2. THE MODEL

Consider a single product, periodic review, two-stage production–inventory system with a single capacitated manufacturer and a single retailer facing independent demand and using an order-up-to inventory policy. In practice, many retailers place orders periodically (e.g., every week or every month). Thus, we assume that the retailer places an order every T time periods (e.g., 7 days) to raise his inventory position to a target level. The manufacturer receives demand information from the retailer every τ units of time (τ ≤ T). For instance, the retailer places an order every week but provides demand information every day. We refer to the time between successive orders as the ordering period and the time between successive information sharing as the information period. Of course, in most supply chains, information can be shared almost continuously (e.g., every second) while decisions are made less frequently (e.g., every day). Thus, information periods really refers to the time interval between successive uses of the information provided.

The sequence of events in our model is as follows. At the beginning of an ordering period, the retailer reviews her inventory and places an order to raise the inventory position to the target inventory level. The manufacturer receives the order from the retailer, fills the order as much as she can from stock, and then makes a production decision. If the manufacturer cannot satisfy all of a retailer's order from stock, then the missing amount is backlogged. The backorder will not be delivered to the retailer until at least the beginning of the next ordering period. Finally, transportation lead time between the manufacturer and the retailer is assumed to be zero. Similarly, at the beginning of an information period, the retailer transfers POS (Point of Sales) data of previous information period to the manufacturer. Upon receiving this demand information, the manufacturer reduces this demand from her inventory position although she still holds the stock, and then she makes a production decision.

To simplify the analysis, we assume that demand is stationary across the ordering periods, whereas the demand distribution may vary across different information periods within one ordering period. For example, grocery retailers typically face much higher demand during weekends than week days. At the same time, these retailers are replenished once a week. Since grocery retailers cannot anticipate future shortages from the manufacturer, they are not able to inflate order-up-to levels to protect themselves from future shortages. Hence, we assume in our model that the retailer's target order-up-to level is constant for all the ordering periods; that is, every ordering period, the retailer raises her inventory position (outstanding order plus on-hand inventory minus backorders) to a constant level. Any unsatisfied demand at the retailer is backlogged; thus, the retailer transfers external demand of each ordering period to the manufacturer. The manufacturer has a production capacity limit (i.e., a limit on the amount the manufacturer can produce per unit of time) and the manufacturer runs her production line always at this capacity limit.

Let N = T/τ be an integer which represents the number of information periods in one ordering period. We index information periods within one ordering period 1,2,…,N, where 1 is the first information period in the ordering period and N is the last. Let C denote production capacity in one information period τ, c is the production cost per item, and 0 < β < 1 is the time discounted factor for each information period.

Since we calculate inventory holding cost for each information period, we let h be the inventory holding cost per unit product per information period. Evidently, one unit of product kept in inventory for n information periods, n = N,N − 1,…,1, will incur a total inventory cost h_n = h(1 + β +···+ βⁿ⁻¹).

Due to the finite production capacity, the manufacturer produces to stock. For every item backlogged at the end of each ordering period, the manufacturer pays π as a penalty cost. It is easy to see that the earlier the manufacturer produces during a single ordering period, the longer she will carry this inventory, thus the higher the inventory holding cost. On the other hand, postponing production too long may result in heavy penalty cost due to the finite production capacity.

We use D_i to denote the end-user demand in the information period i, i = 1,…,N. D_i is assumed to be independent, its distribution only depends on i, and its mean is defined as ED_i. To simplify the analysis, we assume that costs do not change from information period to information period; later, we will demonstrate that our results can be easily extended to include cases where costs change periodically within one ordering period but are the same across different ordering periods.

We start by considering a finite time horizon model with M ordering periods. Index the ordering periods from 0 to M − 1, where the 0 ordering period is the first one and the M − 1 ordering period is the last one. The finite horizon starts in the first ordering period at the beginning of the jth information period, 1 ≤ j ≤ N. Consider the ith information period, i = 1,2,…,N, in ordering period m, m = 0,1,…,M − 1. Of course, mN + i ≥ j. We refer to this information period as the mN + i information period. For instance, information period mN + 1 is the first information period in the mth ordering cycle. We refer to this indexing convention as a forward indexing process.

Define x to be the inventory position at the beginning of the ith information period, and S_i to be the state space for x. Let y − x be the amount produced in that information period, y ∈ A_x, and A_x be the set of feasible actions. Let ξ(x,y,D) be the transition function and D be the demand. In our case, A_x = [x,x + C] and ξ(x,y,D) = y − D.

