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Journa l of t ,he Ope rat ions ResearchSociety of J a p a n

Vol. 39 , No. 1, March 1996

CAPACITY DESIGN IN A FLEXIBLE MANUFACTUR ING SYSTEM

WITH LIMITED WAREHOUSE CAPACITY

AN D THE LIMITED NU MB ER OF PALLETS

Hiroyuki NagasawaOsaka Prefecture University

(Received F ebruary 21, 1994; Revised J anu ary27, 1995)

Abstract Capacity design problems in a flexible manufacturing system (FMS) with a Poisson arrival

process and with l imited system-capacity are formulated to determine the number of boxes to store arrivaljobs in a su pplei nenta ,ry wa,rehouse, called t8he"warehouse capac it ,~ ," nd the num ber of pallets available inthe F MS . Th e through put of the FM S is performed through a,n appro xim ate closed queueing network model.Th e limits, and the first- a,nd the second-order properties of the th rough put function with respect t o thewa,rellouse ca,pa,cit,ya,nd t

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Capacity Design in an F M S

[I /O stationU

Supplementary warehouse

Loading/unloading station

Transport er

Figure 1. A typical FMS with a supplementary warehouse

T he wa,rehouse ca,pa,city ancl the numb er of pall ets, therefore, sho uld b e carefully chosento ma,ximize the thro ughp ut of the FMS. Th e optimal design of t he FMS with respect t othese capacities will be addressed in this paper.

Up t o now, the re are numerous pa,pers analyzing blocking pheno mena o ccurring a t eachindividual station wit,h finite local buffer [6] [7], but there is littl e liter ature dealing with th eblocking in this type of FMS with a supplementary warehouse. Shan thik umar and Stecke91 deal with th e case in which the warehouse capacity is infinite. Yao [l51 an d Yao andShanthikumar [l71 formulate some storage models for the FMSs as lot sizing models inwhich a batch of jobs periodically arrive at the FMS. Since in our FMS model as well asin th e previous models the equilibrium probabilities of qu eue length a t each statio n a nd t hethroughput of the FMS can not be exactly calculated, we use an approximation similar toth e one given in these references [g] [l51 [17].

In section 2, we formu la,te a, ma,thema,tica,l model of the FMS using a closed queue-ing network model a,nd give a heuristic derivahion of the equilibrium probabilities and the

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3 8 II . Ngasa wa

throughput of the FMS. Section 3 gives the limits, and the first- and t he second-order prop-erties of th e thro ughpu t function with respect to both th e warehouse capacity (t he numberof boxes) a,nd th e numb er of pallets. In section 4, we propose solution meth ods t o severalcapacity design problems in the FMS model exploiting these properties.

2 . Model formulation

Consider an FMS consisting of M ma,chining sta,tions, a, mater ial ha,ndling syste m (MHS),a loading/unloading station an d a supplementary autom ati c warehouse. A typical exampleof thi s FMS is illustra ted in Fig. 1. T h e conditions of this system ar e as follows:

Each station has c, servers (i = 0,1, - . , M + 1) and jobs are served in a first-come-first-served (FC FS ) order. Service times a t each stat ion a re exponentially distributedwith queue-length dependent service rate, ^(ni), i = 0 ,1 , - , M + 1, where ni denotesqueue length at statio n i = 0 denotes the loading station (for palletking/ refixturingoperations), i = h4 + 1, the MHS, an d i = 1 , 2 , . , M, the machining stations.Job s follow a Ma,rkov routi ng with r (i , j ) , i(or j ) = -1 ,0 ,1 , - -, M , denoting the prob-ability that a job is routed from station i to station j. The notations r(i , 0) andr ( i , -1) denote the routing probability from station i to the loading station for pal-letizing/refixturing operations an d unloading sta tion for depalletizing operations, respec-tively. We model the loa~ding/unloacling tation as the two stations: station 0 modelingth e loa,ding opera tion a nd station -1 modeling the unloa.ding opera tion. We assume th atth e opera,tiona,l ime a,t the unloading statio n is small enough t o b e negligible.

There axe N S pallets simult,a,neously a,va,ila~ble n tJhe shop a,nd th e num ber of local bufferspaces at each sta,tion s unlimited (or there are N s spaces). Th e warehouse has N H boxes(spaces) to s tore arriving jobs until a pallet becomes available. Define N = N S + N H ,called th e "ma ,ximum population in th e FMS." An ac cepted job will b e temporaril ystored in the warehouse a,nd supplied in th e F CFS order to t he shop whenever a palletbecomes empty.

Consider a Poisson arrival process with rate A. If an arriving job finds the num ber ofjobs in t he FM S being just AT, the job is refused entry and lost.Although t he FMS can b e represented by a restricted open queueing network as shown

in Jackson [4], we construct an equivalent closed queueing network ( CQ N) as shown in Fig.2, so that man y results for CQN s can be direc tly a.ppliec1. In Fig. 2, the notation "1/07'represents the inp ut /o utp ut process a,nd is normally called th e "inpu t/outp ut7' statio n withrate A . This I/ O statio n in the CQN represents the "blocked-and-lost" mechanism becauseth e "service" (corresponding to the "a,rrivalV process) at this station is not implementeduntil any "customer" (corresponding to any "empty position" in the FMS) arrives at this

station. The notation "H" a n d the arcLL+'" denote th e automatic warehouse and th e"blocked-and-hold 0" mechanism, respectively - the warehouse is considered as a stationwith infinite service rate a nd we assume tha t the blocking occurs at service initiation (thismeans "blocked-and- hold 0"). The arrow with "m " denotes the transportation through theMHS (station M + 1).

