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Measuring and designing flexibility as a generalized service degree
European Journal of Operational Research 112 (1999) 98±106
Theory and Methodology
Measuring and designing ¯exibility as a generalized service degree Ch. Schneeweiss a
a,*
, H. Schneider
b
Universit at Mannheim, Lehrstuhl f ur Unternehmensplanung, insb. Operations Research, D-68131 Mannheim, Germany b Louisiana State University, Baton Rouge, LA 70803-6316, USA Received 1 October 1996; accepted 1 October 1997
Abstract The paper designs and measures ¯exibility as the availability of a dynamic system in an uncertain environment. After a brief discussion of the literature, ¯exibility is de®ned essentially as the service degree of a system's dynamic technology. As a further step, this measure is extended to incorporate a system's planning, forecasting, and implementation ability. To obtain a deeper insight into the complex nature of the measure the design of ¯exibility is discussed within the framework of hierarchical planning. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Flexibility; Inventory theory; Service degree; Hierarchical planning; Multi-criteria decision making
1. Introduction Flexibility can be characterized as the ability of a system to cope with unforeseen changes. This seemingly simple characterization, however, is rather dicult to be put into operational terms. Furthermore, similar to ideas such as ``human power'', ``personal freedom'' or ``quality of life'', ¯exibility has an intuitive meaning which cannot easily be cast into decision theoretic terms. Typically, ¯exibility is designed at a higher managerial level than it is actually used. Furthermore, in establishing a ¯exibility potential one is generally less informed than at the time when the change actually occurs. Thus ¯exibility can be
*
Corresponding author. Fax: +49-621 2 92 1270; e-mail: [email protected]
considered as a property of a hierarchical system consisting of a top-level being responsible for the design of a ¯exibility potential and a base-level which makes use of it. This base-level, of course, should be capable of using its ¯exibility potential in order to cope with dynamic and stochastic changes. Some of the numerous attempts that have been made to de®ne a quantitative measure of ¯exibility will brie¯y be reviewed in Section 2. All of these approaches have their own particular diculties. In many cases they are restricted to a small class of models and problems or they do not provide an ordinal level of scale and thus do not allow systems to be compared as to their degree of ¯exibility. The measure we de®ne is closely related to the concept of a service level as it is used in stock control. Such a measure seems to be adequate since a service level is measuring the system's
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 3 8 0 - 9
C. Schneeweiss, H. Schneider / European Journal of Operational Research 112 (1999) 98±106
ability to cope with unforeseen changes (i.e. ¯uctuations of demand). Furthermore, since general inventory problems are stochastic and dynamic they provide an ideal background in discussing ¯exibility. (For a recent example see Jordan and Graves, 1995.) A service degree-oriented ¯exibility measure has particularly three desirable properties: 1. The measure is at least of an ordinal scale and general enough to comprise a large number of dierent ¯exibility components. 2. As with stock control it accounts for the loss of goodwill and hence it is an additional indicator which cannot easily be replaced with a monetary criterion. 3. Management is provided with a measure which, particularly in the production and operations management area, has a long tradition. The paper is organized as follows. After a brief survey of the existing literature, Section 3 will introduce the main ¯exibility components which a general measure has to consider. The measure is then derived in Section 4. The problem of assigning a numerical value will be discussed in Section 5. Having introduced the de®nition, we proceed in Section 6 to incorporate more advanced properties of ¯exibility such as planning and leadership capabilities. These components will be analyzed within the framework of designing a ¯exibility potential which, as mentioned before, calls for taking into account principles of hierarchical planning. Hence, in Section 7, we apply some concepts of hierarchical planning to obtain a deeper insight into the nature of the proposed measure. Section 8 concludes the investigation by a discussion. 2. Concepts and measures of ¯exibility in the literature There is a vast amount of work addressing the subject of ¯exibility. In this review we brie¯y focus on the most common ¯exibility concepts and follow up by a summary of its measures. Since many quantitative measures being proposed in the literature are related to production and operations management problems, let us con®ne our review to
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these two areas. Two aspects can be distinguished. Flexibility is de®ned: 1. with respect to speci®c problems in production and operations management, 2. within the context of dierent managerial levels. (1) For the various types of problems speci®c ¯exibility measures have been proposed. (For comprehensive reviews see Sethi and Sethi, 1990 or Gupta and Goyal, 1989.) To give only a few examples, in operations one de®nes volume, product, machine, routing, and process ¯exibility. ``Volume ¯exibility'', e.g., is de®ned as the ability of a production system to adjust its amount of output to (substantial) ¯uctuations of demand. Similarly, ``machine ¯exibility'' de®nes the ability of a machine to be set up quickly and to handle product variety. (2) For the same manufacturing system ¯exibility at dierent management levels has been considered in the literature. For example, ¯exible manufacturing systems (FMS) possess an operational ¯exibility expressed, for instance, by their ability to process a large number of dierent items. Simultaneously, however, they possess a tactical ¯exibility as they can easily cope with newly designed product families (see e.g. Ettlie and PennerHahn, 1994). A problem typically assigned to the strategic level is the ¯exibility of the design of a ¯exibility potential, i.e. the ability to adjust the type of ¯exibility. This is in fact a ``meta-¯exibility'' and is a property of the design level. Although there are numerous types of ¯exibility, there is even a larger number of measuring devices. Mainly three dierent types of measures can be distinguished. One can measure ¯exibility: 1. by the number of a system's possible reactions, 2. by technical indicators, 3. by economical indicators. (1) Measuring ¯exibility by simply counting the number of possible reactions is common practice. For routing ¯exibility one simply counts the number of dierent routings a production system allows for. For measuring the ¯exibility of working time one counts the number of feasible shift plans. Generally, ¯exibility of a (design) decision is de®ned by the number of the (operational)
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decisions that can still be taken (see e.g. Benjafaar et al., 1995). Sometimes this property is also called robustness (see Rosenhead et al., 1972). (2) Technical measures try to relate ¯exibility to ``technical'' indicators. For mixed model lines, e.g., ¯exibility can be measured by mean throughput time; and for the ¯exibility of working time agreements one can take the range of permissible weekly working time ¯uctuations (Wild and Schneeweiss, 1993). (3) Economic indicators introduce a new aspect into the notion of ¯exibility. Here the costs of an adjustment are taken into account. The lower these costs, the more ¯exible a system is (see e.g. Webster and Tyberghen, 1980). Consequently, rather than talking of ¯exibility one sometimes uses the term ``cost elasticity'' (Son and Park, 1987). In summarizing, most of the concepts of ¯exibility are restricted merely to particular situations and consider only a limited number of components ¯exibility generally consists of. In addition, measures which do not take into account the dynamics of a system and the stochastics of its environment are obviously not suited to capture comprehensively the idea of ¯exibility. In deriving a comprehensive measure of ¯exibility these observations will be taken as a guideline. 3. Elementary components of ¯exibility Flexibility can be considered as the result of various components which contribute to a system's availability. Since availability can be measured by a service degree, it seems to be natural to measure ¯exibility by an appropriate generalized service degree. As more elementary components let us ®rst consider the (a) action volume and the (b) reactivity. The action volume comprises all (temporary) actions a system has available to change its state. Reactivity, on the other hand, describes the speed of this change. Both properties de®ne a set of (dynamic) strategies which will be called a dynamic technology H. It determines all states a system can reach within a certain period of time, Z Z H.
