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Flow optimization in flexible manufacturing systemsJOSEPH KIMEMIA a & STANLEY B. GERSHWIN ba AT&T Information Systems Laboratories , Room IE203, 50 Cragwood Road, South Plainfield,New Jersey, 07080, U.S.A.b Laboratory for Information and Decision Systems , Massachusetts Institute of Technology ,35–433, 77 Mass Ave., Cambridge, MA, 02139, U.S.A.Published online: 24 Oct 2007.

To cite this article: JOSEPH KIMEMIA & STANLEY B. GERSHWIN (1985) Flow optimization in flexible manufacturing systems,International Journal of Production Research, 23:1, 81-96, DOI: 10.1080/00207548508904692

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Flow optimization in flexible manufacturing systems

JOSEPH KIMEMTAT and STANLEY B. GERSHWINf

The problem ofoptimal part muting i n s flexible manufacturing system is solved by a network flow optimization approach. Mathematical methodn which exploit the structure of the pmhlem to generate manufacturing paths are outlined. Numerical examples show that the method produoes muting policies which yield good resulte when applied to a simulation model of a flexible manufacturing system.

Introduction It has been estimated that between 50% and 75% of the US. annual expenditure

on manufactured parts is for items with an annual demand of leas than 100000 units (Cook 1975). There has been concern recently about the low productivity of the medium sized production plants that produce many of these parta.

Transfer lines have long been used for the manufacture of large volumes of single items. Changing a transfer line to the production of a different item is costly in terms of lost production. Job shops which produce many different parts in small quantities have, in the last 20 years, turned to mumerically controlled (NC) machine tools, especially in the aerospace industry. Numerically controlled machines are versatile and can perform a wide range of operations on a part under computer control.

Flexible manufacturing systems (FMS) have recently been introduced in an effort to increase the productivity of those sectors of industry that produce a medium volume of related parts. Annual production ranges from 200-20000 parts per year (Hughes et al. 1978), which is not high enough to warrant the use of a set of dedicated transfer lines. On the other hand, job shops suffer from a low utilization of expensive machines and a high in-process inventory (Cook 1975). An FMS increases productiv- ity by simultaneously reducing inventory and increasing the utilization of the machining centres.

An FMS consists of a number of workstations at which operations are carried out and a materials handling system which transports parts to and from the work- stations. Depending upon the production level and the variety of parta to be produced, the stations may be made up of standard NC machining centres or more process-specific units (Hughes el al. 1978). The stations are flexible in the sense that they can perform different operations on a variety of parts in a random sequence. In the metal cutting industry, one of the factors that limit the number of operations that a machine can perform is the number of tools that can be accommodated in the tool magazine.

Overall control is exercised by one or more computers which control the transportation system and the scheduling of operations at the workstations. The workstations are equipped with stored program controllers which direct local operations.

Received May 1983. tAT&T Information Systems Laboratories, Room IE203. 50 Cragwood Rod, South

Plainfield, New Jersey 07080, U.S.A. iLaboratory for Information and Decision Systems, Massschusettn Institute of

Technology, 35-433, 77 Maw Ave., Cambridge, MA 02139, U.S.A.

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82 J. Kimemia and S. R. Cershwin

An FMS produces a se t of parts simultaneously. The parts are fed into the system at a loading station and undergo a specified sequence of operations at the workstations before leaving a t an unloading station. The flexibility of the system allows the parts the choice of one or more stations for each operation. This allows production to continue even when a workstation is out of service because of failure or maintenance. ..

Examples can be found of FMSs that produce transmission casings for tractors (Hughes el al. 1978, Stecke and Solberg 198l), components for duplicating machines (SchaiTer 1978) and integrated circuit devices (Brnnner et al. 1981). Flexible aeaembly systems are being designed and implemented in the automobile industry (Beecher and Dewar 1979, The Economist 1980). For a survey, see Dupont- Catelmand (1982).

An important problem which has afundamental effect on the production rate and utilization in an FMS is that of part routing. If the production mix of parts is specified and the location at which all the operations can be carried out is known, a sequence of workstation visits or manufacturing paths needs to be chosen for the parts dispatched into the system. Olker (1978) and Solberg (1977) have pointed out that a common industrial practice is to route parts in such a way that the workloads at the workstations are equal. It is shown later by way of an example and by Stecke and Solberg (1982) that this is not always optimal. The manufacturing paths can either be found in advance, or a set of decision rules can he established whichallow the sequence of workstation visits for each part to be determined in real time as the part makes its way through the system (Buzacott 1982). Real time control policies areespecially useful when themachinesin theFMS aresubject to failures (Olsderand Suri 1980, Hildebrant 1980, Kimemia and Gershwin 1981).

