Heavy-traffic inventory—production systems

283

international Journal of Production Economics, 26 ( 1992) 283-29 I Elsevier

Heavy-traffic inventory-production

systems

E.V. Bulinskaya MoscowState University, 119899 Moscow, Russia

Abstract Multi-commodity inventory-production systems with two classes of customers are investigated in heavy-traffic conditions. For the single commodity case the deteriorating items are considered as well as non-deteriorating and optimization is carried out.

1. Introduction We consider a system consisting of a single or several warehouses and a supplier (production centre manufacturing N types of goods). The supplier has ni servers (operators) at the station processing the items of the ith type. Each warehouse holds items of different types to meet the demand. In general the customers demand is described by a continuous-time multi-dimensional integer-valued random process. There are two classes of customers. The ordinary ones, belonging to the second class, demand a single type of item, so this class is formed by N disjoint subclasses. If the stocked items can deteriorate at a given rate or have random life-times, the oldest items are the first issued to satisfy demand. In case of stock-out the ordinary customers wait for the replenishment of inventory (provided by the supplier ) . The most frequently used replenishment policies are the so-called (s,S)-policies (see e.g. [ 1-5 ] ), Thus, from time to time, each warehouse issues an (ordinary) order to the supplier for a single item (in case of (S- I$)-policy) or several items of a given type i. All the orders arriving at the ith station form a queue served according to FCFS (first-come-first-served) discipline. The processing times are random variables (with known distribution). The customers of the first class (impatient ones) arrive with a list of demands. The list in-

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eludes the types of necessary items, that is, a subset z= {i, ,..., ikcz)) ofthe set (1,2 ,..., w, as well as the demanded quantities AZ,, (i~z), Impatient customers can use only new items. Therefore, their arrivals result in emergency orders to the production centre. Such an order is accepted if at least A,, free servers are available at each station i (iez). If this order cannot be accepted, an impatient customer leaves the system immediately. It is clear that the system operation incurs losses, such as production costs, ordering, transportation, inventory holding costs, penalties for unsatisfied demand and customers waiting etc. Our aim is to investigate the system and obtain its optimal performance. As usual, it is possible to choose different objective functions. For example, we can consider the total average costs during a given time-interval or discounted mean costs. The calculation of these objective functions involves the knowledge of the systems transient behaviour. On the contrary, the long-run costs per time-unit take into account only the steady state. The objective function being ascertained, we specify the decision variables (control parameters). There exist many possibilities. Sometimes the ordering policies of warehouses are prescribed including their parameters, occasionally one considers them as variable. Warehouses can cooperate in satisfying the demand or handle their customers separately. As to the production centre, its structure is assumed either fixed or

0 1992 Elsevier Science Publishers B.V. All rights reserved.

284 flexible. In the former case one part of servers at the ith station (i= 1,...,N) is assigned for processing of emergency orders, the rest are dealing with ordinary orders. Then our aim is to obtain the optimal partition for the given set {n,, i= l,...,N) of servers. In the latter case all the servers at a station are available for both classes of orders. Since the emergency orders can be lost it is reasonable to ascribe them high priority. The system structure can be mixed, that is, only at some stations the servers are assigned to the various classes of orders. It is interesting to compare the performance of systems with different structure. One can also vary the demand and/or production rates. For the optimal design of the supply system it is important to study its operation in heavy-traffic conditions. Since our main concern is a large inventory-production system, we suppose that the numbers n, (i= l,..,N) of servers increase together with workload. This assumption enables us to obtain the diffusion approximation. The paper is organized as follows. In Section 2 we tackle the single-warehouse-single-commodity systems for deteriorating and nondeteriorating items. In Sections 3 and 4 we analyse multicommodity systems from the view-point of supplier. The warehouses choose their replenishment policies independently. We control only the operation of the production centre and adjust its facilities to inflowing streams of orders. Section 3 deals with the fixed structure systems, while the flexible systems are treated in Section 4. In Section 5 we discuss other applications of the investigated mathematical models and their various modifications.

