May 1, 1998 - and NSERC Grant A4619. Thanks are due to Youhua (Frank) Chen, Hans-Joachim Girlich, Arie Harel and. Dmitry Krass for helpful comments.
Beyer, D., Suresh P. Sethi: The Classical Papers of Iglehart (1963) and Veinott and Wagner (1965) Revisited.
Statement of Contribution Capital tied up in inventories represents a large portion of a rm's assets. Most rms face demands that are characterized by some level of uncertainty. It is no surprise that stochastic inventory problems represent an important paradigm in Operations Research. Average-cost stochastic inventory models date back to classical papers of Iglehart (1963) and Veinott and Wagner (1965). While these papers are widely assumed to have solved the problem, this in fact is not the case. In "The Classical Papers of Iglehart (1963) and Veinott and Wagner (1965) Revisited" Dirk Beyer and Suresh Sethi provide a careful analysis of what is missing in these papers and what is required to complete their analysis. Unsubstantiated non trivial assumptions made in the classical papers are veri ed. The relationship between the stationary distribution approach of Iglehart and other approaches to average-cost problems are examined. These are important issues in the analysis of inventory problems, and it is about time that they are pointed out. Moreover, it is expected that the clari cation of the widely-cited classical papers would contribute to further research in the area of average-cost stochastic inventory models.
The Classical Average-Cost Inventory Models of Iglehart (1963) and Veinott and Wagner (1965) Revisited y
D. Beyer and S. P. Sethi
z
May 1, 1998
Abstract This paper revisits the classical papers of Iglehart (1963) and Veinott and Wagner (1965) devoted to stochastic inventory problems with the criterion of long-run average cost minimization. We indicate some of the assumptions that are implicitly used without veri cation in their stationary distribution approach to the problems, and provide the missing (nontrivial) veri cation. In addition to completing their analysis, we examine the relationship between the stationary distribution approach and the dynamic programming approach to the average-cost stochastic inventory problems.
This research was supported in part by a Feodor Lynen-grant provided by the A. von Humboldt Foundation and NSERC Grant A4619. Thanks are due to Youhua (Frank) Chen, Hans-Joachim Girlich, Arie Harel and Dmitry Krass for helpful comments. y Technical Sta Member, Hewlett-Packard Laboratories, Palo Alto, California. z Professor, School of Management,The University of Texas at Dallas, Richardson, Texas.
1
1 Introduction In the context of inventory problems, Iglehart (1963) and Veinott and Wagner (1965) were the rst to study the issue of the existence of an optimal (s; S )-policy for average cost problems with independent demands, linear holding and backlog costs, and ordering costs consisting of a xed cost and a proportional variable cost. Iglehart obtained the stationary distribution of the inventory/backlog (or surplus) level given an (s; S )-policy using renewal theory arguments (see also Karlin 1958a, 1958b), and developed an explicit formula for the stationary average cost L(s; S ), s S , associated with the policy. Iglehart assumed that the function L(s; S ) is continuously di erentiable and that there exists a pair (s; S ), ?1 < s < S < 1, which minimizes L(s; S ) and satis es the rst-order conditions for an interior local minimum. While he does not specify these assumptions explicitly and certainly does not verify them, he uses them in showing the key result that the minimum average cost of a sequence of problems with increasing nite horizons approaches L(s; S ) asymptotically as the horizon becomes large. With a short additional argument suggested by C. Derman, Veinott and Wagner were able to advance the Iglehart result into the optimality of the (s; S ) policy; see also Veinott (1966). It is important to point out that Veinott and Wagner deal with discrete demands. They assume without proving that Iglehart's results derived for the continuous demand case hold as well for their discrete demand case. It should also be mentioned that Derman's short additional argument applies also to the continuous demand case treated in Iglehart and would prove that an L(s; S )-minimizing pair (s; S ), if there exists one, provides an optimal inventory policy; see Section 5.. A good deal of research has been carried out in connection with the average cost (s; S ) models since then. With the exception of Zheng (1991), however, most of this research devoted to establishing the optimality of (s; S ) strategies uses bounds on the inventory position after ordering. Examples are Tijms (1972), Wijngaard (1975), and Kuenle and Kuenle (1977). On the other hand, quite a few papers are concerned with the computation of the (s; S ) pair that minimizes L(s; S ), and not with the issue of establishing the optimality of an (s; S ) policy. Some examples are Stidham (1977), Zheng and Federgruen (1991), Federgruen and Zipkin (1984), Hu et al. (1993), and Fu (1994). For other references, the reader is directed to Porteus (1985), Sahin (1990), and Zheng and Federgruen (1991). Furthermore, this literature has generally assumed that together the papers of Iglehart (1963) and Veinott and Wagner (1965) have established the optimality of an (s; S ) policy for the problem1. But, this is not quite the case, however, since the assumptions on L(s; S ) implicit in Iglehart, to our knowledge, have not been satisfactorily veri ed.
