Optimal production and inventory policies of priority and price-differentiated customers.
Publication: IIE Transactions
Date: Saturday, September 1 2007
1. Introduction
Flexibility is essential for businesses in order to deal with variability, uncertainty, and changes in the business environment. Manufacturing flexibility can be achieved in many ways including labor force, machinery, product mix, product design, or new products. Increasingly,
Differentiated service levels based on delivery time allow customers with an immediate need (e.g., businesses) to receive expedited product, while flexible customers receive incentives for their patience. An example of a company using differentiation is Amazon.com, where consumers can choose expedited shipping or free shipping. In the latter Amazon.com receives increased flexibility, since the stated leadtime exceeds the actual processing and transportation time. Customer segmentation by time, whether in manufacturing or the airline industry, provides a mechanism for balancing the supply and demand requirements of the system (e.g., shifting leisure travel from Friday to Saturday), which allows more efficient use of existing resources. A key example of a manufacturing company that employs flexibility in managing customer demand is Dell Inc. Customers are segmented according to type (e.g., business versus personal), and prices of products change regularly (Agrawal and Kambil, 2000).
The primary goal of this research is to provide tools for managing production and inventory tactically when customers differ in their willingness to pay and their willingness to wait. The key questions we address are how much to produce and how to allocate scarce resources (either current inventory or future limited production capacity) dynamically among different customer classes. We incorporate a firm's tactical inventory decisions, which we define to mean inventory or capacity allocations in one time period to serve customer demand in another time period. Specifically, we allow the firm to reserve inventory to satisfy future demand (sometimes called "discretionary sales"), and to plan backlogging, where the firm can accept orders in a period to be delivered in the future.
For example, many manufacturing companies face the following problem: some customers are willing to pay high prices to receive faster fulfillment, while other customers are willing to accept a lower priority for fulfillment, but they demand low prices. The manufacturer has limited production capacity, and in order to maximize profit, he needs to allocate the capacity effectively. With an advanced strategy, the manufacturer can separate the customers into multiple classes according to priority levels and then manage the production and the inventory appropriately; we refer to this as a differentiated strategy.
In this paper we study the Priority Differentiation Strategy (PDS), where we assume the first class pays a premium to have higher priority in the current period over production and inventory resources compared to the second class. We assume that the manufacturer can or is willing to prioritize demand classes. That is, the manufacturer makes a decision on higher priority demand before he accepts or rejects the lower priority demand requests. This situation might occur in practice when requests are submitted electronically and are handled in batches, or it could result from any working environment where a manufacturer may temporarily ignore requests from second-class customers. Studying the general model also allows us to analyze several situations that are special cases or extensions of it. For example, in some circumstances the manufacturer is not able or not allowed to differentiate the customers and will deal with them as a single class.
We assume demand in each period is a general function of price, is continuous and differentiable, and is lost if rejected; we do not make restrictive assumptions regarding the stochastic demand arrivals and the production process. We focus on a periodic-review environment where prices are predetermined but not known by customers until the current period. We initially assume backordered demand is fulfilled in the next period and extend our results to allow backorders until the end of the time horizon.
2. Literature review
One stream of literature related to our work is inventory theory, especially when there are multiple classes of customers. Two seminal papers in this area are Veinott (1965) and Topkis (1968). Veinott shows some conditions under which a base stock policy is optimal for the production decision when cost minimization is the goal. When parameters are time varying and the classes have different priorities, the demand from a higher class should be satisfied before demand from a lower class, and further restrictions are necessary on the costs. A related topic is considered by Topkis who extends the work of Veinott so as to be able to decide a set of critical levels that determine when to satisfy a particular class of demand. Topkis outlines some assumptions under which the optimal policy has a set of critical numbers (e.g., one assumption is that penalty costs must be cheaper now than in the future). In both Veinott (1965) and Topkis (1968), the classes of demand are essentially the same except for priority. In our case, there may be inherent differences between the classes of demand (e.g., willingness to wait or pay), and we may intentionally backlog customers or reserve inventory for future customers, which further distinguishes how the different classes may be served. In addition, we assume production capacity is limited, we do not make any assumptions on costs over time, and we allow revenue to depend upon customer class.
More recent research in inventory that is relevant includes Sobel and Zhang (2001). In this work, the authors study an inventory problem with fixed plus linear production costs and two demand classes. The deterministic demand class must be satisfied immediately, and the stochastic demand can be backlogged if there is not enough inventory. The main result is that a modified (s, S) policy is optimal. In our case, our production costs are simpler (linear only), but demand for both classes is stochastic and we allow tactical inventory.
Frank et al. (2003) add to the work, again considering one deterministic and one stochastic demand class. They allow the firm to specify how much of the stochastic demand to satisfy; this is somewhat similar to using discretionary sales. Their main result is that a state-dependent optimal policy exists but is quite complex, so they propose a heuristic policy of the form (s, k, S), where the rationing policy k specifies the amount of on-hand inventory to reserve for deterministic demand before ordering; thus, k also determines the inventory available to satisfy stochastic demand. Katircioglu and Atkins (1996) also consider production and allocation problems with multiple classes of customers. In this work, customer classes require different service levels, and they propose a heuristic that solves the problem myopically and is easy to implement. For our problem, the optimal policy has a simple structure and includes explicit decisions for reserving and backordering (other differences are as outlined above).
One stream of research that considers multiple classes of customers with stochastic demand in manufacturing focuses on rationing (see for instance, Moon and Kang (1998) or Dekker et al. (2002) as well as Topkis (1968) reviewed above). The term "rationing" is generally used to refer to the allocation of a resource such as capacity or inventory between competing customer classes. The results in this research area often describe threshold or critical levels that indicate the resource to be allocated to each class. This critical-level policy is optimal for some cases and is used as a heuristic in others. These papers generally focus on dynamic control of a single machine, and they do not consider production problems that span a number of periods with non-stationary parameters. In our case we find threshold values of this type (see the nesting policy for PDS), and we also incorporate resource allocations across time periods.
