ECONOMIC ANALYSIS OF PRODUCTION BOTTLENECKS

The management of bottlenecks has become a central topic in the planning and control of production systems. In this paper, we critically analyze bottlenecks from an economic perspective. Using a queueing network model, we demonstrate that bottlenecks are inevitable when there are differences in job arrival rates, processing rates, 
or costs of productive resources. These differences naturally lead to the creation of bottlenecks both for facilities design and demand planning problems. To evaluate bottlenecks from an economic perspective, we develop the notion of an “economic bottleneck,” which defines resources as bottlenecks based on economic, rather than 
physical, characteristics.


INTRODUCTION
In this paper we critically analyze production bottlenecks from an economic perspective, addressing important facilities-design and demand-planning problems. Using a queueing network model, we demonstrate that production bottlenecks are inevitable when there are differences in job arrival rates, processing rates, or costs of productive resources. We further analyze the impact of capacity and demand decisions on the location and characteristics of bottlenecks.
The results of this analysis have a number of important managerial implications. We demonstrate that bottlenecks necessarily arise when costs (profits) are minimized (maximized) and that attempts to control bottlenecks by balancing production will usually be economically counterproductive. We introduce the notion of an "economic bottleneck," which defines resources as bottlenecks based on economic, rather than physical, characteristics. This definition provides the basis for making recommendations for managing and relieving bottlenecks in order to improve the economic performance of productive capacity.
Thus we address the capacity allocation problem from a different perspective than that of most work previously done. Although our Jackson network model of the shop-floor dynamics is simpler than some others, we are able to obtain closed-form expressions, 342 A. H. BUSS AND S. LAWRENCE rather than simply to present an algorithm. This allows us to obtain a much clearer picture of the nature of the solution and thus provide more insight. In particular, the economic bottleneck concept allows easy, managerially useful prescriptions for allocation of additional resources.
A survey of the research and pedagogical literature finds that there exists no clear consensus as to the definition of a "bottleneck" resource. Several of these definitions are: • Congestion points, or bottlenecks, primarily occur when manufacturing resources required in a given time period are unavailable. [30] • A bottleneck is defined as any resource whose capacity is less than the demand placed upon it. A bottleneck, in other words, is a process that limits throughput. [8] • Production bottlenecks are generally considered to be temporary blockades to increased output ... (where) inventories build up at different places and different times. [31] • A facility, function, department, etc., that impedes production ... [12] • A bottleneck operation ... is any operation that limits output. [21] Umble and Srikanth [33] make a distinction between "bottlenecks" and "constrained capacity resources," where the former is defined as "any resource whose capacity is equal to or less than the demand placed upon it" and the latter as "any resource which ... is likely to cause the actual flow of product through the plant to deviate from the planned product flow." Goldratt and Cox [13] and Schonberger and Knod [32] make similar distinctions. Summarizing, there appear to be three principal definitions for bottleneck resources currently in use: A bottleneck resource is one for which (1) short-term demand exceeds capacity; (2) work-in-process (WIP) inventory is maximum; or (3) production capacity is minimum, relative to demand (i.e., capacity utilization is maximal). We assign each definition its own name: the short-term definition, the inventory definition, and the production definition, respectively.
Short-Term Definition. In the long run, demand cannot exceed capacity~ither work will increase without bound or there will be sufficient loss of business to reduce the demand rate below capacity. But demand can, and often does, exceed capacity in the short run. When these short-term bottlenecks occur, shop-floor control techniques (such as finite scheduling, priority sequencing, overtime, subcontracting, and so forth) must be exercised to alleviate the bottleneck.
Inventory Definition. In practice, bottlenecks are often identified by simply walking out on a shop floor and observing where the most work is waiting in queue for processing. The second definition of a bottleneck thus focuses on relative levels of work-in-process (WIP) inventories-that resource with the largest current WIP is defined to be the bottleneck. In a stochastic environment, the workcenter with the most WIP changes from time to time, and so the bottleneck is observed to randomly shift from workcenter to workcenter-a phenomenon which bedevils many managers (for an analysis of shifting bottlenecks, see [23]). Since work-in-process inventories are related to the busy period of the workcenter, they shift more slowly than do instantaneous arrival rates. Inventory bottlenecks are therefore more long-lived than are short-term bottlenecks, making them relevant over a middle-term time horizon.
Production Definition. Finally, for long-range planning, bottlenecks are those resources which, on average, are the greatest impediment to increased output or throughput. For long time horizons, capacity utilization is the most useful measure with which to identify these long-run bottlenecks, since those resources which are most highly utilized are those which over time most severely limit output and slow production.
These three definitions can be operationalized as follows. Let -y be the arrival rate of jobs to a workcenter or machine, and let f.1 be its capacity, expressed as a pmcessing rate. Then a bottleneck exists when: (1) demand temporarily exceeds capacity (short-term definition); or (2) the number of jobs waiting in queue L = L(-y,f.1) is maximum (inventory definition); or (3) the long-run utilization p = -y/f.1 is maximum (production definition). An alternative to utilization is long-run capacity cushion defined as f.1 --y. We use capacity cushion and utilization interchangeably as the context warrants. Note that these definitions are not mutually exclusive and that a particular workcenter may satisfy one or more of them at any given time. Since none of these definitions considers costs, revenues, or profitability of the firm, but focuses solely on the output of the process, we will refer to them as production bottlenecks. In contrast, when cost and revenues are taken into account, we can identify those resources which limit profitability. We will call such resources economic bottlenecks. As with production bottlenecks, we can identify short-, medium-, and long-term economic bottlenecks in a manner similar to the three types of production bottlenecks. Since the nature of the decisions we consider in this paper are long-term, we will use the long-term bottleneck definitions for both the production and economic bottlenecks. In this paper we demonstrate that the economic bottleneck and the production bottleneck do not necessarily coincide.
The remainder of this paper is organized as follows. The next section surveys the literature, and section 3 introduces the general profit maximization model for determining capacities or demand rates. Section 4 investigates optimal bottlenecks in facilities design where the design variable is production capacity. In section 5 we tum to the demand planning problem of determining optimal demand volumes when resource capacities are fixed. We introduce the notion of an economic bottleneck in section 6 and discuss its implications in managing bottlenecks. Section 7 presents a numerical example of our results, and concluding remarks are made in section 8.