It is easy to verify that g_mN+i(x,y), i = 1,2,…,N, m = 0,…,M − 1, the expected inventory and production cost in information period mN + i, given that the period starts with an inventory position x, and y − x items are produced in that period, can be written as

where E(·) is the expectation with respect to D_N and

Because the items produced in the ith information period are carried over until the end of the ordering period, an inventory holding cost equal to h_N−i(y − x) is charged in this information period.

Since g_mN+i(x,y) depends only on i and not m, we can write g_mN+i(x,y) = r_i(x,y), ∀m = 0,1,…,M − 1 and ∀i = 0,1,…,N. Furthermore, r_i(x,y) can be written as the sum of a function of x (φ_i(x)) and a function of y(φ_i(y)), where

Finally, we assume the salvage cost function g_MN+1 = 0.

We define the optimal policy under the discounted cost criterion as follows. Let σ = {σ₁,σ₂,…} be any feasible policy where σ_k is a function depending on the initial inventory position in period k (i.e., σ_k = σ_k(x) and σ_k(x) ∈ A_x for all k). Let Π be the set of all feasible policies. Define the expected discounted cost from the ith information period in the first ordering period until the end of the horizon when M → ∞, as

where E(·) denotes the expectation with respect to demand in all information periods and x_i is the initial inventory position of the ith information period for i = 1,2,…,N. A policy σ* = {σ₁*,σ₂*,…} ∈ Π is called optimal under the discounted cost criterion if for all x ∈ S and i,

Similarly, the optimal policy in an infinite time horizon under the average cost criterion can be defined. Following Heyman and Sobel [14], the performance measure for any feasible policy δ = {δ₁,δ₂,…} ∈ Π under the average cost criterion is defined as

A policy δ* is optimal if it minimizes G_i(x,δ) for all x and i over Π.

It is easily seen that in the information sharing model, except for those periods in which orders are placed, the cost function φ_i(y) tends to negative infinity as y → −∞ (y → −∞ implies that the manufacturer allows for infinite backorders). This is true because although demand is observed by the manufacturer in every information period, it only needs to be satisfied at the end of every ordering period. Thus, the penalty cost is charged only at the end of ordering periods, whereas the inventory holding cost is charged in every information period. Clearly, this issue was not addressed in the models analyzed by Aviv and Federgruen, and Kapuscinski and Tayur since they assumed φ_i(y), for all i, were bounded from below.

To further explain the difference between this model and previous models, let us consider the following three cases. First, if the manufacturer only has inventory holding cost but no penalty cost for all periods, then the optimal policy is not finite; that is, the optimal policy does not have a finite order-up-to level because producing nothing in all periods is clearly the optimal policy. Second, if the manufacturer has both inventory holding cost and penalty cost for all periods, then we have the models studied by Aviv and Federgruen [1] and Kapuscinski and Tayur [16]. Finally, if in some periods the manufacturer only has inventory holding cost, whereas in other periods she has both holding and penalty cost, then it is not clear whether there exists an optimal policy with a finite order-up-to level.

Thus, the objective of this article is twofold: First, characterize the optimal policy for the information sharing model under both discounted and average cost criteria; second, quantify the benefits and identify the conditions under which information sharing is most beneficial for the manufacturer in an infinite time horizon; that is, characterize how frequently information should be shared and when it should be shared so that the manufacturer can maximize the potential benefits.

We show that a cyclic order-up-to policy is optimal under both discounted and average cost criteria. Although the optimal policy has the same structure as those of Aviv and Federgruen [1] and Kapuscinski and Tayur [16], our analysis of the average cost criterion, which is based on the vanishing discount method (Heyman and Sobel [14]), is quite different from theirs.

One of the major difficulties in proving the average cost criterion is to characterize the Markov chain induced by the optimal policy under the discounted cost criterion. To address this issue, we introduce a new method to construct and relate Markov chains in a common state space. Together with the Foster's criterion and the Lyapunov function, this method allows us to show that each of the Markov chains induced by any finite cyclic order-up-to policy has a single irreducible positive recurrent class and finite long-run average cost under certain nonrestrictive conditions (Section 3). Then, in Section 4, we provide a simple proof for the cyclic order-up-to policy to be optimal under the average cost criterion. Finally, extensive computational study is conducted in Section 5, using IPA (Infinitesimal Perturbation Analysis), to quantify the impacts of frequency and timing of information sharing, with a particular focus on the differences between finite and infinite time horizons, and nonstationary external demand.

