Manufacturing capacity and its measurement: A critical evaluation

Computers Ops Res. Vol. 18, No. 7, pp. 615-627, 1991 Printedin Great Britain. All rights reserved

030s.0548p1s3.00+ 0.00 Copyright 0 1991 Pcrgamon Press plc

MANUFACTURING CAPACITY AND ITS MEASUREMENT: A CRITICAL EVALUATION SALAH

E.

ELMAGHRABY*

North Carolina State University, Raleigh, NC 276957913, U.S.A.

Abstract-The report reviews and analyzes the concept of productive capacity, identifies four capacities: nominal, available, actual, and idle, defines them precisely, and suggests approaches to their measurement. 1. INTRODUCTION

If one were to open any operations research (OR) oriented book on production/operations planning (see, for example, Refs [l] or [2]) one shall find some elaborate models on “capacity planning” under a variety of assumptions on the demand pattern, the cost factors, the reliability of the productive facility, the length of the planning horizon, and so forth. Yet, it comes as a shock to many operations researchers in the field of production planning and control to realize that most manufacturing firms cannot measure capacity. This basic data is simply not available, and when available it usually measures the wrong entity! And yet, operations researchers glibly talk about it, and insert it into their sophisticated mathematical models assuming that it is “there”. It usually comes as a rude awakening to most to realize that the understanding of what is meant by “capacity”, and the determination of its value, are no minor feats. The correct understanding of what is meant by capacity, and its accurate and precise measurement, are of vital importance to all concerned: investors, workers, engineers, and managers alike. The following sections shall reveal that I cite the following uses of measures of capacity: for purposes of loading, scheduling, adding to/deleting from resources, evaluating the production control function, determining the “opportunity cost” of various restrictions imposed on the production facility, determining the advisability of acquiring semi-finished products, setting (probabilistic) bounds on the output for global planning of resources, determining idle capacity for purposes of investment, and comparing intra-firm management relative to direct as well as support productive facilities (such as maintenance, repair, and renewal). Therefore, it would be fair to say that manufacturing capacity, which has long been relegated to the “obvious” with little thought to its meaning or its measurement, is worthy of careful attention since a great deal depends on its proper understanding and correct measurement. Perhaps it is appropriate at this point to explicitly state what this paper does not purport to do: it does not intend to add to the huge literature on capacity typically found under the rubrics “capacity planning”, “lot sizing” (dynamic or otherwise), the “impact on capacity” of various factors of production; etc. Rather, this paper is devoted to a discussion of the very concept of capacity and the issues related to its measurement; in a nutshell, it attempts to respond to the question: what are we talking about? The analysis is couched in non-mathematical terms, and for a good reason: I do not wish to clutter this report with symbols and formulae; rather, it is my objective to discuss, in as simple *Salah E. Elmrgbraby is University Professor of Operations Research and Industrial Engineering at the North Carolina State University. He was Director of the Graduate Program in Operations Research in 1967-1990 and previously he was Associate Professor at Yale University (1962-1967) and Research Leader at the Western Electric Engineering Research Center (1958-1962). He has been a Visiting Professor at Cornell University (Spring 1967) and the Catholic University of Leuven, Belgium (19744975). Dr Elmaghraby obtained a B.Sc. in Mechanical Engineering from Cairo University (Egypt), an M.Sc. in Industrial Engineering from Ohio State University (1955) and a Ph.D. from Cornell University (1958). He has 12 years of industrial experience: one in Egypt, five in Western Europe, four in the U.S., and two in Kuwait. His international academic experience includes being a guest lecturer in Aachen (Germany), Alexandria (Egypt), Eindhoven (Holland), Birmingham (England), Dhahran (Saudi Arabia), Grenoble (France), Louvain (Belgium), and Linkoping (Sweden). His areas of research interests include the design and operation of production systems, project planning and scheduling via activity networks, discrete optimixation and dynamic programming, and quality enhancement in manufacturing. Has written four books, edited two books, and authored or co-authored over 65 papers published in scientific journals. 615

SALAH E.ELWAGHRABY

616

language as I know how and with numerical examples, the issues addressed above. I hope that the OR community will forgive this deviation from the norm;just this time. Some ~e~evuff t quotations

