Rigid and flexible automation for manufacturing systems: Characterizing economically-reasonable investments

Annals of Operations Research 15(1988)21-35 21

RIGID AND FLEX/BLE AUTOMATION FOR MANUFACTURING SYSTEMS: CHARACTERIZING ECONOMICALLY-REASONABLE INVESTMENTS*

Michael C. BURSTEIN

Manufacturing Systems Economics Group, Center for Social and Economic Issues, Industrial Technology Institute, P.O. Box 1485, Ann Arbor, Michigan 48106, USA

Abstract

As flexible manufacturing technology has become available across a broad range of applications, an increasingly large number of firms have confronted decisions about the adoption of flexible automation versus transfer lines versus some combination of these technologies. While the work of previous authors has provided some guidance to such decision making, the modular character of flexible capacity and the indivisible character of a transfer line have not been the basis, previously, for the development of formal decision rules. This paper makes use of a mixed- integer mathematical programming model to generate formal decision rules which, in turn, become an instrument for analyzing several fundamental hypotheses about the introduction of flexible and/or dedicated automation. The firm is assumed to have the objective of maximizing its present value and must account for the inter- action of its markets with technological opportunities, present and anticipated.

Keywords

Flexible manufacturing systems, transfer lines, stationary demand, economic assessment, mathematical programming, Kuhn -T uc ke r conditions.

1. Introduction

As flexible or programmable automation has become available for manufacturing systems, the task of deciding among technological approaches ranging from hard or rigid automation through equipment possessing various degrees of

*The author gratefully acknowledges financial support of this work by the W.K. Kellogg Foundation, and the invaluable assistance of Karen Seever in the preparation of the manuscript. This paper is an extension of an earlier work from the Proceedings of the Second ORSA/TIMS Special Interest Conference on Flexible Manufacturing Systems, Ann Arbor, MI., August 1 2 - 1 5 , 1986 (Elsevier Science Publishers, B.V.).

© J.C. Baltzer AG, Scientific Publishing Company

22 M. C. Burstein, Rigid and flexible automation

programmability to manual or manually-assisted automatic equipment has become increasingly complex. The difficulty of decision making presented by the broad range of options is further exacerbated by expectations of continuing technological advance in manufacturing processes, and by the possible advantage from using several of these technological approaches simultaneously and/or sequentially.

An important step in addressing this complexity has been to identify the various types of manufacturing flexibility and to determine the significance of these for the firm's economic opportunities. Browne et al. [1] fulfilled the identification task in a comprehensive way, while other authors have focused on the opportunities for flexible automation. Economic assessment of flexibility in the sense of modular capacity and of new product accommodation was performed by Hutchinson [7,8]. His earlier paper highlighted the advantage of modularity over indivisibility (e.g. transfer lines) in capacity expansion when real interest rates are high. His later paper with Holland emphasized the value both of modularity and of convertibility from current products to new ones in the face of uncertain demand. Incorporating modularity and convertibility, Burstein and Talbi [4] presented a mixed-integer mathematical programming model for planning the evolution of a manufacturing system. The currently installed configuration of this system was posited to have only produce A capability, but demand for product A and for a new product B was foreseen from time zero throughout the planning interval.

With the objective of maximizing present value, this model captured the inter- action of technological options and demand functions along with the effects on current manufacturing investment of anticipating further technical advance in the near future. A characterization of optimal behavior for a somewhat simplified version of this model was provided by Burstein and Ezzekmi [2]. The simplified model was applied to three cases in terms of capabilities for newly available, best-practice technology: first, transfer line focus only on product A; second, transfer line focus only on product B; third, flexibility to manufacture any mix of products A and B. In all cases, the firm was assumed to anticipate the availability of a flexible technology in the near future which would have an investment and operating cost profile superior to any current manufacturing technology.