3. PROPERTIES OF CYCLIC ORDER-UP-TO POLICY

In this section, we study the Markov processes associated with any cyclic order-up-to policy and identify conditions under which they are positive recurrent and have finite steady state average cost. The conditions are similar to those identified by Aviv and Federgruen, and Kapuscinski and Tayur, but the analysis is quite different.

At this point, it is appropriate to remind the reader of the definition of a cyclic order-up-to policy in a capacitated production system. In such a system, every time period the manufacturer produces up to the target inventory position if there is enough production capacity, otherwise the manufacturer produces as much as the production capacity allows. Of course, the target inventory position may vary periodically.

Consider the information sharing model with the cost function r_i(x,y) ∼ O(|x|^ρ) + O(|y|^ρ), where ρ is a positive integer. Define a cyclic order-up-to policy as a policy with different order-up-to levels for different information periods, but these levels are the same for the same information period in different ordering periods; that is, the order-up-to level in information period mN + i is the same for all m but may be different for different i, i = 1,2,…,N.

Let D_i, i = 1,…,N, be the random variable representing demand in information period mN + i for all m; demand is assumed to take discrete values. Consider a cyclic order-up-to policy with levels a₁,a₂,…,a_N, and define the shortfall processes {s_mN+i,m = 0,1,…} for different i = 1,…,N as s_mN+i = a_i − y_mN+i if we are in period mN + i and y_mN+i is the inventory position at the end of this period before demand is realized. Clearly, the shortfall may be negative. When it is positive, it represents the difference between what the manufacturer likes to produce and what the production capacity allows the manufacturer to produce. When the target inventory level, a_i, is lower than the initial inventory position at the beginning of the period, the shortfall is negative. The dynamics of the shortfall is

If excessive stock is returned when the inventory position is higher than the order-up-to level, then the dynamics of shortfall processes {s_mN+i^r,m = 0,1,…} for i = 1,…,N is s_mN+i+1^r = (a_i+1 − a_i + s_mN+i^r + D_i − C)⁺. We refer to this policy as an order-up-to policy with returns.

Proving that the shortfall process induced by a constant order-up-to policy has a finite steady state average cost is known to be difficult (see, e.g., the appendix of Kapuscinski and Tayur [15]). Proving that each of the shortfall processes induced by a general cyclic order-up-to policy has finite steady state average cost and a single irreducible positive recurrent class is even more difficult because of the more complicated transition matrices.

The following methods have been applied in the literature to address these issues. Aviv and Federgruen showed that if (1) E(D_i^l) < ∞ for all positive integers l ≤ ρ + 1 and i = 1,…,N and (2)

, then for any finite order-up-to policy, the shortfall process has a finite set of states such that it can be reached with finite expected cost from any starting state. Kapuscinski and Tayur's method has the following two steps: They first characterized the shortfall process under the order-up-to zero policy. Then, they related a cyclic order-up-to policy to an order-up-to zero policy and proved that in steady state, E(|x_i|^ρ) and E(|s_i|^ρ) are finite for a cyclic order-up-to policy under similar conditions as those proved by Aviv and Federgruen, namely (1) E(D_i^2ρ+2) < ∞ for i = 1,…,N and (2)

.

In this section, we introduce a new approach to prove that {x_mN+i,m = 0,1,…}, the inventory positions at the beginning of mN + ith information period, {y_mN+i,m = 0,1,…}, the inventory position at the end of period mN + i but before demand is realized, and {s_mN+i,m = 0,1,…}, the shortfall in period mN + i, induced by any finite cyclic order-up-to policy give rise to Markov chains with a single irreducible and positive recurrent class and finite steady state average cost under the same condition as Aviv and Federgruen; that is, we require (1) E(D_i^l) < ∞ for all positive integers l ≤ ρ + 1 and i = 1,…,N and (2)

.

This is done as follows. First, using the Foster's criterion and the Lyapunov function, we show that the Markov chain induced by an order-up-to constant policy has the following properties: (i) single irreducible and positive recurrent class and (ii) finite steady state average cost. The same properties hold true for the Markov chain induced by a cyclic order-up-to policy with returns. Then, we relate Markov chains induced by a cyclic order-up-to policy and a cyclic order-up-to policy with returns in a common state space. This allows us to prove, under mild conditions, that properties such as positive recurrence and finite steady state average cost hold true for systems using a cyclic order-up-to policy if and only if they hold true for systems using a similar policy with returns.