LITERATURE REVIEW
The importance of understanding and managing production bottlenecks has been the focus of the proponents of the "OPT" philosophy (for example, [13]), more recently called the Theory of Constraints [14] or Synchronous Manufacturing [33]. This literature argues that bottlenecks are inevitable (and even desirable) in many manufacturing settings (see [24]; [28]; [22]), and typically uses simple analogies and appeals to common sense to support its analyses. While provocative, this qualitative approach is difficult to evaluate objectively. Further, much of the original OPT technology is cloaked in commercial secrecy, further confounding a balanced appraisal of its efficacy.
Another body of literature significant to this paper concerns queueing networks since our model utilizes a Jackson network [17]. (The background literature on the use of queueing networks to model manufacturing systems is vast, so we refer the reader to [6] for a summary.) [19] developed fluid approximations to discrete flow networks and used their results to define and characterize network bottlenecks. Other work related to ours is that of [20], who chooses capacities for an open Jackson network to minimize expected time in the system subject to a linear budget constraint. While Kleinrock deals with a problem similar to our facilities design problem of section 4, his sojourn-time criterion is different and his use of a budget constraint produces a different solution. [35] subsequently generalizes Kleinrock's model to queueing networks with arbitrary service times, utilizing a heavy traffic approximation of [15]. Wein's simulation results show that the approximation gives good results. Neither of these works directly address the issue of production bottlenecks.
Some work in management of machine capacity has used somewhat more sophisticated network models in their analysis, such as Bitran and Tirupati [3] and Boxma, Kan, and van Vliet [4]. This research differs from these two papers in several respects. First, in these papers there is either a target level of WIP given exogenously, with the cost of the machines being minimized, or WIP is minimized for a given budget of either capital or capacity. In this paper, there are no exogenously given constraint; rather, our results are more of a "natural" economic equilibrium, that is, what the rational levels of both capital investment in capacity and in WIP would be in the absence of these exogenously given targets. A second difference is that, with the simpler Jackson network model, we are able to give explicit forms of the solutions. This is important to our study, not because of the fact of obtaining closed-form results, but because of the insights we can obtain, particularly related to the important concept of the economic bottleneck. The above mentioned papers provide algorithms for the solutions, rather than analytic expressions. Finally, the prior research along these lines has not focused on the bottleneck issue, that is, providing means of identifying the workcenters that are constraining flows. In this paper, the economic bottleneck concept allows the framing of capacity and demand decisions in terms of identifying which workcenters constrain cash flows, rather than the more traditional product-flow bottlenecks.
Bottlenecks are also an important topic in the control of production systems, as evidenced by the increasing research interest in bottleneck management. Adams, Balas, and Zawack [1] developed a shifting bottleneck algorithm which iteratively scheduled the bottleneck resource in a classic jobshop scheduling problem to minimize makespan. Chang, Matsuo, and Sullivan [7] examined the makespan scheduling of a flexible manufacturing system, and used the alternative routings available with an FMS to develop a beam-search solution procedure which scheduled around bottleneck machines. Pence, Meegeath, and Morrell [27] describe a cycle-scheduling algorithm for scheduling a single bottleneck when set-up times are sequence dependent.
While this summary illustrates the growing research interest in the management of production bottlenecks, there has been relatively little investigation of the economic consequences of production bottlenecks. Banker, Datar, and Kekre [2] demonstrate how production bottlenecks can affect cost accounting and pricing decisions in stochastic production systems. Morton and Singh [25] generalize these results to deterministic systems and develop a resource pricing methodology based on the busy-period of the resource. Morton, Lawrence, Rajagopolan, and Kekre [26] used a related busy-period resource pricing methodology in the development of a production scheduling algorithm. However, none of these papers has explicitly examined the economic consequences of production bottlenecks, a contribution of this paper.

