© 1995 American Accounting AssociationAccounting HorizonsVol. 9 No. 4December 1995pp. 48-61

Integrating Activity-Based Costing With theTheory of Constraints To Enhance

Production-Related Decision-Making

Robert Kee

Robert Kee is Associate Professor at the University of Alabama.

SYNOPSIS: Activity-based costing (ABC) and the theory of constraints (TOC) represent alternativeparadigms for modeling a firm's production structure. This paper discusses how aspects of the TOCmay be integrated with ABC. The resulting model captures the interaction between the cost, physi-cal resources, and capacity of production activities. The model enables an optimal production mixto be determined from simultaneous evaluation of ABC data and physical attributes of the produc-tion process. Equally important, it facilitates identifying a bottleneck activity that constrains the firm'sproduction opportunities and may lead to excess resources in the firm's other production activities.Sensitivity analysis may be used to estimate the benefits that may accrue from relieving a constraintand identifying the subsequent set of activities that will become a bottleneck as prior constraints arerelieved. Integrating ABC with principles from the TOC provides an expanded framework for under-standing the economic consequences of production-related decisions.

Activity-based costing (ABC) and thetheory of constraints (TOC) represent alter-native paradigms for modeling a firm's pro-duction structure. Both paradigms are de-signed to aid managers in understanding thefirm's production processes and to provide in-formation for resource allocation decisions.While their ohjectives are similar, the meansused to achieve these ohjectives differ signifi-cantly. ABC represents an extension of tradi-tional cost systems and provides more accu-rate product cost information. Firms adopt-ing ABC report an improved understandingof the profitability of their product lines andcustomer hase (Cooper et al. 1992, 55). In ad-dition, many firms report that ABC has stimu-lated improvements in their production pro-cesses (Cooper et al. 1992, 57). Conversely, theTOC represents a theory for optimizing pro-duction that provides insights into the manu-facturing process that traditional cost systemsignore. Firms adopting the TOC indicate thatit has aided in reducing lead time, cycle time.

and inventory while improving productivityand quahty (Goldratt and Fox 1987, 22).

One of the questions confi-onting many man-agers today is that of deciding which paradigmto select for production-related decisions. Asnoted earlier, ABC and the TOC represent radi-cally different approaches to modeling the pro-duction process. However, as noted hy (aoldratt(1990,110) in comparing the TOC with just-in-time and total-quality-management:

Most of these new approaches usually pro-vide a significant contribution. They mayemerge from different angles, they may bebased on different facets of the establishedbase of knowledge.... It is no wonder tbat af-ter some time tbe consolidation process mustbegin. People start to explore ways in wbicbto mold tbe new ideas, whicb bave passed tbetest of reality, into a new and xiniform bodyof knowledge.

The author gratefully acknowledges the helpful com-ments provided by Robin Cooper, Thomas Albright, theassociate editor, and anonymous reviewers. Financialsupport for this paper was provided by the CulverhouseSchool of Accountancy, University of Alabama.

Submitted February 1995Accepted June 1995

Integrating Activity-Based Costing With the Theory of Constraints 49

Similarly, ABC and the TOC may representdifferent aspects of the same underl5ang phe-nomenon.

The purpose of this paper is to discuss howthe principles of ABC and the TOC may beused in conjunction with one another. Thepaper demonstrates that ABC and the TOCreflect different aspects of the production pro-cess and that concepts of both models may beintegrated to provide deeper insights into afirm's underl5dng production process. The re-mainder of the paper is organized as follows:the next two sections evaluate many of theconceptual aspects and limitations of ABC andthe TOC, respectively. This is followed by adiscussion of how the principles of ABC andthe TOC may be integrated within a commonframework for modeling a firm's productionstructure. A methodology is then proposed forimplementing the model. The next section pre-sents a numerical example to illustrate theapplication of the model and the methodologyused for its implementation. The summaryand conclusions of the paper are presented inthe final section.

ACTIVITY-BASED COSTINGABC was developed independently by Gen-

eral Electric and other firms to improve theusefulness of accounting information (Johnson1992, 27). Cooper, Kaplan, and others stud-ied organizations that developed innovativeaccounting systems and introduced ABC to theacademic literature. ABC differs from tradi-tional cost systems in two important respects.First, it traces indirect costs to cost objectssuch as products and customers on the basisof factors (cost drivers) that cause or corre-late highly with indirect costs. The use ofmultiple-cost drivers results in product coststhat more accurately reflect the quantity anddiversity of resources used by products in themanufacturing and support processes used intheir production. Second, ABC traces indirectcosts on the basis of the structural or hierar-chical level at which costs are incurred in theproduction process. For example, many indi-rect costs are incurred at the batch, product,and facility levels (Cooper 1990). The use ofmultiple-cost drivers and tracing cost at the

hierarchical level at which it is incurred en-able ABC to more accurately model the rela-tionship between resources used in productionactivities and the products they are used toproduce. ABC, thereby, provides a better esti-mate of product cost as well as the cost of theindividual activities used in its production.

