International Journal qf Production Economics, 28 (1992) 309-319 Elsevier

309

The optimal mix of flexible and dedicated manufacturing capacities: Hedging against demand uncertainty

Diwakar Gupta”, Yigal Gerchakb and John A. Buzacott”

a McMaster University, Hamilton Ont. L8S 4M4, Canada b Universit), of Waterloo, Waterloo, Ont. N2L 3G1, Canada ’ York Universit!>, North York, Ont. M3J IP3, Canclda

(Received 28 April 1991; accepted in revised form 24 July 1992)

Abstract

A firm that faces uncertain demands for several product groups needs to decide how much of relatively inexpensive dedicated capacity and how much of more expensive flexible capacity to acquire. We model this situation as a two-stage decision problem:

investment and allocation. We pay particular attention to the dependence of acquisition policy on existing capacities, as most

firms facing the problem are not likely to be entirely new. We show that if initial capacities are lower than the levels that would be

optimal in absence of initial capacities, the investment decision is a simple “acquire-up-to” (optimal levels) for each capacity type.

However, if some initial capacity is “too high”, the optimal additions to others depend on its value in a non-linear. yet intuitive,

fashion. We also pose the issue of choosing the degree of flexibility, as the realized product mix will affect the performance of

partially flexible machines.

1. Introduction

A firm produces several items for which the demand is uncertain. Prior to the demands’ realizations, it has to decide how much product-dedicated capacities (machines) and flexible capacity to acquire. Units of flexible capacity are more expensive than those of dedicated capacity, but they can be used to produce various product mixes with negligible changeover times [l, 23. It seems intuitive that these economies of scope would tend to outweigh the higher cost when demands are highly random and not too positively correlated, but it is not a priori clear when exactly the firm should rely solely on flexible or dedicated capacity, and when it should use a mix of the two. This paper proposes and analyzes a model to aid in such decisions. Two previous studies with somewhat similar goals are first summarized, and our approach is then contrasted with theirs.

Fine and Freund [3,4] have considered a firm which is a monopolist in the product markets. Its revenue from selling a product is concave in the quantity sold, but the exact revenue function is revealed only after the investment decision. Prior to that decision, these revenue functions can assume one of a finite number of forms, with given probabilities which are the same for all products. Once the revenue functions are realized, the firm chooses feasible production quantities to maximize the total revenue net of linear variable production cost. After observing how this optimal operating profit depends on the capacities for each realization, the capacities which maximize the expected overall profit can be chosen.

Correspondence to: Prof. D. Gupta, McMaster University, 1280 Main Street West, Hamilton, Ont. L8S 4M4, Canada.

09255273/92/$05.00 c 1992 Elsevier Science Publishers B.V. All rights reserved.

310 D. Guptcc rt d/The oprind mir of’,@e.uihle und dedicuted munufacturing capacities

Chakravarty [S] has considered a situation where, after capacities are in place and demands are realized, capacity ‘rationing’, if required, takes place in a manner that will be consistent with constraints on allowable stockouts. Then, assuming independent and uniformly distributed de- mands, the mix of capacities minimizing investment cost and satisfying the service level constraints is found.

We study a single period investment and capacity allocation model in which the quantities of the products demanded have a general joint distribution. The revenue per unit sold (net of variable production costs) is product-dependent but quantity-independent. Thus, after demands are realized, the revenue-maximizing (“allocation”) problem is a linear program with a simple solution. As such, the allocation portion of our model is essentially a special case of Fine and Freund’s allocation (concave-maximization) problem [3,4], though here sales cannot exceed a realized demand. After observing how the optimal revenue depends on the capacities for every set of realized demands, we solve the investment problem, where the expected profit is maximized. In doing so, we allow the joint demand distribution to be arbitrary; in the general analysis we use a continuous joint distribution.