In most of the described results in the rationing area, a key assumption is that demand is Poisson (see for example, Balakrishnan et al. (1996) and Melchiors et al. (2000)). In some, there is also an assumption that the production time is exponential (Ha, 1997). The most relevant work in this stream is Ha (2000), who assumes demand is Poisson and the processing time is Erlang. The key contribution is that the optimal policy has critical levels with monotonic properties. This policy is most similar to the one we find for PDS in this paper, although in our case we have limited production capacity and tactical inventory. We also consider leadtime differences explicitly and allow planned backlogs.
An important paper that allows tactical inventory is Scarf (2000), who introduced discretionary sales into a problem with fixed production setup costs and one customer class. In his case, a base stock type of policy is optimal for production, but unlike the production decision, the optimal discretionary sales decision should be decided after demand is revealed in a given period in order to achieve the maximum profit. The use of discretionary sales is also analyzed in Chan et al. (2006), which considers a single-class stochastic inventory model with multi-period pricing and production decisions under limited capacity when demand is a general stochastic function.
In the current paper, we build on our work in Chan et al. (2006), where we found that a modified base stock policy with a production and reserving decision pair was optimal, in which the optimal values do not depend on the demand that arrives if price is decided in advance. A fundamental difference in the current research is that we add multiple classes of customers who differ in their willingness to wait (and pay), and we allow delayed fulfillment. The current work also builds on Liu and Simchi-Levi (2003), who extended Chan et al. (2005) to allow delayed fulfillment until the end of the horizon.
The rest of this paper is organized as follows. In Section 3, we introduce and analyze the PDS and non-differentiation strategies. We perform computational analysis to compare expected profits under the two strategies in Section 4 to explore the effectiveness of market segmentation in manufacturing. Conclusions are contained in Section 5.
3. Models and results
We focus on a single product sold at a single manufacturer over a multi-period time horizon, where the manufacturer has limited production capacity in each period. The manufacturer serves two customer classes, whose demand is ordered by class (i.e., sorted by priority). This means that in any period, first-class demand is fully known by the manufacturer before he has to make a decision regarding second-class demand. The customers of these two classes differ in their priority level and willingness to pay. The first-class customers are willing to pay a premium over the price of the second-class customers in order to have priority access in the current period to both on-hand inventory and backlogging availability. Thus, by paying the premium, first-class customers are satisfied first with the inventory and backlogging resources available to the manufacturer in the current period, and the demand of the second-class customers is addressed with the remaining resources.
The main model that we will consider throughout this paper is the PDS, where we assume that the manufacturer has the ability to differentiate the customer classes. We seek to optimize the allocation of limited inventory and production capacity, considering the possibility of reserving inventory to satisfy future demand and allocating future production capacity by backlogging current demand. We show that there is an optimal set of production, backlog, and reserve inventory decisions that allocates current and future resources between customer classes. Considering the general model (PDS) also allows us to analyze other models; for instance, we consider one in which the manufacturer cannot differentiate the customer classes and treats every customer equally (see the Non-Differentiation Strategy (NDS)). This extension and others are described in Section 3.3.
3.1. Notation and assumptions
The manufacturer makes decisions over a multi-period time horizon, t = 1, 2, ..., T, with T representing the end of the horizon. The production in each period t is limited by the capacity, [q.sub.t], and the manufacturer pays a production cost per unit of [c.sub.t]. Inventory holding cost is linear, and a charge per unit, [h.sub.t], is assessed to carry inventory from t to t + 1. Throughout the paper, the superscripts of "1" and "2" will be used for the first and second classes, respectively.
The manufacturer has predetermined prices, [p.sup.1.sub.t] and [p.sup.2.sub.t], for the customers of the first and second classes, respectively, that may be different in each period. Separation of pricing and production decisions is very common in current practice. In some companies, pricing decisions are made by the marketing department before the start of a selling season, while production decisions are made by the operations department.
We assume that each first-class customer is charged a higher price than a second-class customer in the same period; that is, [p.sup.1.sub.t] > [p.sup.2.sub.t] for each t, although we make no restrictions on prices between different time periods. This even allows [p.sup.1.sub.t] < [p.sup.2.sub.t+1], in case there is a significant change in demand curves over time. The salvage value of any units left at the end of the horizon is v, and [p.sup.1.sub.T] > [p.sup.2.sub.T] > v. For classes i = 1, 2, the cost per unit for demand in class i that is rejected and lost is [l.sup.i.sub.t], and [[beta].sup.i.sub.t] is the cost per unit for demand in class i that is backlogged. We assume that [l.sup.1.sub.t] > [l.sup.2.sub.t] and [[beta].sup.1.sub.t] > [[beta].sup.2.sub.t] for each t, since losing or delaying the fulfillment of the first-class customers is more costly than for the second-class i from current inventory as [p.sup.i.sub.t] + [h.sub.t] + [l.sup.i.sub.t]; similarly, the net revenue from backlogging is [p.sup.i.sub.t] - [[beta].sup.i.sub.t] + [l.sup.i.sub.t]. Holding cost, [h.sub.t], is assumed to satisfy [p.sup.1.sub.t] - [[beta].sup.1.sub.t] + [l.sup.i.sub.t] > [p.sup.2.sub.t] + [h.sub.t] + [l.sup.2.sub.t] in each period t, which ensures that backlogging one first-class customer is more expensive than rejecting a second-class customer to save a unit of inventory for the future.
Each customer belongs to only one demand class, and demand from one class is assumed to be independent of the other class. Each demand function is a general non-stationary stochastic function, [D.sup.i.sub.t], with known probability and cumulative distribution functions [[phi].sup.i.sub.t] and [[PHI].sup.i.sub.t], respectively. We assume that the demand function in each period is continuous and differentiable, but no other assumptions are made on the shape of the demand function, so a wide variety of demand models could be used.
