Managing Global Transitions: Globalisation, Localisation, Regionalisation
Proceedings of the 8th International Conference of the Faculty of Management Koper 20–24 November 2007 · Congress Centre Bernardin · Portorož, Slovenia
HOW TO CONSIDER THE COST OF UNUSED CAPACITY IN MANAGERIAL DECISIONS?
Dr. Zoltan Sebestyen, Budapest University of Technology and Economics, Hungary sebestyen@mvt.bme.hu
Dr. Tamas Koltai, Budapest University of Technology and Economics, Hungary koltai@mvt.bme.hu
ABSTRACT
For the last century the most widely used two measures to describe usage of resources in production processes have been two volume-based, conventional measures; the utilization and the efficiency. During this time the ratio of fixed cost to the variable cost and the ratio of overhead to the direct cost increased in the total cost structure. Today the operation is not so labor and material intensive as it was decades ago. It is obvious that the conventional, volume-oriented capacity measures are out-of-date. Planning production with linear programming models based on conventional capacity measures raises several problems, and can provide irrelevant and misleading data for management decisions. In 1998 Kaplan and Cooper introduced a new measure, the „cost of unused capacity”, which seems to be able to handle the problem discussed. The cost of unused capacity is able to characterize resource usage even in non-technical systems and it has economic content. This measure is practically superior to the conventional measures, but there are several questions to answer. The paper shows how to consider the cost of unused capacity in the models for analyzing an existing, on-going system and how to use it for planning a new production facility before its operation.
The suggested policy is illustrated with a capacity extension problem in a sugar production process and with a production planning problem in a manufacturing system.
Keywords: Cost of unused capacity, fixed cost, production planning INTRODUCTION
Capacity is one of the most important parameters of resources used in production processes.
Its definition and analysis is therefore one of the key areas of production management. There are two widely used measures that can be used for the analysis of capacity usage. Capacity usage relates the actual output to the quantity produced under ideal conditions (designed capacity), while efficiency compares the actual output with the maximum output possible according to the work schedule (effective capacity) (Waters, 1991).
Variable cost based operations management methods dominated the production decision making process for several decades (Johnson, 1991; Kaplan, 1990). For example, product mixes were calculated based on contribution margin maximization or variable cost minimization. When fixed cost was incorporated in production planning models, it was considered as stepwise variable cost, which means it is fixed only in a certain interval of the decision variable, so the fixed cost stays constant within a certain range of the required capacity (Johnson – Montgomery, 1974; Basso – Peccati, 2001). Within the intervals he cost is really fixed, but considered as sunk cost.
This approach was questioned by Cooper and Kaplan by introducing the cost of unused capacity. They said that costs are fixed only if managers consider them fixed. Based on this new concept, the management will make an effort to collect data about fixed costs and to utilize this information. Later Cooper and Kaplan divided the fixed cost of resources into two parts; into cost of used capacity and into cost of unused capacity (Cooper – Kaplan, 1992).
The higher the cost not covered by production the stronger motivation of management is required to solve the problem.
Variable cost based approaches were effectively used for a long time, but the change of cost structure of production and service processes has changed in the last decade (Johnson, 1991;
Kaplan, 1991; Wiersema, 1996). While the ratio of variable cost was about 30-70 percent in the total production costs, this ratio decreased to 10-30 percent by now. The increasing ratio of fixed cost within the total cost of manufacturing and service operations directs attention to fixed cost oriented methods (Ferrara, 1995; Johnson, 1991; Koltai, 1995).
It is clear that the utilization of an inexpensive resource (obsolete automatic lathe) and an expensive one (newly purchased computer controlled, programmable manufacturing center) mean different and need different approaches. In parallel, the cost of unused capacity of these machines at the same utilization level can be totally different.
The paper is organized as follows. First, the paper presents how to consider the cost of unused capacity in the models for analyzing and adjusting an existing, on-going system. A capacity extension model in modularly built continuous production is presented and illustrated with the help of a sugar production process. Next, the use of the cost of unused capacity is described for planning a new production facility before its operation. The suggested policy is illustrated with a production planning problem in a flexible manufacturing system. Finally, conclusions about the different implications of the cost of unused capacity based production planning models are presented.