As mentioned above, many ¯exibility concepts simply take the action volume ± often only the number of its elements ± as a measuring device. Similarly one takes just the reactivity to measure ¯exibility, i.e. ¯exibility is simply measured by, e.g., the change over time or, as in stock control, by the lead time of replenishment orders. If the ¯exibility of the entire dynamic technology is measured, one often counts the number of reachable states (Rosenhead et al., 1972; Benjaafar et al., 1995) or one takes the dynamic technology H itself, that is the action space (Lasserre and Roubellat, 1985). Using this simple kind of measure one readily encounters diculties in comparing dierent ¯exibility potentials. As an example, the question of which technology is more ¯exible ± that with a high variety of possible actions and a long reactivity time or that with a small variety and a short time for reactivity ± cannot be easily answered. Obviously, the ¯exibility of H can only be measured at an ordinal scale if, e.g., for two technologies H1 and H2 one of the relations Z H1 Z H2 ; Z H1 Z H2 or Z H1 Z H2 holds. This, however, will only be the case for very simple systems. Consequently, particularly in view of a design task, one cannot simply take the volume of Z H, or H itself, as a ¯exibility measure. In fact, in measuring the ¯exibility of H, it is necessary to evaluate its eectiveness in reacting to a change. Such an evaluation, however, has signi®cant consequences, i.e., ¯exibility is no longer a simple technological property of a system but depends on the management's attitude towards the performance of a technology. In measuring the performance let us introduce a discrepancy D which measures the ``physical goodness of ®t''. D denotes the distance between a desired state zd and the ``nearest'' state z H 2 Z H being reachable with H: D D zd ; z H. As an example, take ``lost sales'' or ``missing the production due date by a certain amount of time'', or, more generally the gap between needed and providable capacity. A mere discrepancy will usually not be sucient to measure the quality of a system's adjustment. In fact, the management has to evaluate when a
C. Schneeweiss, H. Schneider / European Journal of Operational Research 112 (1999) 98±106
discrepancy actually might result in a loss. Let us therefore introduce as a second major component of a ¯exibility measure a loss measure L L D. As an example consider a missed due date. In many situations a small deviation might not be considered as a real problem whereas larger deviations might be taken more seriously. Hence, let L D be a function of D that measures a discrepancy which implies a loss of goodwill, i.e. the fear of losing in the long run a market share. In guaranteeing a system's availability one obviously has to take into account as a further component of a ¯exibility measure the uncertainty of its environment which will be denoted as the system's information status I. Consequently, ¯exibility turns out to be not only a property of the considered system but of its environment as well (see also Malek and Wolf, 1991; Rosenhead et al., 1972; Jordan and Graves, 1995).
and (under H) providable resources. For inventory problems, e.g., L can be identi®ed with non-satis®ed demand. In such situations a normalisation of F turns out to be straightforward. A system is de®ned to have the lowest ¯exibility F ÿ if it is unable to react at all, thus demand for resources is entirely accumulated. Hence, one may normalize F by F : 2 Fÿ Obviously, the (normalized) ¯exibility is zero for not reacting and it takes on its highest value 1 if no mean loss remains, thus F n 0 for F F ÿ and F n 1 for F 0. In stock control F ÿ represents mean total (nondelivered) demand for a speci®ed period of time. Since F describes expected non-satis®ed demand, the expression F ÿ ÿ F can be interpreted as mean satis®ed demand. Hence, the normalized ¯exibility measure F n results in F n : 1 ÿ
Fn 1 ÿ 4. A general measure of ¯exibility In deriving a ¯exibility measure we now combine the eect of the dierent components discussed before. To obtain a well-de®ned measure let us evaluate the conjoint eect of all components by the best adjustment they can achieve and de®ne a (non-normalized) measure of ¯exibility by F min EfL D a j Ig a2H
1
with I denoting the system's information status and a 2 H being a dynamic strategy. F aggregates a constellation fH; I; Lg in just one value. Thus it becomes meaningful to compare dierent ¯exibility potentials as to their reaction with respect to their environment. F describes a mean loss that, in applying the dynamic technology H, cannot be avoided. That is, in view of possible disturbances, H is only capable of transferring the system (in a speci®ed period of time) into a state z H which is usually not the desired state zd at which no loss of goodwill is assumed to occur. Particularly in the area of production and operations management in many situations L describes the de®cit between desired
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F F ÿ ÿ F Efsatisfied demandg ; ÿ F Fÿ Eftotal demandg
which is exactly the de®nition of the well-known b-service degree in inventory theory (see e.g. Schneider, 1981). 5. Numerical speci®cation of ¯exibility Although the measure F (or F n ) looks straightforward, for a design purpose one is still left with the problem of assigning numerical values to it. Indeed, designing ¯exibility one has to consider all the indicators which evaluate a system. Besides F these indicators comprise costs K and all remaining quantitative and/or non-quantitative criteria (denoted by C in Eq. (3)) which might be of relevance. In addition to the costs for establishing and maintaining a ¯exibility potential, K describes the expected operational cost incurred by reducing the expected loss. Hence, for design purposes, the management has to solve the multicriteria decision problem, fF ; K; Cg ! opt:
3
Strictly speaking, in the long run, an economic system has to follow some monetary goal. The
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reason why the management tries to meet some intermediate goals, like ¯exibility, is caused by the complexity of the problem it has to solve. It is important to ®rst summarize the aspects that might in¯uence the long term goals and then to put this quantity into a ``wider'' perspective. As a medium term measure, F has to evaluate the loss of goodwill which, in the long run, might have an eect on the viability of the system. To better appreciate the complex nature of the multi-criteria decision problem (3), let us again consider the various evaluation steps that had to be taken: 1. Determining a discrepancy D, which might be caused by some change. This discrepancy measures the system's deviation from some desired state zd . Being in zd the system is assumed to suer no loss of goodwill. (For a stock control problem zd might be taken to be zero inventory and D describes unsatis®ed demand.) 2. Determining a loss measure L(D), which evaluates the loss of goodwill a non-removed discrepancy might cause. (For an inventory problem only a long lasting and substantial unsatis®ed demand, e.g., might be considered to cause a loss of goodwill.) 3. Assigning a numerical value to F (or F n ) in solving the multi-criterion evaluation (3). The value has to be assigned, in view of the incurred costs K and other important criteria C, resulting in an acceptable loss of goodwill. (For an inventory problem a service level is speci®ed in view of the expected behavior of the market and the costs necessary to maintain that level.) In principle, one encounters two important steps of evaluation: 1. the determination of F (as a function of H in view of I), and 2. the assignment of a numerical value to F. Accordingly, one has to cope with two major uncertainties: · the medium (or short) term occurrence of a change and · the possible consequences one has to face if the system does not adjust to that change. As to the occurrence of a change it will often be possible to assign at least subjective probabil-
