Optimal Control of an Inventory System with Joint Production and Pricing Decisions

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1 Optimal Control of an Inventory System with Joint Production and Pricing Decisions Ping Cao, Jingui Xie Abstract In this study, we consider a stochastic inventory system in which the objective of the manufacturer is to maximize the long-run average profit by dynamically offering the selling price and switching the production on or off. The demand process is non-homogeneous Poisson with a price-dependent arrival rate. System costs consist of switching costs, inventory holding and backlogging costs. We show that an s, S, p policy is optimal. Moreover, we characterize the structural properties of the optimal profit function and pricing strategy, and show that the optimal price is a quasiconcave function of the inventory level when the production is off and is a quasiconvex function when the production is on. Index Terms production/inventory systems; average criterion; optimal s, S, p policy; limited capacity. I. INTRODUCTION With the rapid development of e-commerce and new technologies e.g., big data, could computing and electronic funds transfer, a number of industries have begun to adopt dynamic pricing strategies and sell products directly to customers through their websites. For example, Dell sells its products through its website, offering promotions every week and even changing prices daily. The Alibaba Group operates leading online and mobile marketplaces in retail and wholesale trades and provides technology and services to enable buyers and suppliers to conduct commerce more easily. These developments have resulted in an increased interest in the construction of models to integrate production decisions, inventory control, and pricing strategies to improve the profitability of companies. In this study, we consider an inventory model with joint production and pricing decisions, in which the demand process is non-homogeneous Poisson with a price-dependent arrival rate, the production rate is constant and the production can be switched on or off with a fixed cost. When the production capacity is unlimited, this problem becomes an inventory control problem with pricing and replenishment decisions, which has been well studied in the literature. For example, Chen and Simchi-Levi [3] consider a periodic-review model with price-dependent demand and a fixed ordering cost, and they show under some conditions that the optimal policy for the infinite-horizon problem is an s, S, p policy. In [4], Chen and Simchi-Levi consider a continuous-review model in which the inter-arrival time and the demand size depend on the selling price. They show that optimal stationary s, S, p inventory policies exist for both discounted and average profit problems and prove that a higher inventory holding and shortage cost result in a School of Management, University of Science and Technology of China, Hefei, , China smaller selling price for the average profit problem. Chao and Zhou [2] consider an infinite-horizon, continuous-review stochastic inventory system in which the demand process is Poisson with a price-dependent arrival rate. They show that the optimal price is a unimodal function of the inventory level. However, none of them consider the case in which production capacity is limited. Several papers also consider a limited production capacity in inventory models. Federgruen and Zipkin [5] consider a periodic-review inventory model with a finite production capacity and show that under a few additional unrestrictive assumptions, a modified base-stock policy is optimal under the average-cost criterion. Foreest and Wijngarrd [6] consider a continuous review production-inventory system with compound Poisson demand and fixed switching costs and show that s, S policy is optimal if the holding cost is convex. However, they do not incorporate pricing decisions into their models. In this study, we consider a stochastic inventory system in which the manufacturer can dynamically adjust the selling price and switch the production on or off. To the best of our knowledge, this study is the first to incorporate pricing and production decisions into the inventory control. We assume that the production rate is large. Under this assumption, we show that an s, S, p policy is optimal for a discretized model. An s, S, p policy says that if the inventory level falls to s or below then the production will be switched on; if the inventory level reaches to S or above then the production will be switched off; and if the inventory level is x s < x < S, then the status of production