GENERAL MODEL
The general problem we address is that of determining capacities for workcenters and determining the demand rates for each product manufactured in a facility such that long-run average profits are maximized. We subsequently show that production bottlenecks naturally arise from this optimization.
Consider a production facility which manufactures n distinct products (or product classes) with product j having demand rate ~ items per unit time. These n products are processed in one or more of m workcenters, each with production capacity of !lk units per unit time. Workcenter capacities !lk and demand rates Aj are thus the decision variables of this problem; let !I ~ C!il' ... , !1m) denote the vector of processing rates and A ~ (AI'"'' An) denote the vector of demand rates. The sequence of processing in the facility can be random (a job shop), sequential (a flow shop), or mixed.
Relevant costs include the period costs K(!f) of providing capacity at level !lk for workcenter k and the period flow costs F(!f,A) arising from congestion. Note that all costs are the per-period (e.g., annualized) expenses incurred using a common time unit. Capacity costs include amortized equipment expense, plant costs, labor, and maintenance expense. Congestion costs include warehousing and storage costs, materials handling expense, insurance costs, inventory tracking and expediting charges, capital opportunity costs, lost customer goodwill, quality expenses arising from deteriorating in-process inventory, and other similar inventory holding expenses. Marginal period profits M(A) consist of product contribution (gross revenue less variable production costs) net marketing expenses associated with producing demand at rate A. The general form of the profit function llC!i, A) can thus be written as (1) The firm's objective is to maximize the long-run or steady-state average profit per unit time; that is, to maximize llC!i, A). Implicit in this formulation is the assumption that the facility can, and does, reach steady state.
The focus of this paper is not the solution of (1) per se; rather, it is to investigate the consequences of capacity and demand decisions on the congestion in the facility as exhibited by the bottlenecks. Therefore, we concentrate on the special cases obtained by finding the optimal capacities for given demand, and then on the optimal demand for given capacities. The former problem we call the Facilities Design problem, and the latter the Demand Planning problem. The joint problem of simultaneously determining optimal capacity and demands is an important and difficult one, but is beyond the scope of the present paper. The reader is referred to [5] which addresses this joint problem in a somewhat different context.