ABC has been criticized for its failure toidentify and remove constraints that lead todelay, excess, and variation in the productionprocess (Johnson 1992, 32). Johnson notesthat removing constraints is one of the pri-mary avenues firms should exploit to reducecosts and become more globally competitive.Production constraints also play a significantrole in many of the decisions in which ABC isused. For example, a bottleneck activity in thefirm's production structure plays a critical rolein decisions such as the optimal-productionmix, pricing, make-buy, and special order.Similarly, a production constraint plays a keyrole in determining the opportunity cost of thefirm's resources and in determining whereprocess improvement would be the most ben-eficial to the firm.

Cooper and Kaplan (1992, 12) note thatABC measures the usage of resources withrespect to the demand placed on a productionactivity. If demand for an activity is below thelevel of services supplied, ABC assigns the costof these excess resources to "unused capacity"(Cooper and Kaplan 1992a, 1). Under ABC,the cost of unused capacity is used for resourceallocation decisions to better match the sup-ply and demand for an activity's resources.The excess capacity of production activities isdetermined, in part, by a constraint or bottle-neck activity in the firm's production struc-ture. A constraint or bottleneck restricts pro-duction, thereby limiting resource usage bynon-constrained activities and leading to ex-cess or unused capacity. A constraint, there-fore, plays a pivotal role in understanding whyunused capacity exists in the firm's produc-tion structure and in decisions involving itsdisposition. For example, a constraint repre-sents an activity where resources may beadded to expand production and use the ex-cess capacity of non-constrained activities.Conversely, for a resource reduction decision.

50 Accounting Horizons / December 1995

a bottleneck may be used to determine thelevel of unused capacity that may be elimi-nated without adversely impacting produc-tion. Consequently, identif5dng a constraintand understanding its impact on the firm'sproduction opportunities is crucial for resourceallocation decisions using ABC to maximizethe firm's production and profitability.

THEORY OF CONSTRAINTSThe theory of constraints was developed

by Goldratt as a process of ongoing improve-ment (Sheridan 1991, 48). The objective of theTOC is to maximize the goal of an organiza-tion which is limited by a constraint (Goldratt1990, 4). The system is managed with respectto the constraint or bottleneck, while resourcesare expended to relieve this limitation on thesystem (Goldratt and Cox 1992, 301). Whenthe constraint is removed and the firm movesto a higher level of goal attainment, a newbottleneck will appear, and the cycle of man-aging the system with respect to the new con-straint is repeated.

The simplicity and compact nature ofGoldratt's theory conceal many of the under-lying assumptions that make it so powerful.First, the TOC recognizes both the interde-pendent nature of production activities andhow the most limited of these activities con-trols the performance of the larger system.Equally important, the TOC focuses attentionon constraints fi-om an organizational perspec-tive, i.e., how does removing the constraintimpact the goal of the firm. Consequently, theassumptions underljdng a constraint and itscore problems rather than symptoms may beaddressed. Finally, the TOC's process of con-tinuously removing bottlenecks will impactsuccessively larger subsystems of the firm andpromote change in the firm's managementculture. For example, the TOC encouragescommunication and problem-solving acrossfunctional areas based on an organizationalrather than a local perspective.

The TOC is implemented through threemeasurements: throughput, the rate at whichthe system generates money through sales;inventory, all money the system invests inpurchasing items the system intends to sell;

and operating expenses, all the money thesystem spends in turning inventory intothroughput (Goldratt and Fox 1986, 29). Un-der the TOC, direct material is treated as avariable cost, while direct labor and all othercosts are treated as fixed. The objective of theTOC is to maximize throughput subject to thecapacity of the individual production activi-ties of the firm.

Kaplan has criticized the application of theTOC to production decisions as an extremeform of direct costing or the contribution mar-gin approach to decision-making (Robinson1990, 3). Contribution margin, in its tradi-tional and the TOC forms, has been criticizedfor its short-run focus. Shank suggests thatthe contribution margin approach to decision-making will lead a firm to never drop a prod-uct, always make instead of buy, charge inad-equate prices, and support the short-run sta-tus quo (Robinson 1990,19). Supporters of thecontribution margin approach suggest thatthese problems are the result of inappropri-ate applications of the method rather than themethod itself. However, Shank notes that useof the contribution approach is pernicious andthat it's "a snare, a trap, and a delusion"(Robinson 1990, 19). Finally, the use of con-tribution margin in either its traditional orTOC forms may lead to a series of short-termmaximization decisions that fail to maximizethe long-term profitability of the firm. Forexample. Shank suggests that the contribu-tion margin approach led the airlines andtrucking industries to the verge of bankruptcyafter deregulation (Robinson 1990, 17).

Another limitation of the TOC concerns itsuse of global operational measures to modelthe relationship between resources used in theproduction process and output. As noted ear-lier, the TOC assumes a short-term decisionhorizon. However, even in the short run, manydecisions involve trade-offs between increasesin throughput, inventory, and operating ex-penses. Therefore, using throughput maximi-zation as a decision criterion may lead to sub-optimal decisions in some circumstances. Forintermediate and longer-run decisions inwhich management has discretionary powerover direct labor or overhead cost items, the

Integrating Activity-Based. Costing With the Theory of Constraints 51

global operational measures of the TOC ignorefactors relevant to the decision process.