Combinatorial techniques have been used to solve the job-shop scheduling problem (Coffman 1976). The computatorial requirements for solving job shop problems grow rapidly with the number of jobs and machines. The periodic scheduling algorithm (Hitz 1979) is a heuristic combinatorial technique for evaluatingschedules that maximize the production rateof an FMS. However, before the periodic schedule can be calculated, the routes for all the parts must be established.

Simulation techniques allow detailed investigation of the effects of parameter variations and routing policies (Hutchinson 1977, Lenz and Talavage 1977, Shanthikumar and Sargent 1980). Simulations can be costly in terms of compu- tation, particularly when the number of options to be tested is large.

Analytical techniques based on network of-queues analyais have been used to study the effects of routing policies on the throughput and in-process inventory of an FMS (Solberg 1981, Secco Suardo 1979, Buzacott and Shanthikumar 1980, Stecke and Solberg1982). Suri (1981, 1983) has argued for the validity of this approach when applied to such systems.

In this paper we use a network flow optimization approach to determine optimal pnrt routing in an FMS modelled by a network of queues. We develop solution Lechniques which generate routes for each part type and the rate at which parta should be dispatched on the routes. It is therefore unnecessary to enumerate in advance all of the possible routes for each part type.

Workpiece routing is only part of a much larger decision making problem (Hutchinson 1977, Nof et al. 1979). At the higher manage&al levels, the group of parts to be manufactured and the quantities are first chosen (Nof and Solberg 1980,

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Plow oplimiurtia inflezible manufwluring syslema 83

Solberg 1981). Once the part mix is known, the configuration of operational capabilities a t the workstations must be selected. Stecke and Solberg (1982) and Stecke (1982) term this the loading problem and suggest a mixed integer program- mingapproach for i t s solution. Factors such as tool magazine capacity, which limit the flexibility of a worhtation, are incorporated as problem constraints a t this level of decision making. The flow optimization problem solved in this paper is a t the lower operational levels of the decision hierarchy. The solution techniques of this paper are intended to allow the control computer8 in an FMS to make effective routing decisions. A survey of analytic methods for many problems in the larger production control hierarchy appears in Buzacott and Yao (1982).

In an FMS with failure-prone machines, the production control problem becomes too large to be tackled effectively with a singleset of decision rules or routing policies. Hildebrant (1980), Hilderbrant and Suri (1982) and Kimemia and Gershwin (1981) have developed on-line hierarchical control algorithms for FMS with unreliable workstations. I n Hildebrant's approach, the top level of the hierarchy chooses the part routing for each failure condition of the system by means of a non-linear optimization algorithm. Dynamic feedback control policies are adopted by Kimemia and Gershwin (1981); the top level of the control hierarchy chooses the part mix as a function of the failure state of the machines and current production levels. The solution methods developed in this paper are useful in either case because they allow the routing problem to be solved without enumerating manufacturing paths in advance. Where the part mix is subject to change because of dynamic control policies, the additional effort needed to compute new part routing is reduced. An alternative representation of machine failure appears in Olsder and Suri (1980). Related issues in the operation of automated production systems are discussed by Buzacott (1982).

In the following sections, the model is presented, the part routing problem is formulated, and solution techniques are presented. Network-of-queues models, in general, result in non-linear models and optimization problems. Our discussion concentrates on linear programming techniques because flow generating linear programming steps are essential to external flow algorithms used tq solve non-linear network flow optimization problems (Defenderfer 1977).

As an example of the application of the network flow approach to the routing problem, numerical results for two- and four-workstation systems aiw presented. The effect of changing some of the system parameters on the optimal routing, production rate and workstation utilization are investigated. Optimal routing policies, cal- culated by the network flow method, for a four-workstation system are tested on a discrete simulation. Flow optimization is found to give good prediction of throughput and workstation utilization. It is also shown that i t is fairly easy to generate part loading schedules from the optimal flow-rates. The flow optimization method is therefore applicable for on-line control of an FMS.

Modelling and optimization of flexible manufacturing systems '

Physicul description of an FMS and the routing problem An FMS consists of M workstations producing N different part types. Each part

of type i requires K i operations for its completion. A particular operation can be done a t one or more different workstations. The time to complete operation k on a type i part at workstation j is a random variable with mean r b

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Drill, bn. 1' d*.