ities. Far more dicult is the determination of probabilities for the possible consequences one has to face. These consequences are often of a long term character. Not only are the probabilities unknown, but also the events for which probabilities have to be assigned are often not known. Hence the multi-criteria decision problem (3) is of a highly subjective character. This dicult situation, however, is not new for management. For instance, a debt±asset ratio or a risk measure is of the same nature. It is not that dicult to measure these quantities, but it is dicult to evaluate their signi®cance for a system's viability. An ``intermediate measure'' like ¯exibility, however, gives the management a certain reference point. Like a debt±asset ratio or a service degree it can be objectively measured and its numerical value can be used as a benchmark. Since it combines dierent ¯exibility components it can more easily be dealt with than Z H. Indeed, for H1 and H2 one always has one of the three relations F H1 > F H2 , F H1 < F H2 or F H1 F H2 . No such statement can generally be found for H or Z H. An ordinal relationship, however, is important, otherwise one would not even be able to derive ecient values for the multi-criteria decision problem (3). Consequently, one would not be able to evaluate the usefulness of ¯exibility. 6. Planning and implementation ability as further components of ¯exibility Up to now in determining the ¯exibility of a system we have simply relied on its dynamic technology H as it has been de®ned in Section 3. Since such a technology, however, does only describe the set of feasible strategies it is obvious that one would only take into consideration a ``hardware-oriented'' view of a system's entire ¯exibility. In fact, most of the measures derived in the literature (see Sethi and Sethi, 1990 or Gupta and Goyal, 1989) are restricted to this more technical aspect. This is, as we have shown, too narrow a point of view. We therefore introduced a loss measure L D and considered the information status. Planning, communication, and implemen-
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tation aspects, however, are further components that need to be taken into account. Planning and communication ability can be considered as the system's ability to fully exploit its technology. Since the technology typically is dynamic and the environment is stochastic, optimal planning can only be performed for simple settings. Approximating heuristics (i.e. ecient software) will have to be used and, obviously, the better the approximation, i.e. the better the planning ability, the lower will be F and hence (see Eqs. (1) and (2)) the higher will be the (normalized) ¯exibility. Clearly, the same holds true for the forecasting methods to be applied, i.e., for the system's ability to correctly perceive the probability of a change; or, to put it more generally, the better the communication within a system the more ¯exible it will be. Implementability represents another important component of ¯exibility, since a plan is only as good as its implementation. Often a decision a is not implemented as it was planned to be. Instead one has to accept some deviation d. One may therefore operationalize implementability as the additional stochastics induced by a non-exact implementation. Clearly, leadership activities could reduce this additional uncertainty. (See, e.g. Wild and Schneeweiss, 1993). Hence, ¯exibility does not only depend on the system's capability to plan and to forecast but also on its leadership skill. To obtain a better understanding of these three properties and, in general, to gain a deeper insight into the nature of ¯exibility and the proposed measure let us ®nally discuss the problem of how to design ¯exibility.
7.1. Some basic concepts of hierarchical planning Consider a hierarchical system as depicted in Fig. 1. This system consists of a ``top-level'' described by a top-decision model with preference structure (criterion) C T and decision ®eld (action space) AT , and a ``base-level'' with C B and AB , respectively. Generally, the top-level will only make a decision after receiving information from the base-level. Therefore it anticipates the baselevel in estimating C B and AB by C^B and A^B . Assuming the base-level to behave rationally, the toplevel applies a (tentative) action aT 2 AT to the base-level and calculates its anticipated optimal 0 reaction a^B aT . Let us call this anticipated reaction as a function of aT an anticipation function 0 AF aT : a^B aT . Considering AF aT within the top-criterion C T , a best action aT can be calculated resulting in an instruction IN IN aT which is communicated to the baselevel (see Fig. 1). In many situations one simply has IN aT .
7. The design of ¯exibility Both, the design itself and the action taken within a given design, can be viewed as two hierarchically subordinated activities. Hence before discussing the design of ¯exibility let us ®rst review some concepts of hierarchical planning (see Schneeweiss, 1995). In the second step we shall then relate their signi®cance to the proposed ¯exibility measure.
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Fig. 1. Hierarchical interaction.
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More formally, one has the following relationships: (1) Top-level: In view of the design problem we have in mind, let us take as a top-criterion a vector of two criteria resulting in the following optimization problem: n aT arg opt EfC TT aT j ItT0 g; aT 2AT o EfC TB AF aT j ItT0 g 4a with ItT0 being the information available at the design (top-)level at time t0 when the design decision has to be made and arg f x x. The ``private'' criterion C TT comprises all components of C T that do not explicitly depend on the base-level. This dependence on the base-level is evaluated by the ``top-down'' criterion C TB C TB AF with AF aT being determined by dIN EfC^B ^ aB j I^B g: AF aT a^B aT arg opt 0
a^B 2 A^BIN
4b (2) Base-level: The base-level being in¯uenced by the top-level is now left to solve the following decision problem in time t1 > t0 :
aB arg optIN EfC B aB j ItB1 g aB 2ABIN
4c
with ABIN , ItB1 and optIN being the ``real quantities'' the base-level is working with. Note that the top-decision is taken subject to its information status ItT0 . For the anticipated base-decision the information status I B generally has to be estimated, i.e. I^B is the information status of the base-level from the view point of the top-decision maker. In addition, like A^B the optimization operator (see Eq. (4b)) can be in¯uenced by the instruction IN IN aT of the top-level. Hence the design level not only endows the operational system with a dynamic technology H but it also provides ``handling intelligence'', thus taking into account all aspects described above by the components (4) and (5). The solution of the multi-criterion decision problem (4a) is then communicated to the baselevel (see Fig. 1).
7.2. The design of ¯exibility as a hierarchical planning problem Putting the design of ¯exibility into the framework of hierarchical planning is now straight forward. Clearly the top-model has to be identi®ed by the design level and the base-level turns out to be the system that has to be endowed with a ¯exibility potential. For the discussion to follow let us ®rst consider the case that the design level and the operational level form a team. In particular, let us assume that all constituents of the operational level which are not determined by the design decision are known. Hence one can drop all ``hats'' in Eq. (4b) except that of I^B which is the information status of the base-level estimated by the design level. Note that there is uncertainty about I^B as described by the top-level's information status ItT0 . The information state I^B will usually be a probability distribution. In some situations, however, I^B might describe a deterministic knowledge of the change. Taking C TB C B , Eqs. (4a) and (4b) may be written in the compact form 8 > < aT arg opt EfC TT aT j ItT0 g; aT 2AT > :
E
8 > < > :
min EfC B aB jI^B gjItT0
IN aB 2ABIN
99 > => = > ;> ;
:
5a Turning to the notations of the preceeding sections one readily recognizes the following correspondence: (a) Loss measure LC ^ B. (b) Dynamic technology HA ^ BIN . (c) Planning, forecasting, communication, and leadership ability: `` minIN ''. (d) Non-normalized ¯exibility measure F min EfC B aB jI^B g: IN aB 2ABIN
Note that in suitably de®ning C TB one could easily replace F with F n .