COMPETING ANDCOMPLEMENTARY ASPECTS OF

ABC AND TOCABC and the TOC model different aspects

of a firm's production structure. ABC modelsthe economic aspects of how resources at theunit-, hatch-, and product-level activities aretransformed into the firm's products. ABC,thus, represents a long-term perspective ofhow costs vary with production. Conversely,the principles of the TOC reflect how thephysical resources consumed by productionactivities and their production capacity playa critical role in the production process. Theglobal operational measures used to imple-ment these principles reflect a direct-cost ap-proach to decision-making. Unlike ABC, theyrepresent a short-term perspective of the re-lationship between a change in cost and pro-duction.

As noted earlier, one of the limitations ofABC is its failure to explicitly reflect the physi-cal usage of resources by production activitiesand the capacity of these activities.^ The in-ability to incorporate physical measures ofresource usage and the production capacity ofactivities fosters an inability to identify con-straints and predict their effect on the firmas noted by Johnson (1992, 32). Conversely,the TOC is based, in large part, on managingproduction constraints. One of the limitationsof the TOC is its use of global operationalmeasures to guide production decisions. Asnoted earlier, the global operational measuresof the TOC will lead to optimal decisions un-der very restrictive circumstances. Alterna-tively, ABC provides a comprehensive frame-work for modeling the economic attributes ofthe production process. It thereby providesmanagers with a means of predicting the eco-nomic consequences of alternative resourceallocation decisions. The strengths of ABC andTOC are complementary in nature. Thestrengths of each model overcome a majorlimitation of the other.

Many of the conceptual aspects of ABC andthe TOC may be integrated to form a larger

model that simultaneously links the cost andphysical attributes of a firm's production struc-ture.^ This objective may be accomplished byexpanding the framework provided by ABC toincorporate the resource usage of individualproducts and the capacities of the processesused in their production. However, the costand physical usage of resources by productionactivities must be modeled at the level atwhich an activity contributes to the produc-tion process. Resources used by unit-level ac-tivities vary proportionally with productionand may be represented with a linear rela-tionship. Batch- and product-level activities,however, use resources in large discrete quan-tities at periodic intervals as production isincreased. Consequently, they must be mod-eled with a step function to reflect the non-

' ABC ineorporates the physical resources and capac-ity of an activity indirectly into the analysis of aproduct's cost. An activit^s costs are divided by a mea-sure of its practical capacity to compute a cost driverrate for tracing costs to products using an activity'sresources. Over a short to intermediate time horizon,an activity's production capacity may be fixed. Thatis, the firm may have trouble adding or deleting ca-pacity. Consequently, an activity's production capac-ity represents a potential constraint on the firm's pro-duction opportunities. However, the cost driver ratesused by ABC fail to refiect this limitation. In the longrun, the capacity of production activities is not re-stricted. The firm can add or delete capacity as needed.However, even then, cost driver rates are predicatedon specific levels of production capacity. IntegratingABC with the capacity levels used to compute its costdriver rates may be useful for understanding the pro-duction opportunities inherent in cost driver rates andevaluating whether these capacity levels are optimalfor the firm.

2 Spoede et al. (1994,43) have suggested that ABC maybe used "to generate the data necessary to supportthe Theory of Constraints." ABC cost data, in effect,would be used as an input into the TOC for makingproduct-mix decisions. 'The Spoede et al. (1994) sug-gestion for using ABC with T'OC contrasts with theproposal presented in this paper that expands the ABCmodel to explicitly recognize the physical usage of re-sources and the capacity of production activities. It isdemonstrated that the expanded ABC model over-comes limitations of the traditional ABC model andthat it may be used in conjunction with principles fi"omthe TOC to make product-mix and other resource al-location decisions. However, unlike Spoede et al.(1994), product mix and other resource allocation de-cisions are made within the framework of the ABCmodel.

52 Accounting Horizons / December 1995

linear manner in which their resources areused in the production process.

A METHODOLOGY FORINTEGRATING RESOURCECONSTRAINTS INTO ABC

Mixed-integer programmingTo integrate ABC with the physical usage

and capacity of production activities, a mixed-integer programming model can be used. Abrief review of mixed-integer programming isprovided in Appendix A. A mixed-integer pro-gramming model may be used to representunit-level cost and resources as continuousvariables, while batch- and product-level ac-tivities are represented as discrete variables.^The resulting model captures the interactionamong the cost, physical resources, and capac-ity of production activities. It thereby enablesmany of the principles of the ABC and the TOCto be implemented within a common framework.

The solution to the mixed-integer program-ming model gives the optimal-production mixsubject to the capacity of the individual activi-ties comprising the firm's production structure.The solution identifies non-constrained activi-ties and their excess resources. These are theresources that Cooper and Kaplan (1992) notemay result in excess spending. However, iden-tification of these activities prior to actual pro-duction may assist management in reallocatingthese resources to other uses. The solution tothe mixed-integer programming model also maybe used to identify the constraint that limitsproduction and leads to excess spending. Con-straint identification provides a starting pointfor the application of Goldratt's principles (1990)for managing production bottlenecks. Sensitiv-ity analysis of the mixed-integer model may beused to estimate the expected increase in pro-duction and profit from relieving the bottleneckactivity. Sensitivity analysis also may be usedto identify the activity that will become a con-straint when the current bottleneck is relievedas well as the economic benefits of its removal.

