Finishing, need direct
labor, and Machining activity needs 1/2 ␭i1 labor hour for one unit of product i;
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Product (i)
Maximum demand
Selling price
Direct material
1
2
3
Di
pi
ai1
ai2
1,000
130
8
5
1,000
90
6
2
2,000
110
4
3
␭i1
␭i2
2
3
2
1
1
2
␦i3
␴i3
␦i4
␴i4
1
120
2
10
1
120
2
10
1
60
4
30
T3 = 60
20
10
30
V5 = 55
m=1
m=2
Activity driver
Machine hours
Labor hours
c1 = 3
c2 = 4
Cost/driver dj
$2
$1
Production
Orders
Setup
Hours
$100
Drawings
$400
␳i5
Facility-level cost
Cost
Machine hrs
F0 = $40,000
H0 = 3,000
F1 = $55,000
H1 = 4,000
F2 = $70,000
H2 = 5,000
Direct labor constraint
Cost
Labor hrs
Wage rate
L1 = $8,400
G1 = 7,000
r1 = $1.2/hr
L2 = $12,000
G2 = 9,000
r2 = $1.8/hr
Unit-level activity
Machining
Finishing
j
1
2
Batch-level activity
Scheduling
3
Setup
Product-level activity
Design
4
5
$150
Available Capacity
Q1 = 15,000
Q2 = 12,000
T4 = 50
A Mathematical Programming Approach to Analyze the Activity-Based Costing
Table 1. Example Data.
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these two activities utilize the same group of multi-function workers. Thus, Eq. (13)
will be:
n 1
␭i 1 + ␭i2 X i = G 1 ␮1 + G 2 ␮2
(31)
2
i=1
By using Eqs (11)–(31), the product-mix decision model for the example is
formulated as follows:
Maximize
␲ = 79X 1 + 59X 2 + 82X 3 − 8,400␮1 − 12,000␮2 − 100B 13 − 100B 23
− 100B 33 − 300B 14 − 300B 24 − 600B 34 − 8,000R 1 − 4,000R 2
− 12,000R 3 − 40,000␪0 − 55,000␪1 − 70,000␪2
Subject to:
Unit-Level Direct Material Constraints:
8X 1 + 6X 2 + 4X 3 ≤ 15,000
5X 1 + 2X 2 + 3X 3 ≤ 12,000
Piecewise Unit-Level Direct Labor Constraints:
4X 1 + 2X 2 + 2.5X 3 − 7,000␮1 − 9,000␮2 = 0
␮0 − ␩1 ≤ 0
␮1 − ␩1 − ␩2 ≤ 0
␮2 − ␩2 ≤ 0
␮0 + ␮1 + ␮2 = 1
␩1 + ␩2 = 1
Stepwise Facility-Level Machine Hour Constraints:
2X 1 + 2X 2 + X 3 − 3,000␪0 − 4,000␪1 − 5,000␪2 ≤ 0
␪0 + ␪1 + ␪2 = 1
Batch-Level Activity Constraints:
Scheduling
X 1 − 120B 13 ≤ 0
X 2 − 120B 23 ≤ 0
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X 3 − 60B 33 ≤ 0
B 13 + B 23 + B 33 ≤ 60
Setup
X 1 − 10B 14 ≤ 0
X 2 − 10B 24 ≤ 0
X 3 − 30B 34 ≤ 0
2B 14 + 2B 24 + 4B 34 ≤ 500
Product-Level Activity Constraints:
Design
X 1 − 1,000R 1 ≤ 0
X 2 − 1,000R 2 ≤ 0
X 3 − 2,000R 3 ≤ 0
20R 1 + 10R 2 + 30R 3 ≤ 55
where Xi , B ij ≥ 0, i = 1, 2, 3, j ∈ B; ␮0 , ␮1 , ␮2 ≥ 0; Ri , ␩1 , ␩2 , ␪k = 0, 1, i = 1,
2, 3, k = 0, 1, 2. This is a Mixed-Integer Programming (MIP) model. We solve
this problem by utilizing the software, LINDO (Schrage, 1987), and obtain the
following optimal solution:
X1 = 875
␮0 = 0
␩1 = 0
B13 = 8
B14 = 88
R1 = 1
␪0 = 0
X2 = 0
␮1 = 0.25
␩2 = 1
B23 = 0
B24 = 0
R2 = 0
␪1 = 1
X3 = 2,000
␮2 = 0.75
B33
B34
R3
␪2
= 34
= 67
=1
=0
Accordingly, the optimal product-mix is (X1 , X2 , X3 ) = (875, 0, 2000), which
requires 15,000 units (=8 × 875 + 6 × 0 + 4 × 2,000) of the first kind of
material, 10,375 units (=5 × 875 + 2 × 0 + 3 × 2,000) of the second kind of
material, 3,750 (=2 × 875 + 2 × 0 + 1 × 2,000) machine hours, and 8,500
(=4 × 875 + 2 × 0 + 2.5 × 2,000) direct labor hours. The total profit ␲ is
$76,225. This means that the machine capacity is expanded to 4,000 hours (i.e.
there are 250 excess machine hours) by renting machines, and that the direct labor
capacity is expanded to 8,500 labor hours by adding 1,500 overtime hours.
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5. 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, 1990).
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.
6. 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: (1) unit-level direct material constraints; (2) unit-level piecewise
direct labor constraints; (3) batch-level activity constraints (e.g. scheduling and
setup activities); (4) product-level activity constraints (e.g. product design);
and (5) 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.
A Mathematical Programming Approach to Analyze the Activity-Based Costing
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ACKNOWLEDGMENT
This research was supported by the National Science Council of the Republic of
China under grant NSC90–2416-H-008–001.
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