Another possible approach views continuous demand distributions as the limit of a sequence of discrete demand distributions. Such a limiting procedure could lead to a proof of optimality of ( )-policies in the continuous demand case. However, this is by no means a trivial exercise. 1
s; S
2
The primary purpose of this paper is to provide results that are missing in Iglehart (1963) with respect to the model under his consideration. Without these results, the Iglehart analysis cannot be considered complete. Moreover, these results are by no means trivial2. For this purpose, we need to precisely specify Iglehart's model. For this model we must rst show that there exists a pair (s; S ), ?1 < s S < 1 that minimizes L(s; S ). In order to accomplish this, we establish in Section 5, a priori bounds on the minimizing values of s and S . We should caution that any veri cation of these assumptions on L(s; S ) should not use arguments that rely on the optimality of an (s; S ) policy. With the bounds established in Section 5, the continuity of L(s; S ) provides us with the existence of a minimum. Continuous di erentiability of L(s; S ) follows from the de nition of the surplus cost function L(y) and the assumption of a continuous density for the demand. We then show that if (s; S ) with s = S is a minimum, then there is another minimum (s; S ) with s < S . It is then possible to assume an interior solution always to exist, and obtain it by the rst-order conditions of an interior minimum. A secondary purpose of the paper is to compare the stationary cost analysis of Iglehart and Veinott and Wagner and the vanishing discount approach used in Beyer, Sethi and Taksar (1996); see also Sethi and Cheng (1995) and Beyer and Sethi (1997). Both approaches prove the optimality of (s; S )-strategies, but for somewhat di erent notions of the long-run average cost to be minimized. It turns out that the optimal (s; S )-strategies are optimal for either of the minimization criteria. Tijms (1972) uses the theory of Markov Decision Processes (MDP) to prove the optimality of (s; S )-strategies for a modi ed inventory problem with discrete demand. In particular, he imposes upper and lower bounds on the inventory position after ordering. These bounds provide a compact ( nite) action space as well as bounded costs. Under these conditions, standard MDP results yield the optimality of an (s; S ) strategy. Zheng (1991) has provided a rigorous proof of the optimality of an (s; S ) policy in the case of discrete demands for the model in Veinott and Wagner (1965). He was able to use the theory of countable state Markov decision processes in the case when the solution of the average cost optimality equation for the given problem is bounded, which is clearly not the case here since the inventory cost is unbounded. Note that this theory does not deal with the continuous demand case as it would involve an uncountable state MDP. Zheng relaxed the problem by allowing inventory disposals and since the inventory costs are charged on ending inventories in his problem, he obtained a bounded solution for the average cost optimality equation of the relaxed problem, which involves a dispose-down-to-S component. But the dispose-down-to-S component of the In fact, the motivation to write our paper arose from the example we present in Appendix. The example shows that even for a well-behaved demand density function satisfying Iglehart's assumptions, the derivation of equation (12) crucial for the subsequent analysis requires additional arguments not given in Iglehart's paper. The example also shows that in some cases a base stock-policy can be optimal even in the presence of a positive xed ordering cost. 2
3
optimal policy would be invoked in the relaxed problem only in the rst period (and only when the initial inventory is larger than S ), which has no in uence on the long-run average cost of the policy. It follows therefore that the (s; S ) policy, without the dispose-down-to-S component, will be optimal also for the original problem. While Zheng has solved the discrete demand case, it might still be useful to complete the analysis of Veinott and Wagner. Recall that their analysis uses the results of Iglehart derived in the continuous demand case. Moreover, our completion of Iglehart's analysis in Section 5 assumes the existence of a demand density, and therefore does not cover the discrete demand case of Veinott and Wagner. In Section 4, we provide what needs to be done to complete the Veinott and Wagner analysis. The plan of the paper is as follows. In Section 2, we state the problen under consideration using the notation of Iglehart (1963) and Veinott and Wagner (1965). Section 3 summarizes the results of Iglehart relevant for the average cost minimization problem. Furthermore, we point out exactly which implicit assumptions have been used by Iglehart without veri cation. In Appendix we analyze an example to show that even under the quite restrictive assumption of the existence of a continuous demand density, the assumptions implicit in the Iglehart analysis are not necessarily satis ed. In Section 4 we review the analysis contained in Veinott and Wagner (1965) that is devoted to the solution of the average cost problem in the case of discrete demands. To the extent that they use Iglehart's analysis for their solution, we show how their paper is not quite complete and how it can be completed. Section 5 contains the proofs needed for the completion of Iglehart's analysis and for establishing the optimality of an (s; S ) policy in the continuous demand case. Section 6 lists results that connect the stationary distribution approach and the vanishing discount approach, both undertaken to prove the existence of an optimal (s; S )-strategy for the average cost inventory problem. Section 7 concludes the paper.
2 Statement of the Problem In this paper we consider a stationary one-product periodic review inventory model with the following notation and assumptions:
The surplus (inventory/backlog) level at the beginning of period k prior to ordering is denoted by xk . Unsatis ed demand is fully backlogged.
The surplus level after ordering but before demand in period k is denoted by yk . Orders arrive immediately.
The one period demands k , k = 1; 2; : : :, are i.i.d. and the demand distribution has a density '(). Let denote the mean demand. Assume 0 < < 1. 4
The holding and shortage cost function L(y), where y is the surplus level immediately after ordering, is given by
L(y) = E (h([y ? ]+ )) + E (p([y ? ]?)); where h() and p() represent holding and shortage cost functions, respectively. Furthermore L(y) is assumed to be convex and nite for all y.
The ordering cost when an amount z is ordered is given by c^(z) = (z)K + cz; c 0; K > 0; where (z) = 1 if z > 0 and (z) = 0 otherwise. Given an ordering policy Y = (y1; y2; : : :), the inventory balance equation is as follows:
xk+1 = yk ? k : Let fn(xjY ) denote the expected total cost for an n-period problem with the initial inventory x1 = x when the order policy Y is used, i.e.. (X ) n fn (xjY ) = E [^c(yk ? xk ) + L(yk )] : k=1
The objective is to minimize the expected long-run average cost 1 f (xjY ) a(xjY ) = lim inf (1) n!1 n n over the class of all nonanticipative or history-dependent policies Y . In what follows we use fn (xjs; S ) and a(xjs; S ) instead of fn(xjY ) and a(xjY ), with a slight abuse of notation, if Y is a stationary (s; S )-strategy. The model described above was investigated in a more general setting of Markovian demand and polynomially growing surplus cost by Beyer, Sethi and Taksar (1996) for the average cost objective function (2) J (xjY ) = lim sup n1 fn (xjY ): n!1 They were able to establish the average cost optimality equation and prove the optimality of an (s; S ) policy by using the vanishing discount approach. A policy minimizing either of (1) or (2) does not necessarily minimize the other. However, if an optimal policy Y with respect to (1) is such that limn!1 (1=n)fn (xjY ) exists, then this limit is less than or equal to both objective functions associated with any policy Y 2 Y . On the other hand, if a policy Y is optimal with respect to (2) and if limn!1 (1=n)fn (xjY ) exists, then Y may still not minimize (1). For this reason, Veinott and Wagner (1965, p. 530) consider the objective function (1) to be the stronger of the two; often the term more conservative is used 5
instead. In Sections 5 and 6, we complete Iglehart's stationary state analysis and use Derman's short additional argument to obtain an (s; S ) policy which is optimal with respect to (1) and has the limit (1=n)fn (xjs; S )) as n ! 1. Thus, this (s; S ) policy also minimizes both objective functions (1) and (2). In addition, by combining the stationary approach with the dynamic programming approach, we show in Section 6 that any (s; S ) policy that is optimal with respect to (2) is also optimal with respect to (1).