Th e flow equa.tions corresponding t o Fig. 2 axe given a.s follows:

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Capacity Design in an F M S

Figure 2. A queueing network model of the FMS with a supplementary warehouse

where ei denotes the visit ra tio of an arriving job to st ation i , i = - 1 , 0 , - - , M + 1.Eqns ( l a ) through (I e) have the unique solution, denoted by e:, i = - 1 , O , - . , M + 1,

which represents th e expected visit times of a n arriving job t o each statio n.

If eqn. (I e) is removed, th e remaining equations provide a set of solutions, denoted

by ei,i

= -1,0, - - - ,M +

1, called th e "rela,tive visit ra,tio" of an arriving job. In thi scase, multiplication by an a,rbit rary constan t does not affect t he solution. For instance , the

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Figure 3 . A simplified queueing network model of theFM S

following relative visit ratios, denoted by qi,i=

- 1 , 0 , - - , A4 + 1, a lso sat isfy eqns ( la)through (I d ) (eo corresponds to qo+ q- 1 in this case):

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Capacity Design in an FMS 41

h4Using these ratios which sa,tisfy q; = 1, we ca,n t ransform th e or iginal FM S m odel

i=-lshown in Fig. 2 to th e equiva,lent one shown in Fig . 3. Th is simplified CQ N is helpful tointui t ively understand the beha,vior of th e FMS because the MHS (s ta t ion M +l ) s explicitlyrepresented and there is a one-to-one relationship between th e s ta t ions in th e model a nd th e

physical system. In Fig. 2 , there is no loop at each s ta t ion , tha t is , the dest inat ion ofeach t ranspo rtat ion through the MHS cann ot be the just depart ing s ta t ion. However, thesimplified CQN shown in Fig. 3 permits a loop through the MHS at s ta t ion i ,i = 0 M ,that is , any job can return t o the just depart ing s ta t ion through th e MHS , denoted by M+ 1in Fig. 3. This is no problem for performing th e equilibrium probabili ty of any queu e lengthsand then the th rough put of th e FM S, because eqns (2a) through (2d) provide the rela t ivevisit ratios satisfying eqns ( l a ) through (I d) ; these equat ions hold for e~= (qQ+ q-l)eM+land ei = q m + l , # 0. Similar CQN models for an FM S with no w arehouse are presentedby Solberg [l41 a,nd Naga,sa,wa, Jeon g a,nd Nish iyam a[S].

Since the 11 0 s ta t ion has the sa ,me vis it ra t io as s ta t ion-1 has a nd s ince s ta t ion -1 hasan infinite service rate, we combine these two station s t o ma,ke a single statio n w ith servicera te A a,nd visit ra,tio e-1. T he st atio n is enclosed by the da shed line in Figs 2 an d 3. T he(blocked-a,nd-lost" mec hanism in the a,rriva,l process is com pletely represented by this1/0sta t ion.

T he difficulty in ob ta,ining he equilibrium proba,bili tyof the queue lengths a t each s ta t ionlies in incorporating th e "blocked-a,nd-hold 0" mecha,nism at th e w arehouse. Unfortunately,we can not give an exact formulation of the equilibrium probabili ty in this case but derivea,n app roximation.

Firs t , we consider th e ca,se NH = 0, trha,t is, the re is no w areho use a,nd no "blocked-and-hold 0" mechanism. T h e equi libr ium probabi l itywhere 11 E 1 2 - ~ , H O , - , 7 , ~ - f + ~ ) , s exa,ctly derived a,s

of the queue leng ths denoted by P ( n ) ,follows:

where e (e-1, eo, - - , a nd ( k ) Am i n ( k , l ) for k 2 .

I t should be noted that the ei7S in these equa,tions axe relative visit rakios so t ha t theefficient co m pu tat ona,l algorith ms provided by Buze n 121 can be explo ited for ca lculatingthe value of G ( M +3, N , e , A); Dubois [3] , and Buzaco t t and Yao[ l ] se the unique value e*(instea d of e i) , which makes it h ard to tu ne th e value of ei 's in orde r to avoid overflow an dcom putat ional error.

Th e th roughput of s ta t ion i , TH ;(N ,A ) , the th roughput of the FM S, THF M s(N, A), andthe th roughput o f the CQN ,THcQn( N , A ) are derived as follows:

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where the relative visit ratios, denoted by e, are scaled asE

go , , EM+l) andhl+l M+lE ei/{ V e j } so tha t Ei = 1 holds.

j=-l %=-lTherefore, we get

tha t is, the throughput of th e FMS isL l t imes as large as the throug hput of th eCQN.We sha,ll provide m ot he r representation of t he FMS model using th e following function:

where

Th e func tion T H ( n ) denotes the th roughput of the FMS provided tha t th e populat ionin the FMS is always ?%(inhis case, N H = 0 a,nd therefore the population in the shop isalways 12). Using this function, we can simplify the FMS model as shown in Fig.4, where

stations in the FMS are a,ggregated intoa

single sta,tion with service ra te governed by th eexpression of min{TH(n), TH (Ns)}.From th e product form solut ion given by eqns (3a ) throug h (3c) , th e equi libr ium prob-

ability tha t t he num ber of jobs in the FM S is exactly12 , denoted by P (n ) , is given by

Similar results have been derived by B uzacott an d Yao[ l ] ,nd Shanthiku mar and Stecke

[g]; the only difference is that they defined the func.t,ionTH{n) as the throughput of theCJQN instead of the throughput of the FMS.