CAPACITY BALANCING MODEL
The cost of unused capacity can be calculated when the fixed cost of the resource, the actual resource usage and the effective capacity are known. The applied resources, the fixed cost of the
Used capacity (hi) Fixed cost of capacity (Fi)
Cost of unused capacity (Pi)
Cost of used capacity Unit cost of
capacity
ci
Effective capacity (ui)
Figure 1: Interpretation of the cost of unused capacity
resources, and the actual capacity usage require the development of sophisticated management information (Cooper – Kaplan, 1992). The analysis of the cost of unused
capacity is based on the following simple formula: Activity Availability = Activity Usage + Unused Capacity (Figure 1).
The higher demand of information for the calculation is provided by an Activity Based Costing (ABC) system. ABC approximates an ideal situation where the overhead is only incorporated in the product cost to the extent it actually exploits the resources. Costs must be collected to cost centers, and a characteristic cost driver must be assigned to each cost group in the center. The ABC method works on the basis of this principle (Cooper, 1988).
PRODUCTION PLANNING USING CAPACITY BALANCING MODEL
The analysis can be done for the actual resource, for a part of the resource, but also in an aggregate way for a set of resources belonging to more than one activity center. The question is how to take the cost of unused capacity into consideration for production planning decisions and what results and benefits can be obtain with that. The time horizon of making decisions basically affects the objective function of the model.
Planning and Decision Making for an Ongoing Operations
Continuous flow type operations are used for the production of a single product or small groups of related products in large quantities. Such processes use highly specialized equipment which can operate for 24 hours a day with no change or interruption. The process is capital intensive, but when working, it needs a very small workforce. This type of production is typical in the chemical and food industry (petrol refineries, paper mills, sugar refineries). Continuous flow operation is different to assembly lines, as the product emerges as a continuous flow rather than discrete units.
The optimal capacity extension model for continuous flow operation evaluates the tradeoff of increased bottleneck capacity and capacity underutilization of non-bottleneck operations. It is assumed that capacity of each operation (or a group of operations) is determined by modularly built resources. The production system consists of N independently extendible group of resources. Each resource group originally has a given capacity (Ki) and can be extended modularly. The capacity of the extension modules is ΔKi. If the capacity of resource group i is increased by implementing an extension module, the fixed cost of the production system increases by bi. The original capacity of the bottleneck of the production system is BB0, which is the minimum of the capacity of all the resource groups. As a consequence of the implementation of several extension modules the capacity of the system is increased to B. In case of non-bottleneck operations capacity is underutilized (si), and the unit cost of underutilized capacity is ci. A maximum quantity for production is given (Q), and the decision-maker decides on how much capacity extension modules for each resource group (xi) should be implemented. The objective of production is to satisfy demand and to generate revenue to cover the variable costs and the fixed costs. The unit contribution margin of the product is f. The notations used in this model are summarized in Table 1.
Table 1: Notations
N number of resource groups
f unit contribution margin of the product K i capacity of resource group i, i=1,…,N
bi fixed cost increase caused by one extension module of resource group i, i=1,…,N
ci unit capacity cost of existing resource group i, i=1,…,N Q maximum production quantity
B0 bottleneck capacity of the original production system B bottleneck capacity of the extended production system si idle capacity of original resource group i, i=1,…,N
xi number of extension modules (integer variable) of resource group i, i=1,…,N
It might be economic to implement as many capacity modules as many is required by the production of the maximum quantity (Q). If, however, the cost of unused capacity of non- bottleneck operations is overly high, the decision maker may decide on producing less than the maximum quantity. The increase of contribution margin of the product motivates decision-maker to produce as much as possible, but fixed cost increase (xibi) and the cost of unused capacity (cisi) motivates decision-maker to be cautious with capacity extension. The linear programming formulation of the capacity extension problem is the following,
∑ ∑
= =
−
−
= N
i
N i
i i i
is bx
c Bf z
1 1
) (
Max (1)
Subject to
N i
K x K
B i i i
i [ ] 1,...,
Min + Δ =
= (2)
N i
B K x K
si = i+ iΔ i− =1,..., (3)
Q
B≤ (4)
;
≥0
B si ≥0; xi ≥0; i = 1,…,N (5)
xi , i = 1,…,N are integers (6)
Objective function (1) evaluates the tradeoff of increased contribution margin and increased fixed cost and unused capacity. Contribution margin increase is equal to f(B–B0). Since fB0 is constant, this element can be omitted from the objective function. Constraints (2) find the bottleneck of the system, and constraints (3) calculate the unused capacity of the non- bottleneck resource groups. Constraint (4) limits production quantity. Constraints (5) summarize non-negativity constraints, while constraints (6) impose integrality constraints for the decision variables. Constraints (2) can simply be transformed into linear inequalities, that is,
N i
B K x
Ki+ iΔ i ≥ =1,..., (7)
Since fB is maximized in objective function (1), the value of B will be the highest possible.