Production is a decision made at the beginning of each period and the production leadtime is zero. The net inventory (on-hand--backlogs) at the beginning of period t is [I.sub.t], and let [S.sub.t] represent the net inventory plus production in period t. In our initial analysis we restrict ourselves to delivering backordered items one period later, and we assume previously accepted orders are fulfilled before new orders are accepted, which is possible since we restrict backorders in each period to be no more than the capacity in the next period. In some extensions we allow the delivery date to be the end of the horizon.
The sequence of events in every period is as follows. At the beginning of a period, the manufacturer checks the inventory level [I.sub.t] and decides the production quantity; products arrive immediately, and the manufacturer fulfills the backorders carried from the previous period with the available inventory. Then the demand in the current period is revealed and the manufacturer decides the amount to reserve, [R.sup.i.sub.t], and the amount of future capacity to make available to current customers (i.e., the amount to backlog), [B.sup.i.sub.t]. [R.sup.1.sub.t] is the amount of inventory to protect from (not sell to) classes 1 and 2, and [R.sup.2.sub.t] is the additional amount of inventory to protect from class 2; thus, the total amount to protect from class 2 is [R.sup.1.sub.t] + [R.sup.2.sub.t]. The amount of future capacity to make available to classes 1 and 2 now is [B.sup.2.sub.t], and [B.sup.1.sub.t] is the additional capacity for class 1; thus, the total capacity for backlogging class 1 is [B.sup.1.sub.t] + [B.sup.2.sub.t]. The demand is satisfied according to the [S.sub.t], [B.sup.i.sub.t] and [R.sup.i.sub.t] values. The notation that we defined in this section is provided in Table 1 for ease of reference.
Table 1. Notation
[q.sub.t] production capacity in period t
[c.sub.t] production cost per unit in period t
[h.sub.t] inventory holding cost per unit from period
t to t + 1
[p.sup.1.sub.t] price charged to first-class customers in
period t
[p.sup.2.sub.t] price charged to second-class customers in
period t
[upsilon] salvage value of any item left at the end
of horizon
[l.sup.i.sub.t] cost per unit for demand in class i that is
not satisfied
[[beta].sup.i.sub.t] cost per unit for demand in class i that is
backlogged
[D.sup.i.sub.t] demand realization of class i in period t
[I.sub.t] net inventory at the beginning of period t
[S.sub.t] net inventory plus production in period t
[R.sup.1.sub.t] amount of inventory to protect from classes
1 and 2
[R.sup.2.sub.t] amount of additional inventory to protect
from class 2
[B.sup.2.sub.t] amount of future capacity made available to
classes 1 and 2
[B.sup.1.sub.t] amount of additional future capacity made
availableto class 1
3.2. The PDS
In the PDS, we assume that the first class is willing to pay a premium to receive priority over all available inventory and backlogging in the current period. The result is that the first and second classes may be fulfilled now or in the next period, depending on the status of the system. Thus, the manufacturer has increased flexibility to match supply and demand.
For the purpose of clarity, we introduce some additional notation in Table 2. Due to our assumption of the ordering of demand classes, we satisfy the first-class demand before the second-class demand. Consequently the available inventory for the second class is limited by the first-class demand that is realized. We define the amount of inventory available after the first-class demand is satisfied as [S.sup.2.sub.t]. Since the first class has higher priority in the current period, we use as much of [B.sup.2.sub.t] as necessary to backlog the first-class demand. Then we use the remaining part of [B.sup.2.sub.t] (if there is any left) to backlog the second-class demand. We call this remaining backlog availability [B.sup.2,ef.sub.t], or the effective backlog amount after first-class demand. For ease of presentation in the paper, we further define the actual backlogged orders from first and second-class customers after demand is satisfied as [A.sup.1.sub.t] and [A.sup.2.sub.t], respectively, and inventory carried forward due to [R.sup.1] and [R.sup.2] as [I.sup.[R.sup.1].sub.t+1] and [I.sup.[R.sup.2].sub.t+1], respectively. If demand is low enough so that there is leftover inventory at the end of the period, we denote this additional inventory as [I.sup.low.sub.t+1] (see Table 2 for a summary of the additional notation).
Table 2. Additional notation
[S.sup.2.sub.t] = ([S.sub.t] - [R.sup.1.sub.t] - available inventory
[D.sup.1.sub.t][).sup.+] after first-class
demand is
satisfied
[B.sup.2,ef.sub.t] = [[B.sup.2.sub.t] - effective backlog
[[D.sup.1.sub.t] - [[S.sub.t] - amount after
[R.sup.1.sub.t][].sup.+][].sup.+][].sup.+]) first-class demand
[A.sup.1.sub.t] = min([B.sup.1.sub.t] + actual backlogged
[B.sup.2.sub.t], [[D.sup.1.sub.t] - [[S.sub.t] - orders from first
[R.sup.1.sub.t][].sup.+][].sup.+]) class
[A.sup.2.sub.t] = min([B.sup.2,ef.sub.t], actual backlogged
[[D.sup.2.sub.t] - [S.sup.2.sub.t] + orders from second
[R.sup.2.sub.t][].sup.+]) class
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] inventory carried
forward due to
[R.sup.1] decision
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] inventory carried
forward due to
[R.sub.2] decision
[I.sup.low.sub.t+1] = [[S.sup.2.sub.t] - inventory carried
[R.sup.2.sub.t] - [D.sup.2.sub.t][].sup.+] forward due to
demand
We model the PDS problem as a Markov decision process, where the state of the system is represented by the net inventory. For clarity of exposition, we present the model with the [R.sup.i.sub.t] and [B.sup.i.sub.t] decisions given ex ante. However, in our analysis we show that the optimal [R.sup.i*.sub.t] and [B.sup.i*.sub.t] decisions are the same whether they are made before or after demand revelation. Let [J.sub.t]([I.sub.t]) be the expected profit from period t forward to the end of the horizon, or the profit-to-go. Let [G.sub.t]([S.sub.t]) be the expected profit-to-go with [S.sub.t] units of product available after production. The first and second derivatives of [J.sub.t]([I.sub.t]) are denoted, respectively, as: [J'.sub.t]([I.sub.t]) and [J.sup.".sub.t]([I.sub.t]); the derivatives of other functions are indicated similarly. We can now write the optimal expected profit in period t and onward for the PDS problem as the following recursive equation:
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where
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subject to [B.sup.1.sub.t] + [B.sup.2.sub.t] [less than or equal to] [q.sub.t+1], [R.sup.1.sub.t] + [R.sup.2.sub.t] [less than or equal to] [S.sub.t].