It seems trite to state that definitions are, at best, hazardous since they are among the most difficult statements to make. Perhaps it is best to start with some quotations from recent texts on production/operations management, written by recognized authorities in the subject, on the definition of “capacity”. These should forcefully demonstrate, if such demonstration is needed, the quandary one finds oneself when one desires to define “capacity”. (All emphasis, presented in italics in the following quotations, is mine.) Buffa [3] states (p. 127): “Capacity is the limiting capability of a productive unit to produce within a stated time period, normally expressed in terms of output units per unit of time. But capacity is an illusive concept, because it must be related to the intensiveness with which a facility is used , . . , When the units of output are relatively homogeneous, the capacity units are rather obvious . . .’ (but) when the units of output are more diverse, it is common to use a measure of the availability of the limiting resource as the capacity measure. . , . A jobbing machine shop has many different types of equipment performing a wide variety of machining operations, and the outputs may be unique parts that are never repeated. The value of the labor and material of the outputs could vary widely. Therefore, the capacity of the shop is normally stated as the capacity of the limiting resource, the availability of labor hours. Labor hours are used rather than machine hours, because there is usually a ratio of two or three times as many machine hours available as labor hours; that is, the skilled machinist is the limiting resource.” Chase and Aquilano [4] state (p. 152): “A dictionary definition of capacity is ‘the ability to hold, receive, store, or accommodate’. For (operations management) purposes, it may be defined relative to resource inputs, for example, labor and machine hours available; or relative to outputs, for example, volume of units that can be produced or customers that can be served over a particular period of time. Clearly, true capaciry is dependent upon what is to be produced. For example, a firm that makes multiple products inevitably can produce more of one kind than of another with a given level of resource inputs. . . . An operations management view also emphasizes the time dimension of capacity. This is evidenced in the common distinction drawn between long-range, intermediate-range, and short-range capacity planning.” Finally, Manipaz [23 states (pp. 56-58): “Capacity is the designed rate of output from the process. This characteristic is measured in units of output per unit of time. Capacity is easy to define and hard to measure. It is often possible to determine the theoretical maximum capacity of a process -the most output it could generate under ideal conditions over some short period of time. For planning purposes and management decisions, it is more useful to know the e$ective capacityof a process. To measure effective capacity, it is necessary to know a great deal about the process, and to analyze carefully the particular situation at hand. Quite often, managers believe that the capacity of a process is an absolutely fixed quantity. This is not true. The capacity of a process can change for many reasons. . . . For example, a steel mill may be designed for some ideal capacity of x tons of steel per year, However, the actual capacity may be more or less than x, due to such factors as the nature of the raw materials being utilized, the mix of products in the output, and the quantity and nature of the labor input. A facility accommodates several processes. A facility’s capacity is the rate of its

Manufacturingcapacityand its measurement

617

output, indicating how many products or m (using a common measure) are turned out by the facility per unit of time. An effort is made to express the rate of output without regard to specific products. . . . When expressing capacity, one has to include the time dimension. . . . One also has to be careful not to confuse actual capacity with design capacity. The actual capacity may be lower than the design capacity due to a lower demand for products or services. The actual capacity may change from one time period to another, whereas design capacity, which is the potential rate of output, is constant over a longer period of time. The design capacity may be changed only through expansion or displacement of the available facilities. . . . In discussing capacity, there is one final comment that should be made. This is that the actual capacity may, at times, reach a maximum level that will be maintained for a short time and that is termed peak capacity.” It is not difficult to see the imprecision embedded in these presumed definitions: the authors define capacity in terms of other capacities, which themselves are never defined! The above quotations result in the following crop of capacities (all of which I have highlighted above in italics): 0 rue capacity time dimension of capacity design capacity theoretical maximum capacity efictioe capacity actual capacity ideal capacity facility capacity, and 0 potential capacity.

l 0 l 0 0 l l

I do not doubt that if I had continued my quotations from other sources, the list would have grown even longer! My guess is that operations researchers have tried to emulate the engineering concepts of physical capacity, such as the capacity of a water tank or of an automobile battery, into the domain of production systems, such as manufacturing or distribution systems. Judging by the resultant confusion and imprecision, it would be fair to say that the effort has not been successful. Still, if capacity is to be measured in order to be used in the various functions of production planning and control, one must have some idea about what it is that one is measuring. Therefore there is no escape from some attempt at defining what it is that is sought. I shall try to do so in the following paragraphs, bearing in mind that alternate definitions may be equally valid. What is really important is not exactitude in an absolute sense, but rather agreement on entities that are measurable, and a demonstration of the operational utility of these measures. But first I present further discussion of why one would be interested in the measurement of capacity, i.e. some of its possible uses.

2. SOME TYPES

AND USES OF CAPACITY

As will become amply clear in a moment, there is not one but seoeral capacities, which are used in production planning and control, manpower hiring and firing, physical plant expansion and contraction, short term job scheduling, and the identification of idle capacity for purposes of investment and/or correction. The basic premise is simply the following. A factory (or a shop) is installed with a given nominal (or theoretical or maximum) capacity. When a factory is run in practice one does not achieve the rated nominal capacity, since different factors (to be detailed below) enter into play that inhibit the maintenance of the nominal output except for a very short time. Over a planning horizon of one year, say, the factory must plan according to some realistic available capacity that takes into account these factors. Assuming that the factory was correctly planned, its available capacity should be fully utilized. But when production is realized, the utilization of the productive facility may fall below the available capacity. This under-utilization may be intentional, in which case we speak of planned capacity. On the other hand, the