According to constraints (7) this highest possible value is the minimum of the capacity of the resource groups. The resulting model is a linear integer mathematical programming model which can be solved with any of the commercially available solvers (e.g. Lingo, XA).
1. Arrival of sugar beet
2. Cleaning
4. Diffusion
5. Saturation
7. Distillation
8. Thick juice is cooked in vacuum 10. Warehousing,
packaging 3. Cutting
6. Juice filtering
9. Spin-dry 1. Arrival of sugar beet
2. Cleaning
4. Diffusion
5. Saturation
7. Distillation
8. Thick juice is cooked in vacuum 10. Warehousing,
packaging 3. Cutting
6. Juice filtering
9. Spin-dry
Figure 2: The process flow of sugar refinery
The application of the model was illustrated with a sugar production process (Koltai – Sebestyen, 2002;, Koltai – Sebestyen, 2005) (Figure 2). Major data of the problem are changed for confidential reasons, and the process is slightly simplified, but the major characteristics of the model reflect the problem of capacity extension in case of continuous flow operations.
If the cost of unused capacity is ignored, the required numbers of capacity extension units are higher, and the bottleneck operation is juice filtering. In this case the maximum quantity can be produced, and the capacity underutilization of non-bottleneck operations are higher. These results show that using the cost of unused capacity in the objective function leads to different capacity extension policy. When the contribution margin is high, the increase of contribution margin compensate both for the increased fixed cost of higher capacity and for the cost of unused capacity. If the contribution margin is low, the increase of contribution margin does not compensate for the increased fixed cost and for the cost of unused capacity.
Planning and Decision Making Before Operations
In Flexible Manufacturing Systems (FMS) the machining methods, machine tools, handling equipment, control systems and computer systems are used in an integrated way. The capacity analysis of FMSs is very complicated for two reasons. First, products may follow several routes during the manufacturing process, using several resources. Second, the resources used in FMSs are very expensive; therefore their application should be efficient and effective.
A flexible manufacturing system is a group of machines capable of completing one or more operations and usually connected to an automatic material handling system, all controlled by a central computer (Inman, 1991). A flexible manufacturing system is capable of manufacturing various products. The products can be produced by using different machines.
production process. In a flexible manufacturing system the products can be usually produced on several different routes. The routes are specified by the technologists of the manufacturing system based on the characteristics of the necessary operations to be carried out on the products and the properties of the available machines. Let us assume that P different products must be manufactured in a flexible system. The products are denoted by the index p. The number of the routes of product p is Rp (p =1,…,P). The routes are denoted by the index r.
The manufacturing system contains M machines, indicated by the index m. The planned capacity of the machine is indicated by km (m=1,…,M). In the illustration the unit of measure of planned capacity is minute. The planned capacity indicates the capacity that the machine can reach under ideal conditions. The operation time shows how many minutes the given product spends on the applied machine using the given manufacturing route. The operation time of product p on route r using machine m is indicated by tprm (p=1,…,P; r=1,…,Rp; m=1,…,M). If a product needs a machine in several different stages of the manufacturing process, the operation time will be the total of the times spent in the individual stages. The operation time also includes the time necessary to change the tools and the pieces. However, these times are significantly less than the working time, therefore they can be ignored. Order means the product quantity to be manufactured in the given manufacturing system. The order of product p is indicated by xp (p=1,…,P). The amount of product p manufactured on route r is indicated by xpr (p=1,…,P; r=1,…,Rp). The primary result of the models set up with different purposes is the product quantity to be manufactured, indicated by xp. The applied notations in this model can be found in Table 2.
