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Int. J. Production Economics

Int. J. Production Economics 121 (2009) 121–129

0925-52

doi:10.1

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foundat� Tel.

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journal homepage: www.elsevier.com/locate/ijpe

Theory of constraints and the combinatorial complexity of theproduct-mix decision$

Alexandre Linhares �

Brazilian School of Business and Public Administration, FGV, Praia de Botafogo 190/509, Rio de Janeiro 22257-970, Brazil

a r t i c l e i n f o

Article history:

Received 29 March 2008

Accepted 28 April 2009Available online 15 May 2009

Keywords:

Theory of constraints

Product mix

Product margin heuristic

TOC-derived heuristic

NP-hardness

73/$ - see front matter & 2009 Elsevier B.V. A

016/j.ijpe.2009.04.023

s research has been funded by a grant

ion and by the PROPESQUISA program of FGV

: +55219996 2505; fax: +55212553 8832.

ail address: alexandre.linhares@fgv.br

a b s t r a c t

The theory of constraints (TOC) proposes that, when production is bounded by a single

bottleneck, the best product mix heuristic is to select products based on their ratio of

throughput per constraint use. This, however, is not true for cases when production is

limited to integer quantities of final products. Four facts that go against current thought

in the TOC literature are demonstrated in this paper. For example, there are cases in

which the optimum product mix includes products with the lowest product margin and

the lowest ratio of throughput per constraint time, simultaneously violating the margin

heuristic and the TOC-derived heuristic. Such failures are due to the non-polynomial

completeness (NP-completeness) of the product-mix decision problem, also demon-

strated here.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The theory of constraints (TOC) is a remarkablysuccessful operations philosophy, centered on the idea offocusing managerial attention to the local constraints thatinhibit the global performance of an entire system(Goldratt and Cox, 1984; Goldratt and Fox, 1986; Goldratt,1990a, b). Over the last two decades it has gathered muchmomentum, with the creation of organizations such as theGoldratt Institute and the TOC Center in Dayton, Ohio. Ithas also spawned a host of products, such as the OPT (foroptimization) software and an innovative bestsellingmanagement novel. Blackstone (2001) reviews some ofits core ideas and fields of application.

The focus of our study will be on the problem ofselecting the optimum product mix under the TOC, whichis deemed as an improvement over traditional practices(Gupta et al., 2002; Kee and Schmidt, 2000; Wahlers and

ll rights reserved.

from the FAPERJ

.

Cox, 1994). Consider a facility with a set of products tobuild, but without the capacity (i.e., a fixed time horizon)required to meet the demand for all of them. Let ussuppose that this facility must deal with integer quantitiesof final products. In this case a product-mix decision mustbe made, with the obvious tradeoff of prioritizing someproduct lines at the expense of others. A traditionalmethod for selection of the product mix is given byselecting the products having highest individual productmargins with higher priority, regardless of the timespent on the bottleneck(s) (BN(s); Goldratt 1990a, b;Goldratt and Cox, 1984; Blackstone, 2001; Lea andFredendall, 2002; Patterson, 1992). Let us name thismethod as the margin heuristic. The TOC, however,proposes that product lines should be selected accordingto their ratio of throughput per time spent on the systemconstraint(s). Let us refer to this approach as the TOC-

derived heuristic. The heuristic has been formally stated innumerous TOC publications (Goldratt, 1990a, b; Goldrattand Cox, 1984; Blackstone, 2001; Lea and Fredendall,2002; Patterson, 1992). Fredendall and Lea (1997) suggest,after Goldratt and Cox (1984), the following product-mixheuristic.

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A. Linhares / Int. J. Production Economics 121 (2009) 121–129122

1.1. TOC-derived product-mix heuristic

‘‘Step 1: Identify the system’s constraint (s):

(a)

Calculate the required load on each resource toproduce all the products. The constraint or bottleneck(BN) is the resource whose market demand exceeds itscapacity.