In Equation (1), the maximization of profit is over the target inventory decision. The first term of the function is the production cost; the production also covers any backlogged orders from the prior period. The second term is the profit in the remainder of the period (and horizon) starting with the available inventory after production is completed and backorders are fulfilled.
In Equation (2), the function [G.sub.t], the profit-to-go after production, is maximized over the reserve inventory and backlogging decisions. The first element of the function is the revenue from first-class customers, including both physical inventory and backlogged orders. The second term is revenue from the second-class demand with available inventory and backlogged orders. The third piece is the inventory holding cost to be paid for all inventory not sold. The fourth and fifth terms represent the inventory holding cost that is incurred for all inventory reserved for the future. The sixth and seventh terms are the rejection penalties for demand not satisfied for the first and second classes, respectively, and the eighth and ninth terms are the delay penalty associated with the backlogged demand for the first and second classes, respectively. The last term in the equation represents the profit in future periods, sending forward any leftover physical inventory and backlogged orders. For period T, the final term is replaced by the salvage cost of leftover inventory, namely [upsilon]([S.sub.T] - [D.sup.1.sub.T] - [D.sup.2.sub.T][).sup.+]. Finally, the constraints ensure that the manufacturer does not sell more future capacity than he has or reserve more inventory than is available.
3.2.1. Problem simplifications
For each demand class, the manufacturer decides the amount of inventory to reserve and the amount of backordering. To simplify the problem at hand, we show that in an optimal policy for a class and a time period, at least one set of these decisions must be zero.
The first of these conditions says that if it is good to protect items for the future from class 1 and lose some of the current demand, then it is not reasonable to backorder items from class 1 or the lower-revenue class 2 (the contrapositive is also true). Likewise, the second condition says that if it is good to backorder demand from even the (lower-paying) second class in the current period, then it will not be reasonable to protect items from (and lose demand from) the second class or the higher-paying first class in the current period (the contrapositive is also true). The formal proof can be found in the Appendix.
By Lemma 1, the structure of the optimal policies can be simplified. In each period there are three candidate policies, of which the best policy will be chosen; this choice will be dependent on the state of the system. The possible options are to reserve-inventory ([R.sup.1.sub.t] [greater than or equal to] 0, [R.sup.2.sub.t] [greater than or equal to] 0), to backlog-demand ([B.sup.1.sub.t] [greater than or equal to] 0, [B.sup.2.sub.t] [greater than or equal to] 0), or to reserve-and-backlog ([R.sup.2.sub.t] [greater than or equal to] 0, [B.sup.1.sub.t] [greater than or equal to] 0). Thus,
[G.sub.t]([S.sub.t]) = max {[G.sup.1.sub.t]([S.sub.t]), [G.sup.2.sub.t]([S.sub.t]), [G.sup.3.sub.t]([S.sub.t])},
where [G.sup.1.sub.t]([S.sub.t]), [G.sup.2.sub.t]([S.sub.t]), and [G.sup.3.sub.t]([S.sub.t]) represent the profit-to-go with [S.sub.t] units of products available after production under the reserve-inventory policy, the backlog-demand policy and the reserve-and-backlog policy, respectively. These three policies are given by
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In each of the three cases, the starting inventory after production is completed and backorders are fulfilled is [S.sub.t]. The first function, [g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]), indicates the profit-to-go when inventory may be protected from both classes ([R.sup.1.sub.t], [R.sup.2.sub.t] [greater than or equal to] 0). In this case the manufacturer will not backlog orders of current customers because the backlog orders will reduce the future capacity available to customers (therefore [B.sup.1.sub.t] = [B.sup.2.sub.t] = 0). The profit from this policy is represented as
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The function [g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) indicates the profit-to-go when backorders for each class may be desirable ([B.sup.1.sub.t], [B.sup.2.sub.t] [greater than or equal to] 0). However, the manufacturer will not protect inventory from either class ([R.sup.1.sub.t] = [R.sup.2.sub.t] = 0). The resulting formulation is
[g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) = [integral][integral] {[p.sup.1.sub.t] min ([D.sup.1.sub.t], [S.sub.t] + [B.sup.1.sub.t] + [B.sup.2.sub.t]) + [p.sup.2.sub.1] min ([D.sup.2.sub.t], [[S.sub.t] - [D.sup.1.sub.t[].sup.+] + [B.sup.2, ef.sub.t]) - [l.sup.1.sub.t]([D.sup.1.sub.t] - [S.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t][).sup.+] - [l.sup.2.sub.t] ([D.sup.2.sub.t] - [[S.sub.t] - [D.sup.1.sub.t][].sup.+] - [B.sup.2, ef.sub.t][).sup.+] - [[beta].sup.1.sub.t] min ([[D.sup.1.sub.t] - [S.sub.t][].sup.+], [B.sup.1.sub.t] + [B.sup.2.sub.t]) - [[beta].sup.2.sub.t] min ([[D.sup.2.sub.t] - [[S.sub.t] - [D.sup.1.sub.t][].sup.+][].sup.+], [B.sup.2, ef.sub.t]) - [h.sub.t][[S.sub.t] - [D.sup.1.sub.t] - [D.sup.2.sub.t][].sup.+] + [J.sub.t+1]([I.sup.low.sub.t+1] - [A.sup.1.sub.t] - [A.sup.2.sub.t])}d[[PHI].sup.1.sub.t]([D.sup.1.sub.t])d[[PHI].sup.2.sub.t] ([D.sup.2.sub.t]).