At about the same time as the Burstein and Ezzekmi paper, Gaimon [6] presented a continuous time model of investment by a price-determining monopolist in flexible manufacturing capacity. Her model related degree of attained flexibility to total quantity demanded of the firm's products; and it provided the monopolist with the opportunity to acquire flexible capacity with or without the retirement of original, non-flexible capacity. Flexible capacity also was stipulated to possess an operating cost advantage over the original, less flexible capacity.

As Hutchinson and Holland [8] demonstrated, manufacturing flexibility can be an economically appropriate response to demand uncertainty. More recently, this relationship between flexibility and uncertainty was explored in two papers. One of these, by Fine and Freund [5], characterized optimal investment behavior for a two-

M.C. Burstein, Rigid and flexible automathgn 23

product firm which was uncertain about the future demand functions for its products. Current options included dedicated product A capacity, dedicated product B capacity, and flexible capacity to handle both products. In their two-period model, the authors assumed that uncertainty would be resolved just prior to the production decisions of the second period, and that purchased capacity would come on-line in time for second period production. Fine and Freund demonstrated that an increase in the unit price of dedicated product A (product B) capacity, ceteris parabus, would increase the optimal level of investment for flexible technology and decrease the optimal level of investment for dedicated product B (product A ) capacity.

A related result was obtained by Burstein and Genadis [3] in a model which extended the earlier work of Burstein and Ezzekmi to the situation of uncertainty about the demand function for a new product. Where current best-practice technology competed for investment with anticipated best-practive technology of the flexible type, an early increase in optimal dedicated capacity for the new product led to a reduction in the optimal, planned purchase of future flexible capacity and to an increase in the optimal, future use of dedicated capacity for product A. Actually, Fine and Freund had additional results with complementary or substitution effects, as did Burstein and Genadis. To contrast between the models of these authors, the expected present value criterion was the basis of the Fine and Freund work, while a utility function over expected present value and the variance of present value provided an objective function for the Burstein and Genadis paper. Their more complex objective function permitted Burstein and Genadis to analyze the manufacturing systems investment implications of attitudes towards risk. This analysis presented a possible explanation for some of the reluctance which U.S. companies have shown towards the adoption of new manufacturing technology.

Interestingly, published work after 1984 on economic models for considering flexible manufacturing technology has not embodied the contrast between modularity of capacity of flexible technology and "lumpy", or indivisible, capacity for dedicated automation, even though Hutchinson showed this contrast to be of economic significance to planners of manufacturing systems. In the sections to follow, a mixed-integer, mathematical programming model will be presented which allows the planner to purchase current best-practice technology in the form of indivisible transfer lines (for either the original product A or the new product B, or both) and/or modular capacity with the capability of producing both products. Like the Fine and Freund model, the "battle" of technologies takes place from the opening moment of the planning interval; but the "battle" is tempered by the anticipation of near-term advances in flexible technology. Our formulation of the model and characterization of optimal investment behavior has the primary goal of providing insights as to the validity of the following hypotheses:

(1) Modular capacity expansion is economically preferable to the addition of indivisible transfer lines when the strength of demand for a new product is sufficiently low.

24 M.C. Burstein, Rigid and flexible automation

(2) If the strength of demand for a new product is sufficiently high, capacity expansion through a combination of transfer lines and modular automation can be economically justifiable.

(3) Increased modular capacity in response to a new product introduction will tend to inhibit the replacement of old transfer lines with current best- practice equipment for the dedicated production of established products.

(4) Anticipation of near-term advances in flexible automation will reduce purchases of additional capacity to support new product introduction from myopic levels, although the attractiveness of current best-practice in terms of modular, flexible equipment will be less affected by this than indivisible transfer lines.

To address these hypotheses, we will introduce a model of the firm, through a description of the relevant scenario and by translating elements of this scenario into mathematical terms. Then, the Kuhn-Tucker conditions for the resulting mixed- integer programming model will be presented. These first-order conditions are generated under the assumption that values of the integer variables are parameters for purposes of optimizing over the continuous variables. Drawing on the first-order relationships, we then will state two theorems, each of which provides necessary and sufficient conditions for the purchase of manufacturing technology.