C. Schneeweiss, H. Schneider / European Journal of Operational Research 112 (1999) 98±106
(e) Expected costs of designing and operating the ¯exibility potential: K EfC TT jItT0 g. (f) Range of design decisions AT . With these explanations, except for the criteria C and the expectation EfF jItT0 g, Eq. (5a) turns out to be identical with the multi-criteria decision problem (3). Obviously, C could have also been formulated in the hierarchical framework as a further component of C TT or C TB or of both. What is exhibited by the hierarchical formulation, however, is the information situation. Clearly, the non-speci®ed I in Eq. (1) is now the information I^B one has about the operational level at the design stage. Often there will be no ambiguity about the estimate I^B . In that case EfF jItT0 g reduces to F. What is left to be done at the operational level is to minimize expected loss employing the ¯exibility potential and the speci®c operational information status:
aB arg min
IN aB 2ABIN
EfC B aB j ItB1 g:
5b
This ¯exibility potential represents the result of the design eort and is given by the (above) properties (1), (4), and (5) condensed in the pair fABIN ; minIN g. The potential is measured by the pre-assigned loss measure C B L in view of an uncertain environment of which the design level assumes that it is perceived by the operational level through I^B . Remark. In case of an antagonistic situation ideas of agency theory (see Spremann, 1987) could be applied using incentives to avoid cheating. In particular, C TB would now induce a reevaluation showing, e.g., that the loss measure being employed by the operational level is not in accordance with the top-level. In fact, the operational level might judge a loss of goodwill in a dierent way than the top-level would prefer it to be done. Furthermore it might be necessary to provide incentives in order to motivate the base-level to correctly perceive possible changes. 8. Concluding discussion Measuring ¯exibility as a generalized service level may indeed be the starting point for a com-
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prehensive theory of ¯exibility. We characterized ¯exibility as the ability of a potential to cope with unforeseen changes. This potential consists of various components. Besides the system's dynamic technology we considered its planning, forecasting, communication, and implementation abilities. Furthermore, the capability of a ¯exibility potential was shown to depend on the employed loss function and the uncertainty the system is exposed to. Choosing dierent kinds of loss functions may indeed result in various kinds of ¯exibility measures. In particular, L D could be vector valued such that the optimization in Eq. (1) would have to be replaced with a multi-criteria optimization. If, in addition, in the design stage it is not yet clear which type of discrepancy actually may cause a loss of goodwill, one would need to endow the ¯exibility potential with the capability of easily switching from one discrepancy to the other. Clearly, ¯exibility was introduced as an ex ante property, i.e. as a property to be planned. Measuring ¯exibility empirically would only result in an ex post measure which, of course, will generally be of little signi®cance. For complex situations it may not be possible to determine F and one would have to encumber problem (3) directly, with the dicult task of evaluating a ¯exibility potential without the preliminary step of Eq. (1). Hence, de®ning a ¯exibility measure as an intermediate step might turn out to be obsolete. This will particularly be the case for highly complex strategic ¯exibility potentials. It should be clear that the loss function measures the loss of goodwill which for certain unforeseen scenarios might, in the long-run, result in a ®nancial loss. Hence, within the multi-criteria problem (3) management has to balance short term costs for establishing and maintaining ¯exibility with the long term costs that possibly have to be incurred because of a lack of sucient ¯exibility. For a comprehensive theory of ¯exibility it would particularly be necessary to further explore the multi-criteria decision problem of Eq. (3). Since ¯exibility is conceived to be a reactive property, one is not proactively trying to in¯uence the origin of a change. Hence, for a general theory,
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besides ¯exibility one would have to consider in Eq. (3) further reactive and proactive properties of a system to guarantee its long-term viability. As an additional reactive property, one should take into account a system's nervousness (or stability) (Inderfurth, 1994; Sridharan et al., 1987) shedding some light on the negative consequences of being ¯exible. A proactive behavior is in¯uencing the sources of change and is thus reducing the necessity of a (¯exible) adjustment and should therefore be considered as well. References Benjaafar, S., Morin, T.L., Tavalage, J.J., 1995. The strategic value of ¯exibility in sequential decision making. Eur. J. Oper. Res. 82, 438±457. Ettlie, J.E., Penner-Hahn, J.D., 1994. Flexibility ratios and manufacturing strategy. Management Sci. 40 (11), 1444± 1454. Gupta, Y., Goyal, S., 1989. Flexibility of manufacturing systems: Concepts and measurements. Eur. J. Oper. Res., 43, 119±135. Inderfurth, K., 1994. Nervousness in inventory control: Analytical results. Oper. Res. Spektrum 16, 113±123. Jordan, W.C., Graves, S.C., 1995. Principles of the bene®ts of manufacturing process ¯exibility. Management Sci. 41 (4), 577±594.
Lasserre, J.B., Roubellat, F., 1985. Measuring decision ¯exibility in production planning. IEEE Trans. Automat. Control AC 30 (5), 447±452. Malek, L.A., Wolf, C., 1991. Evaluating ¯exibility of alternative FMS designs ± A comparative measure. Int. J. Prod. Economics 23, 3±10. Rosenhead, J., Elton, M., Gupta, S.K., 1972. Robustness and optimality as criteria for strategic decisions. Oper. Res. Quart. 23 (4), 413±431. Schneeweiss, Ch., 1995. Hierarchical structures in organisations: A conceptual framework. Eur. J. Oper. Res. 86, 4±31. Schneider, H., 1981. The eects of service levels on order-point or order-levels in inventory control models. Int. J. Prod. Res. 19, 615±631. Sethi, A.K., Sethi, S.P., 1990. Flexibility in manufacturing: A survey. Int. J. Flexible Manufacturing Systems, 2(4), 289± 328. Son, Y., Park, C., 1987. Economic measure of productivity, quality, and ¯exibility in advanced manufacturing systems. J. Manufacturing Systems 3, 193±206. Spremann, K., 1987. Agent and principal. In: G. Bamberg, K. Spremann (Eds.), Agency Theory, Information and Incentives. Springer, New York, pp. 3±37. Sridharan, V., Berry, W.L., Udayabhanu, V., 1987. Freezing the master production schedule under rolling planning horizons. Management Sci. 33 (9), 1137±1149. Webster, D.B., Tyberghen, M.B., 1980. Measuring ¯exiblity of job-shop layouts. Int. J. Prod. Res. 18 (1), 21±29. Wild, B., Schneeweiss, Ch., 1993. Manpower capacity planning ± A hierarchical approach. Int. J. Prod. Economics 30/31, 95±106.