Jackson Network Model
To investigate the problem outlined above, it is necessary to specify more precisely the characteristics of the production facility. We do this by modeling the shop as a Jackson network consisting of m single-machine workcenters [17]. The Jackson network is very flexible and can model a wide range of situations from a pure job shop to a flow line. We adopt the standard Jackson network assumptions (see, for example, [18]). Each work center k has a service time distribution that is exponential with rate J.1k ' We thus assume that the processing rates of all productsj have the same mean processing time lIJ.1k on machine k. Jobs arrive to the workcenters as independent Poisson processes, with A.k the mean arrival rate to workcenter k, and are distinguished solely by the workcenter at which they first arrive. Our model can thus accommodate as many products or product classes as there are workcenters-additional products can be included by the addition of dummy workcenters.
Upon completion of processing at workcenter i, a job moves to workcenter j with (known) probability Pij and leaves the shop with probability Pi,m+I' The routing matrix P == (Pij) is thus an (m + 1) X (m + 1) stochastic matrix. The matrix P is assumed to be irreducible with Pi,m+1 > 0 for at least one i. Denote the m X m submatrix of in-shop routing probabilities by Po. The aggregate arrival rate vector "I is given by " . . , A. m ) is the arrival rate vector (we let A. j == 0 if there is no job whose initial workcenter is j). The condition for steady state is "Ik < J.1k for all workcenters k. In this case, the probability of n jobs at the workcenters is identical to that for independent MIMII queues with arrival rates "II' ... , "1m and service rates J.11' ... , J.1m' This is a result for the marginal probabilities of the number of customers in queue only; the workcenters do not behave dynamically as independent MIMII queues (see [11]).
To specify the congestion cost function F{J.1, A.), we assume that it can vary by workcenter, but is proportional to sojourn times at each workcenter. Denote the unit flow cost at workcenter k by F k , where Fk is the holding or flow cost per unit time for a customer at station k. The long-run average flow cost per unit time is given by (see Appendix): The profit function in (I) is thus (3) A critical assumption in our Jackson network model is that changes in arrival rates A. j do not affect routing matrix P. This is not a problem under the assumptions of an ideal Jackson network, but may not strictly hold when an actual production facility is approximated using a Jackson network. In the latter case, changes in A. may necessitate changes in P which in tum greatly complicates marginal analysis of n{J.1, A.) with respect to A., Simulation experiments on shops having jobs with fixed routings have demonstrated the robustness of our assumption, but we acknowledge this is a potential limitation of our model.
A second assumption of the model is that different products can be distinguished by their initial workcenter. Our modeling objective is to capture the impact of different products on flow costs, which will be related to the congestion induced by the arrival of jobs to the facility. For a given arrival rate vector A, the expected ftowcosts for a job arriving at workcenter k will be ~k Fk 'Y'/{J1k -'Yk)' Since we are concerned with the capturing impact of ftowcosts, we may sometimes consider products to be "identical" if they have comparable value, have similar routings, and require approximately the same processing times. Note that we can introduce dummy workcenters at the beginning of processing to further distinguish the different products.

FACILITIES-DESIGN PROBLEM
The Facilities Design problem is that of determining capacities 11 such that the profit in Expression (3) is maximized for given (fixed) demand rate vector A. We use the term "design" because the capacity decisions for a shop or facility are typically made during the design stage. We model the cost of providing capacity 11k at workcenter k as a continuous linear function of capacity, so that the aggregate cost per unit time K{J1) is given by K{J1) = ~k KkIlk' where Kk is the cost (per unit time) of providing one unit of capacity at workcenter k. Since demand is fixed, maximizing the profit in (3) is equivalent to minimizing costs; that is, the facilities design problem amounts to minimizing Our use of a linear capacity cost function is consistent with previous literature (e.g., [16]; [20]; [3]; [35]) and is justified as follows. While it is unlikely that an actual capacity cost function will be linear over its entire range, a linear model can closely approximate a "true" cost function in the neighborhood of any chosen capacity 11 and so will be a good representation of capacity costs in the relevant range of near-optimal capacities. It is also unlikely that actual capacity cost functions will be continuous since capacity is typically acquired in discrete units. A linear function is therefore an approximation, but a useful one, since it allows marginal analysis to be conducted resulting in important managerial insight, as we subsequently demonstrate.