SALAHE. ELMAGHRABY

618

under-util~ation may be unintentional because of many reasons which are also detailed below. The difference between available capacity and actual utilization represents idle capacity which may be utilized if the causes of output decrement are eliminated. It is natural for the firm to wish to identify the idle capacity and its causes to make available to the decision makers the facts concerning future investment and the degree of its impact for possible improvement. Apart from questions of investment and expansion, the concept of capacity plays a central role in the (long term) planning of production as well as in the (short term) scheduling of activities.‘As was pointed out in the opening paragraph of the Introduction, the whole discipline of operations planning and control is based on the concept of capacity and the ability to measure it. For instance, one must plan production to a given capacity, and it is the very limitations imposed by capacity availability that give rise to the “make or buy” type of problems. Altematively, short term scheduling of operations must take into account the available number of facilities and the length of time in which they are supposed to be operating; i.e. their productive capacity; etc. The above discussion delineates some of the uses of capacity measurement. Additional uses shall become apparent as the discussion progresses, and shall be summarized at the end of the paper, For the moment, it is high time to offer the definitions of four capacities: nominal, available, planned, and actual utilization which, I submit, are sufficient to perform the functions defined in the preceding and subsequent paragraphs. 3. PROPOSED

DEFINITIONS

OF CAPACITIES

In what follows I am assuming that the different capacities are defined relative to a specified average workday, which may vary from factory to factory, or within the same factory from season to season. Should the workday be extended or contracted for any reason, this fact should be noted, otherwise erroneous conclusions will be drawn. (I ). Nominal capacity

It is the productive capability assuming continuous availability of process (or machine) and all its support facilities (such as labor, maintenance, material, electric power, warehousing facilities, transportation, etc.), when the process is devoted to the production of a single “standard” product or activity. It is worth noting that there exist production facilities that produce only one product, such as electric power generators and water management systems, in which case the nominal capacity is quite a meaningful entity. (2). A~ailabIe (or operat~ona~~ capacity It is the productive capability available after subtracting from the nominal capacity the expected and unavoidable loss in production due to: age of facility, required maintenance and overhauls, optimal change-over time between products as dictated by the product mix, standard reject allowance, etc., but still assuming that all support facilities listed in (1) are present. (3). Planned capacity It is that portion of the available capacity that is expected to be used in the planning horizon. (4). Actuaf capacity ut~iization It is the capacity actually utilized in the realization of the units of product(s) (actually) produced. Discussion

A few words of explanation are in order relative to these definitions. I have lumped “theoretical “, “maximum” and “nominal” capacities under the single heading of nominal capacity. This is because I see no point in making these fine distinctions within the context of the objectives mentioned in this paper. Furthermore, the definition I have adopted for nominal capacity insists on assuming continuous availability of the productive facility. This may seem odd, and indeed it has been argued against by some who prefer to exclude the required maintenance and overhauls, on the basis that they consume time that is prescribed not to be available for production from the outset. My rejection

Manufacturingcapacity and its measurement

619

of this argument is based on the fact that required meirptenance and overhauls can sometimes be performed during planned non-productive intervals. For instance, maintenance is scheduled by some firms to be performed during the weekend shutdowns, and overhauls are scheduled during the firm’s shutdown for the annual vacation. (In my personal experience, I have worked in a company that schedules its major overhauls during the two-week shutdown for vacation in July.) Consequently, if we are to have a base datum for all firms in the same industry, the definition cannot provide a ready-made loophole that allows such firms to gain an advantage over others by showing an improved status of available capacity relative to nominal capacity. There is another advantage to maint~ning the de~ni~on of nominal capacity to coincide with the absolute maximum that is attainable under the best possible circumstances, namely, that now the determination of the available capacity will assist an analyst who is interested in the measurement of productivity to distinguish among firms (or among shops within the same firm) more easily and more rationally. In other words, if two firms have exactly the same nominal capacity, but one schedules its maintenance during weekends and the other does not, then the available capacity of the latter will clearly be smaller than that of the former. Such cause of discrepancy can then be more easily isolated. The management of the latter firm may be directed to improve its performance by re-scheduling its maintenance activities, if that is feasible. Such comparison would not have been possible if the nominal capacities of the two firms were.reported to be different in the first place. The definition of available (or operational) capacity includes two elements that require care in their dete~ination. The first is relative to the “optimal setup (or change-over) time among products”, and the second is relative to the “standard rate of rejects”. The optimal setup time for the quantities actually produced in a period of time (say a week) may be determined (or closely approximated) by a mathematical programming model that takes product(s) demand(s) and storeroom capacity into account. The issue is not how it is done, but why it should be done: the absence of a benchmark in the form of what setups should be for a particular production plan results in the blind acceptance of time lost in setups, no matter how large it is, as a legitimate consequence of having to produce a mix of items. This is wrong, and should be corrected. In fact, when such a benchmark is available, then the performance of the production scheduling function of a firm can be objectively evaluated. To seek optimality in the setup times taking account of all other factors of production, such as inventory a~umulation in both in-process and finished goods, is mandatory to be able to judge the efhcacy of the pr~uction planning function across shops or across firms, or within the same shop or firm over time. It also helps in pinpointing internal or external factors that impact adversely on the available capacity, such as the size of the warehouse or the efficacy of the materials handling system. As to the “standard rate of rejects”, I propose to use the industry standard as the datum, at least as a starter to “prime” the calculations. In other words, if the industry experiences an average rate of 3.5% rejects, that should be the initial allowance in all resources used in the determination of the available capacity. Actually, the picture is somewhat more complicated than that, and I shall return to it later. As stated before, the difference between the available and actual capacities measures the idle capacity. The following are some possible causes for such idle capacity: -reduced demand for the product(s), -shortage of raw material, -shortage of equipment spare parts, -labor absenteeism, --electric current “brownouts” or “blackouts” (in some regions), -water shortage, or the shortage of any resource that is provided by the industrial infrastructure of the region. Finally, one need not plan to utilize all available capacity (because of market or labor conditions, for instance). in which case there is planned utilization, or planned capacity. Clearly, planned capacity may approach but cannot exceed the projected available capacity. Note that planned capacity refers to the future utilization of projected available capacity, while the “actual” capacity usage measures past uti~tion, which may be more, or less, than what was planned. The difference between the “plan” and the “actual” may serve to measure the accuracy of the firm’s forecast of