Theory and Methodology
Measuring and designing ¯exibility as a generalized service degree Ch. Schneeweiss a
a,*
, H. Schneider
b
Universit at Mannheim, Lehrstuhl f ur Unternehmensplanung, insb. Operations Research, D-68131 Mannheim, Germany b Louisiana State University, Baton Rouge, LA 70803-6316, USA Received 1 October 1996; accepted 1 October 1997
Abstract The paper designs and measures ¯exibility as the availability of a dynamic system in an uncertain environment. After a brief discussion of the literature, ¯exibility is de®ned essentially as the service degree of a system's dynamic technology. As a further step, this measure is extended to incorporate a system's planning, forecasting, and implementation ability. To obtain a deeper insight into the complex nature of the measure the design of ¯exibility is discussed within the framework of hierarchical planning. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Flexibility; Inventory theory; Service degree; Hierarchical planning; Multi-criteria decision making
1. Introduction Flexibility can be characterized as the ability of a system to cope with unforeseen changes. This seemingly simple characterization, however, is rather dicult to be put into operational terms. Furthermore, similar to ideas such as ``human power'', ``personal freedom'' or ``quality of life'', ¯exibility has an intuitive meaning which cannot easily be cast into decision theoretic terms. Typically, ¯exibility is designed at a higher managerial level than it is actually used. Furthermore, in establishing a ¯exibility potential one is generally less informed than at the time when the change actually occurs. Thus ¯exibility can be
*
Corresponding author. Fax: +49-621 2 92 1270; e-mail: [email protected]
considered as a property of a hierarchical system consisting of a top-level being responsible for the design of a ¯exibility potential and a base-level which makes use of it. This base-level, of course, should be capable of using its ¯exibility potential in order to cope with dynamic and stochastic changes. Some of the numerous attempts that have been made to de®ne a quantitative measure of ¯exibility will brie¯y be reviewed in Section 2. All of these approaches have their own particular diculties. In many cases they are restricted to a small class of models and problems or they do not provide an ordinal level of scale and thus do not allow systems to be compared as to their degree of ¯exibility. The measure we de®ne is closely related to the concept of a service level as it is used in stock control. Such a measure seems to be adequate since a service level is measuring the system's
0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 3 8 0 - 9
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ability to cope with unforeseen changes (i.e. ¯uctuations of demand). Furthermore, since general inventory problems are stochastic and dynamic they provide an ideal background in discussing ¯exibility. (For a recent example see Jordan and Graves, 1995.) A service degree-oriented ¯exibility measure has particularly three desirable properties: 1. The measure is at least of an ordinal scale and general enough to comprise a large number of dierent ¯exibility components. 2. As with stock control it accounts for the loss of goodwill and hence it is an additional indicator which cannot easily be replaced with a monetary criterion. 3. Management is provided with a measure which, particularly in the production and operations management area, has a long tradition. The paper is organized as follows. After a brief survey of the existing literature, Section 3 will introduce the main ¯exibility components which a general measure has to consider. The measure is then derived in Section 4. The problem of assigning a numerical value will be discussed in Section 5. Having introduced the de®nition, we proceed in Section 6 to incorporate more advanced properties of ¯exibility such as planning and leadership capabilities. These components will be analyzed within the framework of designing a ¯exibility potential which, as mentioned before, calls for taking into account principles of hierarchical planning. Hence, in Section 7, we apply some concepts of hierarchical planning to obtain a deeper insight into the nature of the proposed measure. Section 8 concludes the investigation by a discussion. 2. Concepts and measures of ¯exibility in the literature There is a vast amount of work addressing the subject of ¯exibility. In this review we brie¯y focus on the most common ¯exibility concepts and follow up by a summary of its measures. Since many quantitative measures being proposed in the literature are related to production and operations management problems, let us con®ne our review to
99
these two areas. Two aspects can be distinguished. Flexibility is de®ned: 1. with respect to speci®c problems in production and operations management, 2. within the context of dierent managerial levels. (1) For the various types of problems speci®c ¯exibility measures have been proposed. (For comprehensive reviews see Sethi and Sethi, 1990 or Gupta and Goyal, 1989.) To give only a few examples, in operations one de®nes volume, product, machine, routing, and process ¯exibility. ``Volume ¯exibility'', e.g., is de®ned as the ability of a production system to adjust its amount of output to (substantial) ¯uctuations of demand. Similarly, ``machine ¯exibility'' de®nes the ability of a machine to be set up quickly and to handle product variety. (2) For the same manufacturing system ¯exibility at dierent management levels has been considered in the literature. For example, ¯exible manufacturing systems (FMS) possess an operational ¯exibility expressed, for instance, by their ability to process a large number of dierent items. Simultaneously, however, they possess a tactical ¯exibility as they can easily cope with newly designed product families (see e.g. Ettlie and PennerHahn, 1994). A problem typically assigned to the strategic level is the ¯exibility of the design of a ¯exibility potential, i.e. the ability to adjust the type of ¯exibility. This is in fact a ``meta-¯exibility'' and is a property of the design level. Although there are numerous types of ¯exibility, there is even a larger number of measuring devices. Mainly three dierent types of measures can be distinguished. One can measure ¯exibility: 1. by the number of a system's possible reactions, 2. by technical indicators, 3. by economical indicators. (1) Measuring ¯exibility by simply counting the number of possible reactions is common practice. For routing ¯exibility one simply counts the number of dierent routings a production system allows for. For measuring the ¯exibility of working time one counts the number of feasible shift plans. Generally, ¯exibility of a (design) decision is de®ned by the number of the (operational)
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decisions that can still be taken (see e.g. Benjafaar et al., 1995). Sometimes this property is also called robustness (see Rosenhead et al., 1972). (2) Technical measures try to relate ¯exibility to ``technical'' indicators. For mixed model lines, e.g., ¯exibility can be measured by mean throughput time; and for the ¯exibility of working time agreements one can take the range of permissible weekly working time ¯uctuations (Wild and Schneeweiss, 1993). (3) Economic indicators introduce a new aspect into the notion of ¯exibility. Here the costs of an adjustment are taken into account. The lower these costs, the more ¯exible a system is (see e.g. Webster and Tyberghen, 1980). Consequently, rather than talking of ¯exibility one sometimes uses the term ``cost elasticity'' (Son and Park, 1987). In summarizing, most of the concepts of ¯exibility are restricted merely to particular situations and consider only a limited number of components ¯exibility generally consists of. In addition, measures which do not take into account the dynamics of a system and the stochastics of its environment are obviously not suited to capture comprehensively the idea of ¯exibility. In deriving a comprehensive measure of ¯exibility these observations will be taken as a guideline. 3. Elementary components of ¯exibility Flexibility can be considered as the result of various components which contribute to a system's availability. Since availability can be measured by a service degree, it seems to be natural to measure ¯exibility by an appropriate generalized service degree. As more elementary components let us ®rst consider the (a) action volume and the (b) reactivity. The action volume comprises all (temporary) actions a system has available to change its state. Reactivity, on the other hand, describes the speed of this change. Both properties de®ne a set of (dynamic) strategies which will be called a dynamic technology H. It determines all states a system can reach within a certain period of time, Z Z H.