Another issue we explore, and that does not appear to have been addressed before, is the dependence of the optimal investment policy on previously available capacities. As most firms will not be starting from “scratch”, especially as far as dedicated capacity is concerned, this is an important issue. It may seem that, in order to determine the investment needed, the available capacity should be simply netted out of the optimal capacity (i.e., “acquire up-to” in each type of capacity). But, due to dependencies created by flexibility, one has to be more careful. It turns out that if the initial capacities are respectively lower than the capacities which are optimal in the absence of initial capacities, the optimal additional capacities to acquire CUY~ be found by simply netting out. But, if some initial capacity is “too high” the optimal additions to the other(s) depend on its value in a non-linear, yet intuitive, fashion.

The first Section has provided an introduction to our model. The second Section will describe the formulation of the model followed by an analysis of the properties of the model in the third Section. The latter also contains an illustrative and highly interpretable numerical example. We conclude the paper by posing a problem of determining the optimal degree of flexibility. The more flexible a machine the shorter the changeover times, but the more expensive a unit of capacity. For partially flexible machines both product mix and operating policies affect performance. After mentioning some relevant studies, we propose a potential formulation, and leave the resulting problem as a challenge for the future.

2. The model

Let i index the product group, i = 1,2. The unit revenues from selling the products are ri, and the joint distribution of the demands Di is Fi2, with marginal distributions Fi and corresponding densities fi 2 and f;.

Let j index the type of capacity, where j = 1,2 are the product-dedicated types, and j = 3 is the flexible type. It is assumed that the production rate of the flexible machines is independent of the product mix. The initial capacities, which cannot be reduced, are Ij, and the unit costs of additional capacities are cj. For the problem to be interesting it will be assumed that max(c,, c2) < c3 < cl + c2. The decision variables of the investment problem are the additional capacities Qi to acquire. When the capacity is installed and demands are realized, let Pi and Si be the quantities of product i produced on dedicated and flexible machines, respectively. Starred quantities are optimal values. Without loss of generality assume that rl > r2.

D. Guptu et al./The optimal mix of,fiexible and dedicated manufacturing capacities 311

Finding the optimal investment strategy belongs to the class of hierarchical planning problems, naturally formulated as two-stage stochastic programs [6]. Chronologically, the first stage de- cisions concern the choice of Qj’s. We shall call this the investment problem (IVP). The second stage decisions, concerned with determining the Pi’s and Si’s, are made after demands have been realized. We shall refer to the second stage revenue-maximizing decision problem as the allocation problem (ACP). Mathematically, the contingent ACP is solved first and will thus be presented first.

2.1. The allocation problem (ACP)

max f ri(Pi + Si)) i=l

subject to

di - (Pi + Si) ~ 0, i = 1, 2,

(Zi + Qi) - Pi ~ 0, i = 1, 2,

V,+Q,)-_(S~+SZ)>O, f’~,Pz>o, SI,SZ>O,

where the di’s are the realized demands.

(1)

Since dedicated machines have no alternative use, and since _rl > r2, the solution to this linear program is obvious: Pf = min(di, Zi + Qi), and, denoting di = max [0, (di - Ii - Qi)], ST = min(d1,Z3 + Q3), and S,* = max [0, min(d,,Z, + Q3 - d,)]. We note that if the variable production (operating) costs associated with the machines are different, the above solution would still be valid if the dedicated machines have lower costs. This would be appropriate as what we compare are technologies of the same vintage, for which materials rather than labour are a significant component of variable costs, and production economies of specialization (dedicated machines) are well documented. In addition, the increased complexity of the control mechanism will render the flexible machines more prone to breakdowns than modern dedicated machines, thereby resulting in higher operating costs. We shall thus not complicate the notation by including variable production costs, as the nature of solutions will not change.