ordinary customers is described by a Poisson process with parameter AZ. Suppose that the warehouse uses the (S- 1, S)-policy for its replenishment. Hence, the production station handles two independent Poison streams of orders with given intensities 1, and A*. Let the structure of the system be fixed so that n, operators serve the emergency orders and rz2= n - II, serve the ordinary ones. The processing times for all the items are independent exponentially distributed random variables with the mean p,- ’ for the orders of the ith class. Since two classes of customers are handled independently, in order to analyse such a system we study the corresponding M/M/n, loss and A4/ M/n, queueing systems separately. Intending to use the long-run costs per time-unit as objective function, we are interested in the stationary behaviour of these systems. Denote by c(t) the number of customers at time t in M/M/n loss system with arrival rate II and service rate ,M.(We drop the indices for the simplicity sake. ) It is well known (see e.g. [ 1 ] ) that r(t) is a birth-and-death process. Let (Yk (resp. /&) be the birth (resp. death) rate in the state k. Since ayk=A for k=O,...,n- 1, Pk=k,u for k= l,...,n and oI,=O,/&=O otherwise, the SyStem is ergodic. Its stationary distribution 17,=lim P{C(t)=k} I-00 is as follows Ii’k=17,6k(k!)-’ 17,’ = i ak(k!)-l

system

To clarify the procedure we begin by treating the simplest model. Namely, we consider a system with a single warehouse of capacity S and a production station supplying a single type of nondeteriorating commodity. Let the impatient customers arrive directly to the supplier. Their arrival times form a Poisson stream with intensity AI. Each customer demands a single item. If its processing cannot begin immediately the impatient customer goes elsewhere. The demand of

=

0

l,...,n

(1)

k=O nk

2. A single-commodity-single-warehouse

fork=

otherwise

here 6=;1p-‘. Hence, the mean number g, (n) of the lost customers per time-unit in the steady state is equal to ;117,. Using ( 1) we can write -I

n,W=~$ (k$o$

.I

(2)

Let now r(t) denote the number of customers in M/M/n queueing system at time t. It is also a with &=A, birth-and-death process P k=min( k,n)p for all k>, 0. This system is er-

285

godic if p = &I - ’ < 1. Then its stationary distribution is given by the following expressions: 17k=17,6k(k!)-’

for k
n,=&dn(n!)-lpk--n

for kan

(3)

where c is a Poisson random variable with parameter 6. Under our assumption &CX and 1/2--ta as rz+co, the normalized ran(n-6)6dom variable ([- 6)S- ‘j2 is asymptotically normal with parameters 0 and 1, hence a

ic3k(k!)-1+6”(n!)-1p(l-p)-1

17;‘=

P{[
k=O

It is easy to obtain the mean queue length g, ( n ) in the steady state. Since

-+ J-L

e-X2/2 dx

The result ( 5 ) is an easy consequence of ( 7 ) and the Stirling’s formula. The result (6) follows similarly, since in (4 ) DC’ =esP{C
using (3) we get (4)

Naturally, we desire to choose for each n such a partition (n=nl +n2) that the least possible number of emergency orders is lost, while the ordinary orders are delayed as shortly as possible. Therefore, it is reasonable to take as objective function

where c1 and c2 are the penalties for the lost emergency order and for the waiting ordinary order, respectively. In view of (2 ) and (4)) for fixed A,, ,Ui (i= 1,2) and small n it is easy to carry out the minimization of Y( n 1, n2 ) numerically. For large n it is more convenient to use the asymptotics given by the following theorem. Theorem 1. Let p= 1 -a n -‘I2 for some positive a. Then