will remain unchanged. The price p depends on the current inventory level. Moreover, we characterize the optimal inventory control and pricing strategy. We show that the optimal price is a quasiconcave function of the inventory level when the production is off and is a quasiconvex function when the production is on. The rest of the paper is organized as follows. In Section 2, we describe the inventory model and apply a discretization technique. In Section 3, we show that the s, S, p policy is optimal and characterize the structural properties of the optimal profit function and pricing strategy. In Section 4, we conclude the paper with several possible extensions. II. MODEL Consider a manufacturer who can control its inventory level by production and pricing jointly. The selling price of each product is set as p t at time t. Demand arrives according to a non-homogeneous Poisson process with rate Λ t p t, which c 205 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 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2 2 depends on the price p t. Demand is met immediately if the inventory is available. Otherwise, it is backlogged. At any time t, the production can be switched on off with cost K K 0. Production rate is constant r, i.e., the manufacturer can produce r products in one unit of time if production is on. This dynamic inventory control problem is analytically challenging. In this paper, we apply a discretization technique and solve the corresponding discretized problem. Let t = /r. Assume r is large enough so that t is very small. Due to industrialization and mass production, this assumption is not unrealistic. According to the property of non-homogeneous Poisson process, there is one arrival in time interval t, t+ t] with probability w.p. Λ t t + o t, more than one arrival w.p. o t, and no arrivals w.p. Λ t t. If t 0, then approximately there is one arrival in time interval t, t + t] with probability w.p. Λ t t, more than one arrival w.p. 0, and no arrivals w.p. Λ t t. Now, we can define the discretized problem. Let the length of each period be t. Suppose that price p P is chosen at the beginning of a period, where P is assumed to be a compact set. Let λp := Λ t t. Then with probability λp that a demand arrives, and with probability λp no demand arrives. Demand is met by end-of-period inventory, and unsatisfied demand is backlogged. Production rate is now, i.e., the manufacturer can produce one product in one period if the production is on. At the beginning of each period the state of production is chosen to be on or off. If production is off on at the end of a period, it can be switched on off at the beginning of the next period at the expense of a setup cost K K 0. When production is on off in a period, keeping it on off is free. We suppose that the production cost per item is c. Inventory costs are accrued at the end of a period according to the function h. A typical form for h is hx = hx + + bx, where x + = max{x, 0}, x = max{ x, 0}, and h and b are the holding cost and shortage cost per item per period, respectively. In the remainder of the paper, we take the following assumptions of hx. Assumption II.. hx is quasiconvex in x with its minimum point x = 0, h0 = 0, and lim x hx =. A function fx x Z is called quasiconvex unimodal if for some value m, it is increasing for x m and decreasing for x m. A function fx is quasiconcave if fx is quasiconvex. The system state can be described by its current inventory level x, and its production status y, thereby denoted by x, y, where y = indicates that the production is on and y = 0 indicates that the production is off. Assumption II.2. λp is strictly decreasing and continuous in p; Moreover, λ max p P λp <. Assumption II.3. There exists a positive number M such that max p P pλp = M <. Assumption II.4. K 0 + K > 0. Given a pricing and production policy u and the initial system state x, y, the long-run average profit can be formulated as V c x, y, u = lim inf T T E { T t= p t D t p t hx t+ cδy t = } K δy t =, y t = 0 K 0 δy t = 0, y t =, where x t, y t and p t are the inventory level, production state and selling price at the beginning of period t, respectively, D t p t is the demand in period t, which is w.p. λp t and 0 w.p. λp t. δa is an indicator function, which is if the event A is true, and 0 if A is false. The manufacturer s objective is to maximize its long-run average profit by dynamically adjusting its price and switching the production on or off, i.e., find an optimal policy u such that V c x, y V c x, y, u = sup V c x, y, u, u U where U is the set of all admissible policies. III. MAIN RESULTS A. Existence of optimal stationary policies Assumption II.4 implies that it is optimal to switch the production at most once at each state. Thus, the problem can be casted as a discrete-time infinite-horizon Markov decision problem under average criterion with state space S = {x, y x Z, y = 0, }, action space A = {p, z p P, z = 0, }, one-step reward function at state x, y under action p, z Rx, y, p, z = K z δz y + pλp cδz = λphx + z λphx + z and one-step transition probability from state x, y to x, z under action p, z λp, if x = x + δz =, P x, y, p, x, z = λp, if x = x + δz =. 