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CapacityDesignin an F M S

Figure 4. A simplified expression of the FMS with = 0

Since P ( N ) denotes th e proba,bility tha,t th e 1/0 station is idle, the throughput of theFMS is also given by

THFMs{N, \)=\{l - P(N)} , (10)which is equivalent to the expressions in eqns (5a) through (5c).

Let us consider the case NH > 0, next. It is intractable to formulate the "blocked-and-hold 0" mechanism exa,ctly. Ther e a,re numerous a pprox imatio n meth ods dealing wit hvarious blocking phenomena between each connected pair of stat ions with limi ted local buffercapacities [6] [7]. In our case, blocking occurs only at the warehouse when the population inthe shop reaches the limit NS and no blocking occurs at an y station in th e shop because ofthe unlimited local buffer capacities. This type of blocking has not been dealt with by anyresearcher.

We present the following approximation, denoted by ~ ( n ) , o get th e equilibrium prob-ability P ( n ) for t he ca,se of NH > 0:

Using ~ ( n ) , e define the approximate throu ghput of t he FM S as

While these equations give the q pr ox im at e values for th e equilibrium probabilities an dthe throughput of the FMS, th e exact va,lues a,re also obtaine d thr oug h t hes e equations in

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4 4 11. Nagasawa

Table 1. Accuracy of the approximation compared with the SIMAN simulation run

SIMAN0.2930.3470.3470.6410.6970.6971.3671.3851.3850.2600.3480.3500.5890.7010.6991.3121.3901.3920.548

0.6970.6931.2611.3841.407

SIMAN : the throughput of the FM S obtained by the simulation runCQN : the ap proxim ate throughput of the FMS calculated usingeqn.(12)

CQN0.296

Error0.9

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CapacityDesign in an F M S

Table 2. Accuracy of the approximation compared with the SIMAN simulation run

( A / { E ^ ~ < - i / E - ~ ( ~ ; / P ; ) } 0.9 )

( N s ,N n ) SIMAN CQN Error( 5, 0) 0.340 0.341 0.3

Er ro r s100 X (CQN-SIMAN)/SIMAN [%l

SIMAN:

the through put of the FM S obtained by the s imulat ion runCQN : the approximate throughput of the FMS calculated using eqn.(12)

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4 6 11. N;~gasa a

Table3 . Accuracy of the approximation comp ared with t he SIMAN simulation run

Error~lOO (CQN-SIMAN)/SIMAN[% lSIMAN: the throug hput of the FM obtained by th e simulation run

Error1.21.00.61.00.01.21.00.70.9

CQN: the app roxima te throug hpu t of th e F'MS calculated using eqn.(12)Figurel . A typical FMS witha supplem entary warehouse

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the following cases: ( l ) he FMS lnoclel which consists of only one machining station withrate / ~ i 72 ) = min{/i,i 12) ) /bi (Ns)} provided th at a, loading/un loading S tation and a MHSlla,ve infinite service ra,tes; (2 ) the FMS model in which th e numb er of available pallets N sis infinite (or very la,rge); (3 ) the FWfS model with ATH = 0 (no supplementary warehouse).

It is lla2rd to invest igate t h e accuracy of t his a ppro xima tion for all cases of t h e FMS.Instea,d) we provide some compa,rison with th e SIMAN simulation r un , where th e conditio nsare the same as given in t.1lis section except that the process route of arriving jobs arepreviously given rando mly so th at each job visits all stat ions only once (ther efore) e: = 1) =l M ) . We set p ~ ( n ) / L A ~ + ~ ( R , ) CO) p - ~ ( n ) = Amin{n) l } and pi(n) = pi min{n) ci}) i =. .l Ad, where e r / ( / i i ~ i ) e : / ( / ~ i ~ j ) ) # j , f,, 1 = l Ad) for ba,la,ncing workload. Th e arrival

114 A4

ra,te A takes the following va,lues: A/{x; / x ( e ; / / l j ) } = 0.7, 0.9 and 1.0, where th e termi=l j=l

A 1 A4

ci / (e;/Pj) represents the limit va,lue of the t l ~ r o u g h p ~ ~ t f th e F'MS because the A *i= l j=l

A I 44

defined in Theorem l ca,n not exceed t he value of c;/ x ( e ; / p j ) for any configuration of

run by dividing 1O)OOO [jobs1 by t he t ime interval between t he arrival of

and the total work

SIMAN simulation

th e 1)OOlst job an dthe departure i f the 11 )000 th ob. Tables l through 3 summarize the results) making it cleartha t th e approximation gives the precise values (only 1.2% error at most) for the thro ughp utof t he FMS wit11 ba,la,nced oading. We a,lso investigated t he accu racy fo r some cases of theFMS with unbalanced loa,cling. The resul ts) omit ted here) are similar t o th e case of balancedloading (we cannot generalize th e results because we can not perform all cases of t he FM S).

Similar approx imati on metho ds lla,ve been presented by S han thi ku mar a nd Stecke [g] forthe case NH = CO and suggested by Yao [l51 and Yao an d S han thi ku mar [l71 for the caseATH > 0.

3. P r op e rt ie s o f t h e t l ~ r o u g l ~ p u t

We derive propert ies of the t l ~ r o ~ ~ g h p ~ i t ~ f th e FMS giveb in section 2. These propertiesare exploited for ma,king opti111a1 design a,s will be sl~ own ater.