The remaining function, [g.sup.3.sub.t]([S.sub.t], [R.sup.2.sub.t], [B.sup.1.sub.t]), indicates the profit-to-go when the manufacturer may backlog orders of the first class for future fulfillment ([B.sup.1.sub.t] [greater than or equal to] 0) and may also protect inventory from the second class for future use ([R.sup.2.sub.t] [greater than or equal to] 0):
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In each period one of these three policies will be chosen, and this choice also impacts the future state of the system. Intuition gives us some idea of when each policy will be selected, which we establish more formally in our results below. We expect that the reserve-inventory policy will be selected in a period where the marginal expected profit from selling each of the reserve units in the future is better than the net revenue of selling a unit now out of inventory. For the backlog-demand policy, intuition suggests that it will be best when the net revenue of backlogging in the current period is better than the marginal expected profit from selling each of the units in the future. Finally, the reserve-and-backlog policy will be optimal when the net revenue of backlogging to the first-class customers is significantly greater than the marginal expected future profit of the backlogged units, but the second class has a lower net revenue when selling from inventory than the marginal expected future profit of sending forward reserved units.
3.2.2. Results
Under the PDS, we can show that all the profit-to-go functions have a nice structure (quasi-concave or concave), thus yielding easy to implement decisions. These results are summarized in the following theorem (see the Appendix for the full details):
Theorem 1. Under the PDS:
(i)[g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]) is a quasi-concave function of [R.sup.1.sub.t] and [R.sup.2.sub.t], for all t = 1, ..., T.
(ii) [g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) is a quasi-concave function of [B.sup.1.sub.t] and [B.sup.2.sub.t], for all t = 1, ..., T.
(iii) [g.sup.3.sub.t]([S.sub.t], [B.sup.1.sub.t], [R.sup.2.sub.t] is a quasi-concave function of [B.sup.1.sub.t] and [R.sup.2.sub.t], for all t = 1, ..., T.
(iv) [G.sub.t]([S.sub.t]) is a concave function of [S.sub.t], for all t = 1, ..., T.
(v) [J.sub.t]([I.sub.t]) is a concave function of [I.sub.t], for all t = 1, ..., T.
(vi) The unconstrained optimizers ([R.sup.1*.sub.t], [R.sup.2*.sub.t], [B.sup.1*.sub.t], and [B.sup.2*.sub.t]) for functions [g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]), [g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) and [g.sup.3.sub.t]([S.sub.t], [B.sup.1.sub.t], [R.sup.2.sub.t]), are independent of inventory level [S.sub.t] and demand realizations [D.sup.1.sub.t] and [D.sup.2.sub.t], where:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In Section 3.1, while explaining the sequence of events, we assumed that the [R.sup.i.sub.t] and [B.sup.i.sub.t] decisions are made after seeing the demand. In Theorem 1 we show that these decisions are independent of the demand in the current period; thus, the manufacturer can decide their optimal levels before the demand is revealed for the period. The theorem implies the optimal policy for the PDS; thus, we have the following corollary.
Corollary 1. Given a vector of prices, there exists an optimal modified base stock policy for the PDS with an optimal order-up-to level ([S.sup*.sub.t]), and for i = 1, 2 optimal reserve-up-to levels ([R.sup.i*.sub.t]) and optimal backlog-up-to levels ([B.sup.i*.sub.t]).
We refer to the policy as modified base stock because it may be limited by capacity or available inventory. If there is not sufficient capacity to bring the inventory level up to [S.sup.*.sub.t], then as much as possible should be produced. Similarly, the [R.sup.i.sub.t] and [B.sup.i.sub.t] decisions are limited by [S.sub.t] and [q.sub.t+1], respectively. The form of the optimal decisions are apparent from the concavity and quasi-concavity of the profit functions. At each stage in the problem, the manufacturer trades off the current net revenue against the marginal future contribution in terms of cost or revenue and chooses the best allocation of resources.
Additional insight may be gained by looking at the optimal decisions in more detail. The optimal decisions are defined by the following (1):
[S.sup.*.sub.t] = max{S : [c.sub.t] [less than or equal to] [G'.sub.t](S)} if [c.sub.t] [less than or equal to] [G'.sub.t](0), [R.sup.1*.sub.t] = max {I : [p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t] [less than or equal to] [J'.sub.t+1](I)} if [p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t] < [J'.sub.t+1](0), [R.sup.1*.sub.t] + [R.sup.2*.sub.t] = max {I : [p.sup.2.sub.t] + [l.sup.2.sub.t] + [h.sub.t] [less than or equal to] [J'.sub.t+1](I)} if [p.sup.2.sub.t] + [l.sup.2.sub.t] + [h.sub.t] < [J'.sub.t+1](0), [B.sup.1*.sub.t] + [B.sup.2*.sub.t] = min {I : [J'.sub.t+1](-I) [greater than or equal to] [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t]} if [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t] > [J'.sub.t+1](0), [B.sup.2*.sub.t] = min {I : [J'.sub.t+1] (-I) [greater than or equal to] [p.sup.2.sub.t] + [l.sup.2.sub.t] - [[beta].sup.2.sub.t]} if [p.sup.2.sub.t] + [l.sup.2.sub.t] - [[beta].sup.2.sub.t] > [J'.sub.t+1](0).
In Fig. 1(a-c) we show the marginal expected profit in period t + 1 as a function of inventory. According to the decisions described above, an optimal decision (e.g., the reservation decision [R.sup.1*.sub.t]) equals the inventory level where the relevant prices and costs (e.g., [p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t]) cross the marginal expected profit curve. Figure 1(a) illustrates the reserve inventory decisions, which correspond to the reserve-inventory policy in the previous section. Also observe that the optimal policy is nested, similar to a type of revenue management strategy in the airline industry. Such a nested decision policy is, of course, no longer optimal in situations without the prioritizing of demand classes.