620

SALAH E. ELMAGHRABY

its sales and its productive capabilities. Alternatively, it may measure the effort of the sales department in increasing the firm’s share of the market. 4. CAUSES

OF

INACCUARACY

IN THE

MEASUREMENT

OF

CAPACITY

The following details the reasons for the presence of errors, minor or gross, intentional or otherwise, in a firm’s estimation of the various capacities available to it, to the point of vitiating the utility of the data at all levels of management. The first five causes mentioned below are technical in nature, and are therefore amenable to scientific analysis. The sixth is partially nontechnical, being anchored in the social and cultural environment of the firm. I hope that its inclusion shall help direct attention to these aspects of the production system that are typically ignored by operations researchers. (I)

The problem of prodttct mix

Some firms profess their inability to measure their capacities because such capacities are clearly dependent on the product mix that happens to be produced in any period. And since capacity is defined in terms of a hypothetical “standard” product which the firm may not produce, at least does not produce all the time, nor does it even produce the same mix of products period after period, its capacities are not fixed, and therefore cannot be accurately measured! It is a fact, which I readily admit, that capacity (any of the four capacities listed above) is dependent on the product mix. As to why this is so, the answer lies in the usual existence of a “bottleneck” stage of production that depends on the item produced. That is, one product may be restricted in its production by the available capacity on a particular machine, while another product may be restricted by the available capacity on a different machine. In fact, if all products possess the same bottleneck operation then the capacity, any of the four capacities defined above, is identical to that operation’s capacity, and no problem of measurement should arise. Consequently, when the firm is considered as a whole, one does not possess one figure as a correct representation of the capacity of the firm, but one has several figures that depend on the items produced in any period, say in a month. And since the product mix varies from month to month, one really has 12 product mixes per year, and the determination of the so-called “capacity” appears to be an insurmountable problem whose solution defies the firm’s analytical abilities. The problem is a real one, and its resolution rests on the concept of “expected” capacity. The concept is best illustrated by an example. Consider a firm that produces three different items, labeled A, B, C. For simplicity of exposition, it is assumed that at any time one and only one item is produced by the firm and all shops are set up to accommodate its production. I shall use available capacity as the vehicle of illustration. (The following section details how the numbers used can be changed to represent other capacities.) The firm’s production facilities are composed of four “shops”, whose capacities are as given in Table 1. The interpretation of this information is as follows. Consider item A: its production requires the use of all four shops. The productivity (i.e. available capacity) of shop 1 is 20 tons/week, of shop 2 is 15 tons/week, etc. Clearly, if the firm is devoted wholly to the production of item A, it cannot produce more than 15 tons/week, which is the “bottleneck capacity*’ of shop 2. Continuing along the row of item A, Table 1 indicates that 4.5 tons were produced in week 1, 5.25 tons in week 2, etc. Other rows in the table can be interpreted in a similar fashion. Next, one must translate the actual production of the different items in tons into the proportion

Table 1. Available capacity and production information (in tons/wk) Bottleneck Shop Item A B C

1

20 14 32

2

IS 16 38

3

21 12 36

Production

4

32 21 43

Cap.

Shop

IS 12 32

2 3 I

Total production (tons) =

WkI

Wk2

Wk3

Wk4

4.3 4.8 9.6

5.25 I .80 16.00

9.75 1.80 6.40

3 3 17.6

18.90

23.05

17.95

23.60

Manufacturing capacity and its measurement

621

Table 2. Determination of pro~rtioss of firm’s available capacity devoted to em91ptudtxt Proportion of capacity Item

Bottleneck cap. (tons)

WkI

Wk2

Wk3

15 12 32

0.30 0.40 0.30

0.35 0.15 0.50

0.65 0.15 0.20

A

Wk4

0.20 0.25 0.55

Table 3. Demand charncteriaics PdUCt

Meandemand

Raage

SD 1.33 0.67 2.0

A

7

3-11

B C

4 12

2-6 6-18

of total firm’s capacity used by each product in each week. This translation leads to the data given in Table 2. The calculation of this table is simple. For instance, 4.5 tons were produced in week 1 of item A. Since the capacity of the firm is 15 tons per week, we must have run it for (4.5 + 15) = 0.30 weeks. In week 2, we produced 5.25 tons of A, whence we must have used (5.25 + 15) = 0.35 of the week, etc. Note that, in any week, the sum of the proportions of all three products must add up to 1.00, since we assumed that there was no idle time. We now come to the crucial point of calculating the average available capacity over the planning horizon of 4 weeks. Of course, the calculations can be extended in a similar fashion if data were available on production for 20,40 or even all 52 weeks of the year. Week Week Week Week