T l ~ e o r e i ~ ~. C o n s i d e r t h e F M S m (o de 1 w i t h t h e w a r e h o u s e d e f in e d in s e c t i o n 2. S u p p o s es e r v ic e r a t e o f e a c h s t a t i o n s a t is f ie s /Li(n) = /li(ci) f o r n 2 ci a n d p i ( n ) 5 Pi(ci) f o rn < c i ) i = 0 , l ) - A 4 + l . T h e n , t h e a p p r o x i m a t e t h r o z ~g h p u t f t h e F M S de f ined zn eqn .( 1 2 ) ha s t h e fo l lo w zng l i mz t s :

(i) lim TTFAlS(Ars, H) ) = T H ( N S ) :A d m

( i i ) lirn lim T ~ ~ ~ ~ ~ ~ ( ~ ~ T ~ , A T ~ , A )A *;Ns-m A-m

( ii i) l im =n~ in( A,A* );Ns-m

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A* E min {/Li(ci)/e:}.O

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CapacityDesjgn inan FMS

if ps < 1) then T ~ ~ ~ ~ ~ ( N ~ ,H) A) +- A ( < TH(Ns)) as NH -+ m ;

if ps > l ) then F H ~ ~ ~ ( N ~ ,H, A ) + Alps(= TH(Ns) < A ) as NH +- m .

Therefore, for any ps) TxFMs(NS) H) A ) --+ min{A) TH(Ns)} as NH +- m . Q. E. D.Although t he exact throug hput of t he FMS is not analytically tracta ble) considering the

practical and physical meaning of these lim its) we can verify th at t he exact through put ofthe FMS approaches these limits shown in Theore111 l . It is therefore imp ort an t t o note th a tboth th e approximate and exact throughp uts of th e FMS have the sa me limits as given inTheorem l . This remark implies th at the approximation provides a precise measure for th ethroughput of the F'MS if a t least o ne of t he following conditions holds: ( l ) ery large arrivalrate; (2) a sufficiently large amount of pallets available in the shop; or ( 3 ) sufficiently largecapacity of the warehouse.

Th e limiting behavior of th e throughp ut with respect t o Ns an d NH can be determinedusing the first- and second-order properties given as follows:

Tlieorem 2. Consider the approximateFMmodel definedin sec t ion2. uppos thatservice rate of each stat ion,P ~ ( I ~ ~ ) ,s nondecreasing and concavein the queue lengthn i , i =0, l , . ) M + l . T h e n ,( i ) T ~ ~ ~ ~ ( N ~ )H) A ) is nondecreasing concavein NH;(ii) TxFMS(Ns, N - Ns) A ) is nondecreasing concavein Ns;

(iii)T ~ ~ ~ ~ ~ ( N ~ )H) A ) is nondecreasingin Ns.

(Proof)

Consider a CQN which consists of two nodes. Node l is the approximate FMS withmean service rate y(n) such that y(12) = min{TH(n), TH( Ns) } for any n 2 . Node 2 isthe 110 node with service ra te A. This two-node CQN represents the ap proxim ate CQN withNH > 0, where the equilibrium probability is defined by eqns (l l a ) and ( l l b ) .

Consider, first) two networks (p = l , 2) such th at N; = N an d N& < N& = N&+ 1) hatis, N1 = N; + N& < N; + N& = N2 = N1 + l . Th e queue-length dependent service ratesat node l in these networks are the salne, i. e., ?l(n) = -y2(n) = y(n), because N = N ;.Th e two networks differ only in job populat ion.

Since all stations in the FMS have service rates) pi(n), i = 0 , l ) ) M + 1) which arenondecreasing concave in 12, the tllrougl~put of the FMS with population n, denoted byT H ( n ) ) is also nondecreasing concave in n as shown in Sha nthiku mar an d Yao [l21 [13].Tha t is) $n) is nondecreasing concave in n. Hence, th e throughput of t he CQN underdiscussion is also n~ndecrea~sing oncave in N.

The t l~ rou g l ~pu t s f nodes l and 2 in the CQN are the same and equal to a half of thethrough put of the CQN - "throughput of a CQN)' is usually defined as th e sum of through -puts of all nodes included in the CQN like th e definition given in eqn. (6a) . Therefore, the

throughput of node l in the CQN, corresponding to TxFMS(Ns) NH7 ) ) is nondecreasingconcave in NH.

To prove (ii )) consider two networks (p = 1) 2) such that N; < N = N; + l andN > N$ = Nk - 1) that is) N1 = N; + N& = N; + N = N2. These networks differfrom each other only in the queue-length dependent service rates. If we set T H ( n ) = n p )that is) ~ ( n ) p min(n) NS)) then we can find th at the throug hput of t he CQN ) and hence-THFMs(Ns, N - Ns) A )is nondecreasing concave in Ns as shown in Shanthi kumar an d Yao[ l l ] . Based on th e proof of this special case) we can derive the same result for the generalcase ~ ( 1 2 )= ~l ii n{T H( n)) H(N s)} . Th e proof is given in Tll eore ~n in appendices.

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N-NH-l N-NH N- NH +~ N - l N N + l

The number of available pallets, Ns

Figure 5. Illustration for explaining the second order property of T H F M s Ns , PTH, A)with respect to N s

Property (iii) is directly obtained from (i) and (ii). That is,

where the first and second inequalities are due to properties (ii) and (i), respectively.Q. E. D.