[FIGURE 1 OMITTED]
In the other parts of Fig. 1, we show the marginal expected profit curve compared to the costs relevant to the other optimal decisions above. The optimal backlogging decision is portrayed in Fig. 1(b); this decision corresponds to the backlog-demand policy. Finally, we show the optimal decision that results from the reserve-and-backlog policy in Fig. 1(c). A similar picture could be drawn for the target inventory decision comparing the production cost ([c.sub.t]) with the derivative of the [G.sub.t] function; this is left out for brevity. In all of the decisions, we note that the manufacturer is trading off the certain net revenue in the current period (e.g., [p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t]) with a marginal expected profit in the future. Clearly there is some risk with betting on the future, but such trade-offs are made regularly in many situations.
3.3. Special cases and extensions
We are also interested in situations in which manufacturers cannot differentiate customers and treat them as a single class. We denote this situation as the NDS, which is a special case of the PDS. We assume that the manufacturer takes the second-class customers' reservation price, [p.sup.2.sub.t], as in the selling price to all customers. Since the lower price is charged to both classes, customers in both classes are willing to wait one period if the item is not available to them, as in the PDS. The difference between the NDS and the PDS lies not in the customers' preferences, but in the manufacturer's treatment of the customers. First-class customers would be willing to pay extra if the manufacturer could differentiate, but he is not able or willing to differentiate. If we set [D.sup.2.sub.t], [R.sup.2.sub.t], and [B.sup.2.sub.t] to zero and replace [D.sup.1.sub.t] with total demand in the formulation of PDS, we get NDS. Thus, the optimal policy is of the form (S, R, B) as in PDS.
Initially, we analyzed the PDS problem for two customer classes under the assumption that all backlogged orders are filled within one period. However, there are several more general extensions that easily follow from our initial proof. Some of these extensions are outlined below.
* Multiple classes: Our results for PDS hold for a problem with more than two customer classes. As before, it is necessary to assume that each class has priority in the allocation of inventory and production capacity over the lower priority classes in the current period. With this assumption, the nesting structure of the tactical inventory decisions is still optimal. To be more specific, one could have a menu (price, priority ranking) for each customer class. If there are many customer classes, it might be difficult for customers to choose from the sets, and at the firm level, priority ordering of many classes would also be difficult. However, it may be reasonable for three to five classes, which can occur in some applications.
* Time-differentiated customers: It is also possible to extend the models to cover situations where some classes are always served immediately while others receive immediate or delayed fulfillment. An example for this is a time differentiation strategy, used when the first-class customers are not willing to wait and are served immediately, while the second-class customers can be served immediately or in the next period. For this problem, the optimal policy is in the form of (S, [R.sup.i], B) for i = 1, 2, which is a critical threshold policy as before. See Duran (2007) for details.
* Long leadtime: The fulfillment leadtime in our analysis is assumed to be one period. However, it is also possible to allow for planned backlogs where the orders can be delivered anytime before the end of the time horizon. For the extended analysis, we assume that backlogs must be filled before new orders are accepted, and under this assumption our nested threshold policies are still optimal. See Duran (2006) for details.
If there is a leadtime 1 < [l.sub.t] < T - t in each period that specifies orders must be delivered in period t + [l.sub.t], then the problem is structurally more complex. (2) In particular, the state space increases since previous orders must be tracked so that they are fulfilled in the correct time period. Furthermore, even if the expected profit is concave, the optimal policy may be complex and not easy to implement.
4. Computational analysis
In this section we report on a computational study conducted to obtain insights about the benefits of customer differentiation and tactical inventory use in PDS and NDS. Our goal is to examine the relative performance of the policies of the (S, R, B) form in different problem settings and identify the situations where this type of policy can provide significant increases in profit.
The benchmark we use for each of our strategies is a traditional base stock policy where the manufacturer uses the modified order-up-to policy (S policy) and serves all customers as in a single class. We assume that sales are lost if there is insufficient inventory on hand or if customers are rejected. We compare the performance of the (S, R, B) type policies over the traditional policies using the metric of profit potential, as defined by 100 x (([V.sub.(S, R, B)]/[V.sub.S]) - 1), where V indicates the expected profit of the problem being solved. In both the traditional policy and NDS, we use [p.sup.2.sub.t] as the price charged to all customers to ensure that we serve both of the classes. This implies that PDS may show a big improvement in profit that is due, in part, to the ability to differentiate customers.
The profit improvement of the PDS compared to traditional inventory policies comes from three sources: prioritized demand classes, differentiated pricing, and shifting inventory to the next period, whereas the NDS only has the last source. Thus, by comparing both PDS and NDS to the traditional policy, we can separate the impact of price differentiation versus tactical inventory.
4.1. Experiment details
The total average demand from the first and second-class customers equals 100 in each experiment. We assume that demand uncertainty is additive with a mean of zero. We define the coefficient of variation of demand in a given period as C[V.sup.i.sub.U] = s(D.sup.i.sub.t])/E([D.sup.i.sub.t]), where s denotes the standard deviation, and E denotes the expected value. In all cases shown, the coefficient of variation of demand uncertainty is the same in each period and is equal to 0.2.
Production capacity is constant for a particular instance, while it is allowed to take the values of 60% (low), 80% (med), and 100% (high) of the expected total average demand for both classes over the horizon (denoted by Dem *) in some experiments. The production cost may vary by period, but the production cost vector is the same across instances. (We also ran experiments where the production cost is the same in each period and obtained similar results.) See Table 3 for the exact data; for example, the average markup of [p.sup.2.sub.t] ([p.sup.1.sub.t]) over the cost is about 30% (60%) for the experiments on class proportions.