1: 2: 3: 4:

15 x 0.30 + 15 x 0.35 + 15 x 0.65 + 15 x 0.20 +

12 x 0.40 + 12 x 0.15 + 12 x 0.15 + 12 x 0.25 +

32 x 0.30 = 32 x 0.50 = 32 x 0.20 = 32 x 0.55 =

18.90 tons 23.05 tons 17.95 tons 23.60 tons

Of course, it is immediately recognized that the “average” capacity per week is exactly equal to the total production as given in Table 1, as it should be. The average capacity utilization over the month = (18.90 + 23.05 + 17.95 + 23.60)/4 = 20.875 ton/wk, and the range of capacity usage = 23.60 - 17.95 = 5.65 tons/wk. Next, we complicate the picture slightly by introducing random variation in demand for each of the three products. This is a more realistic scenario. Assume that the demand for the three products follows a normal distribution (approximately) with means and ranges (= 6 x standard deviation) as in Table 3. The production requirements in the 4 weeks of Table 1 may now be viewed as a 4-tuple sample from each of these distributions. This information may be used to place bounds on the firm’s capacity, and thus secure bounds on its production for purposes of future planning. Assuming independence of the demand for the three products, what is the expected load on the shop, assuming that these distributions of demand.are valid for the foreseeable future? Table 4 summarizes the impact of the demand variation on the occupancy of each shop in the plant. For instance, the entry under column “Shop 3” and along the row of product A is the range of the proportion of the capacity of that shop consumed by product A ( = A-&); the entry in the same column and the row of product B is the range of the proportion of the capacity of that shop consumed by product B ( =4-s) ; etc. The two rows of Table 4 with the prefix “Simultaneous” are illuminating. They give the total mean proportion of the available capacity of each shop that would be consumed by the three products if the demands are to be fully satisfied and if all three products are produced in the same week (and hence share the same weekly productive capacities), as well as the 3a limits on those

622

SALAH E. ELMAGHRABY Table 4. Means and ranges al capacity loads Shop Bottleneck cap.

Product A

15 12 32

:

1 3/20-II/20 v-3/7 3/1&9fi6

2

3

4

3/15-11/H l/a-3/8 3/H-9/19

3/21-11/21 l/6-3/6 l/6-3/6

3/32-11/32 2/21-6/21 6/43-m/43

Simultaneous Avg. 30

1.0107 0.3091

1.0324 0.3342

l.OtW 0.3030

0.6883 0.2102

Sequential

0.3369 0.1768

0.3441 0.1832

0.2539 0.1746

0.2294 0.1199

Avg. 30

demands. The mean is simply the sum of the individual means in its column; the Q is the square root of the sum of the individual variances. For example, on the average, shop 1 is occupied 7/20 of its capacity by product A, 2/7 of its capacity by product B, and 6/16 of its capacity by product C, yielding a total mean of 1.0107 as shown. The variance of the total capacity demand in shop 1 is the sum: (0.4/6)2 + (2/7 x 6)’ + (3/8 x 6)2 = 0.002654, which yields a standard deviation ci= JO.@lZ34 = 0.1030; the 30 limits are f0.3091 as shown. On the average shops 1, 2, and 3 are expected to be loaded approx. 100% of their available capacities-the discrepancy from “full load” is well within the permissible error in the data itself (on the average, shop 1 is 1.07% overloaded, 2 is 3.24% overloaded, and 3 is exactly 100% capacity), while shop 4 is underloaded (at only 68.83% of its capacity). But, in an analysis such as this, averages are meaningless since they occur with probability zero! In reality, shops 1,2, and 3 are overloaded approx. 50% of the time. Even worse, there is an appreciable probability that all three shops shall experience a run of 2, 3, or 4 weeks of above-capacity demand. For instance, the probability that these three shops shall experience above-capacity demand for 3 weeks in suecession is P 12.5%, or approximately one-in-eight. How the plant can respond to such overload is beyond the scope of this analysis, but it must be evident to all that basing decisions of production on the average available capacity can spell disaster to the firm that adopts it. The last two rows of Table 4 are labeled “Sequential” to denote the case in which the productive

capabilities of the plant are devoted to one product only each week. This is a highly impractical situation which is included here for illustrative purposes only, in case it is suggested as an alternative to frequent setups within a week. It is evident that all shops are grossly under-utilized, with shop 2 experiencing the maximum occupancy (rO.3441 of its available capacity) among the four shops. The above analysis has been concerned with individual shops within the plant; can anything be said about the overall plant available capacity? It is easy to see from Table 3 that the average demand on the plant capacity is the sum: 7 + 4 + 12 = 23 tons/wk, and its standard deviation is ~r1.33~ + 0.672 + 223”2 = 2.49; whence, with probability SO.999 (corres~nding to approx. f3a limits), it shall range in the interval: 23 + 3 x 2.490 C15.53,30.471 [The corresponding interval for 95% probability (corresponding to approx. f2a limits) is r[18.2,27.98] ton/wk.-j The plant’s available capacity is 15, 12, 32 tons/wk of products A, B, C, respectively. Since they occur in the ratio of 7:4:12, the plant’s available capacity is (7/23) x 15 + (4/23) x 12 + (12/23) x 32 = 23.3478 tonfwk. Again, on the average, the plant is fully loaded (actually, slightly overloaded 50% of the time. (2)

by 0.3478/23 = IS%),

but in reality it is overloaded

some

The problem of setup time

Another argument presented for a shop’s inability to provide accurate information about its capacity is that the presence of a product mix causes a non-measurable loss in productivity due to the need for frequent change-avers (note the emphasis on non-measurability). In particular, since, in any period, different items are produced to satisfy market demands, this necessitates “setting up” the process when production changes from item to item. This setup (or change-over) time reduces the available time for production, and consequently the available capacity. And since

