Volume 5, Number I OPERATIONS RESEARCH LETrERS June 1986

C{" N T R O L O F ARRIVALS AND D E P A R T U R E S IN A S T A T E - D E P E N D E N T I N P U T - O U T P U T S Y S T E M

Refael HASSIN

L~epartment of Statistics, Tel Avio Uniuersity. Tet Aviv 6997~ Israel

Mordechai HENIG

Faculzy of Menagement. Tel Aviv Uniuersizy, Tel Aviu 69978, Israel

Received October 1985 Revised February 1986

At each decision epoch, art offer [or a ur,it to either enter or leave a system ::. ~eceived. These offezs arrive according to a Poissoa process. With each effer is associated a -alue revealed upon the arrival of the offer. The distribution of the value of the offer is given and is a fl,.nctio.n of wl,a~ kind ~,f offer is received (ente~ or leave). The decision to accept or reject and offer s allowed to depend on the current sta'..e and the current value received. The objective is to maximize the. expected discounted

~fere~ice between the sum of the acce~ted output offers and ihe sum of the accepted input offers. The key result of the paper • ,hat under various conditions, the decision to accept or reject an offer depends on whether or not its value is above or

nelow, respectively, a cr'lical va;ae f~t depends on the state of the ~ystem.

optimal control* birth ;'~nd death Drocesses* inventou and queues control

I. Introduction

We consider a system whose input and output can be controlled to maxiznizc profit. At each decision epoch, an offer for a unit to, either enter (input offer) or leave (output offer) the system is received. These offers arrive according to a P~issor. process, with intensity par,'uneters ',hat may depend on the current number of units in tb,~ .,,ystem (i.e., the system state). With each offer there is associated a value: a non-negative random vaTiable whose distribution depends on the type of the offer (input or output). The decision to accept or reject an offer naturally depends on the system state and the offered value. The objective is to maximize the expected d;.~counted profit, wt'dch is the difference between the sum of accepted output offers and the sum of the accepted input offers minus the inventory- hold,;ng costs.

An example to a ~ystem of this type is a dealer facing streams of customers willing to sell or buy quantities of a comn~odity at various pv.'ces.

In this paper we; seek for sufficient conditions under which the optimal acceptance policy is char- acterized by a decreasing function of the system state. In this case, an output (input) offer with a given value will be accepted if and only if the system state is above (below) a critical hhmber."

The monotonicity of optimal policies is an issue in several birth and death proce:;ses described in the literature. For an example and references, see Albright and Winston [1]. Following the seminal work of Naor [5] monotone acceptance policies bare been largely dealt in queneing control models. The paper of Stidham [6] contains a survey of such models. Recently, David and Yechiali [3] applied a similar model where the offers consist of live organs to be transplanted. The items in the system ere the patients and the value of an offer is the expected lifetime of the patient.

An assumption, commoil to our model and to some of the works described above, is that the decision maker is n,)t able, or does not want, to announce a unit price of the item. For related models where a unit price is aEmounced, the reader is referred to Amihud and Mendel~n [2].

0167-637"~/86/$3.S0 © 19~6, Elsevier S¢',¢ace PubiL~her: n.V. (North-Holland) 33

Our model has several distinctive features: The arrival rates may depend on the systemstate, which in queueing models means the queue length. This extends the model to include situations where the storage capacity is finite (so that the input rate becomes zero when the storage is full), and where output offers emerge from the inventory (e.g., a multi-server repair shop). We allow offers to differ in their value, consider inventory holding costs, and include control of output offers. Finally, in a separate section, we consider batch arrivals of input offers. Queueing models with finite capacity can bc obtained from our model by letting the items to be the vacant places of the wait i ,g room.

In Section 2 we pzove that if the arrival rates of the offers are non-increasing and the. inventory holding costs are rron-decreasing functions of the system state, then there exists an interval 0, 1 .. . . . : /where the optir~,~d acx:c.ptance policy is monotone. Moreover:, once in this interval, the system wiI~ stay ia it forever. In Section 3 we generalize this result, allowing input offers to arrive in batches. It is interesting to note that althou~ we consider an infinite horizon problem, our proof is by backward induction.