Figure 1. An example of a part.

An operation, as defined in this paper, is composed of a set of simple prooasses (such as drilling, tapping or milling) that are completed during a single workstation visit. As an example, consider Fig. 1. The part is made from a casting with the correct external dimensions. The only operations required are the drilling, tapping and boring of holes to the required tolerances. The following operations are identified and referred to by the superscript k:

k=l i Drill and tap eight bolt holes. k=2: Drill and tap chambers to required tolerances. k=3: Drill axial through hole. k=4: Drill and tap outlet line. k=5: Drill and tap supply line.

Thus an operation is identified by a part number i and the operation number k. Operation 1 for the part of Fig. 1 normally requires drilling and tapping tools.

Defining it as a single operation means that any station capable of doing this operation must be equipped with both tools. The station may be a multi-head, multi- spindle machine, in which case the time to complete the operation is shorter than a t a workstation which consists of a single-spindle purnerically controlled machine. If the drilling and tapping tools were a t different machines, drilling and tapping would he defined as two different operations.

The part shown in Fig. 1 may need refixturing before all of the operations are completed. A part which has been refixtured is treated here as a different type with its own set of operations.

The flow rate of type i parts to station j for operation k is defmed as y$. The sys&m control computer monitors these variables and can affect them by varying the rate a t which i t sends parts a station for a given operation.

The variables yb are relilted by conservation of flow equations and the production ratio requirement. Conservation of flow states that the total number of type i parts undergoing operation k per unit time is equal to the production rate of that part type. This is expressed as

K , and i = 1 , 2 ,...,

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Flour optimization in flexible manufacturing systems 85

where u, is the production rate of type i parts. The total production rate is given by

The production ratio requirement stipulates that type i parts should comprise a fraction a, of total production. This can he written as

An important performance measure is the utilization pj(y). This is defined to be the proportion of time that a workstation is operating on a part. The utilization is a function of the flow'rates y = (dj) and is given by

Formdat ia of optimization problems Kxodedge of the workstation flow rates y t and the utilizations pj(y) allows us to

use network-of-queues analysis to compute various system performance measures (Shauthikumar and Buzacott 1979, Hildebrant 1980, Solberg 1981). The particular queueing model depends on the statistical characteristies of operation times and part arrival process. Comparison with data from an operating FMS (Solherg 1977) and analytical research (Suri 1983) shows that network-of-queues models a're fairly robust with respect to assumptions made about operation time distributions.

The general flow optimization problem for a flexible manufacturing system can be stated as

NLP: .maximize f(y)

subject to ( I ) , (2), (3), yf,>O and

where y is a vector of flow rates y t . The objective function f(y) represents a performance measure that is to be maximized. For example, the throughput, or a weighted combination of throughput and in-process inventory, might represent some measure of return. The constraint set is linear. It consists of a set of constraints (1) which affect each part separately, a set of coupling constraints (3) and (6).

Feasible direction algorithms (Luenherger 1973) solve NLP iteratively by solving a t the ith iteration the linear program

LPI :

minimize x <]yt 1.j.k

subject to (1). (2), (3), (6) and yk>O. The cost coefficients are defined as

with the gradient being evaluated a t the current point in the iterative process.

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86 J. Kimemia a d S . B. Gershwin

In what follows. we examine a solution techniaue for LPI which eenerates " manufacturing paths and gives the flow rate of parts on each path. The technique is developed by applying the Dantzig-Wolfe decomposition algorithm (Dantzig 1963) to the linear program

Generation of manufacturing paths Let $](a) a= l ,2 , ... . , be a set of workstation flow rates, each representing unit

production rate for each part type. Thus for all a, i, and k, the following relationship is satisfied:

The set of flow rates gj(s) do not necessarily satisfy the workstation capacity constraints (4) nor the ratio constraints (3). Any flow vector y can he expressed in tkrms of a finite number of unit flow rates &s) as

where w, are appropriate weighting factors. This is because the flow vector y is defined in a finite dimensional space.

We can restate the linear programming problem ~ ~ l ' i n terms of the flow rates z,(a)=(&s) k=1,2 ,..., K,, j = 1 , 2 ,..., M) as

subject to

The variables in LP2 are wb, which are weights on each of the routing policies associated with the flow vectors ~ ~ ( 8 ) . Since each flow vector corresponds to a unit production rate, (11) and (12) ensure that workstation capacity and part ratio constraints are satisfied by the solution of LP2.