As mentioned above, many ¯exibility concepts simply take the action volume ± often only the number of its elements ± as a measuring device. Similarly one takes just the reactivity to measure ¯exibility, i.e. ¯exibility is simply measured by, e.g., the change over time or, as in stock control, by the lead time of replenishment orders. If the ¯exibility of the entire dynamic technology is measured, one often counts the number of reachable states (Rosenhead et al., 1972; Benjaafar et al., 1995) or one takes the dynamic technology H itself, that is the action space (Lasserre and Roubellat, 1985). Using this simple kind of measure one readily encounters diculties in comparing dierent ¯exibility potentials. As an example, the question of which technology is more ¯exible ± that with a high variety of possible actions and a long reactivity time or that with a small variety and a short time for reactivity ± cannot be easily answered. Obviously, the ¯exibility of H can only be measured at an ordinal scale if, e.g., for two technologies H1 and H2 one of the relations Z H1 Z H2 ; Z H1 Z H2 or Z H1 Z H2 holds. This, however, will only be the case for very simple systems. Consequently, particularly in view of a design task, one cannot simply take the volume of Z H, or H itself, as a ¯exibility measure. In fact, in measuring the ¯exibility of H, it is necessary to evaluate its eectiveness in reacting to a change. Such an evaluation, however, has signi®cant consequences, i.e., ¯exibility is no longer a simple technological property of a system but depends on the management's attitude towards the performance of a technology. In measuring the performance let us introduce a discrepancy D which measures the ``physical goodness of ®t''. D denotes the distance between a desired state zd and the ``nearest'' state z H 2 Z H being reachable with H: D D zd ; z H. As an example, take ``lost sales'' or ``missing the production due date by a certain amount of time'', or, more generally the gap between needed and providable capacity. A mere discrepancy will usually not be sucient to measure the quality of a system's adjustment. In fact, the management has to evaluate when a
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discrepancy actually might result in a loss. Let us therefore introduce as a second major component of a ¯exibility measure a loss measure L L D. As an example consider a missed due date. In many situations a small deviation might not be considered as a real problem whereas larger deviations might be taken more seriously. Hence, let L D be a function of D that measures a discrepancy which implies a loss of goodwill, i.e. the fear of losing in the long run a market share. In guaranteeing a system's availability one obviously has to take into account as a further component of a ¯exibility measure the uncertainty of its environment which will be denoted as the system's information status I. Consequently, ¯exibility turns out to be not only a property of the considered system but of its environment as well (see also Malek and Wolf, 1991; Rosenhead et al., 1972; Jordan and Graves, 1995).
and (under H) providable resources. For inventory problems, e.g., L can be identi®ed with non-satis®ed demand. In such situations a normalisation of F turns out to be straightforward. A system is de®ned to have the lowest ¯exibility F ÿ if it is unable to react at all, thus demand for resources is entirely accumulated. Hence, one may normalize F by F : 2 Fÿ Obviously, the (normalized) ¯exibility is zero for not reacting and it takes on its highest value 1 if no mean loss remains, thus F n 0 for F F ÿ and F n 1 for F 0. In stock control F ÿ represents mean total (nondelivered) demand for a speci®ed period of time. Since F describes expected non-satis®ed demand, the expression F ÿ ÿ F can be interpreted as mean satis®ed demand. Hence, the normalized ¯exibility measure F n results in F n : 1 ÿ
Fn 1 ÿ 4. A general measure of ¯exibility In deriving a ¯exibility measure we now combine the eect of the dierent components discussed before. To obtain a well-de®ned measure let us evaluate the conjoint eect of all components by the best adjustment they can achieve and de®ne a (non-normalized) measure of ¯exibility by F min EfL D a j Ig a2H
1
with I denoting the system's information status and a 2 H being a dynamic strategy. F aggregates a constellation fH; I; Lg in just one value. Thus it becomes meaningful to compare dierent ¯exibility potentials as to their reaction with respect to their environment. F describes a mean loss that, in applying the dynamic technology H, cannot be avoided. That is, in view of possible disturbances, H is only capable of transferring the system (in a speci®ed period of time) into a state z H which is usually not the desired state zd at which no loss of goodwill is assumed to occur. Particularly in the area of production and operations management in many situations L describes the de®cit between desired
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F F ÿ ÿ F Efsatisfied demandg ; ÿ F Fÿ Eftotal demandg
which is exactly the de®nition of the well-known b-service degree in inventory theory (see e.g. Schneider, 1981). 5. Numerical speci®cation of ¯exibility Although the measure F (or F n ) looks straightforward, for a design purpose one is still left with the problem of assigning numerical values to it. Indeed, designing ¯exibility one has to consider all the indicators which evaluate a system. Besides F these indicators comprise costs K and all remaining quantitative and/or non-quantitative criteria (denoted by C in Eq. (3)) which might be of relevance. In addition to the costs for establishing and maintaining a ¯exibility potential, K describes the expected operational cost incurred by reducing the expected loss. Hence, for design purposes, the management has to solve the multicriteria decision problem, fF ; K; Cg ! opt:
3
Strictly speaking, in the long run, an economic system has to follow some monetary goal. The
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reason why the management tries to meet some intermediate goals, like ¯exibility, is caused by the complexity of the problem it has to solve. It is important to ®rst summarize the aspects that might in¯uence the long term goals and then to put this quantity into a ``wider'' perspective. As a medium term measure, F has to evaluate the loss of goodwill which, in the long run, might have an eect on the viability of the system. To better appreciate the complex nature of the multi-criteria decision problem (3), let us again consider the various evaluation steps that had to be taken: 1. Determining a discrepancy D, which might be caused by some change. This discrepancy measures the system's deviation from some desired state zd . Being in zd the system is assumed to suer no loss of goodwill. (For a stock control problem zd might be taken to be zero inventory and D describes unsatis®ed demand.) 2. Determining a loss measure L(D), which evaluates the loss of goodwill a non-removed discrepancy might cause. (For an inventory problem only a long lasting and substantial unsatis®ed demand, e.g., might be considered to cause a loss of goodwill.) 3. Assigning a numerical value to F (or F n ) in solving the multi-criterion evaluation (3). The value has to be assigned, in view of the incurred costs K and other important criteria C, resulting in an acceptable loss of goodwill. (For an inventory problem a service level is speci®ed in view of the expected behavior of the market and the costs necessary to maintain that level.) In principle, one encounters two important steps of evaluation: 1. the determination of F (as a function of H in view of I), and 2. the assignment of a numerical value to F. Accordingly, one has to cope with two major uncertainties: · the medium (or short) term occurrence of a change and · the possible consequences one has to face if the system does not adjust to that change. As to the occurrence of a change it will often be possible to assign at least subjective probabil-