Manufacturingcapacity and its measurement

623

product mix varies from period to period, it is not Wbie, so the argument goes, to determine the available capacity of the facility! It is true that setups consume valuable machine time that could have been utilized for production. That is why students of production planning and control are taught the principle of minimizing setup times. And I concur with the view that the problem of production planning to satisfy a specified demand and minimize setups and inventory build-up is a very difficult problem indeed. But there is a pitfall in the above argument that must be avoided; to wit, that excessive setup may be self-inflicted since a vicious cycle can easily develop due to poor production planning. In particular, if the quantity produced of a product in any period is not sticient to satisfy the demand, for one reason or another, a new order is placed on the plant to make up the deficiency. This necessitates a new setup, which reduces the available capac$y to produce other products. These latter will now fall short of satisfying their demand, which would necessitate the placement of additional orders on the plant to make up the deficiencies which, in turn, will necessitate additional setups, which further reduce the available capacity, and so on until the plant appears to be in the absurd position of always being set up for minuscule quantities of the different products! Incidentally, if one compares the actual expenditure in time in setups with the optimal setups for the given product mix demanded in each period determined, for instance, by any of the mathematical programming models presented in the literature, then one would gain a clear indication of the adequacy, or the lack of it, of the production planning function in the shop or plant. Adding over the periods should give the total time spent on setups throughout the year, and a measure of the overall efficacy of the planning function. I suspect that the reason for plants, or shops within a plant, not being able to give a definitive measure of the decrement in their available capacities due to setups is either due to the absence of correct records, or simple lack of knowledge of what to do with them if they were available. The reporting of setup information is thus seen to be important from two points of view: the

(a) It permits the evaluation of the production control function in the firm relative to a datum based on its demand pattern. (b) It permits the determination of the “opportunity cost” due to restrictions imposed on inventory, or on the satisfaction of the customer’s demand, if such restrictions are in fact present, which necessitate frequent change-avers in the production lines. (3) The problem of varying eficiency Some firms insist that product mixes that vary from week to week, or from month to month, have a third detrimental effect on their ability to report their productive capacities. In particular, not only do bottleneck operations vary from item to item, but also the efficiency of production varies according to the item produced. Thus the same facility will have different production rates for different items, and therefore it is not possible to report on its capacity! Granted that the efficiency of a machine changes with the item produced*, it should still be possible to determine the net productivity of the facility for each item. This, then, represents the correct capacity of the machine, and it is the figure used in the evaluation of the system’s capacity. For instance, the productivity cited in Table 1 of 20 tons of item A in shop 1 should represent the net productivity after taking efficiency into account of this particular shop producing this particular item. Thereafter, the calculations proceed as detailed earlier. (4) The problem of semi-jinished items (or suba.wemblies) A remarkable phenomenon occurs when the reported actual usage exceeds the available capacity! I say “remarkable” because I have insisted throughout that available capacity forms an upper bound on usage; then how is it possible that actual usage exceeds its bound? Close scrutiny reveals that such an eventuality is due to the insertion of semi-finished products at some point in the process so that the bottleneck operation(s) is circumvented. Production supervisors would gleefully cite this device as one more reason for their inab~~ty to report “correct” figures on capacities. Evidently, subcontracting part of the work is another device used to boost the actual production beyond available capacity, and thus make a mockery of the figures accumulated on available * Due to differentreasons, not the tcast si@ficant of which is varying labor e&icncy with diffkrentproducts.