(n) -pee

a2,2( 1

e-Xz,2dx)-‘nl,2

-cm

(5) a

(i

g,?(n) -a-2e-“2/2

e-x2/2dx

-cc

In order to use the asymptotical expressions (5),(6) instead of g,(n,), g2(n2) for the minimization purposes, it is necessary to ensure ni+CC (i=1,2)asn-+o9.Onecanassume,forexample, n,=cun, n2=( 1 -a)n, O-CCU<1. The optimal value a0 is also obtained numerically. The necessary condition for its existence is 6, + S2I n. Remark 1. It is interesting to mention that although under the assumption of Theorem 1 the loss probability II,, given by ( 1) tends to 0 as ~l--tco, the mean number of lost impatient customers increases at rate y1‘I2 (see ( 5 ) ). Remark 2. We can also take into account the operation of the warehouse, namely, the mean number of stock-outs g,(n) per time-unit or expected unsatisfied demand g, (n ) in the steady state. To evaluate these characteristics it is necessary to deal with the process c(t), inventory level at time t. It is obvious that C(t) =S-r(t), where r(t) is the number of ordinary orders at time t (as yet not fulfilled by the supplier). If C(t)
-1 +a-le-a2/2

1 -p)-’

This completes the proof. q

g*(n)=17,6”(n!)-‘p(l-p)-2

gl

(7)

nl/2

(6)

as n-+c0. Proof. Rewrite (2 ) as g](n)=M”e-“(P{c
with nk given by ( 3 ).

286 Remark 3. It is possible to treat in a similar way the system with deteriorating items. Assume each stocked item to have exponential lifetime with parameter V. We also suppose that demands arrive according to a Poisson process with parameter 1. The replenishment orders are issued on the (S- l,S)-policy basis. Let c(t) denote the number of customers in the queueing system with n servers. As previously the service times are independent exponentially distributed random variables with the mean ,K ’ . But now the inflowing stream is not a Poisson one, since its rate depends on the state of the process c(t) defined in Remark 2 as warehouse inventory level. Nevertheless c(t) remains the bi~h-and-death process with ak=max (S-k,O)v+ll, /&=min(k,rz)p for all k> 0. As usually (see e.g. [ 61) its stationary distribution is given by

Theorem 2. Under assumption (8) the finite-dimensional distributions of the normalized processes~(~)(~)=(~(ff)~~)-~pt~‘))~-”2converge weakly, as n+co, to the linite-dimensional distributions of the diffusion process q( t ) with infinitesimal generator B given by B=p g

-,u min(x,a)

Remark 4. For more complex systems it is not so easy (sometimes quite impossible) to write the objective function explicitly. Fortunately, there exists an alternative method to obtain its asymptotic behaviour for large systems, namely, the diffusion approximation. In order to illustrate this statement, below we obtain the result (6 ) in such a manner. Let us consider a sequence of M/M/n queueing systems (n= 1,2,...) with parameters depending on n in the following way. The service rate p(“) is independent of n and equal to p for all the systems. On the contrary, the arrival rate A(“)= n,up(“‘, where p(n),

1 ._a n-‘l2

(8)

for some positive constant a. Relation (8 ) , being the assumption of Theorem 1, is called the heavytraffic condition for the system. To underline the dependence on n we write also c(“)(t) for the number of customers in the nth system at time t.

(9)

Proof. It is obvious that v(“) (t) is a Markov process with state space g,,={(k-np(“))n-I/*,

k=O,l,... >

In order to prove the convergence of linite-dimensional distributions of q(“)(t) to corresponding distributions of the limit process q(t) with state space 53= [R’it is sufficient to establish the convergence of probability transition functions. The transition function of the process q(t) induces a family of transformations T’,, indexed by t, defined by (r&(x)

The evaluation of the objective function Y can be carried out along the same lines as in nondeteriorating case.

&

=E.J(rl(t)

)

where E, is the expectation given q(O) =x. These transformations form a one-parameter contraction semi-group of operators on the Banach space of bounded measurable functions on ~3. The infinitesimal generator B of T, is defined as Bj’= lim

v

I-0

for the class 9 (B) of all functionsfsuch that this limit exists in the strong topology. For any f~ $3(B) the function u = T,fis the unique solution of

duft,x)

~

dt

=Bu(t,x)

uto+,t) =_f(x) T: and B, associated with 7’“’ (t) (n= 1,2,...) are introduced in the same way.