0. otherwise. Before we show the existence of an optimal stationary policy, the following lemma is required, which comes from Corollary and Theorem in [9]. Lemma III.. Consider a discrete time Markov decision process {xt : t 0} consisting of four-element tuple {S, Ai, i S, pj i, a, ci, a} under average cost criterion: The state space S is denumerable; Each action space Ai is a subset of the finite action space A; The transition probability pj i, a satisfies pj i, a 0, i, j S, a Ai and j S pj i, a =, i S, a Ai; The one-step cost function ci, a 0, i S, a Ai. There exists an optimal stationary policy if the following set CAV of assumptions hold: CAV There exists a z standard policy d with positive recurrent class R d. d is defined to be a i 0 standard policy if the Markov process induced by d, {x d t : t 0} satisfies that for any i S, the expected time m i,i0 d of a first passage c 205 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

3 3 from i to i 0 during which at least one transition occurs is finite and the expected cost c i,i0 d of a first passage from i to i 0 during which at least one transition occurs is finite. CAV2 Given U > 0, the set D U = {i Ci, a U for some a} is finite. CAV3 Given i S R d, there exists a policy θ i R z, i, where R z, i is the class of policies d such that the expected cost c z,i d of a first passage from z to i is finite. Since P is a compact set, it follows from that there exists a positive number M such that Rx, y, p, z M. Thus, the average profit problem can be casted as a discrete time Markov decision process under average cost criterion with one-step cost M Rx, y, p, z. Now we are ready to show the existence of an optimal stationary policy by applying Lemma III. to verify that CAV-3 holds in our model. Proposition III.. There exists an optimal stationary policy for the average profit problem. Proof. Let policy d be an s, S, p policy which is described in the introduction such that λpx, 0 > 0 for x > S, λpx, < for x < s, and 0 < λpx, y < for s x S and y = 0,. It is straightforward to see that for any inventory level z [s, S] Z, the Markov chain induced by d is z, 0 standard with its positive recurrent class R d = {x, y S : s x S}. Thus, CAV holds. Given U, it is straightforward to verify that D U = {x, y S Rx, y, p, z U for some p, z A} is finite by Assumption II. and thus CAV2 holds. Given any system state x 0, y 0 such that x 0, y 0 / R d, define an s, S, p policy denoted by d, such that λp x, 0 > 0 for x > S, λp x, < for x < s, 0 < λp x, y < for s x S and y = 0,, and s minx 0, s and S maxx 0, S. Thus, it is clear that d R z, 0, x 0, y 0 and thus CAV3 holds. B. Optimality of s, S, p policy Based on the results in subsection III-A, we only consider stationary policies, i.e., the decision depends deterministically on the system s current state. Under a stationary policy u, if the system s current state is x, y, then the price is a function p u x, y, and the production decision is y u x, y = +, 0 or y, where y u x, y = + implies that y = 0 and the production is switched on, y u x, y = 0 implies that y = and the production is switched off, and y u x, y = y implies keeping the production status unchanged. Note that the inventory process is two-sided skip free i.e., the inventory level x cannot jump to x + z, z 2 in one period. It follows from [0] that the following result holds. Proposition III.2. There exists an s, S, p policy which is optimal in the class of all stationary policies. Proof. Suppose that a stationary policy u is given and it is optimal. If the initial state is x 0,, define S u = inf{x Z : x x 0, y u x, = 0 } and s u = sup{x Z : x < S u, y u x, 0 = +}. First it is easy to see that S u <. Otherwise, the inventory level will go to and thus the policy u cannot be optimal. Moreover, we have s u >. Otherwise, the inventory level will go to and thus the policy u cannot be optimal. Since the inventory process is two-sided skip free, we know that the inventory level is always between s u and S u, and thus the long-run average profit under the policy u is equal to that