A Numerical ExampleTo illustrate the integration of ABC with

the TOC, consider the example provided in

table 1. XYZ Inc. is a medium-size firm withtwo production departments, assembly andfinishing, and three support departments, set-up, purchasing, and engineering. XYZ is cur-rently considering adding products X , Xg, Xg,or X4 or some combination to expand its cur-rent product mix. ABC cost, price, and ex-pected annual demand for these products aregiven in panel I.' Direct material and laborare traced directly to individual products.Unit-level overhead was estimated using di-rect labor cost as the cost driver. Set-up andpurchasing costs are incurred at the batchlevel, while engineering is incurred at theproduct level. These costs, their cost driver,and batch- and product-level rates are givenin panels II and III of table 1, respectively.Based on how products use batch- and prod-uct-level resources, their costs are convertedto an equivalent unit cost.

As noted earlier, a mixed-integer program-ming model may be used to integrate ABCdata with the physical usage of resources andtheir productive capacities. The mixed-integerequations for modeling the production struc-ture in table 1 are listed in table 2. The firstequation or objective function reflects the goalof maximizing the profit (Z) that may beachieved fi"om the different production mixesthat XYZ may produce. The contribution ofeach product was computed by subtracting

^ Unit-level activities use resources in discrete quanti-ties. However, the requirement of using integer val-ues to represent these activities may be relaxed if therange of the variable is sufficiently large so that round-ing the solution to nearby integer values will lead toapproximately the same objective function value(Harvey 1979, 245).

'' XYZ does not have sufficient capacity in its supportactivities to produce each of the four new products.Consequently, if customers demand a full product line,then XYZ is confronted with outsourcing the new prod-ucts it cannot produce or adding capacity to expandproduction.

5 For example, set-up (see panel II of table 1) is expectedto provide 500 hours of set-up time while incurring acost of $200,000. Set-up costs are, therefore, $400 perset-up hour. A batch of product Xj, 1,000 units, re-quires two hours of set-up time. Therefore, the set-upcost for a unit of Xj is approximately $.80. Purchasingand engineering costs were converted from hatch- andproduct-levels to a unit basis in a similar manner.

Integrating Activity-Based Costing With the Theory of Constraints 53

TABLE 1XYZ Inc.

Revenue, Cost and Operating Structure

Panel I: Activity-Based Cost, Price, and DemandProduct

Assembly Labor HoursFinishing Labor HoursDirect Material CostDirect Labor CostOverhead Cost*

Unit-Level CostBatch-Level Cost"

Set-upPurchasing

Product-Level Cost**Engineering

Total Activity-Based CostPrice

ProfitMaximum Annual Expected

Demand

Xl

1.00.50

$ 4.0012.0036.00

$52.00

.80

.25

1.00

$54.0570.00

$15.95

100,000

X2

1.00.50

$ 7.0012.0036.00

$55.00

.80

.40

1.00

$57.2080.00

$22.80

100,000

X3

2.002.00

$ 15.0032.0096.00

$143.00

3.202.40

10.00

$158.60223.00

$ 64.40

30,000

X4

5.004.00

$ 28.0072.00

216.00

$316.00

10.006.00

25.00

$357.00516.00

$159.00

20,000

200,000180,000

* 300% of direct labor cost*' Converted to an equivalent unit level cost

Panel II: Batch-Level Activities

Set-Up DepartmentBatch Size (units)Hours/batchExpected CostExpected Capacity (hours)Cost Per Set-Up Hour

Purchasing DepartmentBatch Size (units)Orders/batchExpected CostExpected Capacity (orders)Cost Per Purchase Order

Panel III: Product-Level Activities

Engineering DepartmentDrawings/productExpected CostExpected Capacity (drawings)Cost Per Engineering Drawing

Xl

1,0002

4,0005

Xl

100

ProductX2

1,0002

4,0008

Product

X2

100

X3

5004

1,00012

X3

300

X4

2005

50015

X4

500

$200,000500

$400

$160,000800

$200

$1,000,0001,000

$1,000

54 Accounting Horizons /December 1995

direct material, labor, and unit-level costs fromits price. Set-up and purchasing costs are de-ducted at the batch level, while engineeringcosts are deducted at the product level. Theconstraints of XYZ's production processes arelisted below the objective function. Con-straints 1 and 2 reflect the usage of resourcesin assembly and flnishing measured in directlabor hours. Constraints 3 through 7 reflectthe batch level at which set-up activities areused in production. For example, constraint 3computes the number of set-ups (S ) for prod-uct Xj, where the number of set-ups is re-stricted to an integer value. The hours of set-up time used for a batch of each product, sub-ject to available set-up time, are modeled inconstraint 7. Constraints 8 through 12 are for-mulated to reflect the batch-level behavior ofthe purchasing department in a similar man-ner. Constraints 13 through 17 reflect theproduct-level behavior of engineering services.For example, constraint 13 is used to computethe amount of engineering (E ) required toproduce product X , where E^ is restricted toa value of zero or one. Constraint 17 mea-sures the number of engineering drawings re-quired to support each product and the capac-ity of the department. The last four con-straints, 18 through 21, reflect the maximumexpected annual demand for each of XYZ'sproducts.