3 Review of Iglehart (1963) In this section we will summarize the results in Iglehart (1963) relevant for the problem of the minimization of the long-run average cost and point out the implicit assumptions which have been used in his paper without veri cation. Let fn (x) denote the minimal total cost for the n-period problem when the initial surplus level is x, i.e., fn(x) = min f (xjY ): Y n The sequence of functions (fm (x))nm=1 satis es the dynamic programming equation Z1 fn (x) = min [^c(y ? x) + L(y) + fn?1 (y ? )'()d]: yx 0
(3)
Furthermore, it is known that fm(x) is K -convex and an optimal strategy minimizing the total cost is determined by a sequence (sm; Sm)nm=1 of real numbers with sm Sm, such that the optimal order quantity in period m is 8 < zm = : Sm ? xm if xm sm; 0 if xm > sm; where xm denotes the surplus level at the beginning of the mth period.3 It is obvious that limn!1 fn(x) = 1 for all initial surplus levels x. Iglehart investigates the asymptotic behavior of the function fn(x) for large n. Heuristic arguments suggest that for a stationary in nite horizon inventory problem, a stationary strategy should be optimal. Furthermore, it is reasonable to expect that this stationary strategy is of (s; S )-type. Iglehart obtains the stationary distribution of the surplus level and the expected one-period cost under any given (s; S ) strategy satisfying ?1 < s S < 1. He uses the result of Karlin (1958a, 1958b) that the surplus level xn at the beginning of period n converges in distribution Well-known papers dealing with this nite horizon problem are those of Scarf (1960), Schal (1976) and Veinott (1966). 3
6
to a random variable whose distribution has the density 8 m(S ? x) > < 1 + M () ; s < x S; (4) f (x) = > h(; s ? x) : 1 + M () ; x s; where := S ? s, M () and m() are the renewal function and the renewal density associated with '(), respectively, and h(; ) is the density of the order quantity in excess of . Note that by the Elementary Renewal Theorem, we have M (t)=t ! 1= as t ! 1. Also M (t) ! 1 as t ! 1. Furthermore, the one-period cost C (x) given the initial surplus x is 8 < (5) C (x) = : K + L(S ) + c(S ? x); x s; L(x); s < x S: Averaging C (x) with respect to f (x) yields the following formula for the stationary cost L(s; S ) per period corresponding to the strategy parameters s and S :
ZS Zs h i K + L(S ) + c(S ? x) f (x)dx + L(x)f (x)dx L(s; S ) = ?1
s
RS K + L(S ) + s L(x)m(S ? x)dx = + c: (6) 1 + M (S ? s) Remark 3.1 It follows from the convergence in distribution of the surplus level in period n that the expected cost E (C (xn)) in period n converges to L(s; S ). Therefore, we have ! n X 1 1 a(xjs; S ) = nlim !1 n fn (xjs; S ) = nlim !1 n i=1 E (C (xn )) = L(s; S ): In many cases it is more convenient to de ne the stationary cost in terms of and S : K + L(S ) + R L(S ? x)m(x)dx 0 + c; (7) L~(; S ) := L(S ? ; S ) = 1 + M ()
or in terms of s and :
R K + L(s + ) + L(s + ? x)m(x)dx 0 L^(s; ) := L(s; s + ) = + c: (8) 1 + M () Iglehart then attempts to minimize L(s; S ) with respect to s and S . Implicitly he assumes L(s; S ) to be continuously di erentiable and that the minimum is attained for some s and S 7
satisfying ?1 < s < S < 1, i.e., the minimum is attained at an interior point and not at the boundary s = S of the feasible parameter set. Therefore, he can take derivatives of L^(s; ) with respect to s and and obtain necessary conditions for the minimum by setting them equal to zero: @ L^(s; ) = L0(s + ) + Z L0(s + ? x)m(x)dx = 0 (9) @s 0
and 3 2 @ L^(s; ) = (1 + M ())?2 [1 + M ()] 4L0(s + ) + Z L0(s + ? x)m(x)dx + L(s)m()5 @ 0 2 3 ! Z ? 4K + L(s + ) + L(s + ? x)m(x)dx5 m() = 0: (10) 0
Note that if s = S , condition (9) must be relaxed to @ L^=@s 0. Combining the necessary conditions (9) and (10), one obtains 1 0 Z @(1 + M ())L(s) ? K ? L(s + ) ? L(s + ? x)m(x)dxA m() = 0: (11) 0
Assuming further that m() = m(S ? s) > 0 for the minimizing values s and S , Iglehart obtains the following formula, crucial for his subsequent analysis, by dividing by m(): R K + L(s + ) + L(s + ? x)m(x)dx 0 = L(s; S ) ? c: (12) L(s) = 1 + M () Let us recapitulate Iglehart's implicit assumptions here: For his subsequent analysis he requires that (i) there is a pair (s; S ) with ?1 < s < S < 1 (i.e., an interior solution) that minimizes L(s; S ), (ii) L(s; S ) is continuously di erentiable, and (iii) the minimizing pair (s; S ) satis es m(S ? s) > 0. We shall see that there is always a pair (s; S ) satisfying Assumption (i) since K > 0. On the other hand, in general, not every minimizer of L(s; S ) satis es Assumption (i) even though K > 0; see Appendix for an example. Assumption (ii) is satis ed because of the continuous di erentiability of L() and the existence of a continuous density, which implies continuous differentiability of M (). Assumption (iii) is more dicult to deal with. In Appendix we provide 8
an example in which Assumption (iii), i.e., m(S ? s) > 0, is violated for all minimizing (s; S ) pairs. The fact that Assumption (iii) does not hold in general is a problem that can be xed. It turns out that it is not Assumption (iii) itself but rather equation (12) derived with the help of this assumption that is crucial for the subsequent analysis. In Section 5, we prove that there is always a minimizing pair (s; S ) that satis es (12), even if m(S ? s) = 0 for all minimizing pairs. As the horizon n of the inventory problem becomes large, it is reasonable to expect that the optimal strategy parameters (sm; Sm) and (sm+1; Sm+1 ) for small m do not di er signi cantly and that the optimal strategy tends to a stationary one. On the other hand, if a stationary strategy is applied, the inventory level tends towards a steady state. The minimum cost per period one can achieve in the steady state is k := L(s; S ). If the system approaches the steady state fast enough, one could expect the di erence fn (x) ? nk to be uniformly bounded with respect to n for any x. In Section 4 of his paper, Iglehart obtains bounds on fn(x) in terms of an explicitly given solution () of the following equation: Z1 (13) (x) = min [^c(y ? x) + L(y) ? k + (y ? )'()d]: yx 0
He proves that the function () de ned as 8 > y0 < ?cy; 1 R (s + y) = > L(y + s) ? k + (y + s ? )'()d; y > 0 ; :
(14)
0
satis es (13). The pair (s; S ) is a minimizer of L(s; S ) which satis es (12). Brie y, the proof goes as follows. First, Iglehart veri es that the function () de ned in (14) is K -convex.