2.2. The investment problem (IVP)

Let 4(d,,d,, Q1, Q2, Q3) be the maximum attainable revenue from solving the ACP. The investment problem can then be written as follows

ma': '(QI, Q2, QA)= - i CjQj+ E[4(D,, D2, Q1, Q2, Qs)], j=l

subject to: Qj > 0, Vj = 1, 2, 3. (2)

We note that if acquisition costs include a fixed component in addition to the variable one [.5], we can simply add them to the optimal costs associated with using all three types of capacity (if that is what the previously optimal solution called for) and compare them with expected profits (net of l?;;d{;;d variable costs) associated with the restricted combinations {l), {2}, (1, 2}, (1, 3}, (2, 3),

3. Analysis of the IVP

Substituting the value of 4 into the expected profit function, it becomes

Before analyzing the general optimality conditions and properties, let us consider a simple illustrative example.

Suppose that the joint demand distribution is discrete with the following realizations and associated probabilities:

Prob .(Dt = 0, D2 = 0) = p, Prob {Dl = 0, D, = u2) = q.

Prob {Dr = a,, II2 = 0) = t, Prob (D, = ul, II2 = aZJ = I - p - q - t,

where a,, tia > 0 are constants. The correlation coefficient, cr, of the above distribution function is:

cr = P-((P+9)tP+f)

J(p + q) tp + r) Cl - (P + 4)1 Cl - tP + 01. (4)

D. Gupta et al./The optimal mix cf,flexihle and dedicated mangfacturiny capacities 313

Thus, different values of p,q and r give rise to different values of the correlation coefficient. The extreme scenarios are: cr = 1 when q = r = 0, and cr = - 1 when p = 0 and q + t = 1. Assuming that initial capacities do not exist, the IVP objective function can be written as follows:

~(QI,Q~~Qs)= - 2 CjQj + rlC1 -(P + q)l *minCal,Q1 + Q31 + qr2*minCa2,Q2 + Q31 j= 1

+ [I1 - (P + 4 + t)l r2 * min [a2, Qz + max (0, QJ - max CO, ai - Qi I}]. (5)

Note that the objective function is piecewise linear, and therefore has corner point solution. Several numerical examples have been solved (see Ref. [7] for details), within the above simple structure, to study how demand correlation affects the choice of QT. In solving these examples, the capacities are restricted to have integer values only. Since varying p, q and t affects E(D,) and E(D,) as well, the values of a, and a2 have been manipulated to ensure that the mean demands do not change.

The three data sets used are given in Table 1. The results for selected values of the other parameters are given in Table 2.

Optimal choices of type 1 and type 2 capacities when flexible capacity is not available have also been computed. These are denoted by N1 and Nz, respectively. I$ denotes the expected value of the benefit resulting from the availability of flexible machines. It is the difference between the expected profit when the option to purchase flexible capacity is available and the expected profit when only dedicated capacities are available. From the numerical examples, the following qualitative inferen- ces could be made about the effect of correlation on the optimal mix of capacities.

Obviously, there is no incentive to purchase flexible capacity when cr = 1. On the other hand, when cr = - 1, there exists a high mix uncertainty because the realized demands would be either (a,, 0) or (0, uz). Higher mix uncertainty leads to preferential investment in the flexible capacity. In the example we studied, non-zero QT or Qz is obtained in association with cr = - 1 only because the average demands for the two products are not equal.

The preceding intuitive arguments might lead one to expect that investment in flexible capacity should increase monotonically when cr changes from + 1 to - 1. That however, is not the case because the choice of the Qy’s depends on the entire joint probability distribution of demands which is not adequately captured by the value of cr alone. For instance, each value of cr corresponds to several combinations of (p, q, t), not all of which necessarily result in the same investment strategy (see (p, q, r) triplets (0.0,0.4,0.4) and (0.2,0.4,0.4) in Table 2, corresponding to a cr of - 0.66). Conversely, if p + q + t = 1, i.e., non-zero demand realization occurs for at most one product, then, upon relaxing the integrality requirement on the Qj’s, we can conclude the following: either (1) no investment is made, or (2) Q$ = min(u,, u2) and either Qf = [ai - u2]+, Q: = 0 if E(D,) > E(D,), or QT = [a1 - a,]+ , QT = 0 if E(D,) < E(D,), regardless of the correlation coefficient where [.I’ returns its argument if the latter is non-negative and zero otherwise. We conjecture that this is