The solution to the mixed-integer problemin table 2 is given in table 3. The flrst panelof table 3, labeled "Solution Variables," givesthe production mix that maximizes the objec-tive function in table 2. As indicated, the opti-mal-production strategy consists of producing30,000 units of X , 100,000 units of X2, and30,000 units of X3. Variables S , S2, S3, P , Pg,and Pg measure the number of set-up and pur-chase batch-level activities required to sup-port this production strategy. Similarly, vari-ables Ej, Eg, and E3 measure the product-levelsupport required from the engineering depart-ment. Finally, the proflt (Z) projected for theoptimal-production mix is $4,620,000.

The second panel of table 3, labeled "SlackVariables," measures the slack or excess ca-pacity of the non-constrained activities inXYZ's production structure. As indicated, as-

sembly and flnishing are projected to have10,000 (constraint 1) and 55,000 (constraint2) excess labor hours of capacity, respectively.Similarly, constraints 12 and 17 suggest thatpurchasing and engineering will have excesscapacity of 200 purchasing orders and 500engineering drawings. The slack variables forconstraints 18 and 21 indicate that the flrmwill be unable to supply 70,000 units of de-mand for product Xj and 20,000 units of de-mand for product X .

To provide a benchmark for assessing howwell the expanded ABC model performs, thedata in table 1 were analyzed with the ABCand TOC models. The product mix, excess re-sources, and proflt from these models are pro-vided in table 4. The product mix for the ABCmodel was computed by ranking each prod-uct in terms of its profltability and producing

^ If product X, is produced, 100 engineering drawingsare required before the first unit of Xj is mEinufactured.However, once developed, these drawings can be usedto produce one to 100,000 units of Xj without the useof additional engineering resources. Conversely, ifproduct Xj is not produced, no engineering drawingsare required. Consequently, the use of engineering isa product-level cost dependent solely upon whetheror not Xj is produced. Therefore, the use of engineer-ing resources is modeled in constraint 13 with the bi-nary variable Ej that assumes a value of one if Xj isproduced or zero if it is not produced. The manner inwhich Ej is computed can be seen by moving the sec-ond term in constraint 13 (-lOO.OOOEj) to the righthand side of the equation. The revised equation forconstraint 13 states that the units of X, produced isless than or equal to the maximum number of Xj thatmay be produced times Ej. Consequently, if Xj is notproduced, constraint 13 in conjunction with the ob-jective function will assign a value of zero to Ej indi-cating that no engineering resources will be used forproduct XJ. On the other hand, if X, is produced, Ejassumes a value of one indicating that resources for100 engineering drawings will be required. Con-straints 14 through 16 compute the engineering ef-fort required for manufacturing products Xg, X3, andX^ in a similar manner, respectively.

^ The mixed-integer equations in table 2 were solvedusing STORM, a software product of Storm Software,Inc. The STORM package may be used to solve lin-ear, integer, and mixed-integer programs as well asother statistical and quantitative models.

^ The slack variable for constraint 8 measures the num-ber of additional units that could be produced withthe last batch of purchase orders for product Xj. Simi-larly, the slack variable for constraint 13 representsthe additional units that could be produced from theengineering drawings for product Xj.

Integrating Activity-Based Costing With the Theory of Constraints 55

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56 Accounting Horizons /December 1995

TABLE 3XYZ Inc.

Mixed Integer Programming Solution

Panel I: Solution VariablesVariable Optimal Value

Xi

X2

X3

Si

S2

S3

Pi

P2

P3

El

E2

E3

z

30,000

100,000

30,000

30

100

60

8

25

30

11-1

1

4,620,000

Panel 11: Slack VariablesConstraint Slack Explanation

1

2

8

12

13

17

18

21

10,000

55,000

2,000

200

70,000

500

70,000

20,000

Excess capacity in the assemblydepartmentExcess capacity in the finishingdepartmentExcess capacity of last batch ofpurchasing orders for productXi

Excess purchase order capacityExcess capacity of engineeringfor product XExcess capacity in engineeringExcess demand for product XjExcess demand for product X

the products with the highest profitabilitysubject to available resources. As indicated intable 4, this procedure would lead to produc-ing 20,000 units of product X . Producing20,000 units of product X^ uses the resourcesof the set-up department thereby preventingthe production of other products.