Optimal Workcenter Capacities
Since C{J1) is convex in 11 (see Appendix), there exists a unique optimal vector of capacities which minimizes C(J1). Optimal workcenter capacities can therefore be found by solving the first-order conditions: (5) and are given by (6) (6) is a network generalization of Hillier's result for a single server MIMII queue [16]. If 'A j > 0 for one j, then 'Yk > 0 for all k, by the irreducibility of the Jackson network. Thus, the optimal capacities !1 ~ are strictly positive and are equal to aggregate arrival rate 'Yk plus a capacity cushion V Fk'Yk IK k . The capacity cushion at one workcenter is therefore independent of cost parameters Kk and Fk of other workcenterseach workcenter can be considered in isolation from other centers. This property is a consequence of the product-form solution to the Jackson network-in networks without a product-form solution, we would anticipate a greater interaction between the various workcenters.
The optimal capacity utilization for workcenter k, P: £'YJ!1; , is obtained from (6): Flow shops and pure job shops represent two special cases of this general result. Flow Shop. For a flow shop, the routing is deterministic and each job follows the same sequence of workcenters. All jobs begin at workcenter 1 and proceed in ascending order through workcenter m, after which they leave the system. The routing matrix P thus consists of ones immediately above the diagonal and zeros elsewhere. Since all jobs start at workcenter 1, there is only one job type which arrives at rate ' A l' so we may write the optimal work center capacities and utilizations as (8) which is identical to the single-server MIMII result of Hillier [16].
Pure Job Shop. A Jackson network model of a pure job shop assumes that from any workcenter a job may go to any workcenter (including the one it just left) or out of the shop with equal probability and that the exogenous arrival rate of jobs to each workcenter is identical. Thus, the routing sub-matrix Po consists of identical entries, each of which is I/(m + 1). The matrix (I -Po) -1 consists of 2' s on the diagonal and l' s off the diagonal. The aggregate arrival rate 'Yk at each workcenter k is therefore 'Yk = (m + 1)'Ao, where 'Ao is the common exogenous arrival rate to each workcenter. The optimal capacity and utilization of workcenter k are p' = ( 1 + a close analog of the [16] single-server result.

Designing Optimal Bottlenecks
From expression (7) for p~, it is clear that optimal workcenter utilizations will seldom be equal-this occurs only in the unlikely event that two or more ratios F IKk'Yk are equal. By our earlier definition of a production bottleneck as the workcenter with the largest utilization Pk' this result demonstrates that production bottlenecks are the usually inevitable outcome of cost minimization. Bottlenecks therefore do not represent a failure in facilities design, but rather arise naturally from the optimizing efforts of design and production personnel. Expression (7) offers additional insights. First, all else equal, as capacity costs Kk increase, utilization levels P ~ increase-the more expensive is workcenter capacity, the more highly utilized it will be and the more likely it will be a bottleneck. Conversely, a workcenter with large flowcosts Fk will have a reduced capacity utilization and will less likely be a bottleneck. Finally, as job arrival rate 'Yk increases (all else equal), the greater will be capacity utilization-the busier the workcenter, the greater is the likelihood that it is a bottleneck. These outcomes are intuitively satisfying. For both pure flow shops and pure job shops the ratio K/Fk is sufficient to determine the production bottleneck, which is the workcenter with the smallest ratio. For pure job shops and pure flow shops with equal unit costs of congestion and capacity, the optimal design is completely balanced production.
Bottleneck Sensitivity. We now examine the effect of changes in the parameters on the optimal utilizations p~. Differentiating (7) with respect to ' A j , F k , and Kk gives (10) (11) (12) where Rjk £. (l -Po) -ljk is the mean number of times workcenter k is visited by a job initiating service at workcenter j [19].
Thus, P ~ is increasing in ' A j and K k , but decreasing in F k • From (10), increasing any ' A j will result in an increase in all P ~'s for which Rjk > O-this change will be proportional to the expected number of times the job visits workcenter k. Changing ~ will have the greatest impact on non-bottleneck workcenters (those with relatively small utilizations) which are visited relatively often-increases in Aj will disproportionately increase the utilization of these centers. From (11) and (12), changing Fk or Kk affect the optimal utilization of workcenter k only. The impact is greatest for bottleneck workcenters (those with large utilizations) and for those with relatively low aggregate arrival rates 'Yk' Increasing flow costs at a workcenter has the effect of decreasing capacity utilization, while increasing the costs of capacity increase the utilization at that workcenter alone.
Sensitivity of Costs. We now examine the sensitivity of total costs C{J1*) to changes in workcenter parameters. Such information is managerially valuable since it focuses attention on workcenters with the largest potential return on improvement. Substituting the optimal capacities /1* into (3), we have the minimum cost associated with the optimal design: C(u*)=2L YF,J<k"fk + LK('fk' (13) k=i k=i Observe that the cost at the optimum grows linearly in K k , but as the square root of F k • Examining the impact of changing the various parameters on the optimal cost, from (4) we immediately have This result indicates that for fixed demand, cost-reduction efforts (perhaps through improvements in technology or redesign) should focus on those workcenters with the greatest assigned capacities J.l~. Note that these workcenters may well not be those with largest capacity costs K k , or those with the greatest utilization Pk' as intuition might first suggest. Indeed, all else equal, reducing the capacity costs of non-bottleneck workstations (those with lower utilizations) will provide faster paybacks than will reducing costs at bottleneck stations.
Considering changes in flow costs Fk at workcenter k, we have (15) Here, the largest rate of cost reduction is obtained by reducing Fk at the workstation with the highest capacity utilization, as intuition would suggest. More surprising is that, all else equal, the best total cost improvement rate occurs at that workcenter with the lowest value of F k , since this corresponds to workcenters with low utilizations in the optimal solution. Finally, consider changes in total cost due to increases in >"j (the arrival rate of job type j), perhaps due to changes in marketing effort. From (6) and (13) we have: (16) Recalling that Rjk is the mean number of visits of product j to workstation k, we see that, other things equal, products which visit more costly workstations more frequently produce larger cost increases when demand increases. Similarly, products which visit workstations more frequently with higher flowcosts also produce larger cost increases in response to increased demand for those products. Although these two results are quite intuitive, note that (16) provides a means of explicitly evaluating the impact of increasing demand for each product on costs.