Remark 3.2 It should be mentioned that () is also K -convex if we replace k by any value larger than L(s; S ) = L(s) + c. In a next step he shows that the function
G(y) = cy + L(y) +
Z1 0
(y ? )'()d;
which represents the function to be minimized in (13), attains its minimum at y = S , and G(s) = K + G(S ) for () de ned in (14). To show this, it is essential that (12) holds. The K -convexity of G() follows from the K -convexity of (). Now we return to equation (13) written in terms of G(), i.e., (x) = ?k ? cx + min [K(y ? x) + G(y)]; yx 9
and transform its right-hand side. For x s, the minimum is attained for y = S and we get
?k ? cx + min [K(y ? x) + G(y)] = ?k ? cx + K + G(S ) = ?k ? cx + G(s) yx =
?k ? cx + cs + L(s) +
= ?c(x ? s):
Z1 0
(s ? )'()d
For x > s, the minimum is attained for y = x and we obtain
?k ? cx + min [K(y ? x) + G(y)] = ?k ? cx + G(x) yx = ?k + L(x) +
Z1 0
(x ? )'()d:
Therefore, the function () de ned in (14) actually satis es (13). Observe that we can write explicitly 8 > c(s + ) + L(y + s) for y < 0; < R y G(y + s ) = > c(s + ) + L(s ) + L(y + s ) + 0 L(y + s ? )m()d : ?L(s)[1 + M (y)] for y 0: The main result of this section is the following: For given W 2 IR there are constants r and R depending on W such that the inequalities
nk + (x) ? r fn (x) nk + (x) + R; for x W;
(15)
hold. The assertion is proved by induction. First, Iglehart shows that the optimal order levels Sk for the n-period problem are uniformly bounded from above, i.e., the bound does not depend on n. Then he chooses a constant W larger than this bound and proves that the inequality holds for n = 1. Since f1(x) and (x) are both linear with slope ?c for x minfs1; sg, we can set
r = minfsmin ff (x) ? (x) ? kg and R = minfsmax ff (x) ? (x) ? kg: ;sgxW 1 ;sgxW 1 1
1
The induction step for n = N + 1 uses (3) and (13). Note that because x W and Sn W , the minimum in (3) is attained for some y W , and the inequality (15) can be used for n = N . Because () is continuous and therefore (x) < 1 for any x, it follows from (15) that fn (x) (16) nlim !1 n = k: It should be mentioned that the proof of (15) requires only that () solves (13) and that (x) is linear with slope ?c for all x smaller than some nite constant. Moreover, it follows from (16) that there is no such solution for k 6= L(s) + c. 10
In Sections 6 and 7, Iglehart investigates the limiting behavior of the function gn (x) = fn (x) ? nk as n ! 1. He proves that for K = 0, nlim !1 gn (x) =
(x) + A;
where () is given by (14) and A is a constant. For K > 0 he is only able to obtain lim nsup gn (x) (x) + B !1
for a constant B . For K > 0, he also conjectures that liminf n!1 gn (x) (x)+ B , which would lead to limn!1 gn (x) = (x) + B .
4 Review of Veinott and Wagner (1965) Veinott and Wagner (1965) deal with the inventory problem introduced in Section 2 with one essential di erence. They consider the demand i in period i to be a discrete random variable taking nonnegative integer values. They assume that one-period demands 1; 2; : : : are i.i.d. random variables with the distribution
'(k) = P (i = k);
k = 0; 1; : : : ; i = 1; 2; : : : :
(s; S )-strategy Therefore, in the stationary We only recapitulate here the results of Veinott and Wagner that are important in the context of the existence of an optimal (s; S )-strategy. We restrict our attention to the case of the zero lead time for convenience in exposition. The discrete renewal density and the renewal function are de ned as 1 X 'i(k); m(k) = i=1
M (k) =
1 X i=1
i(k) =
k X j =0
m(j );
k = 0; 1; : : : ;
where 'i and i denote the probabilities and the cumulative distribution function of the i-fold convolution of the demand distribution, respectively. Employing a renewal approach or a stationary probability approach, Veinott and Wagner derive a formula for the stationary average cost a(xjs; S ) given a particular stationary (s; S )strategy. pair. Since the unit purchase cost does not in uence the optimal strategy, the formulas are derived for the case c = 0. An extension to c > 0 is straightforward. Veinott and Wagner obtain S ?s K + L(S ) + P L(S ? j )m(j ) i=0 a(xjs; S ) = LV W (s; S ) = 1 + M (S ? s) 11
for integer values of s and S . It should be mentioned that this function does not depend on the initial surplus x. Veinott and Wagner for their discrete demand case claim that, just as in Iglehart's continuous demand density case, a minimizing pair (s; S ) of LV W (s; S ) would satisfy (16), i.e., 1 f (x) = L (s; S ): lim (17) VW n!1 n n Furthermore, in the appendix of their paper Veinott and Wagner establish bounds for the parameters minimizing LV W (s; S ). The proofs of these bounds, derived in their paper for computational purposes, do not require the existence of a stationary optimal (s; S ) strategy in the average cost case, but depend critically on the discrete nature of the demand. Having established bounds on the minimizers of LV W (s; S ), it is clear that the discrete function LV W (s; S ) attains its minimum for an integer pair (s; S ). But Veinott and Wagner's claim, that a discrete analog of Iglehart's analysis yields (17) for the discrete demand case, requires some additional arguments not included in their paper. Observe that a completion of Iglehart's analysis not only requires the existence of a nite minimizer of the stationary average cost function, but it also needs the minimizer to satisfy equation (12), which with c = 0 reduces to L(s; S ) = L(s): (18) In general, the integer minimizer of LV W (s; S ) will not satisfy this condition. Subsequently, in order to establish the dynamic programming equation in the MDP context, Tijms (1972) has shown that there are integer minimizers (s; S ) of L(s; S ) such that
L(s ? 1) L(s; S ) L(s):
(19)
For our purpose, it follows immediately from (19) and the continuity of L(x) that there is an s# < S such that (i) (bs# c; S ) is an integer minimizer of LV W (s; S ), implying LV W (bs# c; S ) = LV W (s; S ) and (ii) LV W (bs# c; S ) = L(s#), where bxc denotes the largest integer smaller or equal to x. Using the pair (s#; S ) and the function L(s; S ) := LV W (bsc; S ) in Iglehart's analysis yields the desired formula (18). Once (18) is established, the short additional argument suggested by C. Derman and used by Veinott and Wagner would provide the optimality of a stationary (s; S ) strategy for the average cost inventory problem. 12
Speci cally, it follows from the de nition of fn that fn (x) fn(xjY ) for all initial values x and all history-dependent strategies Y 2 Y . Thus, 1 f (x) = lim inf 1 f (x) lim inf 1 f (xjY ) = a(xjY ); a(xjs; S ) = L(s; S ) = nlim n!1 n n n!1 n n !1 n n for any history-dependent strategy Y 2 Y . Therefore, (s; S ) is average optimal. We now return to the continuous demand case of Iglehart.
