A STOCHASTIC INVENTORY MODEL WITH STOCK DEPENDENT DEMAND ITEMS

In this paper, we propose a new continuous time stochastic inventory model for stock dependent demand items. We then formulate the problem of finding the optimal replenishment schedule that minimizes the total expected discounted costs over an infinite horizon as a Quasi-Variational Inequality (QVI) problem. The QVI is shown to have a unique solution under some conditions.


Introduction
This paper discusses a single item continuous time stochastic inventory model for stock dependent demand terms.The discussion is motivated by the well known princi- ple in the marketing literature that demand for certain items depends largely on the quantity displayed on the shelf (see for example, Corstjens and Doyle [2] and Schary and Becket [1]).There are some simple EOQ deterministic models such as Datta and Pal [3], Coswami and Chaudhuri [4] but no attempt has been made to incorporate this principle in continuous time stochastic inventory model due to the technical com- plication that arises from the inclusion of the stock dependent demand items.
To formulate the problem, let x(t) denote the level of stock at time t.We assume that the cost structure of the model is the following: (i) The discount factor is a, with a > 0. (ii) The holding cost is px, for x < 0 (shortage cost), f(x) qx, for x > 0 (holding cost), 318 L. BENKHEROUF, A. BOUMENIR and L. AGGOUN with p>0andq>0.
(iii) The setup cost is k, with k > 0. (iv) A cost per unit of item is c, with c > 0. A replenishment policy consists of a sequence (ti, Qi), 1,..., where t is the ith time of ordering and Qi is the quantity ordered at time ti, where I < t 2 < Let Vn-{(ti, Qi)}i=l n' and set < v.
Assume that the variation in inventory is governed by the following stochastic differential equation dx -(g + ax(t)ZI(x(t) > O))dt-(rdw + E Qi5(t -ti)' (1) i>0 where I(A) is the indicator function of set A, 5 is the Dirac function, g > O, (r > O, a > 0, 0 < < 1, nd {wt} i the standard Brownian motion.Note from (1) that when r 0, then (g + ax(t) ) can be interpreted as the demand rate when x(t)> O.
Also note from (1) that it is implicit in the model that when x(t) < 0, shortages have no effect on the demand.If a 0 then the above model reduces to the model found in Sulem [8].However, our treatment is different from that of Sulem in a number of ways resulting in a more general approach.
Assume that V n is fin-measurable.Then, the optimal replenishment schedule re- duces to the problem of finding the sequence V* that solves y(x) -i E x f(x(t))e-Ctdt + E (k + cQi)e-cti (2) o i>_o where the expectation is taken over all possible realizations of the process x(t) under Policy V.
In the next section, we formulate the problem addressed in (2) as a Quasi-Varia- tional Inequality (QVI) problem and show that under some conditions on the discount factor, the unit cost and the holding cost, a unique solution to the QVI exists.We conclude with some remarks on the problem of finding a replenishment schedule that minimizes the total cost per unit of time.
2. The Quasi-Variational Problem and Optimal (s,S) Policy In this section, we formulate the problem addressed in (2) as a Quasi-Variational In- equality (QVI) problem and show that the optimal impulse control policy is an (s,S) policy, where s and S are determined uniquely under certain technicality conditions (see Theorem 1 below).