Cost of Balancing Capacity
A frequently cited goal of production managers is to balance production, meaning that all machines and workcenters are equally utilized, thus "eliminating" bottlenecks. The intent of such a policy is to ensure that capital equipment and fixed labor is not underutilized, thereby incurring supposed opportunity costs from lost production.
A policy of balanced production is equivalent to insisting that traffic intensities Pk = 'Y'/J1k be identical for all machines k. Let P be this common traffic intensity and C B the cost associated with the balanced utilization policy. Cost function (4) becomes: (17) The optimal (common) utilization level can be found by solving the first-order condition .
1 " 1" The cost associated with this balanced policy is (20) As with C{J1*), CB(p*) grows linearly in Kk and as the square root of F k • The difference between C(p*) and C B {J1*) is CB(p ) -C{J1 ) = 2 * * ( (21) This non-negative (and typically positive) quantity represents the additional costs that the firm must pay for adhering to a balanced production policy. In summary, the results of this section show that cost-minimizing behavior leads to the deliberate creation of production bottlenecks. Bottlenecks are thus an inevitable result of the optimization of resource capacities. Rather than attempting to balance production or eliminate bottlenecks, production managers should in fact design bottlenecks into the production facility, with the location of these bottlenecks determined by appropriate cost (or profit) considerations.

DEMAND-PLANNING PROBLEM
Our analysis in the previous section addressed the design of productive capacity with demand rates assumed to be fixed. In this section we consider the analogous Demand Planning problem: determining optimal demand rates for given fixed capacity. This applies to situations in which management desires to maximize profits at an existing production facility by adjusting demand levels to match the capabilities of an existing plant. We assume that demand levels can be modified by the firm through adjustments to advertising expenditures, price incentives, sales effort and focus, and similar marketing initiatives, but note that the marketing initiatives themselves are not decision variables in our model. We further assume that the plant is currently operating in the region of optimal demand levels, so that required demand changes are relatively small.
Capacity costs K{J.t) are now sunk, and so are irrelevant, whereas marginal revenues M(h) are now relevant. Since production capacities Pk are fixed, the relevant range of feasible production volumes hk is relatively narrow. Consequently, we model marginal revenues as a linear function of production volumes: M(h) &. 2: 'j = 1 Mjhj . As with the design problem of section 3, we distinguish products by the workcenter at which they initiate service. If there are no products with workcenter j as the initial workcenter, we set M j = 0, effectively creating a "dummy" product. Clearly these dummy products will have h; = 0, since they only add flow costs and do not contribute to the profitability of the operation.
Taken together with congestion costs, the objective of the demand planning problem is to maximize contribution l1(h) defined as marginal revenues less congestion costs: The demand-planning problem is complicated by the fact that 'Y is a function of h and we must therefore explicitly consider both non-negativity of h as well as feasibility of 'Y. This latter condition arises because the traffic intensity at each workcenter must be less than unity (Le., Pk &. P,)'Yk < 1). Thus, the model for demand planning becomes max IT(h) subject to (23) The first constraint insures non-negative arrivals, while the second ensures the feasibility of 'Y.