Table 1 The three used data sets

Data set

1

2 3

Cl c2 c 3 rl r2 E(D,) E(D2)

6 6 6.5 25 20 4 5 6 6 8.0 25 20 4 5 6 6 6.5 20 20 5 4

314 D. Gupttr et aLiThe optirnul vrlis of,fie.uibie rtnd dedicated mwz~facturing capacities

Table 2 Experimental results

I 0.00 0.50 0.50 - 1.00 8.00 0.00 0.40 0.40 - 0.66 6.67 0.20 0.40 0.40 - 0.66 10.00 0.25 0.25 0.25 0.00 8.00 0.53 0.20 0.20 0.00 14.81 0.50 0.00 0.00 1.00 8.00

0.00 0.50 0.50 - 1.00 8.00 0.00 0.40 0.40 - 0.66 6.61 0.20 0.40 0.40 - 0.66 10.00 0.25 0.25 0.25 0.00 8.00 0.53 0.20 0.20 0.00 14.81 0.50 0.00 0.00 1 .oo 8.00

0.00 0.50 0.50 - 1.00 10.00 0.00 0.40 0.40 - 0.66 8.33 0.20 0.40 0.40 - 0.66 12.50 0.25 0.25 0.25 0.00 10.00 0.53 0.20 0.20 0.00 18.52 0.50 0.00 0.00 1.00 10.00

10.00 0 2 8 8 10 44.00 8.33 0 2 7 I 8 11.17

12.50 0 2 10 10 12 55.00 10.00 0 2 8 8 10 4.00 18.52 0 0 15 14 1 51.04 10.00 8 10 0 8 10 O.Oq

10.00 0 2 8 8 10 32.00 8.33 0 2 I I 8 0.61

12.50 0 2 10 10 12 44.00 10.00 8 10 0 8 10 0.00 18.52 0 0 15 14 1 28.54 10.00 8 10 0 8 10 0.00

8.00 2 0 8 10 8 44.00 6.67 2 0 7 8 I 12.50

10.00 2 0 10 12 10 55.00 8.00 2 0 8 10 8 4.00

14.81 0 0 1.5 0 0 43.31 8.00 10 8 0 10 8 0.00

true in general for any choice of E(l),) and E(D,). It is also observed that ifp + q + t < 1, then only dedicated capacity is purchased for those scenarios where the entire demand is always satisfied. If flexible capacity is purchased and p + (I + t < 1, then not all demand is satisfied.

Finally, the numerical examples indicate that increases in the unit purchase price of the flexible capacity and reductions in the unit profits do not increase (and typically reduce) the number of units of flexible capacity chosen. Investment in flexible capacity is not recommended when V, is zero. This example clearly establishes the need to carry out the analysis advocated here as the investment strategies turn out to be somewhat counter-intuitive in many instances.

3.2. AnaIJtsis qj’general model: Optimul capacity acquislilcn

Returning to the general model, denote the derivatives of 2 with respect to Qj by Zj, denote fj + Qj by Kj and 1 - F by F. Then, the first order necessary conditions that the optimal Qj’s must satisfy are:

zj(Q,,Q~,Q3)+P-j=O~ QjPj=O, /-lj>O, (6)

where /ij’s are the Lagrange multipliers, and

KI+KJ ,3c Kz+K3 Kl+K3

Z1 = - cl + r,Fl(K1 -t- KS) + r2 s I

.fiz(x,y)dydx + ~2 s s

J-1 2 0~ Y) dx dy,

K1 KZ+K3 K2 Kl +Kz+Kj-y

(7)