We can show that the concavity of T ~ % ~ ~ ~ ( N ~ ,H ) A) in Ns is not likely to hold. A s

illustrated in Fig. 5 ) th e coi~cavi ty s expressed by th e concavity of t he curve G - E - C.From Theorem 2 (i) and (ii)) the curves A - B - C , D - E - F an d G - H - I are allconcave, and XD 5 E,- 5 EZ and E T 5 m. Therefore, if ZD 2 E!? E T holds,the curve G - E - C becomes concave. If NH is small enough and Ns is large enough, then- - pthe relation AD 2 BE 2 C F , and therefore, the concavity of T ~ ~ ~ ~ ( N S ,jy, A ) in Ns islikely to hold. However, if NH is very large and N s is very small, the relation does not holdany more.

Combining Theorems l an d 2 yields t he following corollary:

Corollary l . C o n s i d e r t h e a p p r o x i m a t e F M S m o d e l w i t h t h e w a r e h o u s e d ef i ne din s e c t i o n2. Suppose tha t s e rv i ce r a t e pi(72i) i s nondecreas ing concave in 72 i a n d /ii(ni) = pi(ci) forni > ci , i = 0 , l , ) M + l . Th,en,

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Capacity Design in an FMS

-

Th e number of available pallets, N S

Relationships between the approximate throughput of theFMS and i ts bounds

( i) T H ~ ~ ~ ( N ~ ,H , A) i s nondec reas ing concave in NI{ and conve rges t o min{A, T H ( N d }as NH --+ m;

(ii) % F h f s ( ~ S , o , ~ ) ~ F M S ( N S , N H , A ): ~ \ ~ { = F M S ( N S+ N H , ~ , A ) , T H ( ~ s ) ] ;

(iii) T - ~ ~ N ~ , H , A) i s n o n d e c r e a si n g in NS and conve rges t o min(A , A*) w h e r e A* Emino

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W

T he n umb er of available pallets, N S

Figure 7. Relationships between the approximate throughput of the FM S and i ts bounds

(A * < A )

4. Capacity design problems

We provide solution methods to a few examples of ca pacity design problems for determin -ing warehouse capa,city, NH, and t he number of pallets available in th e shop , NS. Supposetha t t he service times of bot h t he loading/unloading station and t he MHS are small enoughto have a negligible effect on the t hrou ghp ut of t he FMS. Machining sta ti on i has ci-parallelservers (machines) with queue-length dep endent service ra te pi{n) = p min(n, ci), = 1 M .

We consider th e two cases of th e FMS: a single-station FMS with one machining st ati onand a multiple-station FMS with several machining stations. The equilibrium probabilitygiven by eqns (l l a ) and ( l b ) is.approximation for the multiple-station FMS but exact forthe single-S ation FMS. Therefore, t he following differences between t h e single- a nd multiple-station FMSs are imp orta nt in developing the solution methods: (1) Th e throughput of theFM S defined by T > ~ ~ ( N ~ , fi A )is exact for t he single-station FMS bu t approxi mate for

the multiple-station FMS; (2) Th e function T H ( n ) defined by eqns (8a) and (8b) does not

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Capacity Design in an FM S 5 3

have a,ny brea,k point except at 72 = NS in th e multiple-station FMS while in the single-stationFMS the function TH(72) = p min(n , c) has the break point a t n = c a n d T H ( n ) = T H ( c )for any n > c if c < NS, where c denotes th e numbe r of parallel servers at t he machin ingstation in th e single-station FMS.

In both cases, Theorems 1 and 2 and Corollary 1 are very useful for obtaining t he o ptima l

solution efficiently.

4.1 A s ing l e - s t a t i on F M S

Consider-first the single-station FMS. Note that P{n} given by eqns ( l l a ) and ( l l b )provide the exact value for P ( n ) in this case. Substituting T H ( n ) = pmin(n ,c ) to eqns( l l a ) and (l l b ), we get th e exa,ct expression for P ( n ) a,s

^ -p (n - ) , if 0 5 n < min(Ns,c) ;(I3a) P ( 7 2 ) =

f&p(72 - 1)) if m in( Ns, c) n< NS + NH;

( 0 ) if Ns+ NH < n ,

where Nc = min(Ns, c) and p = Alp.Then, t he exact expression for the throu ghpu t of th e FMS is also obtained as

Note that in this case we use THpMs(Ns, NH, A ) instead of using ~ F M S ( N S , H , )because the eqn. (12) gives th e exact values for t he t hroug hput of t he FMS. It is obviousth at the throu ghput of the FMS expressed by this THpMsiNs, NH, A) satisfies Theorems 1an d 2 and Corollary 1. From Corollary 1 ( i i ) , the lower and up per bound s of th e throughp utare obta ined as follows:

In this single-station FMS, we consider the following design problems:P I : max {THFMS(NS, H,A) NS + NH 5 N and NS O,NH > O},

N S J HP 2 : ma,x {THFus(Ns, NH,A) a N s + bNH 5 A and NS 2 0, NH 2 }.

N s , N H

Constraints in these two design problems repre sents th e two kinds of tradeoff between th ewarehouse capacity an d the number of available pallets. Th e constraint in t he first problemmeans the space tradeoff under the fixed total spaces N, and the constraint in the secondone, th e cost tradeoff u nder th e fixed tot a,l available investment A for installing the spaces.

On t he basis of Theo rems 1 an d 2 and Corollary 1, we provide solution methods to theseproblems below.