Table 2: Notations
xp the amount of product p manufactured in the given period (p=1,…,P),
xpr the amount of product p manufactured on route r in the given period (p=1,…,P), (r=1,…,Rp),
km the capacity of resource m available in the given period (m=1,…,M),
tprm the time necessary to manufacture product p on route r on machine m (p=1,…,P), (r=1,…,Rp), (m=1,…,M),
up the maximum amount of product p that can be sold on the market (p=1,…,P),
lp the minimal amount of product p to be manufactured (p=1,…,P), fp the contribution margin of product p (p=1,…,P),
cm the unit cost of resource m (m=1,…,M).
The sum of the orders manufactured on the various routes of the product. The lower limit of the product is the minimal amount of the market demand. This product quantity must be produced in any case. The minimal amount of the product to be manufactured can come, for example, from a contract with a customer, or the mother company may have already ordered a given quantity from one of its own plants for the next period. The amount of the product to be manufactured (lower limit) of the product is indicated by lp (p = 1,…,P). The upper limit of the product is the maximum amount of the product that can be sold on the market and is indicated by up (p = 1,…,P).
Production planning of flexible manufacturing systems is made complex by the larger number of manufacturing possibilities. The differences in the machines are reflected mostly
in the value of the machines and in the support activities therefore fixed costs are different (Koltai, 1995).
Table 3 illustrates in a general form the operation time data requirement of the production planning of a manufacturing system. The columns of the resource-product (machine-product) table show the resources, the rows show the products and routes while the cells show the operation times.
Table 3: Resource-product table
Machine 1 Machine 2 … Machine m
Product 1 r11 t111 t112 … t11M
r12 t121 t122 t12M
…
r1R1 t1R11 t1R12 t1R1M
Product 2 r21 t211 t212 … t21M
r21 t221 t222 t22M
…
r2R2 t2R21 t2R22 t2R2M
… … … …
Product P rp1 tP11 tP12 … tP1M
rp2 tP21 tP22 tP2M
…
rpRP tPRP1 tPRP2 tPRPM
In addition to the usual goals of management, the minimization of the cost of unused capacity has also emerged (Kaplan – Cooper, 1998). We set up two models to evaluate the effect of the cost of unused capacity on the production plan. In both models we apply the same constraints, and the differences are expressed in the objective functions. The first condition (8) represents the limit stemming from the capacity of the machines. The second condition (9) limits the minimal amount to be produced and the maximum marketable amount. The third condition (10) defines the amount to be manufactured by product category, by adding up the amounts manufactured on each route.
∑∑
≤p r tprmxpr km (8)
M m
R r
P p
p
,..., 1
,..., 1
,..., 1
=
=
=
p p
p x u
l ≤ ≤ p=1,...,P (9)
p r
pr x
∑
x =(10)
Rp
r
P p
,..., 1
,..., 1
=
=
In the first case the objective is to make a production plan that maximizes contribution while fulfilling manufacturing and market requirements. The objective function includes the contribution to the profit (11).
⎥⎦
⎢ ⎤
⎣
= ⎡
∑ ∑
p r
pr
p x
f
z Max
The result of the model is a product mix, which maximizes the contribution to profit. The machines, however, represent different values and manufacturing characteristics. The result of the model does not take into account the fixed costs of the machines the products are manufactured on.
The objective of the second model is to create a production plan that not only maximizes contribution but also helps maximize the cost of unused capacity.
⎥⎦
⎢ ⎤
⎣
⎡ + −
=
∑ ∑ ∑ ∑∑
p m p r
pr prm m
m r
pr
p x c k t x
f
z Max ( )
(12)
This objective function (12) contains both the contribution margin and the cost of unused capacity. At a first glance it might seem strange to maximize the cost of unused capacity, but from a management point of view it makes sense. If production can be done in several ways, then that production program should be chosen, which requires the smallest amount of resources expressed in financial measures. In this case, the under-utilized capacity will have high value. For example, if technologically possible, an order will be assigned to a conventional machine, instead of an expensive manufacturing center.
We illustrated the use of the presented production-planning model with an example (Sebestyen et al., 2004). The data and the characteristics of the production environment are from a plant producing hydraulic and pneumatic parts. The products are manufactured on various types of CNC machines. The machines are versatile and can perform many operations necessary for production, and the products can be manufactured on several routes.
The first model maximizes only the contribution to profit. The solution provides the ordered amounts of the individual routes, that is, how many of the products have to be manufactured on the given route in the given period (1 month). Besides contribution, the second model also takes into account the cost of unused capacity.