The product mix for the TOC in table 4was computed using procedures proposed byPlenert (1993, 126).9 As indicated, the TOCwould suggest producing 50,000 units of prod-uct X , 100,000 units of Xg, and 25,000 unitsof X3. The product mix for the expanded ABCmodel was taken from table 3. The excess re-sources resulting from the production mixunder the ABC, TOC, and expanded ABC mod-els are given in the second section of table 4.The ABC model resulted in more excess re-sources than the TOC and expanded ABCmodels. This result was to be expected sincethe TOC and expanded ABC models explicitlyincorporate resource constraints into the de-

termination of an optimal-production mix. Theincome of each model is given in the last sec-tion of table 4. The projected income for theABC, TOC, and expanded ABC models are$3,180,000; $2,280,000; and $4,620,000, re-

^ A product mix is selected under the TOC, based onthroughput maximization. Operationally throughputis defined as a product's price less its direct materialcost. Direct labor and overhead are treated as a fixedoperating expense and therefore irrelevant to the prod-uct-mix decision. Fox (1987) outlines a set of proce-dures for selecting a product mix that maximizesthroughput. However, Plenert (1993, 126) suggeststhese procedures may be inefficient when the firm'sproduction structure contains multiple constrainedresources. Plenert (1993,126) demonstrates that lin-ear-integer programming overcomes the limitationsof Fox's procedures. With unit-, batch-, and product-level activities mixed-integer programming would beappropriate for selecting a product mix rather thanthe linear-integer model. A mixed-integer model wasused to select the product mix for the TOC in table 4.The objective function, Z = 66Xj -1- 73X2 + 208 Xg +488X^-13,520,000, was maximized subject to the con-straints listed in table 2.

Integrating Activity-Based Costing With the Theory of Constraints 57

TABLE 4Comparative Analysis

ABC, TOC, and Expanded ABC Models

ABCModel

Product Mix:

2

X3X4

Excess Resources:Assembly (labor hours)Finishing (labor hours)Set-up (hours)Purchasing (orders)Engineering (drawings)

Profit:Projected IncomeCost Saving Available

From Excess Resources*Available Income

-0-0-0

20,000

100,000100,000

-0200500

$3,180,000

-0$3,180,000

TOCModel

50,000100,00025,000

-0

- 055,000

- 0235500

Expanded ABCModel

30,000100,00030,000

-0

10,00055,000

- 0200500

$2,280,000

2,307,000$4,587,000

$4,620,000

-0$4,620,000

* The cost savings from excess resources was computed by multiplying the excess capacity of each activityby its respective cost driver rate. The cost of excess resources are excluded from income under ABC andthe expanded ABC models.

spectively. The incomes for the three modelsare not comparable since the ABC and ex-panded ABC models exclude the cost of excessresources while the TOC includes these costs.Adding hack the potential cost saving from theexcess resources with the TOC model, itsavailable income is $4,587,000.^°

The performance of the three models listedin table 4 is a result of the opportunity cost ofthe resources used to determine an optimal-product mix. ABC assumes that productioncapacity is unconstrained. Therefore, there areno opportunity costs associated with using thecapacity of production activities. A product'sprice less its ABC costs is used for evaluatingproduction-related decisions. Conversely, theTOC and expanded ABC models assume thatproduction capacity is constrained and thatproducing one set of products precludes theproduction of other products. The TOC and

1" The ABC, TOC, and expanded ABC models discussedthroughout the paper are being proposed as planningmodels for making product-mix and other resourceallocation decisions. Identification of the optimal-prod-uct mix, maximum income potentially available to thefirm, and excess resources are crucial for understand-ing the opportunities and problems facing the firm indeveloping its product-mix strategy. Consequently, theincome of the ABC, TOC, and expanded ABC modelsin tahle 4 exclude the cost of excess resources. Identi-fying and planning how these resources may he usedin more productive activities is an essential aspect ofdeveloping an optimal-product-mix strategy. In caseswhere the excess resources of the optimal-product mixcannot he reallocated or the firm's managementchooses not to reallocate these resources, the coeffi-cients of the mixed-integer set of equations used tomodel the firm's revenue, cost, and production struc-ture may be adjusted to assess the production oppor-tunities available to the firm given the committednature of its resources. For financial reporting pur-poses, the cost of excess resources or unused capacityis expensed in the period incurred. See Cooper andKaplan (1992) for an extended discussion of this andrelated issues.

58 Accounting Horizons I December 1995

expanded ABC models select products with thehighest contribution margin per unit andhighest profit per unit for a bottleneck activ-ity, respectively. The opportunity cost fromusing the resources of the bottleneck activityis reflected in the relative profitability com-puted for each product.

The difference in the income of the TOCand expanded ABC model is the result of thecosts that are assumed to be relevant for theproduct-mix decision. The TOC assumes thatall resources other than direct material arecommitted and therefore irrelevant for deci-sion making. Conversely, ABC assumes thatall costs are relevant for understanding thecosts of resources used to perform the activi-ties used to produce the firm's output (Cooperand Kaplan 1992,12). Based on these assump-tions, the TOC selects products to the pointthat the marginal revenue from the last unitproduced is equal to the cost of the direct ma-terial used, while the expanded ABC modelselects products to the point that the marginalrevenue from the last unit produced is equalto the marginal cost of the resources used inits manufacturing.