5 Existence of Minimizing Values of s and S In this section we rst establish a priori bounds on the values of s and S that minimize L(s; S ). Once that is done, the continuity of L(s; S ) ensures the existence of a solution (s; S ) that minimizes L(s; S ). While Veinott and Wagner have proved bounds on the minimizing s and S , their proofs use the fact that the demands are discrete, and there is furthermore no obvious way to transfer their proofs to the continuous demand case. Additionally, they employ a vanishing discount argument. Since the discounted problem is not within the scope of the original Iglehart paper, we will provide bounds on the minimizing parameters (s; S ) using only the properties of the stationary cost function
Lemma 5.1 There is a constant S such that for all S S and s S , the stationary cost L(s; S ) > L(0; 0). Proof. Let C = 2L(0; 0) > 0. Choose S l and S u such that S l S u and L(S l) = L(S u ) = C ; see Fig. 1. Let S S u. Our proof requires three cases to be considered: s S l, S l < s < S u, and S u s S . We rst take up the case s S l. In this case M (S ? S l) M (S ? s), which we shall use later. >From the cost formula (5) and the stationary probability density (4), we have
ZS
ZS
l
ZS
L(x)f (x)dx C f (x)dx + C f (x)dx S 3 2S s S Z Z = 1 + MC(S ? s) 64 m(S ? x)dx ? m(S ? x)dx75
L(s; S )
s
u
u
h
s
S
l
i = 1 + MC(S ? s) M (S ? s) ? M (S ? S l) + M (S ? S u) " " l ) ? M (S ? S u ) # l ) ? M (S ? S u ) # 1 + M ( S ? S 1 + M ( S ? S = C ?C C?C 1 + M ( S ? s ) 1 + M (S ? S l) # " M (S ? S u) : (20) = C 1+ M (S ? S l) 13
L(x) C = 2L(0; 0)
Su
Sl
x
Figure 1: De nitions of S l and S u In the second case S l < s < S u, we use (5) and (4) to immediately obtain " ZS ZS u) # M ( S ? S C L(s; S ) L(x)f (x)dx = 1 + M (S ? s) m(S ? x)dx = C 1 + M (S ? s) : S
S
u
But in this case S ? s S ? S l. Therefore,
u
"
u) # M ( S ? S L(s; S ) C 1 + M (S ? S l) : (21) Since M (t)=t ! 1= as t ! 1, the expression in the square brackets in (20) and (21), which does not depend on s, goes to one as S ! 1, i.e., M (S ? S u) = lim M (S ? S u) = lim M (S ? S u ) S ? S l S ? S u lim S !1 M (S ? S l ) S !1 S ? S u M (S ? S l ) S ? S l S !1 1 + M (S ? S l ) = 1 1 = 1:
Therefore, there is an S S u, independent of s, such that M (S ? S u )=(1 + M (S ? S l)) > 1=2 for all S S. This means that L(s; S ) > L(0; 0) for all S S and s < S u, which proves the lemma in the rst two cases. Finally, for the third case S u s S , we use (5) to obtain Zs ZS L(s; S ) = [K + L(S )]f (x)dx + L(x)f (x)dx C > L(0; 0): ?1
s
This completes the proof. 14
Lemma 5.2 For S l de ned in Lemma 5.1, L(s; S ) > L(0; 0) for all S S l and s S . Proof. For S S l, it is clear from (5) that Zs
ZS
?1
s
L(s; S ) = [K + L(S )]f (x)dx + L(x)f (x)dx C > L(0; 0): This completes the proof. It is easy to see that together Lemmas 5.1 and 5.2 prove that the minimizing value of S lies in the set [S l; S]. In the next lemma we show that the minimizing value of s is bounded as well.
Lemma 5.3 There is a constant s such that for all s s and S s, L(s; S ) > L(0; 0). Proof. Let S l and S be de ned as in Lemma 5.1. Let s S l. For any S satisfying s S S l or s S l S S , it follows from Lemmas 5.1 and 5.2 that L(s; S ) > L(0; 0). Therefore, we can restrict our attention to the values of S which satisfy S l S S. Then from (5) and (4) we have
2S 3 Z L(s; S ) L(x)f (x)dx C f (x)dx = 1 + MC(S ? s) 64 m(S ? x)dx75 s s s " l) # h i C 1 + M ( S ? S l = 1 + M (S ? s) M (S ? s) ? M (S ? S ) = C ? C 1 + M (S ? s) " ? S l) # 1 + M ( S C ? C 1 + M (S l ? s) : ZS
ZS
l
l
l
(22)
Since M (t) ! 1 as t ! 1, the expression in the square brackets in (22), which does not depend on S , goes to zero as s ! ?1. Therefore, there is an s S l, independent of S , such that (1 + M (S ? S l))=(1 + M (S l ? s)) < 1=2 for all s s. This means that
L(s; S ) > C=2 = L(0; 0) for all s s and S > s, and the proof is completed.
Remark 5.1 The proofs of Lemmas 5.1 and 5.2 can be easily extended to nondi erentiable
renewal functions M by using Lebesgue-Stieltjes integrals. Therefore, the assertions of these two lemmas also hold for demand distributions which do not have densities.
Theorem 5.1 If the one-period demand has a density, then the function L(s; S ), de ned on ?1 < S < 1, s S , attains its minimum. Furthermore, if (s; S ) is a minimum point of L(s; S ), then S l S S and s s S . 15
Proof. Because of Lemmas 5.1{5.3, the search for a minimum point can be restricted to the s s S g. It follows immediately from the existence of a compact set f(s; S ) : S l S S; density of the one-period demand that L(s; S ) is continuous, and therefore it attains its in mum over the compact set.
It remains to show that the minimum is attained at an interior point and that there is a minimum point that satis es (12).
Lemma 5.4 If K > 0, then there is a pair (s; S ), with s < S , that minimizes L(s; S ). Proof. We distinguish three cases:
Case 1. If m(x) is identically zero on [0; "] for some " > 0, it follows immediately that L~(; S ) is constant for 2 [0; "] and any xed S . Therefore, if (0; S ) minimizes L~(; S ), so does ("; S ), which is an interior point. Case 2. Let now m(0) > 0. Any minimal point of L^(s; ) satis es (9). Let s be a solution of (9) for = 0. Then it follows from (10) that
@ L^(s; ) = ?Km(0) < 0 @ =0 and, therefore, = 0 cannot be a minimizer. Case 3. If m() does not satisfy either of the two cases above, it follows from the continuity of m as a sum of convolutions of continuous densities that there is an " > 0 such that m(x) > 0 for all x 2 (0; "]. This property implies that M (x) > 0 for x > 0. For a xed S it holds that R K + L(S ) + L(S ? x)m(x)dx 0 L~(S; ) ? L~(S; 0) = ? (K + L(S )) 1 + M () 1 (23) = 1+M () (?M ()(K + L(S )) + L(S ? )M ()) for some 2 [0; ]. Since M () > 0 for 2 (0; "] and L is continuous, it follows that for suciently small L~(S; ) ? L~(S; 0) < 0; and = 0 cannot be a minimizer.
Lemma 5.5 There are s and S , with s < S , which minimize L(s; S ) and, at the same time, satisfy (12).