Fix and let r be a short interval of time.We then have two cases: (i) If x(t) > 0 and no order is made in the interval [t, t + r), then (1) and (2)  imply that t+r / f(x(t))e-a(s-t)d s + y(t + ')e-ar 1 . (3)  Write x(t+7)=x(t)+Axv, and use the fact that for a standard Brownian motion w(t), E[w(t)] O, and E[w2(t)]t.The Taylor expansion of the right side of (3) gives y(x) <_ f(x(t)) + y(x(t)) + E[Axr]y'(x(t)) + 1/2E[Axr]2y"(x(t)) ay(x(t)) + 2 which leads to Dividing by v and letting v0, gives (g + ax(t)Z)y'(x(t)) / ay( <_ f "((t))+ x (ii) If x(t)< 0 and no order is made in the interval [t,t / 7), then a similar argument to the one used in (i) gives 21cr2y"(x(t)) + gy'(x(t)) + cy(x(t)) <_ f (iii) If an order of size Q is placed at time t, then the inventory level jumps from x(t) to x(t) + Q.In other words, y(x(t Let A and M be two operators defined by Ay(x)-{ 1 2 ,, axf)y cy ), (5) Then the optimal expected costs for the inventory model is given as a solution of the QVI problem L. BENKHEROUF, A. BOUMENIR and L. AGGOUN Ay<_f y<_My (6) (Ay-f )(y-My) 0. For more details on QVI, see Bensoussan and Lions [1].
To find the solution of the QVI problem given by ( 6), we follow Sulem [8] and divide the inventory space into two regions (i) the continuation region where no order is made and where A is defined in (4).
The stopping region C {x ; y(x) My(x)} {x ; x <_ s}, where M is given by ( 5), corresponds to the states where an order is made.
In C, we have y + c(S-) + (S). (s) The solution to the QVI given by problem ( 6) is continuous differentiable and continuity at the boundary point s gives from (8) that y'(s) -c. (9) The infimum in (7) is attained at S. Hence, y'(S) -c.
(10) Also, y is continuous at s, which leads to (s) ()-c(S-). ( 11 Also, we require that lim y(x) x--+ ocf--(-< c. (12) Note at this stage, s must be < 0, otherwise (8)-(11) will lead to s S, which means that k 0, contradicting the assumption that k > 0. The following is the main result of the paper.
Theorem 1: There exists a unique solution to the Q VI problem given in (6) if and only if p + ac) < O.
The proof of Theorem 1 is lengthy and thus is done in stages.In the first stage, we are concerned with the asymptotic nature of y(x) + cx as x+ + ec and of y(s) + cs &S 8--- 00.
There are several ways of finding the coefficients a n but here we use an iterative method.Rewrite (14) as r 2_ g + axle.y,This suggests that Yn(X) may be written as Yn + 1(x) (-1 ) n ( a ) x f Y ' n ( x ) , -5 from which we deduce that y --(-lans(2fl 1)...((n 1)fl -(n 2))x ( 1)n O/nt-" The series E n > oYn is an asymptotic series since Yn +1 --O(Yn) (see Olver [6]).It follows that the )articular solution is p such that Hence (i) is true.
The argument used to show (i) indicates that Y 4-(X) I(X)yp 4-(X), (21)   where el(X) is the complementary solution and yp+ (x) is a particular solution.
The left side of the above inequality is strictly negative while the right-hand side is strictly positive, which leads to a contradiction.Theorem 1 follows from Theorem 2 and Lemmas 3-8.
Assume now that we are interested in impulse control policies, of the form V {(ti, Qi)}i= 1,'", where the t are the ordering times and Qi, the quantities ordered.The total cost per unit time is given by Ex [ f T o f(x(t))dt +ti < (k + cQi)] lira Yv(Z) T--, T where the dynamics of the process are given by (1) and the expectation is taken with respect to all realizations of the process.Then we say that V* is average cost optimal if yv,(X) ifyy(x ).
Let # yv,(X).Then, it is known (see Lions and Perthame [5]), that the optimal cost y in (2) behaves like (-+ Y0) where Y0 satisfies some QVI problem that can be obtained from (6).Also, it is known that the optimal (s,S) policy obtained from (6) converges to the optimal policy that minimizes the expected average future costs.
In this paper, we proposed a new continuous time stochastic inventory model for stock dependent demand items.We also formulated the problem of finding the optimal replenishment schedule that minimizes the total expected discounted costs over an infinite horizon, as a QVI.The QVI was shown to have a unique solution under some conditions.