The solution of LPI has two steps which are repeated iteratively until the optimality conditions for LPI are met (Dantzig 1963). First, flow assignment $)(a) are selected. Second, LP2 is solved to find the weight w , to be given to each assignment. Additional flow vectors are generated until the problem is solved.

The flow vectors z,(s) are generated by solving N linear sub-problems, one for each part type. Let n1 be the dual variables or Lagrange multipliers associated with constraints (11). Flow vectors are generated by the solution to (Lasdon 1970)

minimize C (ct-n,r$ y t i. k

(14)

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I'low optimidion in flezible manufacturing system 87 '

each polh and laqronqs multiplier for each slotion

Figure 2. The path generating decompoeition slgorithm.

subject to

The sub-problems are easily solved. The dual variables nl allocate a cost ~ ] - n , r ~ for each part of type i that has operation k done at station j. The sub-problems are therefore solved by finding, for each operation, the station with the lowest cost and allocating a unit flow rate of parts to that station. Each vector z,(s) that is generated as asolution to LP3 therefore corresponds to a manufacturing path because all parts go to a single station for each operation. The problem LP2 thus chooses the flow rate on each path generated by LP3. The algorithm stops after a finite number of iterations because there can be a t most Slinearly independent flow vectors z(s) where S is the dimension of the flow vector y = yb.

The overall procedure is summarized in Fig. 2. An initial flow vector &I) can be found by setting unit flows on arbitrary paths for each part.

The decomposition method results in a savings in computational effort. The initial linear program LPI is replaced by LP2 with fewer constraints and easily solved sub-problems LP3. In a dynamic control scheme where the objective is changed periodically, the flow vectors already generated give a good initial starting point for the on-line solution of the linear program.

Numerical results for two- and four-workstation systems

Two-workstalion two-part As an example of the application of network flow optimization techniques to

FMSs, consider the two-workstation system of Fig. 3. The system consists of two machining centres, each with a tool magazine capacity of 60 tools. Two different part types are produced by the system. The first part requires the application of 20 tools and the second requires 70 tools (the 2 parts may have some tools in common). We consider the case where half the tools for the second part are loaded into the first workstation and the other half a t workstation 2. All 20 tools for the first part type are loaded into both workstations. The flow optimization problem in this case is one of

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Figure 3. A two-workstation system

deciding what proportion of the first part typeshould b e ~ e n t to the first workstation. I t should be noted that configuring the tools at the workstations is part of the production planning process tha t takes many factors not considered in this example into account (Stecke and Solberg 1981). The assignment of tools to workstations determines the available routes through the FMS.

The compound operation done by the 35 tools a t workstation 1 for part type 2 is labelled "operation l", and the one performed by the other 35 tools a t workstation 2 labelled "operation 2". Par t type one requires one operation, labelled "operation 1 ", which can be done a t either station. The ratio requirement on the par& is that two type 2 parts should be produced for each type 1 part. Thus a, = 113 and a2=2/3.

Assume that the nature of the operations is such that the time that a part spends a t a station can be modelled - - - - as - - an - - - exponentially . distributed random variable whose mean l/pl=~:l depends only on the station index j. The random description of the operation may account, for example, for adaptive control systems a t the work- stations which continuously adjust feed and spindle rates to account for tool and part condition. I t may also account for the random availability of the machines.

In this case, there only three possible path flow vectors z,(s) (i.e. the solutions of LP3). They are z l ( l )=( l ,O) (unit flow of type 1 parta through workstation l ) , z1(2)= (0 , l ) (unit flow of type 1 parts to workstation 2) and 2, ( I ) = ( l , 1) for the second part type. Any flow vector y =(y:,, y i2 , y:,, y:,) can be expressed ea

with the appropriate choice of thescalars w,,, w12 and w,,. With these assumptions, the system can be modelled as an open network of

queues. The average queue length a t each station is the same as that of an isolated M/M/I queue with the same arrival and service rates (Jackson 1963). The arrival rate a t workstation 1 is (w,, +wZI) and a t workst@ion 2 (w,,+w2,). The average .queue length can thus be expressed as (Kleinrock 1975)

The service rates of workstations 1 and 2 are p, and p, respectively

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Flow aptimizolion in flexible manujacturing syslems 89

The optimization problem is t o maximize the production rate while keeping the average in-process inventory below a level Q. This can expressed as

maximize w,, +w12+w2,

subject t o

~ I ( w ) +qz(w)<Q

The problem is solved with Q = 10. The speed of workstation 2 is fixed at 5 parts per hour, and that of workstation 2 is varied from 2 to 10 parts per hour. Theresults are compared to the asymptotic case when there is no limit on the average in-process

Figure 4. The optimal split.