Proof. It follows from Lemma 5.4 that there is an interior minimizer (s#; S ) of L(s; S ). If m(S ? s# ) > 0, it follows immediately from Iglehart's analysis that (12) is satis ed. 16
If m(S ? s# ) = 0, we de ne 0 = inf f 0 : m(x) 0 on [; S ? s# ]g and
1 = supf 0 : m(x) 0 on [S ? s# ; ]g: Obviously for 2 [0; 1], the pair (S ? ; S ) minimizes L. Let " > 0. Then we have
L~(1; S ) ? L~(1 + "; S ) 0: Therefore, it follows that 1 0 Z1 @K + L(S ) + L(S ? )m()dA (1 + M (1 + ")) 0 0 1 Z1 +" ? @K + L(S ) + L(S ? )m()dA (1 + M (1)) 0; or,
0
0 1 Z1 @K + L(S ) + L(S ? )m()dA (M (1 + ") ? M (1)) 0
?(1 + M (1))
Z1 +" 1
L(S ? )m()d 0:
Applying the Mean Value Theorem and dividing by (M (1 + ") ? M (1))(1 + M (1)), which is strictly positive by the de nition of 1 and the monotonicity of M (), we obtain R 1 K + L(S ) + L(S ? )m()d 0 L(S ? ) 1 + M ( ) 1
for some 2 [1; 1 + "]. Since L(y) is continuous, we nd for " ! 0, R 1 K + L(S ) + L(S ? )m()d 0 L(S ? 1): L(1; S ) ? c = 1 + M ( ) 1
If 0 > 0, we nd analogously
L~(0; S ) L(S ? 0):
(24) (25)
If 0 = 0, it is easy to see from (7) that
L~(0; S ) ? c = K + L(S ) > L(S ): 17
(26)
Since L~(; S ) and L() are both continuous, it follows from (24)-(26) that there is a > 0 such that L~(; S ) ? c = L(S ? ); i.e., (s; S ) := (S ? ; S ) is an interior minimizer that satis es (12). Lemma 5.5 nally establishes the existence of an interior minimizer of the stationary cost function that satis es equation (12) as required for Iglehart's analysis. With that analysis completed, the following result is easily established with the help of the same short additional argument that C. Derman suggested to Veinott and Wagner.
Theorem 5.2 The parameters s and S obtained in Lemma 5.5 determine a stationary (s; S )strategy which is average optimal.
Remark 5.2 It follows from (9) that for any minimizer (s; S ) of L(s; S ), we have s argmin L(y). Therefore, for any two minimizers (s1; S1) and (s2; S2) of L(s; S ) that satisfy (12), it holds that
L(s1) = L(s1; S1) ? c = k ? c = L(s2; S2) ? c = L(s2): Since k ? c > minS L(S ), L(x) is convex, and limx!1 L(x) = 1, it follows that s1 = s2.
6 Stationary Analysis versus Dynamic Programming Beyer, Sethi and Taksar (1996) have established the average optimality of an (s; S )-strategy in the more general setting of Markovian demand. They use dynamic programming and a vanishing discount approach to obtain the average cost optimality equation and show that it has a K -convex solution, which provides an (s; S )-strategy that minimizes (2). Furthermore, they prove a veri cation theorem stating that any stable policy (de ned later in the section; see (29)) satisfying the average cost optimality equation is average optimal. More speci cally, Beyer, Sethi and Taksar prove that there is a policy Y that minimizes the average cost de ned by 1 f (xjY ) J (xjY ) = lim sup n n!1 n over all history-dependent policies Y 2 Y . Furthermore, this policy Y can be represented as an (s; S ) policy. In addition, they show that this policy also minimizes the criterion 1 f (xjY ) a(xjY ) = lim inf n!1 n n 18
over the class of all stable policies.4 that Moreover, the completion of Iglehart's analysis in the previous section allows us to drop the stability restriction on the class of admissible strategies and to obtain the stronger result that an optimal (s; S )-strategy also minimizes a(xjY ) over all history-dependent policies Y 2 Y . The following proposition connects the two approaches. For the average cost problem under consideration, the average cost optimality equation derived in Beyer, Sethi and Taksar (1996) is given by Z1 (27) (x) = min [^c(y ? x) + L(y) ? + (y ? )'()d]: yx 0
A pair (; ) such that (x) = min[^c(y ? x) + L(y ) ? + yx
Z1 0
(y ? )'( )d ]
is called a solution of (27). Note that for = k, (27) reduces to equation (13) speci ed by Iglehart.
Proposition 6.1 Let (; ) be a solution of the average cost optimality equation (27). Let
be continuous and let the minimizer on the right-hand side of (27) be given by
8 < y(x) = : S if x s x if x > s
for ?1 < s S < 1. Then, (i) the pair (s; S ) minimizes L(s; S ), (ii) (s ; S ) satis es (12), and (iii) for all history-dependent policies Y 2 Y , it holds that = k = lim 1 f (xjs; S ) lim inf 1 f (xjY ): n!1 n n
n n Proof. To prove (i) we assume to the contrary that (s; S ) does not minimize L(s; S ). Then there is another strategy (s; S ) with L(s; S ) > L(s; S ). Therefore, in view of Remark 3.1, we obtain 1 f (xjs; S ) = L(s; S ) > L(s; S ) = lim sup 1 f (xjs; S ): (28) lim sup n n n!1 n n!1 n Equation (28) contradicts the optimality of (s; S ) proved in Beyer, Sethi and Taksar for the average cost objective function (2). Therefore, (s; S ) minimizes L(s; S ). It is shown in Beyer, Sethi and Taksar that the from the solution of the average cost optimality equation is equal to the minimum of the average cost de ned in (2), and is therefore n!1
Bounds on the action space imposed by Tijms (1972) imply that the admissible policies considered by him are stable. In this case, either of the criteria | lim inf or lim sup | can be used in the MDP context. 4
19
equal to k = L(s; S ). Knowing that, part (iii) of the proposition follows immediately from (i) and Theorem 5.2. For the proof of (ii) we note that can by expressed as in (14). As shown in Iglehart, is continuous if and only if (12) holds. This proves part (ii) of the proposition.
Proposition 6.2 If limx!?1 [cx + L(x)] = 1, then there is a unique (up to a constant) con-
tinuous bounded-from-below K -convex function such that ( ; ) is a solution of the average cost optimality equation (27). Furthermore, is equal to the minimal average cost with respect to either of (1) and (2). Proof. Because of the K -convexity of the solution, the minimizer on the right-hand side of (13) is given by 8 < y(x) = : S if x s x if x > s for some not necessarily nite s S . Since limx!?1 (cx + L(x)) = 1 and is bounded from below, it follows that
h
lim cx + L(x) ? k + x!?1
Z1 0
(x ? )'( )d
i
= 1;
and therefore s > ?1. Since limx!1 L(x) = 1 and is bounded from below, it is obvious that Z1 i h (y ? )'( )d = 1; lim cy + L ( y ) ? k + y!1 therefore S < 1.