2. T h e m o d e l

We consider a firm that wishes to maximize its present value and operates in an industrial environment characterized by the following assumptions:

• the planning interval is comprised of era 1 (years 1, 2 , . . . , T) and era 2 ( years T + 1 . . . . );

• the annual demand function for each product is stationary and deter- ministic;

• the investment function Iv(. ) for flexible technology of vintage v is linear in the annual capacity qv (v -- 1, 2), which is measured in units of product A;

• the revenue functions R A ( . ) and RB(. ) are continuously twice dif- ferentiable in annual outputs Q~ and Q~, respectively, of products A and

tp e N i t e , B with negative Rj~ (Q~) and B(QB) where the superscript e designates era 1 or era 2 as e = 1 or 2, respectively; here, market prices clear outputs;

• operating costs for each type of manufacturing process with reference to any appropriate output of that process are proportional to the quantity of output;

M.C. Burstein, Rigid and flexible automation 25

• equipment is infinitely durable;

• scrapped equipment has no salvage value;

• transfer lines are available at a single level of capacity qAT and qBr for each of products A and B, respectively, although the firm can buy any finite number of transfer lines to support the manufacture of these products;

• the products of the firm are non-durable, so that demand in any year is independent of production in previous years;

• the firm occupies a distinct competitive niche and can behave like a monopolist as a result.

At the present time, the firm has the ability to produce one type of product only (product A ) with its existing equipment, which has capacity qo and unit operating cost CAo. This equipment is called vintage 0 whose investment has already been sunk. We will suppose that the firm earns positive profits when its vintage 0 equipment is used to its full capacity. Also, its marginal revenue is greater than its unit operating cost; i.e. R~(qo ) > CAO.

Initially, a more advanced vintage of product A-dedicated equipment, of product B-dedicated equipment, and of flexible equipment with the capability of producing both products A and B is available. This equipment is designated vintage 1 or current best-practice. If no technical improvements were foreseen after vintage 1, the cost structures of current best-practice are such that some vintage 1 equipment would be purchased; and vintage 0 would either be scrapped or used in conjunction with the vintage 1 equipment. This would be the behavior of a myopic firm. If further technical improvements were expected in the form of more cost-effective flexible automation (i.e. vintage 2), then the firm could reduce its acquisition of vintage 1 equipment in anticipation of vintage 2 purchases.

Before presenting the constraints and objective function of the model, some additional definitions are necessary:

CAT and CBT indicate the unit operating costs to manufacture products A and B on their respective dedicated transfer lines of vintage 1 technology;

CAr and CBv are the unit operating costs to manufacture products A and B by flexible automation of vintage v (v = 1, 2);

I A and I B are the first costs or purchase prices for vintage 1, dedicated lines of capacities qAT and qBT, respectively;

i and ] are integer variables which designate the number of transfer lines purchased for the manufacture of products A and B, respectively;

Q.~T and Q~T are continuous variables which indicate the annual outputs of products A and B, respectively, by dedicated transfer lines of vintage 1 in era e;


I~ is the constant-valued derivative of 10 (.);

Q~v and Q~v are continuous variables which indicate annual output of products A and B, respectively, by dedicated transfer lines of vintage 1 in era e;

m o designates the amount of product A output which is sacrificed when a unit of product B is manufactured by flexible automation of vintage u;

r is the annual interest rate, so that the discount rate a = 1/(1 + i").