under a s u, S u, p policy. If the initial state is x 0, 0, then we define s u = sup{x Z : x x 0, y u x, 0 = +} and S u = inf{x Z : x > s u, y u x, = 0 }. Using a similar argument, we know that the long-run average profit under the policy u is equal to that under a s u, S u, p policy. Therefore, there exists a s, S, p policy which is optimal in the class of all stationary policies. Note that the resulting s u and S u depend on the system s initial state. This proposition tells us that we can restrain our discussion on the s, S, p policies. We mention that the above argument does not work for discounted case. C. Average profit under an s, S, p policy To compute the average profit, we use renewal reward theory. A cycle starts from S,. The manufacturer switches off the production and the inventory level falls from S to s. Afterward, the production is switched on and the inventory level rises from s to S, which ends a circle. Next, we compute the expected cycle length and the expected profit per cycle. Under an s, S, p policy, if the production is off, the length of the time in which the inventory level falls from x to x, denoted by τ 0 x, is geometrically distributed with mean /λpx, 0, i.e., P τ 0 x = k = λpx, 0 λpx, 0 k, k. The expected profit in which the inventory level falls from x to x is R 0 x = E[px, 0 hxτ 0 x hx ] λpx, 0 = px, 0 hx hx. λpx, 0 If the production is on, the length of the time in which the inventory level rises from x to x +, denoted by τ x, is geometrically distributed with mean / λpx,.the expected profit in which the inventory level rises from x to x + is R x = E[px, τ x hxτ x hx + cτ x] λpx, = px, λpx, hx λpx, λpx, c hx + λpx,. It follows from renewal reward theory that the average profit under an s, S, p policy is K + S Rs, S, p = R 0i + R i S /λpi, 0 + / λpi,, where K = K + K 0. By subtracting a dummy profit γ from original profit in each period, we define the following auxiliary total profit function c 205 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

4 4 in a cycle Ls, S, p, γ S = K + γ S R 0i + R i /λpi, 0 + / λpi, S = K + pi, 0 hi + γ + hi hi λpi, 0 pi, λpi, c hi γ + + hi hi + λpi, S = K + pi, 0 hi + γ λpi, 0 pi, λpi, c hi γ + λpi,. 2 Define Lγ = max s,s,p Ls, S, p, γ. Then it follows from basic theory of fractional programming [8] that γ is the maximal average profit if and only if Lγ = 0. It follows from the first equality of 2 that Ls, S, p, γ is a strictly decreasing, convex function of γ. Since both the monotonicity and convexity are preserved under maximization, we know that Lγ is also a strictly decreasing, convex function of γ. It follows from the first equality of 2 and the definition of Lγ that if γ is sufficiently large, Lγ < 0 and if γ is sufficiently small, Lγ > 0. In fact, in Proposition III.3 we show that Lγ = + if γ < 0. Since Lγ is strictly decreasing in γ, there exists a unique γ such that Lγ = 0 and γ is the optimal average profit. D. Analysis of the optimal price given s, S and γ We analyze the optimal price given s, S and γ, which is the optimal solution to the problem Ls, S, γ := max Ls, S, p, γ. p It follows from Assumption II.2 that λp has a strictly decreasing, continuous inverse pλ. Therefore, we can take arrival probability λ as the decision variable instead of p. Define f 0 γ, x, λ = pλ hx+γ/λ and f γ, x, λ = λpλ c hx γ λ. It follows from the last equality of 2 that Ls, S, p, γ = K + + S f γ, i, λpi,. f 0 γ, i, λpi, 0 We first compute the optimal arrival probability λ x, 0 = λp x, 0 when the inventory level is x and the production is off. The following assumption is required for our discussion. Assumption III.. For any 0 < λ λ, λp λ + 2p λ < 0. Note that the condition that pλ is concave decreasing in λ is sufficient for λp λ + 2p λ < 0. Moreover, it follows from Assumption III. that the feasible region of λ is the interval [0, λ]. It follows from dp d = d 2 p 2 p, d 2 = 4 p p that Assumption III. implies that p is strictly concave in for >. Thus, by letting = λ it is known that f 0 γ, x, λ = p hx + γ is strictly concave in and its first order derivative is f 0 = λ2 p λ hx + γ. 3 Now we can claim that it suffices to consider the case γ 0 in our paper. Proposition III.3. If γ < 0, then Lγ = +. Proof. Note that for admissible λ, λ 2 p λ 0. Thus, if γ < 0, then f 0 is increasing in and thus decreasing in λ at x = 0. Therefore, max λ [0, λ] f 0 γ, 0, λ = f 0 γ, 0, 0 = + and thus Ls, S, γ = + which implies Lγ = +. Let g 0 λ = λ 2 p λ, then g 0 is strictly increasing in λ for λ > 0 as g 0λ = λλp λ + 2p λ > 0 according to Assumption III.. Thus, g 0 has a strictly increasing inverse denoted by g0. Moreover, g 00 = 0 and