0
and It is easy to show that the (s; S )-strategy with nite parameters s and S is stable with respect to , i.e., (xn ) (29) nlim !1 n = 0: Given this fact, Beyer, Sethi and Taksar (1996) prove that (s; S ) is an optimal strategy with respect to the average cost de ned in (2) with minimal average cost . It follows from Proposition 6.1 that = k By Proposition 6.1, (s; S ) minimizes L(s; S ) and satis es (12). It follows from Remark 5.2 that s is unique. The K -convexity and the parameter s determine the solution (14) of (13) uniquely (up to a constant). Therefore, ( ; ) is the unique solution of (27) with the desired properties.
Remark 6.1 In the case of a constant unit shortage cost p, the condition limx!?1(cx + L(x)) = 1 is equivalent to the requirement p > c. This condition was only introduced to simplify the
proof and can be dropped altogether.
20
7 Conclusions In this paper we have reviewed the classical papers of Iglehart (1963) and Veinott and Wagner (1965) treating single-product average-cost inventory problems. We have pointed out some conditions that are assumed implicitly but not proved in these papers, and we have proved them rigorously. In particular, we have provided bounds for any pair (s; S ) minimizing the stationary one-period cost L(s; S ), using only the properties of the stationary cost function. The main purpose of our paper has been to complete the stationary distribution analyses of Iglehart and Veinott and Wagner. Therefore, we have stayed with the relatively restrictive assumptions on the demand distribution made by Iglehart. However, these assumptions are not necessary for the results obtained in our paper. Indeed, it can be shown that even for general demand distributions consisting of discrete and continuous components, the corresponding stationary cost function attains its minimum provided the expected values of all the quantities required in the analysis exist. Furthermore, there is a pair minimizing L(s; S ) that satis es (12) and that can in turn be proved to be average optimal. We have also brought out the connection between the stationary distribution approach and the dynamic programming approach to the problem. While dynamic programming is applicable even in problems defying a stationary analysis but with objective function (2), the stationary analysis, when possible, can prove optimality with respect to the more conservative objective function (1). Finally, by combining both approaches we have shown that an (s; S ) policy { optimal with respect to either of the objective functions (1) and (2) { is also optimal with respect to the other.
21
Appendix Example. We will give an example with K > 0, in which (i) s = S is optimal and (ii) m(S ? s) = 0 for all minimizing pairs of parameters (s; S ). To do so, we have to prove two preliminary results.
Lemma 7.1 Let L() be a convex function with limx!1 L(x) = 1. Then for 0 x1 x2, min fL(S ) + L(S ? x1)g min fL(S ) + L(S ? x2)g: S S
Proof. Because of the limit property and the convexity of L(), L(S ) + L(S ? x2) attains its minimum. Let S2 be a minimum point of L(S ) + L(S ? x2). Let us assume rst that L(S2) L(S2 ? x2). Then it follows from the convexity of L() that L(S2 ? x2 + x1) L(S2), and we obtain
min fL(S ) + L(S ? x1)g L(S2 ? x2 + x1) + L(S2 ? x2)g S L(S2) ? L(S2 ? x2) = min fL(S ) + L(S ? x2)g: S
If L(S2) < L(S2 ? x2), we conclude from the convexity of L() that L(S2 ? x1) L(S2 ? x2). Thus, min fL(S ) + L(S ? x1)g L(S2) + L(S2 ? x1)g S L(S2) + L(S2 ? x2) = min fL(S ) + L(S ? x2)g; S which completes the proof.
Lemma 7.2 Let L() be a convex function with limx!1 L(x) = 1. Then lim minfL(S ) + L(S ? D)g = 1: D!1 S
Proof. Fix D > 0 and let S denote a xed minimum point of L(). It is easy to see that L(S ) + L(S ? D) L(S ) L(S + D=2) for S S + D=2:
Also,
L(S ) L(S ? D=2) for S S ? D=2:
Thus,
L(S ? D) L(S ? D=2) for S S + D=2; from which we can conclude that L(S ) + L(S ? D) L(S ? D) L(S ? D=2) for S < S + D=2: 22
Therefore, it follows that lim minfL(S ) + L(S ? D)g Dlim minfL(S + D=2); L(S ? D=2)g = 1; !1
D!1 S
and the proof is completed.
Example (cont'd). For the purpose of this example we assume the unit purchase cost c = 0.
It can be easily extended to the case c > 0. Let the one-period demand consist of a deterministic component D 0 and a random component d. Let '() be the density of d, where '() is continuous on (?1; 1), '(t) = 0 for t 0, and '(t) > 0 for t 2 (0; ") for some " > 0. The density of the one-period demand is then given by 'D (t) = '(t ? D). It is continuous on the entire real line. We denote the renewal density and the renewal function with respect to 'D by mD and M D , respectively. Let Z1 D L (x) = l(x ? )'D ()d ?1
be the expected one-period holding and shortage cost function. Let L() = L0(). Then we have
LD (x) = L(x ? D): Now the stationary cost function given s and S is RS K + LD (S ) + s LD (x)mD(S ? x)dx ; LD (s; S ) = 1 + M D (S ? s)
and in view of = S ? s, we de ne
K + LD (S ) + R LD (S ? x)mD(x)dx 0 : L~D (; S ) = LD (S ? ; S ) = 1 + M D () We will show that there is a constant D0 such that for all D > D0, the pair that minimizes the stationary average cost are of the following form:
(s; S ) is a minimum point of LD (s; S ) if, and only if, S minimizes LD (S ) and 0 S ? s D . It is obvious that because of 'D (t) = 0 for all t D, mD (t) = 0 and M D (t) = 0 for t D. Therefore, for any given D, min L~D (; S ) = K + min LD (S ) = K + min L(S ): S S S 23
We will show that for a suciently large D, min L~D (; S ) > K + min L(S ) for all > D: S
S