Step 2: Decide how to exploit the system’s constraint (s):

(a)

Calculate the contribution margin (CM) of eachproduct as the sales price minus the raw material(RM) costs;

(b)

Calculate the ratio of the CM to the productsprocessing time on the BN resource (CM/BN);

(c)

In descending order of the products’ CM/BN, reservethe BN capacity to build the product until the BNresource’s capacity is exhausted;

(d)

Plan to produce all the products that do not requireprocessing time on the BN (i.e., the ‘free’ product) indescending order of their CM’’ (Fredendall and Lea,1997, pp. 1535–1536).

1 By combinatorial complexity we mean that the problem is NP-

complete. Each new product introduced leads to an exponential increase

in the number of possible product mixes.

It was believed, when originally suggested, that thisTOC-derived heuristic would obtain the best combinationof products for all cases. For example, after following theabove reasoning, Blackstone (2001) suggests extrapolationof the strategy to sales: ‘‘What is the best basis for sales

commissions; that is, the one that will have the salespeople

emphasizing the items that are most profitable to the

company? If you said Throughput per Constraint Minute

give yourself a gold star. But how many companies do you

know which actually use it? Could it be that nearly every

company can make more money by restructuring sales efforts

as well as product mixes?’’ (Blackstone, 2001, p. 1063).In fact, the strategy does indeed obtain optimum

solutions when production is allowed to be fragmented.However, studies of Plenert (1993) and Lee and Plenert(1993) clearly demonstrate (by comparing the productmixes obtained with the TOC-derived heuristic with thoseobtained with integer linear programming) that there is apositive probability of failure of the TOC-derived heuristicin finding the optimum product mix when productionmust be done over integer quantities in the case ofmultiple constraining resources. These interesting resultslaunched an increasing series of studies for improvedpolicies (Fredendall and Lea, 1997; Hsu and Chung, 1998),advanced heuristics such as genetic algorithms (Onwubo-lu and Mutingi, 2001a, b) and tabu search (Onwubolu,2001) providing new product-mix methods for the case ofmultiple constrained resources.

Aryanezhad and Komijan (2004) present an improvedalgorithm for the multiple BN case. Balakrishnan andCheng (2000) question the relation of the TOC heuristic tolinear programming. Bhattacharya et al. (2008) presentresults over their fuzzy linear programming approach tothe problem. Mishra et al. (2005) have developed a tabusearch and simulated annealing hybrid approach. Patter-son (1992) compares the TOC heuristic to classical

accounting practices. Singh et al.’s (2006) approach works‘‘on the principle of artificial immune system andbehavioral theory, namely Maslow’s need hierarchytheory’’. Finally, Wang et al. (2008) also use immune-based approaches, such as self-adaptive regulation andvaccination. The literature on methods for solving pro-blems with multiple BNs is rapidly growing. After all,Plenert (1993) demonstrated over 15 years ago theshortfalls of the TOC-derived product-mix heuristic onthe multiple-constraint resources scenario.

The situation, however, is in fact more complex than itseems. We will see that the problem is of high combina-torial complexity.1 The research objective of this paper isto better understand the nature of the problem. If theproblem is NP-complete, our research question, thiswould justify why many heuristics have been proposedfor this problem, as, in such case, no algorithm cancompute the optimum solution for large instances. Wemust use advanced heuristics. The following facts godirectly against current thought in the TOC literature andare demonstrated in this paper.

FACT 1. There are cases in which the TOC-derived heuristic

fails even with a single BN.

FACT 2. There are cases in which the TOC-derived heuristic

fails to obtain a higher profit than the traditional product

margin heuristic.

FACT 3. There are cases in which the optimum product mix

includes products with the lowest product margin and the

lowest ratio of throughput per constraint time, violating both

the traditional heuristic and the TOC-derived heuristic.

FACT 4. There are strong reasons to believe that an efficient

and optimum heuristic is simply impossible.