We begin our analysis by presenting new proofs for the positive recurrence and finite steady state average cost of the order-up-to zero policy. To simplify the analysis, let us assume Pr(D_i < 0) = 0,∀i.

Proposition 3.1: Given order-up-to zero policy, the inventory positions {x_mN+i,m = 0,1,…} and {y_mN+i,m = 0,1,…} and the shortfall process {s_mN+i,m = 0,1,…} for i = 1,…,N generate discrete-time Markov chains (DTMC) with single irreducible and positive recurrent class if

.

Proof: See the appendix for details. █

The proof that the steady state average cost associated with an order-up-to zero policy is finite is based on the following lemma.

Lemma 3.2: Consider an irreducible and aperiodic Markov chain {X_n,n = 0,1,…} with a single period cost function r(·). r(·) is continuous and bounded from below. Assume that there exists a Lyapunov function V(·) mapping the state space S to [0,∞) and a constant η such that

Then, given an initial state x₀ with V(x₀) < ∞, the Markov chain X_n has finite steady state average cost if X_n is positive recurrent.

This lemma is a variation of the Foster's second criterion by Meyn and Tweedie [23; see Theorem 14.0.1 (f-regularity)]; thus, we omit the proof and refer to Simchi-Levi and Zhao [29] for technical details. The lemma implies the following.

Lemma 3.3: Consider the order-up-to zero policy. If E((D_i)^k) < ∞ for all integers

, then in steady state, the following hold:

Proof: See the appendix for details. █

Proposition 3.1 and Lemma 3.3 can be extended to any finite cyclic order-up-to policy with returns as follows.

Proposition 3.4: Consider any finite cyclic order-up-to policy with returns. If

for any positive integer k ≤ ρ + 1 and ∀i, then the following hold:

Proof: Assume a_i,i = 1,…,N, to be the order-up-to levels. We only need to transform this policy to an order-up-to zero policy and then apply Proposition 3.1 and Lemma 3.3.

Consider y_mN+i^r, the inventory positions at the end of period mN + i before demand is realized. For simplicity, we drop the subscript mN. The system dynamics is y_i+1^r = min{y_i^r − D_i + C,a_i+1} = a_i+1 + (y_i^r − a_i+1 − D_i + C)⁻, where x⁻ = min{0,x}. Let z_i′ = y_i^r − a_i and D_i′ = D_i + (a_i+1 − a_i); then,

. For z_i′ and D_i′, this is order-up-to zero policy, where demand D_i′ can be negative but bounded from below.

Notice that the shortfall processes associated with this order-up-to zero policy have state space {0,1,2,…} even if demand can be negative (due to return). Using proofs similar to those of Proposition 3.1 and Lemma 3.3, we can show that the same results hold for this order-up-to zero policy if there exists a positive constant d_i so that Pr{D_i < −d_i} = 0, ∀i = 1,2,…,N. █

We now introduce a method to construct and relate Markov chains induced by a cyclic order-up-to policy and a cyclic order-up-to policy with returns in a common state space. Consider two inventory systems: One uses a cyclic order-up-to policy and the other uses a corresponding policy with returns. The key idea of the method is to characterize the gap between the inventory position processes without returns, y_mN+i, and with returns, y_mN+i^r, assuming that both inventory systems start with the same initial inventory level and face the same stream of demand.

Lemma 3.5: Consider two inventory systems with cyclic order-up-to levels a₁,…,a_N, for N ≥ 2: one system without returns and the second one with returns. If the two systems start with the same initial state x₀ ≤ max{a₁,…,a_n} and face the same stream of random demand, then the stochastic processes {z_mN+i = y_mN+i − y_mN+i^r,m = 0,1,…} have the following properties for all i:

Proof: The proof is by induction. Clearly, z₁ = 0. Consider period mN + i and assume 0 ≤ z_mN+i ≤ max{a₁,…,a_N} − min{a₁,…,a_N}. We distinguish between the following two cases. In the first case, a_i ≤ a_i+1 and in the second case, a_i > a_i+1.