Designations such as t = 1 refer to the end of the numbered year. Thus, the problem of the firm is posed at t = 0 as it plans for the indefinite future. Vintage 1 technology has appeared, can be immediately adopted for production in the first year, and then can be used as long as is economically advisable. Vintage 2 becomes available at t = T; thus, it can be adopted for production in the f i r s tyearofera2 and continued in production indefinitely. In fact, we will stipulate that CA2 + rI~ < CAx + rI~; i.e. the marginal equivalent uniform annual cost (MEUAC) for purchase of vintage 2 flexible equipment to produce product A is less than that of vintage 1 flexible equipment for the same product. We will assume similarly that CB2 + rm 21~ < C m + rm~ 1~. Consistent with our above comments about vintage 0, let CAO > CA1 + rI~ and CAO > CAr. Since no new transfer line technology is anticipated for era 2, this means that any purchases of additional capacity during era 2 will be of vintage 2 flexible technology. This economic imperative to purchase capacity of a given type only at its first appearance, if at all, obviates the subscripting of capacity variables by era.

From these definitions, we can construct the following relationships:

Q~ = Q~o + Q~, + Q~T ( la)

QB = QB, + QBT ( lb)

Q] = Q2o + Q~I + Q]T + Q]2 ( lc)

The annual output of the product or products from a technology of a particular vintage is limited by the purchase capacity-to-date of that technology a~ ~ vintage. This requirement and the posited convertibility of flexible capacity from t e original product A to the new product B yields the constraints below:

qo >~ Q~o' (e = 1,2); (2)

ql ~ QAI + ml Q~I' (e = 1,2); (3)


q: ~> Q~: + m: Q~:; (4)

iqAT >/ e . Q~T' (5)

/qBT >~ Q~T; (6)

all variables (integer and continuous) must be non-negative. (7)

The objective function Z can be written in terms of various discounted cash flows, as follows:

T

z : % % ) - if,-& + Z a ~ t G ( Q ~ + R~(O-~-CAoGo-%G, t = l

o o

- CB,Q~,] - aTI2(q2 ) + ~. at[RA(Q I ) + RB(Q ~) t = T + l

- - - C 2 - CAoQ~o CA1Q21 CBI Q21 CATQIT - BTQBT

- q : e : : - % Q L ] - (8)

Then, the mixed-integer programming model for our firm requires the maximation of (8) subject to the capacity constraints (2)- (6) , the non-negativity requirement (7), and de f'mitions (la) through (ld).

3. F i r s t -o rde r cond i t i ons

This maximization involves a nonlinear objective function with linear constraints. Since our problem has a concave objective function and convex constraints, the Kuhn-Tucker conditions below are both necessary and sufficient for an optimal solution in terms of the continuous variables. Derivation of these conditions makes use of

T

~. a t= (1 - aT)/r t = l

and o o

Z at-- aT/r T = T + I

28 M.C Burste& Rigid and flexible automation

to transform the objective function into an easily workable form. Note that the multipliers ( "shadow prices") from the Lagrangian are not discounted in their values back to t = 0, unlike traditional multipliers in present value maximization.

1 (9) o30 R;, (o3 ) - CAO --< Xo

O~, R~4(QJ) - CAt ~< X 1 (1{3)

Q11 R~(Q~) -- CB1 ~ m t k I (11)

QBr RB(QB) - CBT <~ XIBT (13)

Q2 0 R]4(Q 2) - CAO < X g (14)

oi, R~(ol) - % < x~ (15)

QB1 ' 2 2 ( 1 6 ) RB(Q~) - CB1 < .71 X 1

Q12 R~ (QI) - CA2 ~< X~ (17)

QB2 R;(Q2B ) - CB2 <<" m2X22 (18)

Q1r RI(QI) - CAr <<" XIT (19)

Q~ R;(o~) - % < x~T (20)

T 2 1 + /7 )k 1 ~ /"11 (21) ql (1 - aT)x 1

2 ~< ri, 2 (22) q2 X2

xl 0~o < qo (23)

Xl QI, + mx QB1 <~ ql (24)

)kiT QIT <~ iqAT (25)

XlT Qlr <" ]qBT (26)

2 2 Xo QAO <~ qo (27)