2. Review of the Blackstone (2001) example

Let us start the discussion with an example fromBlackstone (2001). Fig. 1 presents a hypothetical facilitycapable of producing three products: product X, product Y,and product Z. Product X sells for $90 and has a weeklydemand of 50 units, product Y sells for $100 and has aweekly demand of 75 units, and product Z sells for $70and has a weekly demand of 100. The facility has 5 workstations (A–E) and the products are devised from fourtypes of RMs (RM1, RM2, RM3, and RM4).

Production of X is started with two units of RM2processed at station A for 10 min each. One of them istaken to station C for 15 min of processing while the otheris taken to station D also for 15 min. These materials arethen joined, along with a new unit of RM1, at station E, ina 5 min process. Hence X has $40 of material cost.

Production of Y is started with a unit of RM2 processedat station A for 10 min, and a unit of RM3 processed atstation B also for 10 min. After station A is finished, theresulting material flows to station D for 15 min and then

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RM2$15

RM2$15

RM1$10

RM3$15

RM4$10

A10 min

A10 min

A5 min

B10 min

E5 min

D10 min

C5 min

C15 min

D15 min

E10 min

E5 min

X$90

50/Week

Y$100

75/Week

Z$70

100/Week

RM2$15

RM2$15

RM1$10

RM3$15

RM4$10

Fig. 1. After Blackstone (2001): a hypothetical facility.

A. Linhares / Int. J. Production Economics 121 (2009) 121–129 123

to station E for an additional 10 min. After station Bprocesses RM3, the resulting material is taken to stationsC (5 min) and D (10 min), and finally it is joined with thematerial resulting originally from RM2, in station E, in a10 min process. Material costs for Y equal $30.

Product Z is made out of one unit of RM3 and one unitof RM4. RM3 is initially processed at B for 10 min, thenfollowed by 5 min at C, 10 min at D, and 5 min at E, whereit will be joined with the unit of RM4 processed for 5 minat A. The material costs for Z are $25.

A word of caution—the reader might have noticed thatthere are some incorrect data in Blackstone’s paper. Forexample, in that paper, Fig. 6 reveals that product Zrequires 15 min of processing at station C, while the nextfigure places that number at only 5 min. It is impossible totell from that text which numbers were the correct ones,but this does not affect the arguments in either way. Inhere we are thus referring to the data from the matrix inBlackstone (2001)’s Fig. 7, and not from the facilityscheme of Blackstone (2001)’s Fig. 6.

We thus have in Table 1 a summary of the load on thestations, where it may be concluded that there is notenough capacity in station D, which clearly makes it theonly system constraint. Now a decision must be made—-

which products should be produced? This is the product-mix problem under a BN.

Operating costs of the system are given by $10 of laborper hour and $30 of overhead per hour, which brings a totalof $8000 per week, given 5 stations operating for 40 h.However, if we break these costs per product, differentproduct margins emerge. Table 1 gives Blackstone’s (2001)

labor costs and overhead costs per product, which, whenadded to RM costs, let us obtain the product margin ofproducts X–Z.

Under the traditional viewpoint, Blackstone argues,companies tend to view products as either ‘‘dogs’’ (whichhave low profit margins), or as ‘‘stars’’ (with largemargins). This view is compatible with that of determin-ing a product mix by selecting the products with thehighest margins first. Under this product margin heuristic,a company should prefer to produce all demand of 75units of product Y (margin: $30), followed by 52 units ofthe product with the next best margin, which is Z(margin: $25). Its production of only 52 units of Z, insteadof its whole demand of 100 units, is due to the constraintsof the BN. The BN, station D, should still have 5 min left ofavailable capacity, but we may assume that no productscan be produced in this timeframe, and so this productmix based on margins leads not to a profit, but to a loss of�$410 (Table 2).

Now, this decision (and its associated loss) does notconsider the actual time that products spend on theconstraint. In this case the product with the highestmargin is also the product with the highest utilization ofthe BN. This shortcoming is why the TOC proposes toinclude products based on their comparative ratio ofthroughput per constraint minute. This is illustrated inTable 3, where product Z, with a ratio of $4.50/min, isproduced to meet its total demand of 100 units. Thisdecision consumes 1000 min of the BN. The next product,product X with a ratio of $3,33, is also produced to meetits total demand of 50 units, consuming 750 min of the

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Table 1Summary data of the Blackstone (2001) example.