█

Remark: If x₀ > max{a₁,…,a_N}, then the state space for {z_mN+i,m = 0,1,…} is {0,1,…,x₀ − min{a₁,…,a_N}}, ∀i.

Theorem 3.6: Consider two arbitrary irreducible DTMCs {x_n,n = 0,1,…} and {y_n,n = 0,1,…} starting with the same initial state. If their difference process {z_n = x_n − y_n,n = 0,1,…} has finite state space S_z, then x_n is positive recurrent if and only if y_n is positive recurrent. Also, x_n has certain finite steady state moments if and only if the same steady state moments of y_n are finite.

Proof: First, we show that x_n is positive recurrence if y_n is positive recurrent. Define S_x and S_y to be the state space for {x_n,n = 0,1,…} and {y_n,n = 0,1,…}, respectively. Assume that y_n is positive recurrent; using contradiction, we assume x_n is transient or nonrecurrent for all of its state. Since y_n = x_n − z_n and y₀ = x₀, we have

Since S_z has only a finite number of states and x_n is transient or nonrecurrent, Pr{y_n = i|y₀} → 0 as n → ∞ for i ∈ S_y. This contradicts the assumption that y_n is positive recurrent (see Kulkarni [20, p. 80, Theorem 3.4], and Kemeny, Snell, and Knapp [19, p. 36, Prop. 1-61]).

Second, assume that y_n has finite steady state moments E(|y|^l) for 0 < l ≤ ρ, where ρ is a positive integer. Consider

Taking the limit n → ∞ on both sides, we obtain

where c₁,…,c_ρ are positive finite constants and |S_z| is the size of the state space for z_n.

The necessary condition can be easily proved in a similar way. █

Lemma 3.5 and Theorem 3.6 provide a method to simplify Markov chains with infinite state space and complicated dynamics. Finally, we have the following corollary.

Corollary 3.7: Consider any finite cyclic order-up-to policy σ. If

for all i and positive integers k such that k ≤ ρ + 1, then the following hold:

converges to a finite value independent of initial period i and initial state x.

Proof: Parts (i)–(iv) are the direct results of Theorem 3.6, Lemma 3.5, and Proposition 3.4; thus they do not need a proof.

To prove part (v), notice that

in steady state is due to the assumption that r_i(x,y) ∼ O(|x|^ρ) + O(|y|^ρ). Since the shortfall processes and thus y_mN+i and x_mN+i+1 give rise to Markov chains with single irreducible and positive recurrent class, we let the steady state distribution for (x,y) in the ith information period be p_(x,y)ⁱ, where (x,y) ∈ Ωⁱ (the feasible region of (x,y)), then

must converge since the summation is over at most a countable number of positive values and it is bounded from above. Finally, applying Proposition 1-61 (arithmetic average) of Kemeny et al. [19], the long-run average cost converges and equals the steady state average cost. █

4. A MARKOV DECISION PROCESS

Our objective in this section is to discuss the discounted and average cost criterion and present finite optimal policies for the information sharing model. For the discounted cost criterion, we follow Aviv and Federgruen [1] and specify conditions under which the cyclic order-up-to policy is optimal and the optimal order-up-to levels are finite. Other methods can be found in Heyman and Sobel [14], Federgruen and Zipkin [8], and Kapuscinski and Tayur [16].

For the average cost criterion, Sennott [25] and Federgruen et al. [6] introduced conditions under which there exists an optimal policy for general Markov decision processes (MDPs). Thus, one way to characterize the optimal policy is to verify that these conditions hold in our case. In fact, Aviv and Federgruen [1] and Kapuscinski and Tayur [16] have applied this method in their analysis.

We apply a different approach based on the vanishing discount method. For this purpose, we observe that in our model, the optimal policy under the discounted cost criterion can be characterized by a finite set of critical numbers (i.e., order-up-to levels). Thus, to prove the existence of an optimal policy under the average cost criterion, we only need to specify conditions (see Theorem 4.3) for the optimal order-up-to levels under the discounted cost criterion, instead of conditions on the entire policy.

To specify these conditions, we apply a result from Aviv and Federgruen [1]; namely that under certain conditions, the optimal order-up-to levels in the discounted cost criterion are uniformly bounded for all 0 < ε ≤ β < 1. We extend this result to the information sharing model (see Proposition 4.2), in which φ_i(y) is unbounded from both above and below for some i. This result, together with the results that long-run average cost is finite (Corollary 3.7), implies that a cyclic order-up-to policy is optimal under the average cost criterion (Theorem 4.3). We start by presenting conditions under which the optimal policy in the discounted cost case is a finite cyclic order-up-to policy.