2 2 2 (28) kl Q]I + ml QBI ~< ql

k2AT Q]7" ~ iqAT (29)

k~r Q~T < ]qBr (30)

2 2 2 ( 3 1 ) k2 Qfl2 + m2 QB2 ~< q2 "

Complementary slackness requires that a positive value of any variable on the left be accompanied by a strict equality of the relationship immediately to the right of that variable. Meanwhile, an additional requirement of the Kuhn-Tucker condition is the

2 2 2 2 2 Inter- non-negativity o f t he multipliers Xl l , kll , h i T , klBT , kol , k l l , kAT , k ~ T , k 2 . pretation of the Kuhn-Tucker conditions indicates the following:

• If equipment is used at any time, then the incremental impact on the optimized objective function of marginal output from that equipment must be equal to the rate of accumulation for current profit. This statement refers to each of the pairs (9)-(20) .

• Additional flexible technology is purchased until the incremental impact on the optimized objective function of marginal capacity from thepurchased technology reaches equivalence with the current component of marginal purchase cost. This statement refers to each of the pairs (21) and (22).

• Conditions (23) through (31) call for a positive incremental impact on the optimized objective function of marginal capacity only if the associated capacity source is exhausted.

4. S o l u t i o n

Since there are fourteen continuous variables other than the lamdas, we could have 214 possible combinations of these variables for each of them either zero or positive. However, zero values of new capacity in era 1 restrict other era 1 and era 2 variables to zero values. Hence, a maximum of 256 cases is realizable. Within this smaller group of cases, the parametric restrictions CA1 > CAT, C m > CBT, and CAO > CAT make the problem even more manageable. These restrictions are the operating advantage of transfer lines over flexible automation from era 1. An important implication of this operating advantage is that no product would be manufactured in an era by flexible automation as long as a transfer line is in use for the product and has slack capacity. Thus, flexible automation will be applied only in two types of situ- ations: first, when the required output rate for any product exceeds transfer line capacity but not by enough to justify the purchase of additional transfer lines; second, when new, flexible technology in era 2 is cheaper to use for some products than the era 1 transfer line technology.


Actually, flexibility only makes sense if the associated technology is used for both products. In concept, this can happen either sequentially (some product on the flexible system, then the other) or simultaneously (i.e. both products at the same time). However, use of flexible technology in our model for only one product during era t implies that a transfer line technology handles the other product in that era. Thus, the flexible technology from era 1 would always be at an operating cost dis- advantage to the transfer lines for that other product. This does not preclude the use of superior flexible technology, new in era 2 for one or both products; but it does rule out the stituations of sequential flexibility.

The remaining cases for analysis at this point use flexible technology in one era at least, and manufacture both products with that flexible technology. If the flexible technology is used in conjunction with transfer lines, the latter are at full capacity although the number of transfer lines of a given type is an integer variable. To illustrate the form of the solution, we will present two theorems. The first of these is the only setting for our model in which transfer lines are introduced to produce both products with flexible technology in a complementary era 1 role. The second theorem is, in a sense, a perverse use of the flexible technology in era 1 for a single product and no use of that technology in era 2. The two theorems are rich enough to facilitate consideration of our four above hypotheses.

THEOREM 1

The policy that maximizes (8) subject to the capacity constraints (2 ) - (6 ) , the non-negativity requirements (7), and definitions ( la) through ( ld) with QJT = iqnT for i > 0 and Q~T = / q B r for / > 0 (e = 1,2) is as follows:

Annual total outputs and annual flexible outputs for era 1"

if CA1 > C 4 2 + r f 2 and CA1 + r I ' l / (1 - a T ) < RA(qo )

, i r : i / (1 a t ) then R A(QA) = CA1 + - (32)

with Qm = 0.~ - iqAT- qo for q o "<< Rj (OJ);

QJ - iqA~ for q o > R; (QJ); (33)

if

then

with

t ? C m > CB2+rm2I 1 and C m + m l r I 1 / ( 1 - a

, 1 Z ~ / ( 1 - a T) R~(04) = %1 + rm,

o. 1 = - :qB ,

r) < R;(qo)

(34)

(35)


#Flexible capacity of vintage 1#

qx

Q~ - iqaT - qo + ml(QB - JqsT)

if (32) and (34) apply with Go ~< RA(Q] );

Q1 _ iqAT + m,(Q~ - ]qBT)

if (32) and (34) applywith CAO > RA(Q]);

(36)

Annual total outputs and annual flexible outputs for era 2

t .