2. The optimal poli~

Let m :_: 0 denote the inventory level. Let #s and h,,, be the arrival rates of input and output offers, ',espectively, when the inventory level is m. We assume that 0 _< ?,,,, _<~, and 0 _< p.,~ < g for m -- 0, 1 . . . . . l e t G and H be the probability distribution functions• of the input and output offc,'s, respectively. We assume that output offers have finite expected valuel Let ot > 0 be the discount facior, and h(m) be the holding costs associated with each unit time when the inventory level is m. We as~,ume that h(0) =- 0 and h(m + 1) >_ h(m) for m = 0, 1 . . . . .

Define F+(m, x) to be the expected discounted profit associated with an optimal policy when an input offer with value x is considered .and the inventory level is m. Similarly, define F_(m, x) for an output c~er. Standard application of the principal of optimality and basic results in the theory of Poisson proce,,~es give

F+(m, x) = m a x ( - x +f(m+ I),/(m)}, F_(m, x) = max(x + ] ' ( m - 1 ) , / (m)} ,

r_(o, x) =/(o),

where

m =- 0, .t . . . . .

m = l , 2 . . . . . (1)

f(m) =a,,,jo®F_(m," x)dH(x)+ b,,,Jo°°F+(rn," x)dG(x)- c m ,

a,,,=X,,,/(X,n-~-itrn+ot), bm=l~m/(Xm+#m+a) and c,,,=h(m)/(Xm+It,,+a).

(2)

and

F+(m, x~ = -x+f(m+l), x~l,,,+l, (3b)

-~f(m), X>lm+l,

Thu: f(m) can be interpreted as the expected discounted profit associated with an optim: I policy at some randor., point of time when the inventory level is "m.

Let

i~=f(m)-f(m-1), m = l , 2 ... .

(and for cor-.,enier~ce l o = oo). The-:. by (1)

' " .v)--x+f(m-i), x~l,,,, ~ (3a)

- / ( m ) , x<tm,.

for m ffi O, 1 . . . . . We can substitute (3) in (2) to obtain

f ( m ) = a,.[ f( m )G( I,..) + f( m - l ) (1- G( lm) ) + "..f~xdG( x )] +bm[ f (m+ l)H(lm+,)- fo"+'xdH(x) + (1 - H(lm+,))f(m)] --Cm

~am[i,,,G(i,,,) + f(m- i) + f~xdG(x)]..,., j

+bm[i,,,+,H(I,,,+,) + f (m) - fo t ' ~ ' xdH(x ) ] - c m, m= 1,2 . . . . .

f(O) = aof(O) + bo [ I,H( I, )+ f (O) - fo t 'XdH(x) l .

It can be written also as

f ( m ) f ~ X , , . (x-l,.)dG(x)+~t,.jo (t,.+~ r e = l . 2 . . . . .

(4)

f(o)=-~Ojo (il - x)dH(x).

The integrals in (4) are the expected profit of input and ~,'utput offers, respectively. The term inside the brackets is the expected rate of profit inflow if the system state is m. If this rate of profit is continued forever, then the total discounted profit is f(m).

We turn now to prove that there exists an inte~al. 0, 1 . . . . . M for some M ~ {0, 1 . . . . . oo} such that the critical values I t . . . . . 1M form a decreasing geque~ce, and if the system starts with an inventory level within this interval then it will never go beyond il,-The ~onotonically of the critical values reflects, by definition, the decreasing margi, nal worth of units in the iaventory. This result need not hold if, for example, large inventor,: atracts more off~ts. Thus we assume taat the arrival rates are non-increasing in the system state. The marginal worth of units in inventory becomes negative for m > M both because of the inventory holding costs and the decrease in the arrival rates. It can be shown that M = oo if h(m) = O, 1~,,, = t~, and ~'m = ~" for m = 0, 1 .. . . (except of course ~'0 =0). On the other hand, if the holding costs are high, M may be zero so that all input offers will be rejecte\~even when m = 0. Fast decrease in the arrival rates may also decrease M, but if h(l) -- 0 then M >_ 1 (Hassin and Henig [4]).

Theorem I.. Supaose that )~m and #m are non-increasing, then there exists M ~ {0. 1 . . . . . e¢} such that oO=1o>! 1 > 1 2 > . . . >IM>O>IM+ 1.