Properties of the Solution
Program (23) can be solved using standard nonlinear programming techniques, but it will be useful to establish several of its properties, including the existence of a solution. Since the set S2 = {h E R+: h(l -PO)-l < p} is convex and 11 is convex on S2' there is a Kuhn-Tucker point (h*, eO), where e is the vector of dual variables associated with the non-negativity constraints (see Appendix). Elements e j of the dual vector e represent the PRODUCTION BOTTLENECKS 353 reduced margins of the various products available for production (analogous to reduced costs in linear programming). For products being produced O'j > 0), the reduced margin is zero (e j = 0). For products not being produced Ol.j = 0) the reduced margins are non-negative (e j ~ 0), representing the amount by which margin M j must be increased in order to make product j attractive enough for production.
If revenues M j are not sufficient to cover costs for product j then it should not be produced, and if this is true for all products then it is optimal not to produce at all. The following conditions indicate when it is not profitable for the firm to produce a given product: (1) If M j -2: ~ = I F k Rj/! 11k ::5 0 for all j, then the product initiating service on machine j should not be produced (i.e., A * = 0); (2) Conversely, if M j -2: ~'= I Fk Rj/! J.1k> 0 for some j then A; > 0 for some i (see Appendix).
Henceforth, to avoid the no-production situation (A; = 0 for all j), we will assume that The bottleneck workcenter will be the one for which V Fi/l1k (Mk + e;) is smallest.
Workcenters with large flow costs or small capacities (i.e., small 11k) will tend to have small capacity cushions and therefore be production bottlenecks. One again, we see that production bottlenecks are the inevitable outcome of optimizing behavior.
Although e* cannot be expressed in closed form, a simple condition indicates whether some will be positive: if Mk :::; F i/l1k' then e; > 0, and hence X.; = 0 (see Appendix). To interpret this condition, observe that F i/l1k is the average flow cost accumulated during the processing of job k on its first work center, k. The value of Mk = Mk -Lj Pkj Mj can be interpreted as follows. After processing on workcenter k, the job goes to workcenter j with probability Pij' That workcenter thus has an opportunity cost of M j for processing a job of type k, since no revenue is received. Thus, Mk can be interpreted as the marginal revenue net the expected opportunity cost of processing on the second workcenter. If this is not sufficient to meet the expected flow costs during processing (which is a lower bound on flow costs), then it will clearly not be profitable to take type k jobs. The planning problem is complicated by the possibility of the optimal solution not being an interior point, that is, having X.; = 0 for some j's. However, if we do have an optimal point in the interior, it is of the form: • "Yk = 11k - (28) and (29) with optimal utilization (30) Equation (29) is, again, the network analog of Hillier's [16] MIMII single-server results.

Special Cases
Flow Shop. The profit function for planning problem in a flow shop is: (31) with corresponding first-order condition (32) PRODUCTION BOTTLENECKS 355 There will be a positive optimal ~1 iff M > ~~'= 1 F '/l1k' and the production bottleneck will be the workcenter with the smallest capacity 11k' Uniform Utilization. Unlike the capacity design problem, it may not be possible to have the same utilization at all workcenters for the planning problem. However, the following two conditions are necessary and sufficient for uniform utilization to be possible (see Appendix): (I) I1j > Lk 11k P kj for all workcenters j with an exogenous arrival stream, and (2) I1j = Lk 11k P kj for all workcenters j without an exogenous arrival stream. For those cases in which equal utilization is possible, we can determine the optimal utilization p *.
We obtain the profit associated with balanced utilization, IlB in a manner similar to section 3: The optimal utilization for balanced production is therefore (34) We may then obtain similar results as for the facilities design problem with equal utilization.