The relative superiority of the ABC, TOC,and expanded ABC models for product-mixdecisions is dependent upon the economic cir-cumstances of the firm. When demand for thefirm's products exceeds the capacity of at leastone production activity, the expanded ABCmodel will lead to a more or equally profit-able product-mix decision relative to ABC. ^The expanded ABC model incorporates theopportunity cost of a production constraint,thereby identifying those products with thehighest profit potential given the firm's lim-ited production opportunities. Similarly, whendemand for the firm's products exceeds thecapacity of at least one of its production ac-tivities and the firm has some discretionarypower over the excess resources in its produc-tion structure, the expanded ABC model willlead to a more or equally profitable product-mix decision relative to the TOC. As notedearlier, the expanded ABC model selects prod-ucts to the point where the marginal revenuefrom the last unit produced is equal to themarginal cost of all the resources used in its

production. Conversely, the TOC will continueto produce beyond the point where the mar-ginal revenue is equal to the marginal cost ofthe resources used in production, thereby con-suming additional resources, and reducingavailable income. The expanded ABC modelselects a product mix based on the resourcesused rather than the resources supplied to theproduction process. The expanded ABC model,thereby, measures the maximum potentialincome available to the firm and identifiesresources that are unable to generate a returnsufficient to justify their ^

Further Analysis and InterpretationThe expanded ABC model provides a

framework for implementing many of the prin-ciples of the TOC. As noted earlier, under theTOC a system is managed with respect to aconstraint while resources are expended toremove a bottleneck activity. An examinationof the slack variables in panel II of table 3 in-dicates that the set-up department is the onlyproduction or support activity without excessresources. Therefore, it represents the con-straint in XYZ's production structure limitingfurther production and profitability. The op-timal-product mix for XYZ Inc. listed in table3 was based, in part, upon the limited re-sources in the set-up department. This mayhe seen hy computing the profit per hour ofset-up time for each product. The profit perhour of set-up time is $7,975 for X , $11,400for Xg, $8,050 for X3, and $6,360 for X4.13 The

' When a ranking of each product's profitability is thesame under the ABC and expanded ABC models, bothmodels will lead to identical product-mix decisionswhen there is a single constrained resource.

12 In cases where the firm has no discretionary powerover production resources other than direct material,the TOC will provide a product mix with a higher orequeJ income relative to the expanded ABC model. TheTOC includes only direct material cost in selecting anoptimal-product mix, while the expanded ABC modelincludes all production costs in developing a productmix. Consequently, the expanded ABC model may re-sult in a lower level of production, higher level of ex-cess resources, and lower income relative to the TOC.

^ Each product's profit per hour of set-up time was com-puted fi-om its profit per unit in table 1 multiplied bythe units and divided by the hours of set-up time perbatch.

Integrating Activity-Based Costing With the Theory of Constraints 59

relative rates of profit per hour of constrainedresources are reflected in the optimal-productmix, i.e., demand for products X^ and Xg andpart of the demand for X are met in the opti-mal solution.

The slack or excess resources identified inthe second section of table 3 are the direct re-sult of the limited capacity of the set-up de-partment to support further production. Theimpact of expanding the resources in the set-up department may be evaluated by increas-ing the amount of set-up time in the originalmixed-integer programming model, solvingthe revised model, and comparing the origi-nal and revised solutions. Adding two addi-tional hours to set-up would increase produc-tion of Xj by one batcb, profit by $17,200, andwould decrease excess resources in assemblyand finishing by 1,000 and 500 hours, respec-tively. The TOC would suggest this has sev-eral implications for the management of XYZInc. First, an hour of resources lost in the set-up department is much more critical than anequivalent amount of capacity lost in anotherdepartment. Every pair of hours lost in theset-up department results in one less batch ofproduct Xj, reduced profit of $17,200, and in-creased excess resources in the assembly andfinishing departments of 1,000 and 500 bours,respectively. A comparable inefficiency in as-sembly, finishing, purchasing, or engineeringwould bave no impact on tbe firm's produc-tion and profitability. Consequently, monitor-ing tbe efficiency of tbe set-up department iscritical for maximizing production and profit-ability and minimizing XYZ's excess produc-tion capacity.

A second implication of tbe TOC is tbatadding resources to tbe set-up department willdramatically impact tbe production and prof-itability of tbe firm. However, wben one con-straint is relieved, another activity will be-come a bottleneck. To evaluate bow mucb ca-pacity can be added to tbe set-up department,tbe resource usage firom expanding productionand tbe excess capacity available in otber de-partments must be evaluated. Every addi-tional batcb of Xj produced uses 1,000 and 500bours of capacity in assembly and finishing,respectively, and takes five additional pur-

cbase orders for every 4,000 additional unitsof production. Since tbere are 10,000 bours inassembly, 55,000 bours in finisbing, and 200orders in purcbasing of excess capacity, assem-bly will become constrained after 20 additionalbours of resources are added to tbe set-upfunction. Adding tbese resources will increaseprofit by $170,000, wbile using 10,000 boursin assembly, 5,000 bours in finisbing, and 10purchase orders tbat are currently unused. ^At tbis point, assembly will become a con-straint. Therefore, to expand production andprofit furtber, additional resources would needto be added to tbe assembly and set-up de-partments. Tbe subsequent set of activities inXYZ's production structure tbat will become aconstraint and tbe economic impact of tbeir re-moval can be determined in a similar manner.