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capacity. Of course, this implies that the shop floor personnel interpret available capacity as their productive ava~ability exclusive of any such devices that may be used to augment it. To see the import of these capacity augmenting devices, consider once more Table 1. Suppose that the firm can secure partially finished item A that requires only shops 3 and 4. Then the old bottleneck operation (2) has been circumvented, and the firm’s capacity of item A has now jumped to 21 tons/wk, an increase of 40% [ = (21- lS)/lS] over the old capacity! The fallacy lies in the procedure of data collection itself. There is no inherent problem in determining the correct capacity in the presence of semi-finished products or subassemblies. Of course, the data collection method should permit the specification of such semi-finished products, which are to be treated as new items with their own bottlenecks and their own share of the shop time. In this instance one must be careful lest the total capacity of the plant (or shop) be underestimated, because of the possibility of simultaneously performing production on two “items” which are not really different in terms of the end-product. To exemplify the calculations under these circumstances, consider once more item A in Table 1, and suppose that in fact a semi-finished product A was inserted at shop 3 in week 1. Call it item A’. Then the new bottleneck is shop 3 whose capacity ‘is 21 ton/wk. In a nutshell, the injection of semi-finished products at intermediate steps of production may have the impact of changing the bottleneck capacity; otherwise, the procedure for calculating the average capacity and its variance and confidence limits remains intact. If 30% of the firm’s time was still devoted to the production of item A in week 1, of which 10% was for item A’ and 20% was for item A, then the firm must have produced 5.1 tons (A 0.10 x 21 + 0.20 x 15), and not the mere 4.5 tons previously reported. The difference of 5.1 - 4.5 = 0.6 tons represents the difference between producing A “from scratch” and injecting A’ in the process at shop 3. With this change in data, the process of calculating the mean and variance yields: week 1 average capacity = 19.5 ton (instead of 18.9) average capacity = 21.025 tons/wk variance = 7.504, and standard deviation = 2.739 ton/wk. As can be seen, the increase in the q~ntity produced of A increased the average capacity, and decreased the variance. The problem of the use of semi-finished products and/or subassemblies to augment capacity serves as illustration for the need to exercise great care when designing the data collection form. If such capacity-augmenting devices are possible, and are in fact used, even on an ad-hoc basis, then the data collection system should include the possibility of reporting such ca~city-au~enting devices and distinguishing among inputs in view of the fact that outputs are indistinguishable. (5)

The problem

ofscrap/dropout

The production of a mix of products gives rise to a problem along a different dimension, namely, that of the proportion of scrap/dropout throughout the process and as defective end products. The difficulty is sometimes subtle, especially when it is concerned with the estimation of future performance. Suppose that the industry standards for the different products are known. Then it should be a simple matter indeed to secure an estimate of the expected dropout for each product and of each shop, as well as for the plant as a whole dependent on the proportion of time devoted to the production of each item. I have illustrated the calculations in the example below by the use of a standard scrap rate for product A in shop 1 of 3.5%. The same calculations may be repeated for each item produced and the expected loss in productivity due to scrap/dropout determined for the firm as a whole. The subtle difficulty stems from the fact that, by definition, available capacity is based on industry standards while actual scrap/dropout may be more, or less, than that standard. If the actual scrap/ dropout is more than the standard then there is no possibility of error or misinte~retation since the firm is performing worse than the average. But if the converse, i.e. if the firm performs better than the industry average, hence its scrap/dropout is less than the standard, then one may end up with compensating errors that hide poor performance in some other criterion, or, worse still, with actual usage that exceeds available capacity, which I have ruled out from the outset!

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To exemplify, consider once more product A. Ass&t? that the industry standard of scrap/dropout in shop 1 is 3.5%, and that the firm’s actual performance (in that shop) is 8.5%. The difference: (0.085- 0.035) x 20 = 1 ton/wk represents the degradation in productivity due to excessive scrap/dropout. This is unambiguous. But suppose that -the.shop’s dropout was only 2.5%. Then the firm would have actually produced (0.035 - 0.025) x 20 = 0.20 ton/wk more than was anticipated. If its performance along some other criterion (e.g. setup or loss of power) is below standards then the improvement in quality shall hide the poor performance elsewhere, and the firm may appear to he doing well overall! Such “compensating errors” should be carefully scrutinized, lest complacency sets in and the firm would lose on both counts: (i) it would be oblivious to the areas of poor performance and hence would not take any corrective steps, and (ii) it would be unaware of the areas of high performance and would not take advantage of its know-how in them!

This analysis would be incomplete without a discussion, albeit cursory, of some socio/cultural/ economic reasons for the inability of anafysts to accurately determine the capacity of the productive facility (any one of the four capacities). In particular, in an environment where the faking of figures to hide unpalatable truths from management is endemic, it seems odd to insist on correct data in this particular instance. A plant within a company, or a*-shop within a plant, may possess the correct data and may know how to use that data correctly to measure its capacity, but may not be willing to divulge these parameters to its management lest it be deprived, for example, of bonus payments when its planned performance is raised by the firm’s central planning office to match its true capabilities. (The congnoscenti will identify an age-old argument relative to “time standards”.) Evidently, the eradication of this cause requires leadership at the highest level of management, which is responsible for fostering an atmosphere of trust, understanding, and working for the common good. There is little more I can add to such a dictum, since the course of action varies with the particular circumstances of the firm.

5. THE DETERMINATION OF THE DIFFERENT CAPACITIES In what follows I shall illustrate how the data for the model can be modified according to the capacity being considered. Consider, again, item A in Table 1 and suppose that the facilities in shop 1 were on a one-shift basis; i.e. 8 hr/day for 5 days/wk, and had nominal capacity of 25 tons per wk. Then if we were interested in nominal capacity, the entry in the tabie would have been 25 tons instead of the given 20 tons. The same logic would apply to all other entries in the table. Now suppose that in shop 1, the following statistics are available from past records (still on the basis of 8 hr/day, 5 dayfwk): Maintenance information

Stipulated regular maintena~~ = 1.50 hr/wk, Electric power blackout

Average number of occurrences per week = 2.067, SD = 0.6. Average duration of blackout per occurrence = I S hr, SD = 0.5 hr. Setup information