D. Gupta et al./The optimal mix offlexible and dedicated man$acturing capacities 315

Kl+K3 K2+K3 Kz+KI KI +KJ

z, = - c2 + r$*(K2) - r-2 s s fi2ky)dydx + r2 I s

.I"I~(x,Y)~~~.Y, (8)

0 K2 K2 KI+Kz+K~~F

Kz+K2 KI+K~

23 = - c3 + rlF1(KI + K3) + r2Fl(KI + K3) - r2 J s

fi 2 (7 Y) dx dy

0 0

Kz+K, KI+K~

+r2 j j fi2(x,y)dxdy. (9)

K2 K,+Kz+Kj-y

In Eqs. (7) through (9), Fj = 1 - Fj. In Appendix A we show, that Z is jointly concave in (Q i , Q2, Q3) for all (IL, Z2, I,). The feasible region is convex. Thus, we are dealing with a convex programming problem, and the conditions in Eq. (6) are also sufficient. A local optimum will also be a global optimum. If the Qj’s obtained by equating the Zj’s to zero are all non-negative, then that is the optimal investment strategy. If however, one or more Qj’s are strictly negative, these will be set equal to zero. Since Z(Qi, Q2, Q3) is a concave function, the hill climbing directions at (0, 0,O) will point to the location of the optimum, that is, if Zj(O, 0,O) < 0, then the corresponding Qj will be set equal to zero. A necessary and sufficient condition for the existence of an interior maximum is Zj(O, 0,O) 3 0 Vj = 1,2,3 and that can be obtained in terms of model parameters. Thus, the optimal investment strategy would be obtained by setting any of the negative Qj’s to zero and solving for the others, until a non-negative solution is obtained. For example, if only Q1 is set to zero, we can solve for Qz and Q: from Zj(O, Q2, Q3) = 0, J’ = 2,3.

Denote the optimal solution of the zero initial capacities case by (Qy, Qz, Q!). Now suppose first that Zj d Qg, j = 1,2,3. Since in the first order conditions capacities appear only in the additive form Zj + Qj it follows immediately that in this case Qj* = QJ” - Zj. That is, if the initial capacities are lower than the “ideal” ones, the optimal investment pohcy is “acquire up-to” the ideal levels.

Suppose, however, that Zj < Qg does not hold for allj. Since in practice a firm is likely to already have type 1 and/or 2 capacities, but little, if any, of type 3 capacity, we shall assume that I, < Qt. For concreteness, suppose first that Z2 d Qi also, but Ii > Qy, so that at Q1 = Q2 = Q3 = 0, Z, < 0, while Zz 3 0 and Z3 3 0. Then, QT = 0, while QT and QT are solved from the following equations:

II +Ka Kz+K3 Kz+Kx II+K?

~IFAKI) + r2 s s f12(x, Y) dy dx - r2 s s fi2ky)dxdy = r2 - c2, (10) 0 K2 Kz II+Kz+K~-y

Kr+Kx II+& Kz+Kj 11+K3

h - r2FlU, + K3) + r2 s s

.f~2(x,.ddxdy - r2 s J

.f~2(x,y)dxdy = rl - c3.

0 0 KZ II+KI+K~-~

(11)

Clearly, KT and Kg depend on I,, and may be quite different from Qy and Qi, respectively. To probe the direction of’dependence on Ii, we take the total differentials of the above with respect to

I,. The derivatives are found to be

? ‘f;2(x.K2 + K,)d.v + J

2)

(13)

(14)

are used for notational convenience and have no intuitive interpretation. Similarly,

The above implies that K: (and hence Q:) decreases in I,, whereas r(;T (and hence QT j increases in II. The reason is that as 1 1 becomes larger, a greater proportion of type 1 demand is satisfied by the dedicated facility; this, in turn, implies that it becomes more economicai to meet type 2 demand using a dedicated facility. Eventually, as 1 1 becomes sufficiently large, Q$ will become zero.