Identification of a constraint plays a keyrole in a program of continuous improvement.Tbe TOC indicates tbat process improvementsbould be targeted at a constraint (Sberidan1991, 46). Improving a constrained activityrelieves a bottleneck, tbereby increasingtbrougbput. Conversely, improving non-con-strained activities simply increases tbeir ex-cess capacity. Applying tbese principles to tbedata in table 3, XYZ Inc. sbould target tbe set-up department for improvement. Prior sensi-tivity analysis indicates tbat tbis strategy willincrease production, profitability, and use ex-cess resources in otber departments. However,once set-up time bas been increased more tban20 bours, process improvement must be di-rected at the assembly department to expandproduction and profitability furtber. Tbe ex-panded ABC model, tbus, provides a systemfor prioritizing process improvement activi-ties and estimating tbeir potential economicimpact.

SUMMARY AND CONCLUSIONSABC and tbe TOC incorporate facets of tbe

production process ignored by traditional costmodels. Accordingly, botb paradigms provide

I"* The increase in profit may be computed by executingthe mixed-integer model in table 2 with 520 set-uphours or manually by multiplying the additional unitsby their unit-level contribution less batch- and prod-uct-level costs.

60 Accounting Horizons /December 1995

insights into the production process that en-hance production decisions. An examinationof their cost measures and time frames formaking production-related decisions suggeststhat they are diametrically competing para-digms. However, an examination of their re-spective strengths and limitations suggestthat they are more complementary than theymay first appear. The product costs of ABCmay be combined with the physical resourcesused by production activities and their capac-ity to form a more comprehensive model of afirm's production structure. The resultingmodel provides an expanded framework forunderstanding the economic consequences ofproduction-related decisions and implement-ing the principles of the TOC.

The expanded ABC model may beoperationalized using mixed-integer program-ming to integrate a firm's ABC data withphysical resource used by production activi-ties and their production capacity. The mixed-integer programming model provides informa-tion that may be used for making a variety ofmarketing and production-related decisions.The solution to the expanded ABC model iden-tifies the firm's optimal-production mix fromsimultaneous evaluation of the cost, physicalresources, and marketing opportunities avail-able to the firm. Perhaps equally important,it aids in identifying a production bottleneckand assessing its economic impact upon thefirm. The solution to the expandedABC modelalso facilitates identifying excess resourcesthat can be reallocated to more productiveuses. Finally, the expanded ABC model maybe used to identify activities where a program

of continuous improvement has the highestpotential for enhancing organizational produc-tivity and profitability.

The expanded ABC model may have sev-eral limitations when used in practice. Theexpanded ABC model is formulated in termsof quantitative measures of the firm's revenue,cost, physical resources, and production capac-ity. However, product-mix and other resourceallocation decisions frequently involve numer-ous non-quantitative factors. Consequently,the expandedABC model provides only a sub-set of the information needed for many mar-keting and production decisions. The ex-panded ABC model requires numerous esti-mates and assumptions of the firm's businessopportunities and resources over an extendedtime period. The usefulness of much of theinformation developed from the expandedABC model is dependent, in part, upon theaccuracy of these assumptions and estimates.However, assumptions and estimates are re-quired for other approaches to making prod-uct-mix and resource allocation decisions andimpact the usefulness of their information aswell. Finally, the cost of using the expandedABC model may be relatively high comparedto other approaches because of the substan-tial time required to model the firm's revenue,cost, and production structure and interpretand analyze the resulting mixed-integer pro-gramming solution. The additional costs ofapplying the expanded ABC model may berelatively insignificant, however, compared tothe potential benefit of the information thatit may provide for making more informed mar-keting and production decisions.

Integrating Activity-Based Costing With the Theory of Constraints 61

APPENDIX AMixed-Integer Programming

Mixed-integer programming is a subset of mathematical programming designed to optimize an objectivefunction subject to constraints with continuous and integer variables. A business problem is formulatedas a mixed-integer model identical to that of the more familiar linear-programming technique. An objectivefunction is used to represent an organizational goal sucb as profit maximization. A second set of equationsis used to model constraints that limit the attainment of this objective. Once formulated, a mixed-integerprogramming algorithm is used to solve the resulting set of equations.

The algorithm will provide the variables that maximize the objective function, the slack variablesthat measure the excess resources of non-constrained activities, and the value of tbe objective function.Unlike linear programming, the mixed-integer programming model will not generate information aboutthe sensitivity of the optimal solution to changes in a constraint or to a change in the value of an objectivefunction coefficient. However, sensitivity analysis may be performed by changing the value of one or morevariables in the original mixed-integer model, solving the revised model, and observing the change betweenthe original and revised models.

A variety of computerized algorithms are available for solving integer and mixed-integer programmingproblems. However, an integer or mixed-integer problem requires more computational time than acomparable linear-programming problem. For example, the hranch-and-bound algorithm, which is widelyused for solving mixed-integer problems, takes a prohibitive amount of time to solve if the problem hasover one hundred integer variables and the continuous solution is distant from the optimal integer solution(Budnick et al. 1988, 413). However, even under these conditions, the manner in which a mixed-integerproblem is formulated or structured significantly affects its computational time. For a more in-depthdiscussion of mixed-integer programming and factors affecting its computational time see Budnick et al.(1988) and Eppen et al. (1993).

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