Average optimal setup time allowance per week = 2.20 hr, SD = 0.25. Quality information

Average industry standard = 3.5%. Then one would proceed as follows to calculate the available capacity for item A in shop 1. Since nominal production per working hour is equal to 25/40 = 0.625 ton@, the maintenance requirement subtracts 1.50 x 0.625 = 0.9375 ton/wk. The average hours lost due te electricity blackout = 2.067 x 1.5 z 3.10 hr/wk, whence the average blackout translates into 3.10 x 0.625 = 1.9375 ton/wk, The average pr~uction loss due to setup is 2.20 x 0.625 = 1.375 ton/wk. Finally, an average industry dropout allowance of 3.5% translates into a loss of 0.035 x (25 - 0.9375 1.9375- 1.375)= 0.7263 ton/wk of rejects because of poor quality. Adding the four factors we get

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that, based on averages, available capacity = 25 - (0.9375 + 1.9375 + 0.7263 + 1.375) = 20.02 ton/wk which is approximately the figure of 20 ton/wk quoted in the calculations related to shop 1 for item A. Now to the calculation of actual capacity utilization. Suppose that the following information is available from past performance records: actual: average maintenance and repairs: 4.5 hr/wk average blackout duration: 5.25 hr/wk average setup time: 3.6 hr/wk average reject/week = 8.5%, SD = 1.8%. Then we have that the actual capacity utilization is given by, actual capacity = (1 - 0.085) x [25 - (4.5 + 5.25 + 3.6) x 0.6251 z 15.24 ton/wk. This is the figure that should be inserted in Table 1 (instead of 20 tons) if the table were constructed for the determination of actual capacity utilization. A similar correction should be made to other entries. The difference of 20.02 - 15.24 = 4.78 tons/wk between the available and the actual capacities in shop 1 for product A is identified as idle capacity. Its components are: (i) an increase in maintenance of (4.5 - 1.50) = 3.00 hr/wk, * 3.00 x 0.625 = 1.875 ton/wk. (ii) an increase in electric power blackout of (5.25 - 3.10) = 2.15 hr/wk, -2.15 x 0.625 z 1.344 ton/wk. (iii) an increase in setup time of (3.6 - 2.20) = 1.40 hr/wk, * 1.40 x 0.625 = 0.875 ton/wk. (iv) an increase in reject rate from the industry standard to the actual of 0.085 x [25 - (4.5 + 5.25 + 3.6) x 0.625]- 0.7263 = 0.6895. The sum: 1.875 + 1.344 + 0.875 + 0.6895 = 4.78 ton/wk, which is the discrepancy previously measured between the available capacity and actual capacity utilization. Of course, other reasons for the discrepancy between the three types of capacity can be added with ease. A final note is in order. A more sophisticated analysis may be undertaken that includes probability statements based on the variability of the various elements. For instance, instead of dealing with average electric blackouts, one may opt to determine available capacity on the basis of the 95% upper confidence limit. Indeed, with probability 0.95 the disruption due to blackouts shall not exceed 2.067 + 2 x 0.6 = 3.267 outages/wk, and its duration shall not exceed 1.5 + 2 x 0.5 = 2.5 hr/outage. Hence the weekly blackout, assuming indepencence between the frequency of occurrence of blackout and its duration, shall not exceed 3.267 x 2.5 = 8.1675 hr/wk, equivalent to 8.1675 x 0.625 z 5.1 ton/wk, with probability of at least 1 - 0.052 = 99.75%. The remainder of the calculations proceed with the new figures in a similar fashion to the above analysis. 6. CONCLUSIONS

Operations researchers working in the field of production planning and control talk glibly, and rather freely, of the “capacity of the production system” as if this is a well known entity on which agreement has been established, and whose measurement is “straightforward”. Nothing can be farther from reality: the concept of “capacity” in humans and machines is ill-defined and rather elusive, and its measurement is an art cloaked in mystery and practiced only by the high priests of accounting. This treatise is an attempt, and a modest one, first at formulating a coherent set of definitions of the various forms of capacity, and second at its measurement. I hope that this is the first salvo in a series of studies that establishes this important concept on a more scientific and rational basis. Research is needed on at last three important issues. First, the establishment of a taxonomy of

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what I shall call “applicable capacity concepts” in the various fields of production and logistics. Second, the construction and validation of models for the accurate and precise measurement of these capacities, taking stochastic behavior into account and incorporating the relevant concepts of manufacturing engineering relative to degradation of performance with use and age. Third, the implementation of such paradigms on select prototype enterprises that would demonstrate the subtleties and degree of applicability of the procedures to real life situations. REFERENCES 1. L. A. Johnson and D. C. Montgomery, Operations Research in Production Planning, Scheduling, and Inwntory Control. Wiley, New York (1974). 2. E. Manipaz. EsscnMs ojPr&fion and Operations Management. Prentice-Hall. Englewood Cliffs, N.J. (1984). 3. E. S. Bulia, Ma&n ProducGon/Operations Management. Wiley. New York (1983). 4. R. B. Chase and N. J. Aquilano, Producfion and Operations Management. Irwin, Homewood. Ill. (1985).