Speed of W.S. *I Q. m

1.0-

A .8

.7 0 U - .- a .6- - 0

0, - k - - .- - P .?

2 E .- - 0"

- - -

p2 = 5 piecer/hour

- .4-

- - -

I I I 2 4 6 B I0

Speed of WS. *I 0. I0

PI

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90 J . Kimemio and S . B. Cershwin

I I I I 1 6 0 10

S E E D ff W.S. l p ,

FiECEV HOU)

Figure 5. Workstation utilization.

L t I . LO 4.0 bO L O 10 P I

WEED ff W.S. *I

Figure 6. Queue lengths a t the workstations.

inventory. The optimization problem (19)-(22) with a non-linear constraint function is converted t o the form of NLP by adjoining the non-linear constraint to the objective function by means of a penalty Lagrangian function (Hestenes 1968).

The proportion of type 1 parts sent to workstation 1 (referred to a8 the optimal spliL) is shown in Fig. 4 for Q= 10 and Q = a. The differences between the two are small. There are three operating regimes.

When p , is small compared to &, the optimal split is small and all type 1 parts are sent to workstation 2. Similarly, if )I, is large compared t o p2, the optimal split is

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Flow oplimirnlia injlezible munu/acturing systems 91

. . Figure 7. System production rate.

unity and all type 1 partsaresent to workstation 1. This would seem to indicate that, when the differences in speed between the two workstations is large, i t is not worth while making theslowerworkstation flexible. Even if i t has the capability toperform operations on type 1 parts, i t is not utilized. On the other hand, the flexibility may he valuable when the faster machine is unavailable because of failure or maintenance.

In the range where p , is about 40% of p,, the optimal split changes rapidly from zero a t the lower speed to unity at thk higher speed. The three regions are evident in the effect on utilization and average queue lengths shown in Figs. 5 and 6 for Q = 10. The change in optimal split keeps the utilizations of the two stations roughly equal. For this system, a t least, the optimization produces approximately balanced workloads on the two workstations when their speeds are not too widely different. When one workstation is faster than the other, it is no longer optimal to have balanced workloads. Stecke and Solberg (1981) have also ahown that balanced worksLation loads do not necessarily yield the maximum throughput.

The utilization p , ( y ) of workstation 1 does not decrease monotonically asp,, the speed of the workstation, is increased. I n the speed range where the optimal split is changing, the utilization of workstation 1 increases with the service rate, as is shown in Fig. 5. This is due to the rapid change in the split aa type I parts are switched from workstation 2 to workstation I , resulting in an increase in utilization. At workstation 2 the loss of parts results in decreased utilization.

The graphs of Figs. 5 and 7, which show the effect of workstation 1 speed on utilization and production rate, demonstrate the importance of analysingan FMS as an interconnected system. The results here show the effect of changing two system parameten on a given configuration. In practice, many more parameters can be varied and a trade-off between contlicting requirements is needed before the optimal choice of parameter values can be made. At the operational level, changes in production requirements require that adjustments be made in routing policies even though the system configuration might not have been changed. An analytical

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92 J . Kimemia and S . R. Cerahwin

j-workntation k-operation\ 1 2 3 4

j-workstation k-operation \ 1 2 3 4

f$-part type 3

j-workatation k-operation

1

I:,-part type 4

j-workatation k-operation) 1 2 3 4

1 2.0 2.9 m . 3.0 2 3.7 3.9 3.0 3.9 3 4.9 5.9 6.0 5.9

&part type 5

k-operation) I 2 3 4

1 3.0 3.4 3.2 3.9 2 2.0 2.8 1.9 ' 2.8

type 6

part ratio reauirement

Table I . Production requirements for the aix-part four-machine example.

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Flow oplimizatiun inflezible manujaduring sy8lema 93

technique, such as the flow optimization method, for evaluating the performance of the system is an essential tool in the planning and operation of an FMS.

I n making trade-off studies, i t is important that each candidate configuration have optimal operational amignments; otherwise the results would not be valid. For example, if we want to compare two different machines for workstation 1, one with p , = 4 and the other with p, =6, using a fixed value of A=0.2 would show the system with the faster machine as having a 6% higher production rate than one with thq slower machine. However, using optimal values of A8(A=@IO for p, = 4 and A=0.75 for p , =6) shows the true improvement in throughput to be 22%.