The calculations justified our assumptions (Figure 3). With a minimal change in the product structure and contribution, a significant amount of capacity is freed on the high-value resources. With the resources freed up, we do not have to bear the costs of renting, or if we own these resources, we can sell them, or use them for another product with a high contribution margin. And if we assume that the value of the machines correlate with the complexity of operations they can perform, they can be more easily assigned to other outsourced tasks.
0 5000 10000 15000 20000 25000
machine 1 machine 2 machine 3 machine 4 machine 5 unused
capacity [gó/h] Model_1
Model_2
Figure 3: Utilization of machines
CONCLUSION
The consideration of the cost of unused capacity in capacity extension decisions reflects the awareness of management to the efficient utilization of resources. In many cases, resources are available, and the analysis of cost of unused capacity requires special attention. In this paper, two basic approaches of production planning models were presented to study the effect of the cost of unused capacity on the production plan. First, the objective function of a production planning model contained the cost of unused capacity. If we assume that the high value resources once hired and these resources in an ongoing production can not be dedicated to other works in the market, the model leads to a minimization problem. Second, if we assume that in the planning phase of production, the free capacity on high value resources can be utilized in the market, the linear programming model leads to a maximization problem.
The maximization of the cost of unused capacity (along with the contribution to profit) implicitly assumes that high value capacities will be utilized later. Ideally the higher the value of the capacity, the higher the contribution to profit of the produced parts can be. It is the responsibility of the production manager to evaluate what can be produced by the high value resources, and to ensure that parts which can be manufactured on simple machines, do not use expensive capacities. As a summary, it can be concluded that to develop fixed cost oriented models to substitute variable cost oriented approaches is an unexploited possibility for improving the effectiveness of resource utilization in high fixed cost production and service systems. The models presented in this paper are examples for this opportunity.
REFERENCES
1. Basso, A. and Peccati, L. A. (2001). Optimal resource allocation with minimum activation levels and fixed costs, European Journal of Operational Research, 131, 536- 549.
2. Cooper, R. (1988). The rise of Activity–Based Costing–Part One: What Is an Activity–Based Cost System? Journal of Cost Management for the Manufacturing Industry, Summer, 45–54.
3. Cooper, R. and Kaplan, R. S. (1992). Activity–Based System: Measuring the Cost of Resource Usage. Accounting Horizons, September, 1–13.
4. Ferrara, W. L. (1995). The 21st century paradigm, Management Accounting, December, 30-36.
5. Inman R. A. (1991). Flexible Manufacturing Systems: Issues and Implementation, Industrial Management, 33, July-August, 7-11.
6. Johnson, H. T. (1991). Activity-based management: Past, present, and future, The Engineering Economist, 36 (3), 219-238.
7. Johnson, L. A. and Montgomery, D. C. (1974). Operations Research in Production Planning, Scheduling and Inventory Control. John Wiley and Sons, NY.
8. Kaplan, R. S. (1991). New system for measurement and Control, The Engineering Economist, 36 (3), 201-218.
9. Kaplan, R. S. (1990). Contribution margin analysis: No longer relevant/Strategic cost management: The new paradigm, Journal of Management Accounting Research, Fall, 2-15.
10. Kaplan, R. S. and Cooper, R. (1998). Cost & Effect. Using Integrated Cost Systems to
11. Koltai, T. (1995). Fixed Cost Oriented Bottleneck Analysis with Linear Programming, Omega, International Journal of Management Science, 23 (1), 89–95.
12. Koltai T. and Sebestyen Z. (2005). The consideration of the cost of unused capacity in capacity extension decisions of continuous production processes, 12th International EurOMA Conference on Operational and Global Competitiveness, Budapest, Hungary, 2005, 1630-1639
13. Koltai T. and Sebestyen Z. (2002). New directions of developing a production planning model based on the cost of unused capacity approach, 12th International Conference on Flexible Automation and Intelligent Manufacturing, Dresden, Germany, 654-663.
14. Sebestyen Z, Koltai T. and Juhasz V. (2004). Operations management decisions in FMSs based on the cost of unused capacity. Flexible Automation and Intelligent Manufacturing 2004, Toronto, Canada, 832-839.
15. Waters, C. D. J. (1991). An Introduction to Operations Management. Addison–
Wesley Publishing Company.
16. Wiersema, W. H. (1996). Traditional Costing Methods Just aren’t Making It Today, APICS-The Performance Advantage, September, 32-36.