Capacity requirements per unit Station load (min)

Product X Product Y Product Z Product X Product Y Product Z Total requirement

Station A 20 10 5 1000 750 500 2250

Station B 0 10 10 0 750 1000 1750

Station C 15 5 5 750 375 500 1625

Station D 15 25 10 750 1875 1000 3625

Station E 5 10 5 250 750 500 1500

Demand 50 75 100

Selling price 90 100 70

Raw materials 40 30 25

Labor 10 10 5

Overhead 30 30 15

Product cost 80 70 45

Product margin 10 30 25

Table 2Product-mix decision following the product margin heuristic.

Product mix by product margin

Product X Product Y Product Z

Selling price $90 $100 $70

Raw material $40 $30 $25

Throughput/unit $50 $70 $45 Totals

Production 0 75 52 127

Throughput/product $0 $5.250 $2.340 $7.590

Operating expense $8.000

Plant profit �$410

Minutes of D per unit 15 25 10

Minutes of D per product 0 1875 520 2395

Table 3Product-mix decision following the throughput per constraint time (TOC) heuristic generates greater profit than selection by product margin.

Product mix from a theory of constraints standpoint

Product X Product Y Product Z

Selling price $90 $100 $70

Raw material $40 $30 $25

Throughput/unit $50 $70 $45

Constraint minutes 15 25 10

Throughput/constraint min $3,33 $2,80 $4,50 Totals

Production 50 26 100 176

Throughput/product $2.500 $1.820 $4.500 $8.820

Operating expense $8.000

Plant profit $820

Minutes of D per unit 15 25 10

Minutes of D per product 750 650 1000 2395

A. Linhares / Int. J. Production Economics 121 (2009) 121–129124

BN. With the remaining minutes, the heuristic thenallocates the BN to work on product Y, and there iscapacity to produce 26 units. This new product mixgenerates $8.820, raising profit to $820 per week.

In the words of Blackstone (2001)—‘‘this exampleproves emphatically that products do not have profits,companies do. Making decisions based on ‘product profit’

while ignoring the impact of the product on the constraintis clearly suboptimal. The correct decision variable fordetermining product mix is Throughput per ConstraintMinute’’ (Blackstone, 2001, p. 1062).

In the next section, a new product is introduced intoBlackstone’s example. It will let us look closer at thisheuristic of throughput per constraint time.

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RM2$15

RM2$15

RM1$10

RM3$15

RM4$10

C5 min

RM3$15

RM3$15

RM2$15

RM2$15

RM1$10

RM3$15

RM4$10

C5 min

X$90

50/Week

Y$100

75/Week

RM3$15

RM3$15

RM2$15

RM2$15

RM1$10

RM3$15

RM4$10

A10 min

A10 min

A5 min

B10 min

E5 min

D10 min

C5 min

C15 min

D15 min

E10 min

E5 min

Z$70

100/Week

RM3$15

D1650 min

Alpha$6630 1/Week

RM3$15

Fig. 2. Introduction of product Alpha.

Table 4Introduction of product Alpha places additional pressure on the system constraint.

Capacity requirements per unit Station load (min)

Product X Product Y Product Z Product Alpha Product X Product Y Product Z Product Alpha Total requirement

Station A 20 10 5 0 1000 750 500 0 2250

Station B 0 10 10 0 0 750 1000 0 1750

Station C 15 5 5 0 750 375 500 0 1625

Station D 15 25 10 1650 750 1875 1000 1650 5275

Station E 5 10 5 0 250 750 500 0 1500

Demand 50 75 100 1

Selling price 90 100 70 6630

Raw materials 40 30 25 30

Labor 10 10 5 275

Overhead 30 30 15 825

Product cost 80 70 45 1130

Product margin 10 30 25 5500

A. Linhares / Int. J. Production Economics 121 (2009) 121–129 125

3. Introducing pathological product Alpha

Let us now introduce a new product into Blackstone’s(2001) example, product Alpha. On a first reading, productAlpha will seem very different from the products alreadyoffered. Its parameters are very distinct from those ofproducts X–Z. But this is precisely intended to be the case,in order to show the limitations of the heuristic in thisnew special case. By pathological product(s), we mean aproduct, or a set of products, which, if included, wouldlead the TOC heuristic to suboptimum solutions.