ECONOMIC BOTTLENECK
Our analysis of the facilities design and demand planning problems has shown that bottlenecks naturally arise when firms organize capacity design and demand volumes to minimize costs and maximize profits. Implicit in this analysis is the notion of an economic bottleneck, defined to be that workstation which most severely increases costs or limits profits. Below we offer a formal definition of an economic bottleneck, distinguish it from a production bottleneck, and demonstrate that the two do not necessarily coincide.
Consider a production facility with existing capacity levels 11 and fixed production demands ~. Management of the firm wishes to reduce costs and improve throughput rates by increasing capacity at the appropriate workstation(s). Capacity might be increased by incremental technological improvements, better management of secondary resources such as labor and tooling, reduced setup times, and so forth. We distinguish these types of continuous improvement activities from the initial design of the facility. Given limited resources, at which workstation(s) should management focus its attention, and what are the potential benefits?
Under these assumptions, initial capacity costs K(J1) are sunk (since the plant is in operation), and contribution margins M(~) are constant (since production demands ~ are fixed). Consequently, the objective function facing the firm is simply to minimize congestion costs with respect to /1. We define the economic bottleneck to be that station for which marginal increases in capacity provides the largest decrease in congestion costs; that is, the workcenter k for which (36) is a minimum. This definition of an economic bottleneck provides the marginal benefit or shadow price of marginal increases in capacity at a particular workcenter. This result applies to any production facility, whether or not it is optimally configured with regard to X. and /1. Note also that, as defined in (36), the costs of additional capacity are not explicitly considered.
If the facility is optimally planned, that is X. = X.', then further results can be obtained. The economic bottleneck concept has important consequences for the improvement of facilities. While the production bottleneck is the workstation which constrains capacity, the economic bottleneck is the workcenter which constrains profitability. Clearly the two need not be identical and, indeed, will typically be different. Indeed, Expression (37) shows that the concepts coincide only when Mk + e; are identical for all workcenters k, an unlikely occurrence, Capacity decisions based on production bottleneck considerations may therefore not be the most profitable ones, as shown in the following example,

NUMERICAL EXAMPLE
A simple example will serve to illustrate several of our results. Consider a production facility with 5 single-server workstations having routing matrix given in Table I, and with other parameters given in Table II (the base case).
The facility design problem is straightforward, as discussed previously, since the optimal capacities will always be positive and given in closed form, for linear cost of capacity functions, Using the base case values for F, K, and demand arrivals X. as shown in Table III, equation (6) gives the optimal capacities /1* in Table III. For the demand planning problem, using base case values for F, M, and /1 as given in Table III, equation (26) gives optimal values of X. * as in Table III. Since all X.; are positive, PRODUCTION BOTTLENECKS   Table III shows that station 1 is the production bottleneck since it has the highest capacity utilizations among the five stations (although stations 2 and 4 could be considered secondary bottlenecks). Conventional analysis therefore suggests that station 1 should first be considered for relief before the other stations. However, determination of the economic bottleneck using equation (37) demonstrates that station 3 is the economic bottleneck, with a dual price that is substantially greater than the second largest (station 2) as shown in Table III. To emphasize this result further, Figure 1 shows the impact of increasing or decreasing capacity on profits at each of the workstations, and clearly shows station 3 to be the economic bottleneck. Costs will decline most rapidly by increasing capacity at the economic bottleneck, not the production bottleneck.
Finally, Figure 2 uses equation (17) to compare optimal costs with actual costs balanced production is enforced. For our example, this figure shows that the negative impact of balancing utilization is small when that utilization is optimal, but that there is a stiff penalty if the optimum is missed. Furthermore, as the common utilization approaches unity, the penalty gets increasingly worse. Similar results are obtained when using equation (33) to compare optimal profits with balanced production profits.

CONCLUSIONS
In this paper, we have critically analyzed bottlenecks from an economic perspective. Using queueing network theory, we have demonstrated that bottlenecks are inevitable when there are differences in job arrival rates, processing rates, or costs of productive resources. These differences naturally lead to the creation of bottlenecks both when designing production facilities to meet anticipated demand, and when determining demand loads for existing facilities.
Our results have a number of implications for the management of stochastic production facilities: • Care must be taken when managing a "bottleneck," since several definitions are In current use, each requiring a different management response; • Production bottlenecks are the natural outcome of minimizing costs when designing production facilities, or of maximizing profits when determining demands loads; • Balanced production is rarely optimal, from a purely economic perspective; • The concept of an economic bottleneck provides an economic basis for bottleneck management and improvement.
Our treatment of the demand-planning problem demonstrates that consideration of flow costs due to congestion and the demands different products place on facilities can contribute to the unprofitability of certain items. In those cases, our model gives rise to dual variables which measure the degree of unprofitability for those items.
We have found that while the various production bottleneck concepts are related to the economic bottleneck, they do not necessarily coincide. In our opinion, the economic bottleneck is ultimately a more useful concept for managers, since it shows the way to the best financial return on capacity investment. Thus, relieving an economic bottleneck will bring greater returns to the firm than relieving a production bottleneck. The latter will merely increase output by the greatest amount without regard for profitability. Furthermore, we have found that attempting to balance the shop by equalizing utilization is also not economically the best course. However, for the facilities-design problem, our solution can be considered the economically balanced shop.