Table 3. Specific experimental data
t
1 2 3 4 5 6 7 8 9 10 11
[c.sub.t] 70 90 70 50 70 90 70 50 70 90 70
[p.sup.2.sub.t] 90 110 90 70 90 110 90 70 90 110 90
[p.sup.1.sub.t] 110 130 110 90 110 130 110 90 110 130 110
t
12 Avg
[c.sub.t] 50 70
[p.sup.2.sub.t] 70 90
[p.sup.1.sub.t] 90 110
We study the impact of the percentage of first versus second-class demand in our first set of experiments. In these cases the expected demand from first-class customers over the horizon, E([D.sup.1]), takes the values of 20, 25, 50, 75, and 80, and the expected second-class demand, E([D.sup.2]), equals 100 - E([D.sup.1]). The prices are constant over the set of experiments but may vary by period. Having varying prices increases the likelihood that all of the policies will be optimal in some period of an experiment, since the prices create an incentive to shift capacity. The average ratio of [p.sup.1.sub.t]/[p.sup.2.sub.t] is 1.22 for the experiments studying the proportion of demand. See Table 3 for the prices used in this set of experiments.
We also consider the relative price difference between classes. In these experiments E([p.sup.2]) is fixed over the instances, and the price for the first class is set according to E([p.sup.1])/E([p.sup.2]) = 1.1, 1.2, and 1.3, where E([p.sup.i]) represents the average price over the horizon. We allow the trend of [p.sup.2.sub.t] (and correspondingly, [p.sup.1.sub.t]) to be either linearly increasing or decreasing (we also ran experiments with no clear price trend). Let [gamma] = [p.sup.2.sub.t+1] - [p.sup.2.sub.t], which we assume to be fixed for all t = 1 ... T - 1; [gamma] shows the rate of change of price over time. For the increasing price experiments, [p.sup.2.sub.1] = $70, and for the decreasing price experiments, [p.sup.2.sub.12] = $70 where 12 is the last period.
4.2. Results
For all experiments, the policies in PDS and NDS using tactical inventory have a higher profit than the traditional policy. This is clear because of the usage of [p.sup.2.sub.t] for all customers in the traditional policy. However, note that the profit difference is significant, even when the E([D.sup.1]) percentage is small (see Fig. 2(a)).
[FIGURE 2 OMITTED]
The performance for a given proportion of first-class customers is better under the tactical inventory policies when the capacities are tight. As an example, in Fig. 2(a) the performance of PDS when capacity is 0.6 Dem * is better than the performance of PDS when capacity is 0.8 Dem *. As expected for a given capacity level, the performance of the tactical inventory policy in PDS increases almost linearly as the proportion of first-class customers increases. This profit improvement is due to the additional revenue opportunities that the tactical inventory policies have over the traditional policy including higher revenue from first-class customers and an increased ability to meet demand by shifting capacity.
As expected, the profit under NDS is insensitive to the first-class proportion since it does not differentiate between the classes. However, the significant profit shows improvement over the traditional policy, even though both NDS and the traditional policy offer [p.sup.2.sub.t] to everyone, suggests that the tactical inventory may greatly improve profit. In our experiments, production cost and prices are time varying and capacity is limited. When all parameters are stationary over time and there is sufficient production capacity, the differentiation strategies are unlikely to offer as much improvement over the traditional policy.
For several levels of price proportions (E([p.sup.1])/E([p.sup.2])), we look at the rate of price increase (measured by [gamma]) over the time horizon in Fig. 2(b); the decreasing price trend showed nearly the same results. Whether or not the pricing trend is increasing or decreasing, the performance of the (S, [R.sup.i], [B.sup.i]) policy relative to the traditional policy increases with decreasing [gamma]. To see this for the case of increasing prices, note in Fig. 2(b) that the performance of PDS when [gamma] = 1 is better than the performance of PDS when [gamma] = 4 at every ratio of price differences between the classes. This result is somewhat surprising. Looking at our results more closely, we find that as [gamma] increases, the profits of PDS and the traditional policy are both increasing because the mean prices are increasing. In fact, the absolute profit difference between the two strategies is increasing with [gamma]. However, the percentage profit improvement is not increasing with [gamma]. This seems to be because the total demand in each case is constant and the additional marginal profit from selling one more unit in PDS is small relative to the overall increase in the profit of the traditional policy when [gamma] is large.
The values of the average tactical inventory levels for PDS and NDS are depicted in Fig. 3(a) and Fig. 3(b), respectively, for increasing prices. In the increasing price experiments all three policies in PDS are active, while for decreasing prices (not shown) only the backlog-demand policy resulted. In some cases the magnitude of the average tactical inventory increases with [gamma] (that is, with increasing trend in price), but this is not true in all cases. Note here that Fig. 3(a and b) depicts average tactical inventory over the horizon, not necessarily in each period. When we look at the solutions in more detail for increasing price trend, we find that the reserve inventory is used in periods with lower prices and backlogging is used in periods with higher prices. Thus, for each [gamma] level in the figure, we have positive backlogging. This is also due to the fact that the backlogging decision is comparing the net revenue from a certain current customer with the marginal expected profit from a future customer. In NDS all available tactical inventory decisions are employed, and in same cases (e.g., [gamma] = 6), the best value of R for NDS is approximately equal to [R.sup.1.sub.t] + [R.sup.2.sub.t] in PDS, suggesting that NDS is partially compensating for limited flexibility with high values of tactical inventory for the single customer class.