Product Alpha is expensive and sells for $6630 (Fig. 2).It is created from 2 units of RM3, i.e., it has $30 of materialcost, and it demands 1650 min of processing time instation D. This means that it adds further burdens on thesystem constraint, and only on the system constraint. The

weekly demand for Alpha is a single unit. Thus, as can beseen in Table 4, the load on BN station D skyrockets to5275 min.

Now, how does the heuristic of selecting productsbased on their ratio of throughput per constraint time farein this case? Product Z still leads this parameter, with aratio of $4.50 per constraint minute. It is followed by thenew offering, product Alpha, with a ratio of $4.00; andthen by products X (ratio of $3.33) and finally Y (ratio of$2.80). So the heuristic tells us to start production byfulfilling all demand for product Z. This will in turnrequire the BN to work for 1000 min. After demand for Z isfulfilled, the BN will have 1400 min remaining. But this isnot enough time to produce the product with the nextbest ratio of throughput per constraint time, as productAlpha requires 1650 min of BN time. So the TOC-derived

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Table 5Profit rises to $1100 (or to $2700 if station B is allowed to shut down), violating the TOC-derived heuristic in a single constrained resource case.

Product Alpha: a pathological case for the theory of constraints

Product X Product Y Product Z Product Alpha

Selling price $90 $100 $70 $6.630

Raw material $40 $30 $25 $30

Throughput/unit $50 $70 $45 $6.600

Constraint minutes 15 25 10 1.650

Throughput/constraint min. $3,33 $2,80 $4,50 $4,00 Totals

Production 50 0 0 1

Throughput/product $2.500 $0 $0 $6.600 $9.100

Operating expense $8.000

Plant profit $1.100

Minutes of D per unit 15 25 10 1650

Minutes of D per product 750 0 0 1650 2400

A. Linhares / Int. J. Production Economics 121 (2009) 121–129126

heuristic would either lead to an unfeasible solution thatexceeds the BN availability, or it would have to skipproduction of Alpha and produce the same mix as that onebefore the introduction of Alpha, which, as we have justseen, leads to a profit of $820. If, however, one does notproduce the item with best ratio of throughput perconstraint minute at all, and instead resorts to producingonly product Alpha and the demand for product X, the totalthroughput would grow, turning the higher profit of $1100.(Notice also that this higher profit assumes that operatingexpenses are held fixed at $8000, but according to Black-stone (2001), operating expenses are based on $1600 perstation per week, and this work schedule does not requirestation B to be active at all. So if operating costs under thisschedule drop to $6400, total profit would grow from the$820 seen above to $2700, only by selecting a differentproduct mix than that pointed out by the TOC-derivedheuristic.) This goes against current thought in the TOCliterature and demonstrates FACT 1 given in the introduc-tion: There are cases in which the TOC-derived heuristicfails even with a single constrained resource (Table 5).

So there seems clearly to be a problem with the TOC-derived heuristic. The skeptical reader may not beconvinced and argue that ‘‘the introduction of productAlpha is too unconventional’’, that ‘‘its parameters aremuch too distinct from those of the previous products’’,and that ‘‘it is the inadequacy and unrealistic nature ofthese numbers that somehow make up for this failure ofthe TOC-derived heuristic’’. This is not the case (thoughthe numbers of product Alpha really do seem quiteartificial). The TOC-derived heuristic does not lead tooptimum product mixes in a much larger number of cases.

In order to clarify this issue, let us consider in thefollowing section the simplest possible case of cata-strophic failure of the TOC-derived heuristic. This willhelp us in isolating the problem and in making theunderlying reason clear.