Suppose now that the firm starts with “too much” of both type 1 and 2 capacities, that is the Ij’s are such that I, > Qy,i, > Q: and Z3 < Q:, so at Q, = Qz = Q3 = 0, Zr < O,Z, < 0 and Z, 3 0. Then QT = QT = 0, and Q: depends on f , and f, according to the equation:

Iz+K.3 II +R IL+K~ ll+K3

D. Gupta et a/./The optimal mix qf,fle.uihle and dedicated manufacturing capacities 317

Using implicit differentiation, it is not hard to show that Kz (and hence Qz) is decreasing in both I1 and Z2. This is quite intuitive, as increasing the amount of either dedicated capacities makes flexible capacity less needed.

4. On choosing degree of flexibility and related issues

The above analysis and its predecessors only recognize completely flexible capacity. However, FMSs often come with different physical characteristics resulting in different degrees of flexiblity. For our purposes, the most important determinants of the degrees of flexibility are the changeover times. However, operating decisions such as scheduling also affect the performance of the partially flexible FMS and thus have a direct impact on its effective production capability. Buzacott and Gupta [S] report that for a given set of physical characteristics, the frequency of changeovers and, hence, the available production capability depend on the realized product-mix. Furthermore, the sensitivity of the frequency of changeovers to product-mix is affected by the choice of scheduling policy. Therefore, to capture the joint effect of product-mix and operating decisions, we propose the concept of a “flexibility parameter” a. If Q3(a) units of a-degree flexible facility are available, then the effective capacity constraint is of the form:

(17)

where 0 < r < 1, and the function T has the following properties:

(i) T(SJ(S, + SZ); 1) = 1, f or all S1 and Sz. This is the case of perfect flexibility. (ii) For any Si, S2 > 0, T(S,/(S, + S,);c() is strictly decreasing (and hence effective capacity is

increasing) in X. (iii) For a fixed c(( < l), T(S,/(S, + S,);cz) is concave and symmetrical in S,/(S, + S,). In particu-

lar, it has a maximum at S1 = S2 which corresponds to the case of most frequent changeovers. The flexibility parameter would then become an additional decision variable in the investment problem where the unit acquisition cost C~(CI) of the cc-degree flexible facility will be non-decreasing in CL

Karmarkar and Kekre [9] consider an investment problem where the flexible facility has a non- zero changeover time, and determine the optimum frequency of changeovers that minimize cycle stock induced inventory carrying charges. However, they do not optimize over the possible choices of available changeover times. A composite flexibility parameter like r might permit, in principle, simultaneous optimization over physical and operational characteristics of the FMS, when evalu- ating investment decisions. Other relevant studies that feature partially flexible facilities are summarized in Ref. [lo].

A particular form of the function T has been proposed in Ref. 171. While this results in a tractable allocation problem, the corresponding investment problem turns out to be too complex for analytical treatment. Further efforts to find tractable ways to capture the degree of flexibility as part of the capacity mix problem are most worthwhile.

Flexibility-generated scope economies also include such benefits as the relative ease of adding capacity in a modular fashion (as compared to the “lumpiness” of dedicated systems) and increased competitiveness of the firm in product markets (and related strategic implications). These features affect capacity mix decisions and have been addressed in Refs. [ 1 l] and [ 121, respectively, for deterministic product demands. Such considerations will make flexible capacity more desirable in general.

5. Concluding remarks

While the model presented in this paper ignores many features of actual capacity acquisition decisions, it nevertheless provides useful insights into the circumstances where flexible capacity is

318 D. Gupta et ul./The optimal mix qf,flesihle and dedicated mangfhcturing capacities

preferable to dedicated capacity. Flexible capacity is not chosen because of an inherent advantage in producing a number of products simultaneously. Interestingly enough, the circumstances when it is most likely to be chosen, when the demands are highly negatively correlated, will generally result in the flexible capacity being used primarily for just one product. We prefer to acquire flexible capacity because when we make the decision on capacity we do not know which product will be demanded. But, once demands are realized we envisage that we will be able to dedicate the flexible capacity to a single product. Such flexibility will also enable us to deal with the dynamics of demand, in particular the substitution of new products for old ones over time. This stresses that the value of flexibility lies in its potential to deal with change and uncertainty yet still maintain focus once uncertainties are resolved.