To do so we consider three cases arising when > D for any given D. Case 1. Let 2 f : 0 < M D () 1g. L~D (; S ) min S ) ( R LD (S ) + LD (S ? x)mD (x)dx K + min S 0 = D ( 1 + M () ) D D D L (S ) + x2min fL (S ? x)gM () K + min S [D;] D 1+M ( () ) D D D D D D (1 ? M ())L (S ) + x2min fL (S ? x) + L (S ) ? K gM () (1 + M ())K + min S [D;] = 1 + M D () (1 ? M D ()) min LD (S ) + M D () min f min fLD (S ? x) + LD (S ) ? K gg S S x2[D;] K+ : (30) 1 + M D () Using Lemma 7.1 we have min f min fLD (S ? x) + LD (S ) ? K gg = min f min fL(S ? D ? x) + L(S ? D) ? K gg S x2[D;] S x2[D;] = min f min fL(S ? x) + L(S ) ? K gg S x2[D;] = min fL(S ? D) + L(S ) ? K g: (31) S
On account of Lemma 7.2, this expression tends to in nity as D ! 1. Therefore, there is a D1 > 0 such that min fL(S ? D) + L(S ) ? K g > 2 min L(S ) for all D D1 : S S Thus from (30){(32), we have
L~D (; S ) > K + min L(S ) for all D D1 ; min S S implying that of this case cannot be a minimizer. Case 2. Let 2 f : 1 < M D () 2g. Then K + LD (S ) + R LD (S ? x)mD(x)dx 0 L~D (; S ) = 1 + M D () 24
(32)
K + 12 R (LD (S ? x) + LD (S ))mD (x)dx 0 3 1 D (S ? x) + LD (S )gM D () 1 min fLD (S ? x) + LD (S )g: f L 6 x2min [D;] 6 x2[D;]
In view of Lemma 7.1, taking the minimum with respect to S yields ~D (; S ) 1 minf min fLD (S ? x) + LD (S )gg = 1 minfLD (S ? D) + LD (S )g L min S 6 S x2[D;] 6 S = 61 min fL(S ? D) + L(S )g: (33) S
From Lemma 7.2 we know that this expression tends to in nity as D ! 1. Therefore, there is a D2 > 0 such that min fL(S ? D) + L(S )g > 6(K + min L(S )) for all D D2: S S
(34)
Then from (33) and (34),
L(S ) for all D D2 ; L~D (; S ) > K + min min S S and thus of Case 2 cannot be optimal. Case 3. Let 2 f : 2 < M D ()g. Now we de ne C := 3(K +minS L(S )) = 3(K +minS LD (S )), Sl := minfS : LD (S ) C g, and Su := maxfS : LD (S ) C g. It follows from the convexity of LD () and the fact that limx!1 LD (x) = 1 that Sl and Su are nite. Let D0 := maxfD1; D2; Su ? Slg and choose D D0. Then we obtain K + LD (S ) + R LD (S ? x)mD(x)dx R LD (S ? x)mD(x)dx 0 L~D (; S ) = 0 1 + M D () : 1 + M D () It is easy to conclude from the de nitions of Sl and Su that LD (S ? x) C for x 2 [0; ] n [S ? Su ; S ? Sl]. Because [0; ] n [S ? Su ; S ? Sl] = [0; ] n [(S ? Su)+ ; (S ? Sl)+ ]; we obtain
D + ) ? M D ((S ? Su )+ )) : (35) L~D (; S ) CM () ? C (M ((1S+?MSDl)() Since 0 (S ? Sl)+ ? (S ? Su)+ Su ? Sl D, it is clear that there is almost surely at most one renewal between (S ? Su)+ and (S ? Sl)+ , and therefore M D ((S ? Sl)+) ? M D ((S ? Su)+ ) 1.
Thus,
D () ? 1) C C ( M D ~ L(S ): L (; S ) 1 + M D () > 3 = K + min S
25
Therefore, for D > D0 and all = S ? s > D, we have
L~D (; S ) min LD (S; S ): min LD (s; S ) = min S S S Because LD (s; S ) is constant in s for S ? D s S , it is clear that the minimizing point (s; S ) satis es the desired conditions. Furthermore, because mD (t) = 0 for 0 t D, it holds that
mD (S ? s) = 0 for all minimum points of LD (s; S ):
References Beyer, D. and Sethi, S. P. 1997. Average Cost Optimality in Inventory Models with Marko-
vian Demands. Journal of Optimization Theory and Application, 92, No. 3, forthcoming.
Beyer, D., Sethi, S.P., and Taksar, M. 1996. Inventory Models with Markovian Demand
and Cost Functions of Polynomial Growth. Working Paper, University of Toronto, Toronto, Ontario, Canada.
Federgruen, A. and Zipkin, P. 1984. An Ecient Algorithm for Computing Optimal (s; S )
Policies. Oper. Res. 32, 1268{1285.
Fu, M. 1994. Sample Path Derivatives for (s; S ) Inventory Systems. Oper. Res., 42, 351{364. Hu, J.-Q., Nananukul, S. and Gong, W.-B. 1993. A New Approach to (s; S ) Inventory
Systems. J. Appl. Prob. 30, 898-912.
Iglehart, D. 1963. Dynamic Programming and Stationary Analysis of Inventory Problems.
In: Scarf, H., Gilford, D., and Shelly, M. (Eds.) Multistage Inventory Models and Techniques. Stanford University Press, 1{31.
Karlin, S. 1958a. Steady State Solutions. In: Arrow, K., Karlin, S., and Scarf, H. (Eds.),
Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, 224-269.
Karlin, S. 1958b. The Application of Renewal Theory to the Study of Inventory Policies.
In: Arrow, K., Karlin, S., and Scarf, H. (Eds.), Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, 270{297.
Kuenle, C. and Kuenle, H.-U. 1977. Durchschnittskostenoptimale Strategien in Markov-
schen Entscheidungsmodellen bei unbeschrankten Kosten. Mathematische Operationsforschung und Statistik, Series Optimization, 8, No. 4, 549{564. 26
Porteus, E. 1985. Numerical Comparison of Inventory Policies for Periodic Review Systems.
Oper. Res., 33, 134{152.
Sahin, I. 1990. Regeneration Inventory Systems: Operating Characteristics and Optimization.
Bilkent University Lecture Series, Springer Verlag, New York.
Scarf, H. 1960. The Optimality of (s; S ) Policies in the Dynamic Inventory Problem. In:
Arrow, J., Karlin, S. and Suppes, P. (Eds.), Mathematical Methods in Social Sciences. Stanford University Press, Stanford, CA, 196{202.
Schal, M. 1976. On the Optimality of (s; S ) Policies in Dynamic Inventory Models with Finite
Horizon. SIAM J. Appl. Math., 30, 528{537.
Sethi, S. P. and Cheng, F. 1995. Optimality of (s; S ) Policies in Inventory Models with
Markovian Demand. Forthcoming in Oper. Res.
Stidham, S. Jr. 1977. Cost Models for Stochastic Clearing Systems. Oper. Res., 25, No. 1,
100{127.
Tijms, H. 1972. Analysis of (s; S ) inventory models. Mathematical Centre Tracts, 40, Mathe-
matisch Centrum, Amsterdam, The Netherlands.
Veinott, A. and Wagner, H. 1965. Computing optimal (s; S ) policies. Mgmt. Sci., Vol.11,
525{552.
Veinott, A. 1966. On the Optimality of (s; S ) Inventory Policies: New Conditions and a New
Proof. J. SIAM Appl. Math., 14, No. 5, 1067{1083.
Wijngaard, J. 1975. Stationary Markovian Decision Problems. Doctoral Theses, Technische
Hogeschool Eindhoven, Eindhoven, The Netherlands.
Zheng, Y.-S. 1991. A Simple Proof for Optimality of (s; S ) Policies in in nite-Horizon Inven-
tory Systems. J. Appl. Prob., 28, 802-810.
Zheng, Y.-S., and Federgruen, A. 1991. Finding Optimal (s; S ) Policies Is About As
Simple as Evaluating a Single Policy. Oper. Res., 39, No. 4, 654{665.
27