4. Discussion

Fig. 3(a) presents an extremely simple case of a set ofproducts {#1, #2, #3, #4}, a single work station A, and a

single RM (worth $100). Our planning period, and thecapacity of station A, is a single work day, or 8 h. Theselling price of each final product is a direct functionof the time spent on station A, and the demand for theday is one product #1, one product #2, one product #3,and one product #4. Since the production of theseitems would require 19 h, station A is clearly a systemconstraint and is obviously the only one. Now what is thebest product mix? Let us see how the product marginheuristic and the TOC-derived heuristic fare in thispeculiar case.

Let us first consider the classic method of product mix,by selecting the products with the highest margins first (Fig.3(b)). In this case, since all products are derived from RM1,we need only to select the product with the highest sellingprice, which is product #1. This product would thusconsume 6 h of station A, and the remaining hours wouldnot be sufficient time to produce any of the remainingdemanded items. Thus, the total throughput would equal$500 under this heuristic.

Now, by using the TOC-derived heuristic in Fig. 3(c), wewould start by selecting the products with the highest

relation of throughput/constraint time, which turns out inthis case to be product 2, with a $110/h ratio (a ratio largerthan $100, the ratio of all the remaining items). Thisproduct would consume 5 h of station A time, and therewould not be sufficient time for the production of anyremaining items, leading to a total throughput of $450.This demonstrates FACT 2 that there are cases in whichthe TOC-derived heuristic fails to obtain a higher profitthan the traditional product margin heuristic.

Now, the optimum heuristic for this case is exactly toviolate both the traditional product margin heuristic andthe TOC heuristic and to produce the items with thelowest product margin and the lowest ratio of throughputper constraint time! The simultaneous violation of bothpolicies leads to a total throughput of $600. Thisdemonstrates FACT 3 that there are cases in which theoptimum product mix includes products with the lowest

product margin and the lowest ratio of throughput per

constraint time, violating both the traditional heuristic andthe TOC-derived heuristic. The reason for such unantici-pated results is given in the following section.

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Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Station A4 hours

Product 4$400 1/day

Station A4 hours

Product 3$400 1/day

Station A5 hours

Product 2$550 1/day

RM1$100

Station A6 hours

Product 1$600 1/day

RM1$100

RM1$100

RM1$100

Fig. 3. The simplest case of such failure: (a) a simple case with one bottleneck, (b) result of the product margin heuristic: $500, (c) result of the TOC-

derived heuristic: $450, and (d) optimum solution violates both policies: $600.

A. Linhares / Int. J. Production Economics 121 (2009) 121–129 127

5. The reason for these failures

The simple case presented in the previous section mayhave led the reader to the following obvious conclusion:

Proposition 1. Selection of a product mix in a constrained

facility is NP-hard.

Proof. Reduction from the knapsack problem.&

To demonstrate that selection of a product mix in aconstrained facility is NP-hard, it must be shown how aparticular polynomial-time method for its optimumsolution would also be capable of solving a known NP-hard problem (e.g., Garey and Johnson, 1979).

Consider the following optimization problem: we aregiven a sack and a set of N items of value Vi and weight Wi.The knapsack problem asks us to select a subset of itemsthat maximizes the sum value of items placed in the sack,while the weighted sum of these items lies at most at a

particular threshold T (Garey and Johnson, 1979). Thisfamous problem is NP-hard to solve, which means thatthere is no known method that will produce an optimumsolution under an efficient (polynomial) time frame, as afunction of N.

It is easy to see that any exact method of the product-mix problem under TOC can be used to solve the knapsackproblem. Let us consider a production setting with a singleBN (i.e., station A) and N possible products. The productscorrespond to the items to be (or not to be) placed on thesack; so, for each item, let us create a correspondingproduct. As in the previous example, each productconsists of RM processed in station A, the BN. Let eachproduct have a RM price of $0. The selling price of eachproduct is given by the value of each item Vi. The loadrequired (time in minutes) for each product in the BN isgiven by weight Wi of each corresponding item. Let thecapacity of station A (i.e., available time for processing)equal the sack’s maximum weight allowed, T. After these

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A. Linhares / Int. J. Production Economics 121 (2009) 121–129128

mappings, an optimum solution of the product-mixproblem is also an optimum solution to the originalknapsack problem. Thus any exact algorithm for optimum

product mix under TOC can also be used for the solution of

the NP-hard knapsack problem. This demonstrates thatselection of a product mix in a constrained facility is alsoin itself an NP-hard problem.