Operators

The sets S?n approximate sense that

53, as n-co,

in the

IISlln=~~~lf~~~I-t~~~I~~~~I=ll~II forfbounded at GO.

continuous

on 2 and continuous

287 Convergence from

of the transition

functions comes

llT:f-T,fll.= sup IT:f(x)--T,f(x) I+0 XE9n

(10)

as n+a, where fis a bounded continuous function on 9. The Trotter-Kato theorem (see e.g. [ 71) states C, that from IIB,f-Bfll.-+0, as n&a, for . all ftz core of B, follows ( 10). As easily verified, in our case B nu(tx)=~‘“)[u(t,x+n-I”)--(tx) 7

3

1

+~min[n,(np’“)+xn”*)] x [U(t,X-ti--‘J)

(11)

-u(t,x)]

forx= (k-np’“))n-“2, k= 1,2,... Let f be such a function that u (t,x) has three bounded continuous derivatives in x. Using the Taylor’s formula in ( 11) we readily obtain the required convergence of B, to B given by (9 ). Hence, the proof is complete.

Theorem 2 indeed proof of (6). It is easy to treat systems, n = 1,2, in yields the following

enables us to give another the sequence of loss M/M/n heavy-traffic conditions. This theorem.

Theorem 3. Under assumption (8 ) the limit process q(t) has the infinitesimal generator given by ( 9 ) for x < a and reflecting boundary condition at x=u. The proof is carried along the same lines as that of Theorem 2. Investigation of local times of the limit process q (t ) enables us to establish ( 5 ) .

Remark 5. If instead of (8) we suppose P (n)-+p < 1, the infinitesimal generator B has the form

a2

a

The limit process r](t) has a stationary distribution. Its density n(x) satisfies the relation

P1"s-W$g

B*l7= 0

for all x. Now we consider a flexible system. The operation of supplier is described by n-server system with two customers classes arriving according to independent Poisson processes (with parameter ;li for the ith class). The first (resp. second) class customers receive high (resp. low) priority for service. The priority is of the preemptive-resume type. All the service times are exponential random variables with parameter pi for the customers of the ith class (i= 1,2). The first class customer is lost if at his arrival time all the servers are busy processing the customers of the same class. The second class customers waiting times are unrestricted. In order to evaluate the performance of the system it is necessary to study two-dimensional Markov process (& (t), T2(t) ), where <,(t) is the number of the ith class customers at time t. As in the case of the fixed structure we consider a sequence of systems indexed by n. Let A,(n)= npg,‘“’ , the service rates pi being constant

(12)

where B* is the operator conjugate to B. Writing B* explicitly, we obtain instead of ( 12 ) gy+

& (xD)=O

forxca

(g=o

for x>,a

2 =+ dx*

Solving each equation separately, then using the continuity of nat x= a and the fact that JZg n(x) dx= 1 we find e-x=/2 for xa where

1 c

a

r

p9/2

C-l=

(-&+u

-

I e-&2

-cc

Since

s

(x-u)e-

-co

ax+a=/2+u-2

e-a’/2

288 (i = 1,2 ) . The heavy-traffic lowing form: p!“’ =pi -U;)2-I”,

condition

i= 1,2,

has the fol-

p, +pz = 1

p,, a, being positive. We use the same normalization is,

(13)

as before, that

~~“‘(t)=(~!“‘(t)-np~“))n-“2,

i=1,2.

Theorem 4. Under assumption ( 13) the normalized processes $“) ( t ) = (7;“) ( t), r]j”) ( t ) ) converge in distribution to the diffusion process q(t) = (q, (t), q2 (t) ) with infinitesimal generator B given by

B=

PIPI

$

-111x, I

$ +p2p2$ 1

-p2 min(xz,a,

2

+a2 -xl)

( 14)

& 2

Proof. We only briefly outline the proof, since it has much similarity to those of the preceding theorems. We have

Bnu (tlx1 ,x2) =npulpyyU(t,XI

+n-“2,X2)--U(t,XI,X2)]

+nj.42p$“)[u(t,x,,x2+n-‘/2)-u(t,x,,x2)] +h(x,n

‘l”+npj”))

x [u(t,xl -n-“’

9x2)