R~(Q~) = Ca2 + r l 2 , (37)

with Q~2 QA -- iqAT-- qo

a ] - iqA r

for Cno ~< R~I(Q,~);

for CAo > RA(Q~]);

(38)

R;(Qg) = CB2 + rm2 Z'2 (39)

with Q/~2 = Q~ -iqBT; (40)

#Flexible capacity of vintage 2 #

ql

QA - iqaT -- qo + m2(Q~ - fqBT)

if (37) and (39) apply with CAo

Q~ - iqA T + m2(Q ~ - fqBT )


t 2 < R~(Q,~);

> Ri(e );

(41)

The parametric conditions of (32) state that the unit operating cost to manufacture product A for flexible equipment of vintage 1 exceeds the MEUAC for manufacture of product A throughout era 2 by vintage 2 equipment, and that the MEUAC of the flexible equipment from era 1 in the manufacture of product A only for era 1 is smaller than marginal revenue at capacity for the originally installed equipment. A similar interpretation is available for the parametric conditions of (34).


THEOREM 2

The policy that maximizes (8) subject to the capacity constraints (2)-(6), the non-negativity requirements (7), and definitions (1 a) through (1 d) with QJIT = iqAT for i > 0, QBT" < ]qBT for ] > 0, CAa > CA2 ar rI~, CAT > CA2 + rl~, and CB7, > CB2 + rm2I ~ is as follows:

Annual total outputs and annual flexible outputs for era 1 :

same parametric conditions and marginal condition on

Q~ as (32) above;

same parametric conditions and form of results as (33) above;

Flexible capacity of vintage 1 :

- i q A T - go

if (42) and (44) apply with CAO >1 R~ (Q~4); q l =

- iqAT

if (42) and (44) apply with CAo >1 R~4(Q~4 );

Annual total outputs and annual flexible outputs for era 2:

Q2 _ qo ~< RA (Q,]);

i 2 Q] <

same form as (37) above;

for cA0 Q,]2 =

for CAo

R~(Q~) = CB2 + rm2It2

with Q~2 =Q~;

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)


Flexible capacity of vintage 2:


q2 = Q~ + m2Q ~


(51)

5. Discussion

Consistent with our goal of providing insights as to the validity of the four before-mentioned hypotheses, we will draw upon theorems 1 and 2 to consider each of the hypotheses in turn. The first hypothesis states that weak demand can be a reason for introducing current best-practice in terms of flexible, modular automation rather than dedicated, indivisible capacity. The economic issue is the difference in purchase cost between a complete but potentially slack transfer line and an amount of modular capacity to match quantity demanded versus the operating advantage of the transfer line over the modular technology. Equation (34) specifies Q~ for flexible production, while eq. (44) determines Q~ for dedicated, but slack capacity. Because the former is smaller than the latter by our earlier parametric assumptions, the direction of the following necessary inequality for the use of flexible technology alone to manufacture product B requires weak revenue R B (-):

[ R , , ( Q ~ , ) - CB~,Q~, ] (~ - a ' ) / , - ~' B < - I , ( Q ~ )

* [ R ~ ( Q L ) - c~, o~, ] (~ - a ' ) / , . (52)

This inequality can be restated for era i profits as

IB - I1 (QB1) > [dedicated profit for B- flexible profit for B] (1 - aT)/r. (53)

The second hypothesis holds that strong demand for product B can justify a combination of transfer lines and flexible automation. Theorem 1 provides a demon- stration of this with tight, dedicated capacity and the coverage of excess demand by modular technology. The impact of slack transfer line capacity on the use of flexible automation is evident from theorem 2 for era 1.