The above example and proof have considered the 0–1knapsack problem, where a binary decision either with-drew or placed a specific item on the sack. The situationwith the product-mix decision under a constraint is closer,however, to the traditional knapsack problem that asks forthe quantity of each specific type of item to be placed onthe sack. This change in perspective does not change themajor proposition that the problem is NP-hard and it maybe unpractical to attempt to obtain optimal solutions to allcases.

The reader should have noticed that this resultobviously implies that the corresponding problem undermultiple constraints is also NP-hard. The reasoning is thefollowing. Suppose we have an optimum product-mixalgorithm for the case of multiple constraints. Thisalgorithm can be used to solve the case of a singleconstraint, as for example by adding a new constrainedresource and a new product. Let the value of the productbe zero and the new constrained resource only be used toproduce the new product (and be the only one so).A solution to the transformed multiple-constraint pro-blem is a direct solution to the single constraint one. So analgorithm for the multiple-constraint problem wouldsolve optimally an NP-hard problem, which immediatelydemonstrates that the multiple-constraints case is alsoobviously NP-hard.

6. Conclusion

This paper illustrates forms in which the TOC product-mix method may fail, even in the case of a single BN. Thereason for these failures stems not from the nature of themethod, but from the nature of the problem itself: it is anNP-hard problem, and any algorithm that optimally solvesthe product-mix problem under one or more BN(s) mayalso be used to solve the NP-hard knapsack problem.

Since an exact method for selection of a product mix ina constrained facility would imply that P ¼ NP, this resultdemonstrates FACT 4 that there are strong reasons tobelieve that an efficient and optimum heuristic for theproduct-mix decision under the TOC is simply impossible.It also explains Plenert’s (1993) results and illustrates whyresearchers have not been able to find a simple optimumheuristic for product mix under multiple constrainedresources. As we have seen here, the TOC-derivedheuristic fails also on facilities with a single constrainedresource.

It seems thus more reasonable to expect that the bestpossible that may be obtained for large instances is high-quality approximations given by advanced heuristics(such as genetic algorithms) that have been under studyrecently (see, for instance, Fredendall and Lea, 1997; Hsu

and Chung, 1998; Onwubolu, 2001; Onwubolu andMutingi 2001a, b).

There are two obvious limitations on this result. Thefirst limitation is that we are dealing with cases where wehave perfect information. This supposition does notreflect numerous industrial environments that exhibitrapidly shifting BNs, major deviations in processingtime for each station, etc. While these suppositions areshared with a large part of the literature (e.g., Goldratt,1990a; Fox, 1987), there is still a clear need for furtherresearch of cases with imperfect and non-deterministicinformation.

The other limitation is that if items are allowed to becompleted in part over days and weeks (instead of inwhole numbers during the planning period), then theTOC-derived heuristic is in fact optimal. This last point isworth developing. It is crucial to distinguish the TOCphilosophy from the TOC-derived product-mix heuristic.This paper is in no way a ‘criticism’ of the TOC. TOC is amanagement philosophy that does not require, nevermentions, and is completely independent of the integerassumption. The results presented here only demonstratethat the straightforward extension of the heuristic for theinteger case may be problematic. This does not implyanything for the TOC as a management philosophy (Mabinand Balderstone, 2003; Mabin and Gibson, 1998; Mabinand Davies, 2003). It does, however, give us a newimportant research problem. Since TOC is based onbreaking constraints, how can we properly use the TOCphilosophy to deal with the combinatorial complexitydemanded in the integer production case? After all, we donot want to turn the product-mix decision itself into abottleneck.

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