3. Multi-commodity

As explained above, the investigation of the fixed structure systems is equivalent to the separate study of two systems meant for ordinary and impatient customers. We begin by dealing with impatient customers. Let us consider the N-commodity system. For simplicity we suppose the customers to be indexed by subsets z={i,,...,ikcrj} of the set { 1,2,...,N). That is, all the demanded quantities A,; (iez) are equal to 1. The arrival times of customers indexed by z form a Poisson stream with intensity 2,. The streams with different z are independent. The customer is lost if all the servers are busy at least at one station i (iEz>. Otherwise, a server at each station i (iEz) is occupied for exponentially distributed time interval with the mean ,uu;’ . Denote by t,(t) the number of customers indexed by z (.zeR, the set of the considered demand lists) in the system at time t. It is obvious that

where ni is the number of servers at the station producing the ith type of commodity. As in Section 2 we analyse the asymptotic behaviour of large systems in heavy-traffic conditions. Namely, we consider a sequence of systems indexed by ~1,such that n,=y,n,

--u(m,

,x2)

1

systems with fixed structure

i= l,...,N

A(“) z =np zp(“) z

withp(“’ z

(15) =pz-a

z n-1/2,

(16)

+p2 mint (x2n1’2+np$“)), where yi,pz, a, are positive constants and

(n-p$“)n-x,n’/2)] X[U(t,X,,XZ-n-“2)-~(t,X,,X2)] for xi=

kl=l

,..., n-l,

L:Ts,Pz=Yi,

i= 1,..-,N

(17)

Theorem 5. Under assumptions ( 15 )- ( 17 ) the finite-dimensional distributions of the normalized processes q”” (t) with components

(kj-npj”))n-‘/2, i=1,2,

ZER

k2=1,2 ,...

Once more using in the last relation the Stirling’s formula and assumption ( 13 ) we obtain ( 14)) thus completing the proof.

weakly converge to corresponding distributions of the diffusion process with infinitesimal generator B. Evaluated in the inner points of the set

289 pj”)~y~-a~n-‘/~,

i=l

,**-,N

(19)

Let r.!“) (t) denote the number of customers the ith station in the nth system, operator B can be written as

rl’“‘(t)=(r(“‘(t)-np!“‘)n-‘/* 1 1 1 being the normalized

process.

Theorem 6. Under assumptions ( 15 ), ( 18 ), ( 19 ) distributions of finite-dimensional the q(“)(t)= (q$“)(t),...,qg)(t)) converge weakly to corresponding distributions of the diffusional process with infinitesimal generator B given by

At the boundary

where the kth station is completely have

occupied, we B=

au/aX,l,=O

at

T

F

Pkr

pk

min(Xk?ak)

forzsk

The proof being long and tedious is omitted. Since we consider the operation of the systems from the view-point of the supplier, turning to the ordinary customers, it is reasonable to suppose that they also arrive directly to the production centre according to N independent Poisson processes with parameters Ai (i= 1,...,N). If the number of warehouses is large this fact can be verified under mild restrictions using the properties of Poisson streams (see e.g. [ 8 ] ). Service times at the ith station are independent exponentially distributed with the mean ~~7‘. Note, that under such assumptions we have N independent single-commodity systems investigated thoroughly in Section 2. Suppose in addition that a customer completing the service at the ith station can enter the queue at the kth station with probability Pi,& With probability 1 - CkPik the customer departs from the system. Such situation arises in the case of transportation difficulties. Namely, the arrival of an additional order from the same warehouse can delay the delivery of the completed one. Once more we consider a sequence of systems indexed by n. As previously, we are interested in the study of large systems, therefore the numbers ni of servers (i= 1,...,N) increase with n as in ( 15 ). However the heavy-traffic conditions are different. Instead of ( 16), ( 17) we assume

The proof is also omitted. Remark 6. The limit processes obtained in Theorems 5 and 6 (as well as those of Theorems 2-4) have a stationary distribution with density 17satisfying equation ( 12 ) . Although for these more complex systems we cannot write the solution of ( 12 ) explicitly, we can establish some properties of the solution (or obtain it numerically). In particular, we can investigate its behaviour at inlinity in the following way. It is well known (see e.g. [ 7 ] ) that

T&f= JfT, Bf ds 0

where Ttis the semi-group associated with the diffusion process q ( t ) having infinitesimal generator B. Therefore EJ(rl(t))