A displacement or substitution effect is described by the third hypothesis, which suggests the addition of modular capacity for manufacture of a new product as


a potential inhibitor of equipment replacement. Here, the economic issue is the extent of investment subsidization by the new product of manufacture on the flexible equipment of the established products. In the model of this paper, additional investment is expended in direct proportion to the purchased flexible capacity. Had the constant term of the investment function for each flexible vintage come into play only for qo > 0, subsidization would have been possible.

The fourth hypothesis describes an effect of anticipating more advanced flexible automation in the near future on current purchases of flexible technology. Under myopic expectations, the firm of our model would anticipate the indefinite use of new equipment. Thus, the amount of purchased, flexible capacity would be determined as follows:

I R~(Q~) = C m + rm, I, . (54)

The right-hand side is smaller than that for eq. (34), so the myopic Q~ value will be larger than its "foresighted" equivalent. While this impact of anticipation is of interest for flexible automation, the hypothesis suggests a stronger effect of anticipation on the purchase of current best-practice in terms of dedicated lines. The present value for indefinite use of dedicated lines for the new product is given by the left-hand side of the inequality below:

-/,r + [RB(Q r) - C ' BTQ T]/r > --/I B

+ [ R B ( 0 - C ' - BTQ~T] (1 af )/r, (55)

where QBT is determined from eq. (44). Clearly, the purchase of dedicated lines is far easier to justify for a given RB(. ) in the myopic case, even with both sides of the inequality (52) adjusted for the indefinite horizon.

6. Resea rch d i rec t ions

We have so far considered stationary demand, although this is rarely the situation of a firm. Thus, exploration of shifting demand could be of interest. Similar opportunities for research exist in the relaxation of almost any combination of our assumptions. Another major issue is the representation of convertibility from one product to another of flexible capacity, both in a technological sense and in the sense of economic value. The former issue might be addressed through empirical and simula- tion analyses of actual flexibly-automated processes. Our model only considers substitute processes, whereas such advanced manufacturing technologies as machine vision and computer numerical control are possible complements to each other. This opens still more vistas for investment modelling.


References

[1] J. Browne, D. Dubois, K. Rathmill, S.P. Sethi and K.E. Stecke, Classification of flexible manufacturing systems, The FMS Magazine (April, 1984).

[2] M. Burstein and E. Ezzekmi, Market-based planning to support the time-phased introduction of programmable automation, Proc. 1985 Annual International Industrial Engineering Conference, Los Angeles, CA (1985).

[3] M. Burstein, E. Ezzekmi and T. Genadis, The economic planning of programmable automation under uncertainty about future product mix requirements: The closed-loop case, Spring Joint National Meeting of ORSA/TIMS, Los Angeles, CA (1986).

[4] M. Burstein, E. Ezzekmi and M. Talbi, Economic justification for the introduction of flexible manufacturing technology: Traditional procedures vs. a dynamic-based approach, Proc. 1st ORSA/TIMS Special Interest Conference on FMS, Ann Arbor, MI (1984).

[5 ] C. Fine and R. Freund, Economic analysis of product-flexible manufacturing system investment decisions, Working Paper No. 1757-86, Sloan School of Management, MIT (1986).

[6] C. Gaimon, The optimal acquisition of flexible automation for a profit maximizing firm, Manag. Sei. 31(1985).

[7 ] G. Hutchinson, Production capacity: CAM vs. transfer line, Industrial Engineering 8, 9(1976). [8] G. Hutchinson and J. Holland, The economic value of flexible automation, J. Manufacturing

Systems 1, 2(1982).

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Rigid and flexible automation for manufacturing systems: Characterizing economically-reasonable investments