S o l u t i o n t o P I

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Since THFh4s(Ns, NH, A ) is nondecreasing in both Ns an d NH from Theorems 2(i)and 2(iii), th e optimal values for NS and NH, denoted by Ng an d Nu, respectively, satisfyN$ + NJr = N. Hence, we ca,n transform the original problem P 1 t o

From Theorem 2(ii), N$ = N is clearly one of the opt ima l solutions to P i / and thenP i . On the other hand, when c < NS < A^,substituting Nc = c to eqn. (14) ) we get for anyNs ^ [c, NI

Ns+N~-c-l

( 1 6 4E; P"/"! + (pC/c') E':;!)

THFMs{NS,NH,\) = A NS+NH-c (plc)'~ s ' o'/"+ (pC/c!) E'=!)

This equation implies that any NS G [c, NI is also optimal.If N < c, then NS < c. Substituting Nc = NS to eqn. (14), we get

This equation shows that the useful rela,tion represented by eqn. (16b) does not holdin thi s case. We know, however, that THFhls(NS1 N - NS, A ) is nondecreasing in NS.Therefore, we get N$ = N.

In conclusion, we get the optimal solution to P i , denoted by N$ and NJr, as any NSand NH satisfying

min(c, N) < f i < N and NH = N - ATs.

S o l u t i o n t o P 2

Using an approach simila,r to that used for solving P i , we get Nu = \(A - aNz)/b]where 1x1 denotes the maximal integer being less than or equal to X. Hence, the originalproblem can be transformed t o

There are three possible cases:(i ) a = b

In this case, the constraint aNs + bN H < A is equivalent to NS + NH < M a j . Fromthe results obtained for P 1 we get NZ and Nu as any Ns satisfying

min(c, N) < Ns < N a,nd NH = N - NS, where N = [A/a].(ii) a < b

In this case, Ns + NH = Ns + \ ( A - Ns)/b] < N = \A/a\ and NS = N (and NH = 0)is a feasible solution. From the upper bound of th e throu ghpu t given in eqn. (15) an d thenondecreasing p roperty of THFns(Ns, 0, A ) in NS, we get THFMs{Ns, [(A- aNs)/b\ , A ) bIn this case, the total space NS + N H is maximized when NH = [A/bJ and NS = 0.

However, ATs = 0 implies th at the throug hput of th e FMS is zero. On th e other hand, while

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Capacity Design in an FM S 55

the maximal value of f i s [A/a] ( 5 A / b ] ) and then NH = 0, the to tal space NS +NH canbe increased by sacrificing the space of pallets NS. This fact implies th at th e throu ghpu t ofthe F M S may b e increa,sed in this ma,nner.

Consider the case c 5 [A/a] first.For C 5 Ns 5 [A/aJ , eqn. (1611) holds and Ns + N y 5 c + \(A - ac)/bj with equality

when NS = c and NH = \ (A - ac) /b] . Hence, we get

with equality when Ns = c and NH = [(A - c)/b].For 0 5 Ns < c, we do not know whether th e throughput T HWs(Ns, \( A - Ns)/ b] , A)

is less than THp Msi c, \(A - c) /b j, A). However, from eqn. (15) we get for an y NS < Nl -

where NI = \ T H p ~ s ( c [(A - c)/b ], 0, A)/ ] + 1.I t is clear that NZ exists on [Nb c] and is obtained by solving

P 2 " : max{eqn.(17) \ NI 5 NS c and NH = [(A - Ns)/b)Ns

Consider the case \A/aj < c. Obviously, NS N2 - 1, we get

where N2 = \THFMs( A/a], 0, A ) / ^ ] + 1.It is a,lso clear tl ia,t NZ exi sts on [N2, A/ aJ ] a,nd is obtaine d by solving

P ~ w max{eqn.(17) 1N2

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where THFMs(0, 0, A ) = 0.These recursive equations are stable in the sense that computational erro r is not increased

by the successive iterations as shown in Theorem C (see appendices: T l f o n Theorem Cshould be replaced with p 72 in this case). The optimal solution to problem R P 2 is obtainedefficiently by calcula,ting th e values of THFMs(Ns, \(A - aNs)/ b\ A ) using these recursiveequations for each value of N g on [ N 3 ,N4].

4.2 m,ultiple-station F M SWe consider, nex t, th e following capacity design problems in t he multiple- sta tion FM

denoted by M P 1 and M P 2 similar to P I a,nd P 2 , respectively:

M P 1 : rnax { T I ~ ~ ~ s ( N ~ , A T ~ , A )Ns+NH 5 N a n d Ns >O,NH g } ,N s , N H

MP2 : max { T H F M ~ ( N ~ , , ) 1aN s + bNH 0, 2 ) .N s , N H

In these case, it should be noted tha t eqns ( l l a ) , (l l b ) and (12) provide th e approximationfor the throughput of the FMS, and that TH(n) given by eqns (8a) and (8b) does not havea,ny break point except at 12 = h . onsidering these facts, we provide solution methods toproblems M P 1 and M P 2 below.

Solution to M P 1For problem M P l , the optimal solution is Nt = N a.nd N& = 0; in th e multiple-station

FMS, from the a,bove fact (2 ) , we can not get a,ny relation similar to that shown in eqn.(16b) for c < Ns < N .

Solution to M P 2There are three cases to consider:

(i) a = bFrom the argument similar to tha t in case (i) of P 2 and the result to M P I , we getNZ = N and NE = 0 where N = \A/aJ.