G(X)

for any positivefsuch bysheff’s inequality P,df(q(t))

>f(z))

that Bf< 0.Hence, by Che-


(20)

Often it is possible to take f (x) = exp (Ax,x) with suitably chosen positively defined matrix A. Then

290 from (20) one easily derives the exponential decrease of the stationary distribution at infinity. 4. multi-cornrn~i~ structure

systems with flexibfe

In this section we treat the following mathematical model. Let N be the number of service stations, the station i having ni servers. Two classes of customers are served by the system. The flexibility of its structure means that the servers are not assigned to the various classes of customers. The first class customers have high priority for service of preemptive-resume type. They are indexed by subsets z= (ii,...,ikfz?) of the set ( 1,2,...,Nj and arrive in independent Poisson streams with rates ilZ (zER), here R is the set of the possible demand patterns. A customer leaves the system immediately if at least at one station i (i~z) all the servers are occupied by the customers of the first class. Otherwise, the service time is exponentially distributed with the mean y; ‘, the processing being performed simultaneously at each station i from the subset t by a single server. The second class customers arrive according to N independent Poisson processes with parametersAj (i=l ,...,N). The service times of the customers at the ith station are independent exponential random variables with parameter ,Ui*They have the low priority for service. If all the servers at station i are busy, the customers form a queue served on ~rst-come-~rst-served basis. As in Section 3 we suppose that the customer completing the service at the ith station with probability Plk joins the queue at the kth station, with probability 1 - Ck Pik departing from the system. The heavy-traffic conditions in this case are given by (16),( 18),( 19), however we put yi=pi in ( 19) and instead of ( I7 ) we assume y,=p*+

1 pZ, z: Z3,

i=l,..., N

(21)

The normalization of the multi-dimensional Markov processes CC”)(t ) is also carried out in a different way. Namely, we take ZER

$“‘(+(
f

where
The proof is omitted. The statement of Remark such systems.

6 is also valid for

5. Conclusion The main part of the paper was devoted to the investigation of heavy-traffic situations for large single- and multi-commodity systems. It was shown that the diffusion approximation enables us to evaluate the asymptotic behaviour of some systems characteristics pertaining to optimization. For brevity only the simplest models were analysed. In particular, the class of feasible replenishment rules consisted of (S- I ,S)-policies. The consideration of (s,S)-policies results in the treatment of queueing systems with batch arrival and/or batch service. We can also study the multi-commodity systems with deteriorating items from the view-point of warehouses or as a whole. Some possible modifications of the models such as mixed structure of systems, more general arrival and/or service processes were described in the Introduction. As is well known, various real processes can be described by the same mathematical model. The above considered models are suitable not only for

291 the study of inventory-production systems, but also for repairment centres, communication systems etc. (see e.g. [ 9 ] ) . References Aggaraval, S.C., 1974. A review of current inventory theory and its applications. Int. J. Prod. Res., 12: 443-482. Girlich, H.J., 1984. Dynamic inventory problems and implementable models. J. Inform. Process. Cybernetics - EIK, 20: 462-475. Silver, E.A., 198 1. Operations research in inventory management: A review and critique. Oper. Res., 29: 628-645. Sivazlin, B.D., 1974. A continuous review (s,S)-inventory system with arbitrary arrival distribution between unit demands, Oper. Res., 22: 65-7 1.

Wagner, H.M., 1980. Research portfolio for inventory management and production planning systems. Oper. Res., 38: 445-475. Afanasyeva, L.G. and Bulinskaya, E.V., 1980. Random processes in queueing and inventory theory. Moscow State Univ. (In Russian). Burman, D.J., 1979. An analytic approach to diffusion approximation in queueing. Ph.D. Dissertation, Courant Institute of Mathematics, New York Univ. Cramer, H. and Leadbetter, M.R., 1967. Stationary and related stochastic processes. Wiley, New York. Bulinskaya, E.V., 1990. Asymptotic behaviour of some communication networks. In: Symp. Automatic Queueing Systems, Abstracts Commun., Vinnitsa. (In Russian).