ities. Far more dicult is the determination of probabilities for the possible consequences one has to face. These consequences are often of a long term character. Not only are the probabilities unknown, but also the events for which probabilities have to be assigned are often not known. Hence the multi-criteria decision problem (3) is of a highly subjective character. This dicult situation, however, is not new for management. For instance, a debt±asset ratio or a risk measure is of the same nature. It is not that dicult to measure these quantities, but it is dicult to evaluate their signi®cance for a system's viability. An ``intermediate measure'' like ¯exibility, however, gives the management a certain reference point. Like a debt±asset ratio or a service degree it can be objectively measured and its numerical value can be used as a benchmark. Since it combines dierent ¯exibility components it can more easily be dealt with than Z H. Indeed, for H1 and H2 one always has one of the three relations F H1 > F H2 , F H1 < F H2 or F H1 F H2 . No such statement can generally be found for H or Z H. An ordinal relationship, however, is important, otherwise one would not even be able to derive ecient values for the multi-criteria decision problem (3). Consequently, one would not be able to evaluate the usefulness of ¯exibility. 6. Planning and implementation ability as further components of ¯exibility Up to now in determining the ¯exibility of a system we have simply relied on its dynamic technology H as it has been de®ned in Section 3. Since such a technology, however, does only describe the set of feasible strategies it is obvious that one would only take into consideration a ``hardware-oriented'' view of a system's entire ¯exibility. In fact, most of the measures derived in the literature (see Sethi and Sethi, 1990 or Gupta and Goyal, 1989) are restricted to this more technical aspect. This is, as we have shown, too narrow a point of view. We therefore introduced a loss measure L D and considered the information status. Planning, communication, and implemen-
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tation aspects, however, are further components that need to be taken into account. Planning and communication ability can be considered as the system's ability to fully exploit its technology. Since the technology typically is dynamic and the environment is stochastic, optimal planning can only be performed for simple settings. Approximating heuristics (i.e. ecient software) will have to be used and, obviously, the better the approximation, i.e. the better the planning ability, the lower will be F and hence (see Eqs. (1) and (2)) the higher will be the (normalized) ¯exibility. Clearly, the same holds true for the forecasting methods to be applied, i.e., for the system's ability to correctly perceive the probability of a change; or, to put it more generally, the better the communication within a system the more ¯exible it will be. Implementability represents another important component of ¯exibility, since a plan is only as good as its implementation. Often a decision a is not implemented as it was planned to be. Instead one has to accept some deviation d. One may therefore operationalize implementability as the additional stochastics induced by a non-exact implementation. Clearly, leadership activities could reduce this additional uncertainty. (See, e.g. Wild and Schneeweiss, 1993). Hence, ¯exibility does not only depend on the system's capability to plan and to forecast but also on its leadership skill. To obtain a better understanding of these three properties and, in general, to gain a deeper insight into the nature of ¯exibility and the proposed measure let us ®nally discuss the problem of how to design ¯exibility.
7.1. Some basic concepts of hierarchical planning Consider a hierarchical system as depicted in Fig. 1. This system consists of a ``top-level'' described by a top-decision model with preference structure (criterion) C T and decision ®eld (action space) AT , and a ``base-level'' with C B and AB , respectively. Generally, the top-level will only make a decision after receiving information from the base-level. Therefore it anticipates the baselevel in estimating C B and AB by C^B and A^B . Assuming the base-level to behave rationally, the toplevel applies a (tentative) action aT 2 AT to the base-level and calculates its anticipated optimal 0 reaction a^B aT . Let us call this anticipated reaction as a function of aT an anticipation function 0 AF aT : a^B aT . Considering AF aT within the top-criterion C T , a best action aT can be calculated resulting in an instruction IN IN aT which is communicated to the baselevel (see Fig. 1). In many situations one simply has IN aT .
7. The design of ¯exibility Both, the design itself and the action taken within a given design, can be viewed as two hierarchically subordinated activities. Hence before discussing the design of ¯exibility let us ®rst review some concepts of hierarchical planning (see Schneeweiss, 1995). In the second step we shall then relate their signi®cance to the proposed ¯exibility measure.
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Fig. 1. Hierarchical interaction.
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More formally, one has the following relationships: (1) Top-level: In view of the design problem we have in mind, let us take as a top-criterion a vector of two criteria resulting in the following optimization problem: n aT arg opt EfC TT aT j ItT0 g; aT 2AT o EfC TB AF aT j ItT0 g 4a with ItT0 being the information available at the design (top-)level at time t0 when the design decision has to be made and arg f x x. The ``private'' criterion C TT comprises all components of C T that do not explicitly depend on the base-level. This dependence on the base-level is evaluated by the ``top-down'' criterion C TB C TB AF with AF aT being determined by dIN EfC^B ^ aB j I^B g: AF aT a^B aT arg opt 0
a^B 2 A^BIN
4b (2) Base-level: The base-level being in¯uenced by the top-level is now left to solve the following decision problem in time t1 > t0 :
aB arg optIN EfC B aB j ItB1 g aB 2ABIN
4c
with ABIN , ItB1 and optIN being the ``real quantities'' the base-level is working with. Note that the top-decision is taken subject to its information status ItT0 . For the anticipated base-decision the information status I B generally has to be estimated, i.e. I^B is the information status of the base-level from the view point of the top-decision maker. In addition, like A^B the optimization operator (see Eq. (4b)) can be in¯uenced by the instruction IN IN aT of the top-level. Hence the design level not only endows the operational system with a dynamic technology H but it also provides ``handling intelligence'', thus taking into account all aspects described above by the components (4) and (5). The solution of the multi-criterion decision problem (4a) is then communicated to the baselevel (see Fig. 1).
7.2. The design of ¯exibility as a hierarchical planning problem Putting the design of ¯exibility into the framework of hierarchical planning is now straight forward. Clearly the top-model has to be identi®ed by the design level and the base-level turns out to be the system that has to be endowed with a ¯exibility potential. For the discussion to follow let us ®rst consider the case that the design level and the operational level form a team. In particular, let us assume that all constituents of the operational level which are not determined by the design decision are known. Hence one can drop all ``hats'' in Eq. (4b) except that of I^B which is the information status of the base-level estimated by the design level. Note that there is uncertainty about I^B as described by the top-level's information status ItT0 . The information state I^B will usually be a probability distribution. In some situations, however, I^B might describe a deterministic knowledge of the change. Taking C TB C B , Eqs. (4a) and (4b) may be written in the compact form 8 > < aT arg opt EfC TT aT j ItT0 g; aT 2AT > :
E
8 > < > :
min EfC B aB jI^B gjItT0
IN aB 2ABIN
99 > => = > ;> ;
:
5a Turning to the notations of the preceeding sections one readily recognizes the following correspondence: (a) Loss measure LC ^ B. (b) Dynamic technology HA ^ BIN . (c) Planning, forecasting, communication, and leadership ability: `` minIN ''. (d) Non-normalized ¯exibility measure F min EfC B aB jI^B g: IN aB 2ABIN
Note that in suitably de®ning C TB one could easily replace F with F n .