Appendix A

Proof of concavity of Z(Qi, Qz, Q3): The following expressions are derived assuming linear capacity costs. This, however, is not necessary for concavity to hold. All our results are valid for general concave functions. Twice differentiating Z (using Leibniz’s rule) we obtain the following expressions:

K2 I

a2z -=

aQ: - (~1 - ~2).f;W, + K3) - r2

.i .hzW, + K,,y)dq’ - ~2

s hzW,,y)dy - u < 0, (A.11

0 Kz+K.1

x

a22 -= -_r

aQi 2

?’ .fizk K,)dx - r27’ .f;2(x,K2 + K,)dx - U < 0, (A.2)

Kx+Kj 0

Kl KI

a2z p=

aQS

- 0.1 - f”2) A&, + K3) - f-2 [.f;li& + Kj,_v)dq’ - ~2 j”h2W2 + KJ)dx - u < 0,

0 0

a2z a22

aQ1 aQ2 = aQ2 aQ1

= -u<o,

K2

a2z a2z

aQ1 aQ3 = aQ3 aQ,

= - (rl - r2).fi(K1 + K3) - r2 l.f;2(K1 + K3,Y)dJ

0

Kl

a22 a2z

aQ2aQ, = aQ3aQ2 = - r2 .i

52(.x, K2 + K,)dx - U < 0,

0

where

U = r2 s ,f;2(x,K1 + K2 + K3 - x)dx.

Kl

_ -u<o, (A.5)

(A.3)

(A.4)

64.6)

(A.7)

Recall that Kj = Ij + Qj, Vj. Using the above expessions it can be shown that the determinants of 2 x 2 matrices of second partial derivatives are positive and that the determinant of the 3 x 3 matrix

D. Guptu rt a/./The optimal mix of,flesihle and dedicated mant&acturing capacities 319

is negative. Together with the negativity of second partial derivatives of Z, the above observation implies that Z is concave [13]. Concavity of Z is entirely distribution free.

References

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[S] Chakravarty, A.K., 1989. Analysis of flexibility with rationing for a mix of manufacturing facilities, Int. J. Flexible Manuf. Sys., 2: 43- 62.

[6] Lenstra, J.K., Rinnooy Kan, A.H.G. and Stougie, L., 1984. A Framework for the probabilistic analysis of hierarchical planning systems, Annals Oper. Res., I: 2342.

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[9] Karmarkar, U. and Kekre, S., 1987. Manufacturing configuration, capacity and mix decisions considering operational costs, J. Manufac. Sys., 6: 315-324.

[lo] Fine, C.H., 1990. Developments in manufacturing technology and economic evaluation models, in: S.C. Graves, A. Rinnooy Kan and P. Zipkin, Eds., Logistics for Production and Inventory, North-Holland, Amsterdam.

[I I] Burstein, M.C., 1986. Finding the Economical Mix of Rigid and Flexible Automation for Manufacturing Systems, in: K.E. Stecke and R. Suri, Eds., Proc. of the Second ORSA/TIMS Special Interest Conf. on FMS:OR Models and Applications., pp. 43-54.

[I21 Gaimon, C., 1986. The strategic decision to acquire flexible technology, in: K.E. Stecke and R. Suri, Eds., Proc. of the Second ORSA/TIMS Special Interest Conf on FMS:OR Models and Applications, pp. 69980.

[13] Luenberger, D.G., 1984. Linear and Nonlinear Programming. Addision-Wesley, Reading, MA.