Four-muchine siz-part problem Consider Table 1, which shows the operational requirements for six parts to be

produced on four workstations. All operation times are deterministic. I n this example, the total number of possible paths for all parts is too large to beenumerated in advance as could be done for the two-part two-part problem.

The formulation of NLP results in a problem with 56 y:, variables. There are 4 inequality and 15 equality constraints resulting from flow conservation and ratio requirements.

The problem is solved by a standard linear programming code and the resulta implemented on a discrete simulation of an FMS. The results are shorn in Table 2 and Fig. 8. In the example, few paths have positive flow in the optimal solution. Although there are over 200 possible paths, only 6 are needed to achieve the maximum throughput. It can be shown that the number of paths in an optimal solution depends only on the number of parts and workstations. The result follows from the fact that the number of non-zero variables in a basic feasible solution cannot exceed the number of constraint functions (Dantzig 1963).

The routing of Table 2 is implemented on a discrete simulation of a four- workstation system. The results are shown in Fig. 8. The flow rates are achieved by loading parts at regular intervalson each route (the interval is equal to the reciprocal of the flow rate). The scheduling algorithm is simple to calculate and eesy to implement.

The production rate of the aimulation model is within 4% of that predicted by the optimization result. The optimal assignment satisfies all the workstation capacity constraints as equalities, indicating full utilization for the stations. The simulation results show station utilizations ranging from 0.91 to 097. The difference between the simulation and optimization results may be accounted for by the initial transient period and because the simulation starts with an empty system.

Also illustrated in Fig. 8 are the queue occupancies at the four workstations. Because the processing times are deterministic, the average queue length a t a station depends on the scheduling algorithm. For this reason, queue lengthsdo not enter into the linear programming problem.

hnclusions A mathematical programming approach has been applied to the problem of

routing parts in a flexible manufacturing system. In this method, the aggregated flow of parts is modelled. All possible routes through the system need not be enumerated in advance.

Decomposition techniques are used to break up the optimization problem into path generating sub-problems and a coordinating master problem. For a part, apa th

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Table 2. An optimal routing policy

WQ(KSlA1ION 2 1

R O U R l l O N ' UIIIIZATION .Plb(

ff l lMt n PIIEDICTEO UTILIZATION Pl iCfS IN OUIUI

Figure 8. Queue occupancy during 1500 time step simulation.

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Flow vptimization in Jlexible manufactu~iltg systems 95

is a set of workstations, one -for each operation, tha t are capable of doing all operations required by the part. The result of the optimization is the proportion of parts tha t should be manufactured by each of the available paths.

Results for a two-part two-workstation system show tha t approximately balanced workloads are optimal when the difference in speed between 'the two workstations is not large. When the difference is great, the optimal routing does not produce equal workloads. The optimal routing is found t o be sensitive t o the relative work rates of the stations.

Optimal assignments yield good results when tested on a discrete simulation. Workstation utilizations and system throughput in the simulation model are close to the values predicted by the optimization. In the simulation, scheduling technique based on the optimal flow rates is used. The technique is simple to calculate and easy t o implement.

Acknowledgments This research was done in the Laboratory for Information and Decision Systems

of the Massachusetts Institute of Technology, under NSF grant number DAR78- 17826.

We would also like t o acknowledge the effort put in by the reviewera of this paper. Their comments and suggestions were of great help in rewriting the paper.

Le probkme de I'aoheminement optimal dea pieces dans un sysame i fabrication flexible est &olu par un systAme d'optimalisation de debit du &em. I1 est soulign6 des m6thodes math6matiques qui exploitent la structuredu problbme pour produire des parcoura de fabrication. Des'exemples num6riquea montrent que la mCthode produit dea tactiques d'acheminement qui donnent de bons dsultata loraqu'elles aont appliqu6es B un rnodgle de simulation d'un sy&me de fabrication flexible.

Das Problem der optimalen Teilfertigungsplanung bei einem flexiblem Herstellungssytem wird durch eine NetzfluBoptimierungamethcde gelost. Mathematiache Methoden, die die Struktur des Problems ausnutzen, um Herstellungswege zu eneugen, werden dargestellt. Numeriache Beispiele zeigen, daB dime Methode Fertigungsplane ergibt, die gute Ergebniase enielen, wenn sie suf ein Simulationsmodell eines flexiblqn Herstellungssystems iibertragen werden.

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