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(e) Expected costs of designing and operating the ¯exibility potential: K EfC TT jItT0 g. (f) Range of design decisions AT . With these explanations, except for the criteria C and the expectation EfF jItT0 g, Eq. (5a) turns out to be identical with the multi-criteria decision problem (3). Obviously, C could have also been formulated in the hierarchical framework as a further component of C TT or C TB or of both. What is exhibited by the hierarchical formulation, however, is the information situation. Clearly, the non-speci®ed I in Eq. (1) is now the information I^B one has about the operational level at the design stage. Often there will be no ambiguity about the estimate I^B . In that case EfF jItT0 g reduces to F. What is left to be done at the operational level is to minimize expected loss employing the ¯exibility potential and the speci®c operational information status:
aB arg min
IN aB 2ABIN
EfC B aB j ItB1 g:
5b
This ¯exibility potential represents the result of the design eort and is given by the (above) properties (1), (4), and (5) condensed in the pair fABIN ; minIN g. The potential is measured by the pre-assigned loss measure C B L in view of an uncertain environment of which the design level assumes that it is perceived by the operational level through I^B . Remark. In case of an antagonistic situation ideas of agency theory (see Spremann, 1987) could be applied using incentives to avoid cheating. In particular, C TB would now induce a reevaluation showing, e.g., that the loss measure being employed by the operational level is not in accordance with the top-level. In fact, the operational level might judge a loss of goodwill in a dierent way than the top-level would prefer it to be done. Furthermore it might be necessary to provide incentives in order to motivate the base-level to correctly perceive possible changes. 8. Concluding discussion Measuring ¯exibility as a generalized service level may indeed be the starting point for a com-
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prehensive theory of ¯exibility. We characterized ¯exibility as the ability of a potential to cope with unforeseen changes. This potential consists of various components. Besides the system's dynamic technology we considered its planning, forecasting, communication, and implementation abilities. Furthermore, the capability of a ¯exibility potential was shown to depend on the employed loss function and the uncertainty the system is exposed to. Choosing dierent kinds of loss functions may indeed result in various kinds of ¯exibility measures. In particular, L D could be vector valued such that the optimization in Eq. (1) would have to be replaced with a multi-criteria optimization. If, in addition, in the design stage it is not yet clear which type of discrepancy actually may cause a loss of goodwill, one would need to endow the ¯exibility potential with the capability of easily switching from one discrepancy to the other. Clearly, ¯exibility was introduced as an ex ante property, i.e. as a property to be planned. Measuring ¯exibility empirically would only result in an ex post measure which, of course, will generally be of little signi®cance. For complex situations it may not be possible to determine F and one would have to encumber problem (3) directly, with the dicult task of evaluating a ¯exibility potential without the preliminary step of Eq. (1). Hence, de®ning a ¯exibility measure as an intermediate step might turn out to be obsolete. This will particularly be the case for highly complex strategic ¯exibility potentials. It should be clear that the loss function measures the loss of goodwill which for certain unforeseen scenarios might, in the long-run, result in a ®nancial loss. Hence, within the multi-criteria problem (3) management has to balance short term costs for establishing and maintaining ¯exibility with the long term costs that possibly have to be incurred because of a lack of sucient ¯exibility. For a comprehensive theory of ¯exibility it would particularly be necessary to further explore the multi-criteria decision problem of Eq. (3). Since ¯exibility is conceived to be a reactive property, one is not proactively trying to in¯uence the origin of a change. Hence, for a general theory,
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besides ¯exibility one would have to consider in Eq. (3) further reactive and proactive properties of a system to guarantee its long-term viability. As an additional reactive property, one should take into account a system's nervousness (or stability) (Inderfurth, 1994; Sridharan et al., 1987) shedding some light on the negative consequences of being ¯exible. A proactive behavior is in¯uencing the sources of change and is thus reducing the necessity of a (¯exible) adjustment and should therefore be considered as well. References Benjaafar, S., Morin, T.L., Tavalage, J.J., 1995. The strategic value of ¯exibility in sequential decision making. Eur. J. Oper. Res. 82, 438±457. Ettlie, J.E., Penner-Hahn, J.D., 1994. Flexibility ratios and manufacturing strategy. Management Sci. 40 (11), 1444± 1454. Gupta, Y., Goyal, S., 1989. Flexibility of manufacturing systems: Concepts and measurements. Eur. J. Oper. Res., 43, 119±135. Inderfurth, K., 1994. Nervousness in inventory control: Analytical results. Oper. Res. Spektrum 16, 113±123. Jordan, W.C., Graves, S.C., 1995. Principles of the bene®ts of manufacturing process ¯exibility. Management Sci. 41 (4), 577±594.
Lasserre, J.B., Roubellat, F., 1985. Measuring decision ¯exibility in production planning. IEEE Trans. Automat. Control AC 30 (5), 447±452. Malek, L.A., Wolf, C., 1991. Evaluating ¯exibility of alternative FMS designs ± A comparative measure. Int. J. Prod. Economics 23, 3±10. Rosenhead, J., Elton, M., Gupta, S.K., 1972. Robustness and optimality as criteria for strategic decisions. Oper. Res. Quart. 23 (4), 413±431. Schneeweiss, Ch., 1995. Hierarchical structures in organisations: A conceptual framework. Eur. J. Oper. Res. 86, 4±31. Schneider, H., 1981. The eects of service levels on order-point or order-levels in inventory control models. Int. J. Prod. Res. 19, 615±631. Sethi, A.K., Sethi, S.P., 1990. Flexibility in manufacturing: A survey. Int. J. Flexible Manufacturing Systems, 2(4), 289± 328. Son, Y., Park, C., 1987. Economic measure of productivity, quality, and ¯exibility in advanced manufacturing systems. J. Manufacturing Systems 3, 193±206. Spremann, K., 1987. Agent and principal. In: G. Bamberg, K. Spremann (Eds.), Agency Theory, Information and Incentives. Springer, New York, pp. 3±37. Sridharan, V., Berry, W.L., Udayabhanu, V., 1987. Freezing the master production schedule under rolling planning horizons. Management Sci. 33 (9), 1137±1149. Webster, D.B., Tyberghen, M.B., 1980. Measuring ¯exiblity of job-shop layouts. Int. J. Prod. Res. 18 (1), 21±29. Wild, B., Schneeweiss, Ch., 1993. Manpower capacity planning ± A hierarchical approach. Int. J. Prod. Economics 30/31, 95±106.