(ii) a < bThe derivation of th e optimaJl solution t o t he case (ii) of P 2 is also applicable to this

case as well. We get NZ = N and Nu = 0 where AT = \A/aj .(iii) a > b

Since there is no break point in th e function T H ( n ) for n < NS, we can not specifythe upper bound of NZ less than ~ A / u ] nlike case (iii) of P 2 . Using Corollary 3(ii) indetermining the lower bound of NZ yields

N5 = rnin{Ns 1TH(Ns) 2 T H ~ ~ ( \ A / u ~ ,, A ) and 0 < NS < \A/u]}.The n, th e original problem is transformed into

The value of T > ~ ~ ( N ~ ,TH, A )is easily calculated by th e following recursive equations:

for NH 2 0 and Ns 1,

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Capacity Design iii an FMS

for Ns > 0,a,nd TH~-M~(O), A ) = 0.

Th e function T H ( n ) is defined in eqn. (8b) an d the value of G ( M + 2, n, e ) in TH (n ) isefficiently obtaine d by Buzen's algori thm [2] using any scaling for ei, i = - , 0 , , M + 1.The above recursive equations are stable as shown in Theorem C(see appendices). Theoptimal solution to problem MP21 is obtained by the efficient manner similar to that usedfor solving problem R P 2.

5. Conclusions

We extended the existing FMS models through the addition of a supplementary auto-matic warehouse with finite capacity. Since the equilibrium probabilities of queue lengths a teach station and t he th roughput of th e FMS can not b e exactly formulated in this case, weprovided approximate expressions for the probabilities and the throughput, using a closed

queueing network model. We showed the condition that the approximations yield the exactvalues. Comparing with t he SIMAN simulation run, we also made clear tha t t he accuracy ofth e approximation is very high especially in th e FMS with balancing workload. We derivedthe limits, and the first- and second-order properties of the t hroughpu t with respect t o bot hth e warehouse capacity an d t he number of th e pallets available in th e shop. Exploiting theseproperties, we proposed efficient solution methods to several capacity design problems.

Capaci ty design problems presented here are essential ones and can be applied t o variousversions of these problems; for instance a design problem to maximize a profit functioninvolving bot h t he warehouse capacity and t he nu mber of available pallets. Furthermore,th e following capacities should b e modeled t o b e det ermine d in design problems: (1) capacityat each statio n; the number of machines a t each machining stat ion, th e number of cart s in th eMHS, and th e number of machines (or labors) and fixtures at th e loading/unloading stati on;(2) capacity of local buffer spaces at each stat ion . Solution met hods t o design problems withthese capacities will be topics of future research.

AppendicesTheorem A. C o n s i d e r a c lo se d q u e u i n g n e t w o r k (CQN) w i t h M s t a t i o n s a t w h i c hserv ice t im,es a re expon,en ,t ial ly d is t r ibu ted wi th serv ice ra tes , / ^ (K; ) , i = 1 , 2 , - -, M , w h e r eK; d e n o te s t h e q u e u e l e n g t h a t s t a t i o n i. J o b s f o ll o w a M a r k o v r o u t i n g w i t h y(2, j ) , i(or j) =0,1, - - ,M', w h e r e s t a t i o n 0 d e n o te s a d u m m y s t a t i o n re p re s en t in g a n in p u t / o u t p u t s t a t i o nw i t h a n i n f i n i t e s e r v ic e r a t e . T h e e xp ec te d v i s i t t i m e s o f e a c h J ob ,e L i = 1 , 2 ) - , M , are

o b ta i ne d a s a u n i q u e s o l u t i o n t o t h e f l ow e q u a t i o n s s u c h t h a tM

ei = 7(O,i) + Y, n(j , i ) for i = 1 , 2 , . - . M .j= l

D e fi n e t h e s y s t e m t h r o u g h p u t a sM G ( M ,N - , e*)TH(N) E T H i ( N ) y ( i , 0) =i=1 G ( M , AT, e*)

where N a n d TH;(N) a r e t h e p o p u l a t i o n in t h e C Q N a n d t h e t h r o ug h pu t o f s t at i o n i ,respec t ive ly, and

G ( M , N, e * )

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If li,(12) = pi(Ni) fo r n >. Ni, a n d p;(n) 5 p i ( N ) f o r n < Ni, i = 1 , 2 , - , M , t h e n

lim TH{N)N-.m

(Proof)

Schweitzer [g] has proved the above result for th e special case of p i (n ) = p; for i =

1,2, , M. We shall generalize his result below.Consider two networks, denoted by superscript p = 1, 2, with service r ates ^{n) 's such

thatd ( n ) = pi (Ni ) l{n >. Ni} a n d f i n ) = f i (Ni) , = 1 , 2 , . - , M,

where l{ x] is an indicator function; l{x} = 1 if X is true and 0 otherwise.Since for any n

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CapacityDesign in an FM S 59

it suffices to prove D1(t) + D 4 ( t ) 2' ~ ( t )D3 (t ), where DP(t) = (D?( 7;; then ? l 2 = 7!, and 73 4 = $, a,nd hence A? 2 A \ , A: 5 A!(b ) 7: > 7; and 7; 5 71 ; then 7" = 3 nd 73 4 = 74, and hence A% A} , A: 2 A:;(c) 7; 5 7; and 74 5 7;; then 7" = 7: and i34 = $, and hence A: > A}, A: > At;(d) > 7: and 74 > ?;.