g 0 λ = λ 2 p λ. It follows from 3 that the optimal arrival probability at state x, 0 is { λ g x, 0 = 0 hx + γ, if hx + γ g 0 λ λ, otherwise. Next we compute the optimal arrival probability λ x, when the inventory level is x and the production is on. Let = λ. It follows from d p = d 2 p + p, d 2 d 2 p = 4 p p and Assumption III. thatf γ, x, λ = p c+ hx + γ is strictly concave in for >, and its first order derivative is f = λ λp λ + pλ c + hx + γ. 5 Let g λ = λ λp λ + pλ, then g is strictly decreasing in λ for λ > 0 as g λ = λλp λ + 2p λ < 0 according to Assumption III.. Thus, g has a strictly decreasing inverse denoted by g. Moreover, g 0 = lim λ 0 λp λ + p0 and g λ = λ λp λ. It follows from 5 that the optimal arrival probability at state x, is { λ g x, = c + hx + γ, if c + hx + γ g 0 0, otherwise. 6 Define r 0 γ, x = max λ f 0 γ, x, λ = f 0 γ, x, λ x, 0 and r γ, x = max λ f γ, x, λ = f γ, x, λ x,. Obviously, both r 0 γ, x and r γ, x are strictly decreasing, 4, c 205 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

5 5 convex functions of γ as both f 0 γ, x, λ and f γ, x, λ are strictly decreasing, convex functions of γ. Moreover. we have Ls, S, γ = K + = K + S r 0 γ, i + r γ, i S r 0 γ, i + r γ, i. 7 The following two propositions characterize the relationship of r 0 γ, x, r γ, x with x, which is crucial in determining the optimal sγ and Sγ. Proposition III.4. For any given γ 0, both r 0 γ, x and r γ, x x R are quasiconcave functions of x, and 0 is the maximum point. Proof. It follows from the envelope theorem [7] that dr 0γ,x dx = h x/λ x, 0 and drγ,x dx = h x/ λ x, 0. Since hx is quasiconvex in x, we have h x 0 when x 0 and h x 0 when x 0. Therefore, both r 0 γ, x and r γ, x are quasiconcave functions of x, and 0 is the maximum point. Proposition III.5. For any given γ 0, r 0 γ, x+r γ, x x Z is a quasiconcave function of x, and 0 or is the maximum point. Proof. It follows from Proposition III.4 that r 0 γ, x + r γ, x is increasing on x 0 and decreasing on x. Since x Z, r 0 γ, x + r γ, x takes its maximum at point 0 or and is quasiconcave in x Z. Note that the constraint x Z is necessary for Proposition III.5 to be held as r 0 γ, x + r γ, x might not be quasiconcave for 0 < x <. E. Analysis of the optimal s and S given γ Now we want to find the optimal s and S given γ, which is the solution of max Ls, S, γ. s<s,s,s Z Proposition III.6. Given γ 0, if the set {x Z : r 0 γ, x+ r γ, x 0, x 0} is non-empty, then the optimal sγ and Sγ are given by sγ = min{x Z : r 0 γ, x + r γ, x 0}, 8 Sγ = max{x Z : r 0 γ, x + r γ, x 0}. 9 Proof. Since r 0 γ, x+r γ, x is a quasiconcave function of x Z, it follows from 7 that the optimal solution for max s,s Ls, S, γ should be the s and S such that r 0 γ, i + r γ, i 0 for all s + i S. Thus, the result holds. The set {x Z : r 0 γ, x + r γ, x 0, x 0} might be empty if γ is quite large. In this case, r 0 γ, x + r γ, x < 0 for all x Z. In this case, if r 0 γ, 0 + r γ, r 0 γ, + r γ, 0, then the optimal sγ, Sγ =, 0. Otherwise, the optimal sγ, Sγ = 0,. Proposition III.7. Consider the domain of γ such that the set {x Z : r 0 γ, x + r γ, x 0, x 0} is non-empty. sγ is increasing in γ and Sγ is decreasing in γ. Proof. Note that r 0 γ, x+r γ, x is decreasing in γ. For γ < γ 2, it follows from 8 that r 0 γ, sγ +r γ, sγ < 0 and thus r 0 γ 2, sγ + r γ 2, sγ < 0. Again, it follows from 8 that sγ < sγ 2 +, or sγ sγ 2. Thus, sγ is increasing in γ. Similarly, it follows from 9 that Sγ is decreasing in γ. Proposition III.8. λ x, 0 is decreasing on x 0, and increasing on x 0; λ x, is increasing on x 0, and decreasing on x 0. Proof. It follows from the definition of λ x, 0 that f 0 γ, x, λ x, 0 f 0 γ, x, λ x, 0 f 0 γ, x, λ x, 0 f 0 γ, x, λ x, 0. Summing the above two inequalities, we obtain λ x, 0 λ hx hx 0. x, 0 It follows from the quasiconvexity of hx that λ x, 0 is decreasing on x 0, and increasing on x 0. Similarly, we have λ x, λ x, hx hx 0. It follows from the quasiconvexity of hx that λ x, is increasing on x 0, and decreasing on x 0. Note that pλ is strictly decreasing in λ and the optimal price p x, y = pλ x, y. Thus, the interpretation of Proposition III.8 is as follows: When the production is off i.e., y = 0, the optimal price increases in the inventory level x if x 0, and decreases in x if x 0. When the production is on i.e., y =, the optimal price decreases in the inventory level x if x 0, and increases in x if x 0. Therefore, the optimal price is a quasiconcave function of the inventory level x when the production is off and is a quasiconvex function of x when the production is on. The result is consistent with our intuition. Intuitively, because