Proof. Suppose that for some m > 1 i,,+ ! > l,,+ 2. and ,,,+~' > 0. so thai

a[f(m + 1) - ] ( m ) ] = f l t / r n 4 [ • 0. (5)

By (4), the left-hand side of (5) is the sum of the following two expressions" 0¢ ' ,

(6) m ~, I m

ft.,. 21 . I • ,l,,+2-x)dH(x)-#mL/"~"(i,,+ - x ) d U t x ) - [ h ( m + 1 ) - h ( m ) ] (7) P,,,+lJo

Since h(m + 1) _>. h(m), and t~m+! < #,,,, thei~ (7) is non-positive. To obtain (5) the expression in (6) must be positive, and sin~ X,, > A,,,+~ it folIo~.~s that / , , >/,,,+t. Therefore. if there exists some M such that l~t>0 and IM+t <lu thea the above induction proves that i m is monotone decreasing over re=O.

t

1 . . . . . M. Indeed, let M = sup{ m I 1~ > 0, M = 0, 1 . . . . }. If M < ~ then the theorem is proved by the , m i above induction, since ,M > 0 >_ilM+t. Suppose that M = oo, then lm converges to (~ero since E , ~ ~ = f ( , n )

- f ( 0 ) is bounded. Therefore, for every m > ~ there exists ~ >_ m + 1 such that l,~'>/n~ + 1 > 0, and by the induction l,,, > l,,,+ ~. Hence i m is monotone decreasing over all positive integers. []

3. Batch arrivals of input offers

In Scct~a~ "2 we as~am,~d tha! each offer consists of one unit. Here we let each input offer to consist of several units of equal value x, and show that "l'heorem 1 still holds. We note that the theorem will not necessarily hold if output offers arrive in batches. Suppose, for example, that each output offer consists of two units, and that h ( m ) = h for m = 1, 2 . . . . . so that there is a fixed cost for holding positive inventory, independent of its size. The marginal worth of the first unit of inventory is reduced because of this cost, but not the second unit, since both units will leave simultaneously. Therefore, ! 2 > I t is possible. Theorem 1 may hold m this case under more restrictive assumptions, but a different proof (such as Successive Approximations) will be needed since expression (7) will contain terms with Ira.. i for positive values of i, for which the inductive hypothesis does not apply.

Let p( i + ) denote the probability that an input offer consists of a batch of size i or bigger. Suppose that lm+~ > i,~+~ > . . . > I u > 0 > lm+~, then an optimal acceptance policy for an input offer of size k and value x i.~ as follows: reject the offer completely if x ~ l,,~+ i, accept all of it if x < i,,,+ k, accept i units of it if l,,,+,+; < x < ! ~ , Note that since IM+t -< 0, no more than M-m units will be accepted.

Considerafi~ms similar to those presented in Section 2 yield that for the present case, eq. (4) is replaced by (8),

] ( m ) - - d ~,~ ( x - l , , ~ " 3 ( x ) + l ~ , , ~ p ( i + ) ( l , , , + , - x ) d H ( x ) - h ( m ) , (8) i= l

where i,,,+, is taken as zero for m + i > M. "lhe summation reflects the acceptance policy described abov%where tt,~e ith unit will be accepted only

if x <lm+ i, and then the expected value of x in this range is subtracted while f ( m + i) - f ( m + i - 1) = 1,~+, is added.

The summation term is easily seen to decrease with m if we use the inductive hypothesis that 1,1+ t > ! .... 2 . . . . > lu > 0 and IM+, = 0 for i > 1. Therefore~ as in the proof of Theorem 1, expression (6) must be posmve and the rest of the proof follows identical/ly.

References

[1] S.C. Albfight and W. Winston, "A birth-death model of advertising and pricing". Ad~,. Appl. Prob. It, 134-!5.? (~79). [2] Y. Amihud and H. Mendelson, "Dealership market", Journal of Financial Economics 8, 31-53 (1980). :,' [3] 1. David and U. Yechiali, "A time-dependent stopping problem with application to live organ transplantation", Operations

Research 33, 491-504 (1985). 14] R. Hassin and M. henig, "Optimal inventory control with randomly valued leplenishment-depletion offers", Working Paper

722/82, Faculty of Management, Tel Aviv University, Tel Aviv, 1982. [5] P. Naor, "On the regulation of queue size by levying tolls", Econometrica 37, 15-24 (1969). [6l S. Stidham. Jr.. "Optimal control c~f admission, routing, and service in queues and networks: A tutorial review", NCSU Tech.

Report no. M-4~ 1984.