Case (d) is impossible because if z? > c + 2 then c + 2

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1 1 2 2 2 2and if 24 5 c + l then 71 - 7; = ( z l ) - (z?) 5 7?(z;) - l (z l ) = -yl (zl + ( 2 ; - g))2 2 2 4 2 4 2 3 2 4 3 3 4 4 3 4

71 (2115 7, (21 + (2; - 4 ) ) - 71 (2 1 5 71 (21)- 71 (21) = 71 (21) - 71 ( 2 1 ) = 71 - 71Therefore, we get(a ) A: + Ay - A} - A t

=

1 { ~ ~ ~$-

u k

0;(b) A2 +A; - A} - Af- 1{R13 - 7; - $) 5 U K RI3}- {R" - 7; - $) 5 U & R"} > 0;These results imply that d}+ df 5 d; + 1;

i i ) + l , 0, A ) = A T H ( N s+ 1) for N s >. 0T H ( N s+ 1) + A - % s ( ~ ~ s , , ~ )an d -S(O, 0, A)= 0;

1A(NS, NH)1and - A ( N s , N H - ) 1< -* l A(Ns ,O) l5 - *THpMs(Ns, NH,A ) THpMs(Ns , NH- 1, A) TH FM S(N S, 0, A )

where "*" a n d A ( N s , N u ) d e n o t e t h e e x a c t v a l u e w i t h o u t c o m p u t a t i o n a l e r r o r a n d t h ecomputa t iona l e r ro r f ro m the ca l cu la t ion o f T > ~ ~ ( N ~ , H ,A ), respectively.

(Proof)We show theproof in an argument similar t o tha,t in Yaoand Shan thikum ar [16]. From

the definition of B (N s, N H ), we get

and

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C7apa~,ityesign in anFM S 6 l

Substituting T H F ~ ~ [ N ~ ,H , A ) = A{1 - B ( N s , h ) } nto these equations yields (i)and (ii).

Define A(Ns, NH) m d ~ s , H, A) - r f f d ~ s , ^, A) = ^ B * ( N s , N H ) -B( N s, NH )}. Since both B(ATs, NH ) and B *(N s, N H ) satisfy the above equations, we get

and

Since T & ~ ( N ~ , N H ,A ) ( and then B*(Ns , NH) ) is nondecreasing (nonincreasing) inboth N g and N H , we can show tha t (iii) holds. Q. E. D.

References

Buzacott J. A. an d Yao D. D.: On Queueing Network M odels of Flexible Man ufa ctu ringSyste ms . Queueueing Sys tems , Vol. 1 (1986), 5-27.Buzen J. P.: Co mp uta tion al Algorithms for Closed Queueing Networks with Expon entialServers. Communications of the ACM, Vol. 16 (1973), 527-531.Dubois D.: A Mathematical Model of a Flexible Manufacturing System with LimitedIn-process Inventory. Eur. J. Oper. Res., Vol. 14 ( 1983) , 66-78.Jackson J. R.: Jobshop Like Queueing Systems, Mgmt.Sci. Vol. 10 (1963), 131-142.Nagasawa H., Jeong S. C. an d Nishiyama N.: A Generalized Repre sentatio n of Flexi-ble M anufa cturing Systems Using a Closed Queueing Network Model. Bulle tin of theUniversity of Osaka Prefecture (Series A ) , Vol. 38 (1989 ), 113-132.

Onvural R. 0 . : Survey of Closed Queueing Networks with Blocking, ACM ComputingSurveys, Vol. 22 (1990), 83-121.edited by Perros H. G. and Altiok T .: Queueing Networks w ith B locking, North-Holland(1989).Schweitzer P. J. : Maximum Throughput in Finite-Capacity Open Queueing Networkswith Pr od uc t-F or m Solutions. M gm t. Sci. Vol. 24 (1977), 217-223.Shanthikumar J . G. and Stecke K. E. 1986. Reducing Work-in-process Inventory inCertain Class of Flexible Manufacturing Systems. Eur.J. Oper. Res. Vol. 26 (1986),266-271.Shanthikumar J. G. and Yao, D. D.: The Preservation of Likelihood Ratio OrderingUnder C onvolution. Stochastic Process an d The ir Applications, Vol. 23 (1986), 259-267.Shanthikumar J . G. and Yao, D. D.: Optimal Server Allocation in a System of Multi-server Stations. Mgmt. Sci., Vol. 33 (1987), 1173-1180.Shanthikumar J. G. and Yao D. D.: Second-Order Properties of the Throughput of aClosed Queueing N etwork. M ath. Oper. R es., Vol. 13 (1988 ), 524-534.Shanthikumar J . G. and Yao D. D.: Second Order Stochastic Properties in QueueingSystems. Proceedings of the IEEE, Vol. 77 (1989), 162-170.Solberg J. J.: A Mathematical Model of Computerized Manufacturing Systems. 4thIn ter na tio na l Conference on Prod uction Research, Tokyo (1977), 1265-1275.Yao D. D.: An Optimal Storage Model fora Flexible Manufacturing System. Flexible

Manufacturing Systems; Methods and Studies, A. Kusiak (eds.). North-Holland (1986),113-125.

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[l61 Yao D. D. and Shanth ikumar J . G.: Th e Op t ima l I npu t Ra t e s t oa Sy ste m of Manufac-turing Cells. I N F O R , Vol. 25 ( 198 7), 57-65.

[l71 Ya,o D. D . and Shanth ikumar J. G.: Allocating a Jo in t Se tu p in a Multi-Cell System.Annals of Ope rations Res earch, Vol. 15 (1988), 155-167.

Hiroyuki Na,gasa.waDepartment of Industrial EngineeringCollege of EngineeringOsa,ka Prefecture University1-1 Gakuen-cho, Sakai, Osaka,580, J a p a nE-mail: ng@ie.osakafu-u.ac.jp