of inventory holding costs and backlogging costs, the manufacturer attempts to maintain the inventory level around 0, by keeping the inventory level around 0 as long as possible and moving the inventory level away from S and s as fast as possible. Thus, he will make different pricing strategies according to the current production state. Suppose that the production is off on: if the inventory level is high, the manufacturer should set the selling price low high to attract more demand to sell the product in stock to make the inventory level quickly reach S to switch the production off; if the backlogging level is high, the manufacturer should also set the selling price low high to make the inventory level quickly reach s to switch the production on to meet more backlogged demand c 205 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

6 6 F. Comparative statics We discuss the effect of switching cost. Let γ be the optimal average profit and s = sγ, S = Sγ. Proposition III.9. γ is strictly decreasing in K = K 0 + K ; s is decreasing in K and S is increasing in K. Moreover, p x, 0 is increasing in K and p x, is decreasing in K. Both the expected production-on time length and productionoff time length of a cycle are increasing in K. Proof. It follows from 0 = L γ = max Ls, S, s<s,s,s Z γ 0 S = max K + r 0 γ, i + r γ, i s<s,s,s Z and the fact that both r 0 γ, x and r γ, x are strictly decreasing in γ that γ is strictly decreasing in K. Thus, it follows from Proposition III.7 that s is decreasing in K and S is increasing in K. It follows from 4 and 6 that λ x, 0 is decreasing in K and λ x, is increasing in K. Thus, p x, 0 is increasing in K and p x, is decreasing in K. The expected production-on time length of a cycle is S / λ i,, which is increasing in K as λ x, is increasing in K, s is decreasing in K, S is increasing in K and λ i, > 0 for all i Z. Similarly, the expected production-off time length of a cycle S + /λ i, 0 is also increasing in K. The above result indicates that when the switching cost K increases, switching less frequently is preferred. The impact of holding cost on the optimal policy is unclear, although γ decreases as holding costs increase. Consider a special case in which the holding cost hx at each state x is increased by h i.e., hx + h. In this case, the optimal average profit is decreased by h, but the optimal production-pricing policy i.e., s, S and the optimal price remains unchanged. G. A numerical example In this part, we provide a simple numerical example to illustrate the properties of the optimal pricing strategies. In this example, the production rate r =, and λp = e θp with θ = 0.2. The allowable price set P = [, 0]. The switching costs are K 0 = 50 and K = 00. The holding cost function is hx = hx + + bx with h = 0., b = 0.5. By using value iteration algorithm, we find that the optimal average profit is γ = 0.3, s = 2 and S = 7. Fig shows a sample trajectory of inventory level and price under optimal policy, starting from level 0 with production on, which reflects an asymptotic cyclic behavior. When the production is on, the price first decreases and then increases, the inventory level increases from s to S. The price reaches the minimum when the inventory level is close to zero, so that the demand and production can be balanced while keeping low inventory. When the production is off, the price first increases and then decreases, the inventory level decreases from S to s. The price reaches the maximum p = 0 and keep for a while when the inventory level is close to zero. 20 S = s = Inventory Level Price Time Fig.. A sample trajectory of inventory level and price under optimal policy IV. CONCLUSION In this paper, we consider a stochastic inventory system where the manufacturer can dynamically adjust the selling price and switch the production on or off to maximize the long-run average profit. We show that an s, S, p policy is optimal. Moreover, the structural properties of the optimal profit function and pricing strategies are presented. There are a few topics worthy of further discussion. One can consider a compound Poisson process or batch production, formulating the problem as a continuous model with Poisson process directly. The problem under a discounted criterion, in finite horizon, or with lead times can also be studied. ACKNOWLEDGMENT The authors thank Prof. Xin Chen for proposing this research problem. This work was supported by NSFC under grants 72054, 74059, and 75776, and the Fundamental Research Funds for the Central Universities under grants WK and WK REFERENCES [] Dimitri P. 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