Following convention, we index periods in a reverse order starting at the end of the planning horizon. Let m = 0 be the last ordering period and m = M − 1 be the first ordering period. We set i = N for the first information period and i =1 for the last information period in any ordering period. Thus, period mN + i represents the ith information period in the (m − 1)st ordering period. Finally, let g_mN+i(x,y) = r_i(x,y) denote the single period expected cost in this information period. We refer to this indexing as backward index and we will only use backward index in this section.

Let U_mN+i^β(x) be the minimum expected total costs if there are mN + i periods remaining in the planning horizon, starting with an initial state x. Let the salvage cost U₀^β ≡ 0, and, hence,

where E(·) is the expectation with respect to D_i. Observe that r_i(x,y) can be written as the sum of a function of x (φ_i(x)) and a function of y (φ_i(y)), where

Thus, the following recursion must hold:

Observe that in the very first information period (i.e., information period (M − 1)N + N = MN) of the entire planning horizon, we have to add h_N x⁺ to U_MN^β(x) to account for the holding cost of initial inventory.

The dynamic programming model has the following properties:

2. r_i(x,y) = φ_i(x) + φ_i(y) for all i, and there exists a positive integer ρ so that φ_i(x) ∼ O|x|^ρ and φ_i(y) ∼ O|y|^ρ. From Proposition 3.7, the total expected discounted cost using cyclic order-up-to policy is finite if

for any positive integer l ≤ ρ + 1, ∀i. This implies that U_mN+i^β(x) converges pointwise to a finite value for any finite x and for all i (Heyman and Sobel [14, Thm.8-13]). Let U_i^β(x) denote the convergence point of U_mN+i^β(x).

Theorem 4.1: For the Markov decision process defined in Eq. (19), if

then the following hold:

Proof: Since J_mN+i^β(y) is a convex function of y and lim_|y|→+∞ J_mN+i^β(y) → +∞ for all m and i, the order-up-to policy is optimal for all m and i. Since U_mN+i^β(x) is bounded from above for any m and finite x, order-up-to levels are finite as m → +∞ because of (b) (see the proof of Theorem 2 in Aviv and Federgruen [1]). Notice that U_mN+i^β(x) is nondecreasing and converges to U_i^β(x), which implies J_mN+i^β(y) is nondecreasing and converges to, say, J_i^β(y), due to the monotone convergence theorem. Hence, part 3 is true (also see the proof of Theorem 2 in Aviv and Federgruen [1]). Finally, parts 1–3 imply that cyclic order-up-to policy is optimal under the discounted cost criterion. █

We now extend Theorem 3(b) of Aviv and Federgruen [1] to our model.

Proposition 4.2: For the Markov decision process defined by Eq. (1), if the conditions of Theorem 4.1 are satisfied, then the optimal order-up-to levels y_i*,i = 1,2,…,N, under the discounted cost criterion are uniformly bounded both from above and from below for any 0 < ε ≤ β < 1.

Since the extension follows a similar proof technique to the one in Aviv and Federgruen, we omit the proof and refer to Simchi-Levi and Zhao [29] for technical details. We are ready to characterize the average cost criterion.

Theorem 4.3: Consider the information sharing model and assume the following:

If these conditions are satisfied, then a cyclic order-up-to policy is optimal under the average cost criterion.

Proof: Consider a sequence of β₁,β₂,…,β_j,…↑1 as j → ∞. Since the optimal order-up-to levels under the discounted cost criterion are uniformly bounded from both above and below for all 0 < ε ≤ β < 1, there must exist a finite cyclic order-up-to policy f and a subsequence j_n → ∞ so that f is the optimal policy for all U_i^β_jn(x), ∀i = 1,2,…,N.

Since we can show that for any finite cyclic order-up-to policy σ,

converges to a finite value which is independent of the initial period i and initial state x (see Corollary 3.7); then, by Tauberian theory (Heyman and Sobel [14, p.172]),

The last inequality is justified by the Lemma A2 of Sennott [25]. █

Finally, we combine Theorem 4.3, 4.1, Corollary 3.7 and Proposition 4.2, to get

Corollary 4.4: In the information sharing model, if

then cyclic order-up-to policy is optimal under the average cost criterion.