[FIGURE 3 OMITTED]
In the experiments thus far, we set the regular price to be [p.sup.2.sub.t] (in the traditional policy), and some customers are willing to pay a higher price [p.sup.1.sub.t] for priority service (in PDS) over no priority at regular price [p.sup.2.sub.t]. In this situation PDS clearly offers an advantage over the traditional policy, since the average revenue is higher. However, it is also interesting to see what happens when the regular price is [p.sup.1.sub.t] and some customers are willing to be served at a lower priority for a discount, paying [p.sup.2.sub.t]. We show the results of these experiments with increasing first-class demand in Fig. 4(a), where the total number of expected customers is 100 as before. Note here that PDS does not necessarily provide an improvement over the traditional policy, since the average revenue per customer is less than in the traditional policy. When the first-class demand proportion is more than 50% and capacity is tight, we see that PDS can have a higher profit than the traditional policy, even though the latter has larger average revenue. This result suggests that tactical inventory to shift capacity can overcome the average revenue decrease per customer in some cases. We also consider experiments where the regular price (in the traditional policy) is the average of [p.sup.1.sub.t] and [p.sup.2.sub.t] in each period, and PDS has some customers willing to pay more ([p.sup.1.sub.t]) for higher priority and some customers willing to have a lower priority for a price discount ([p.sup.2.sub.t]). In this case we see that PDS has greater profit than the traditional model in almost all cases, even though the average revenue is less in PDS when the expected first-class demand is less than 50% (see Fig. 4(b)). The main insight from these graphs is that if prioritization of demand classes costs a firm in the average revenue per customer, the benefit of tactical inventory may outweigh the revenue loss.
[FIGURE 4 OMITTED]
5. Conclusions
In this paper we analyzed a multiple-class customer problem where production and tactical inventory decisions must be made in every period and demand is a general stochastic function of time and customer class. We have shown that there are a variety of problems using tactical inventory decisions for which a threshold policy in each period is optimal under a PDS. Specifically, we have a modified base stock policy consisting of the target inventory decision (S), the reserve-up-to levels ([R.sup.i]), and the backlog-up-to levels ([B.sup.i]) for each demand class, or an (S, [R.sup.i], [B.sup.i]) policy. Under prioritized demand this policy is further nested by customer class.
The problem we model and analyze may also have application in other industries. For instance, in some healthcare environments there may be multiple customer classes competing for time on a piece of equipment where priorities are based on the status of the illness. In this problem, there may not be an explicit production decision, but one could still apply backordering and reservation decisions such as promising to service a lower priority class customer in a future time period.
Clearly the analysis in this paper makes assumptions to simplify the problem, such as focusing on a single product. Yet these simplifications allow the development of an optimal policy that is easy to understand, and more importantly, is easy to implement; and the results have extensions beyond those focused on in this article. Furthermore, the simple structure of the threshold policy may give insight for policies to apply to more complicated problems.
Appendix
The following lemma is used in the proof of Theorem 1.
Lemma A1. Given g(x, y) is jointly concave in x and y, G(x) = ma[x.sub.y] g(x, y) is a concave function for x.
Proof. For any [x.sub.1], [x.sub.2] [member of] R, let [y.sub.1] = arg max{y|g([x.sub.1], y)}, [y.sub.2] = arg max{y|g([x.sub.2], y)}. For any [lambda] [member of] [0, 1], let [x.sub.[lambda]] = [lambda][x.sub.1] + (1 - [lambda])[x.sub.2], [y.sub.[lambda]] = [lambda][y.sub.1] + (1 - [lambda])[y.sub.2]. We have G([x.sub.[lambda]]) = ma[x.sub.y] g([x.sub.[lambda]], y) [greater than or equal to] g([x.sub.[lambda]], [y.sub.[lambda]]) [greater than or equal to] [lambda]g([x.sub.1], [y.sub.1]) + (1 - [lambda])g([x.sub.2], [y.sub.2]) = [lambda]G([x.sub.1]) + (1 - [lambda])G([x.sub.2]).
Proof of problem simplification with nesting for PDS (Lemma 1)
We will show this result by contradiction. Let us start with the first condition, ([B.sup.1.sub.t] + [B.sup.2.sub.t])[R.sup.1.sub.t] = 0. Assume that there exists an optimal policy in the form of {([B.sup.1.sub.t] + [B.sup.2.sub.t]), [R.sup.1.sub.t]}, where ([B.sup.1.sub.t] + [B.sup.2.sub.t])[R.sup.1.sub.t] > 0. We will show that there exists an alternate policy, which is at least as good as and sometimes better than the "optimal" policy, which will contradict the assumption of optimality of the policy where both the reserve inventory decisions and the backlogged order decisions are positive. We consider two main market environments: (i) when the current net revenue from selling out of inventory is better than the future expected profit of an additional unit; and (ii) when the future expected profit of an additional unit is better than the current net revenue from backlogging.
Case 1: Since the current net revenue from selling out of inventory is better than the future marginal expected profit, the alternative policy is saving one less item in the current period.
So the alternate policy is {[[[bar.B].sup.1.sub.t] + [[bar.B].sup.2.sub.t]], [[bar.R].sup.1.sub.t]} = {[B.sup.1.sub.t] + [B.sup.2.sub.t], [R.sup.1.sub.t] - 1}. In both policies, decisions for the second class ([R.sup.2.sub.t] and [B.sup.2.sub.t]) are the same. But in the alternate policy, the values of [S.sup.2.sub.t] and [B.sup.2,ef.sub.t] can be higher than the values of the assumed-optimal policy. Let [V.sub.t] and [[bar.V].sub.t] be the expected profit starting from period t under the two policies, respectively. Let us consider the following case.
When [D.sup.1.sub.t] - [S.sub.t] + [R.sup.1.sub.t] > [B.sup.1.sub.t] + [B.sup.2.sub.t] => [[bar.B].sup.2,ef.sub.t] = [B.sup.2,ef.sub.t] = 0 and [[bar.S].sup.2.sub.t] = [S.sup.2.sub.t] = 0:
[V.sub.t] = [p.sup.1.sub.t] ([S.sub.t] - [R.sup.1.sub.t] + [B.sup.1.sub.t] + [B.sup.2.sub.t]) - [h.sub.t] [R.sup.1.sub.t] - [l.sup.1.sub.t] ([D.sup.1.sub.t] - [S.sub.t] + [R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t]) - [[beta].sup.1.sub.t] ([B.sup.1.sub.t] + [B.sup.2.sub.t]) - [l.sup.2.sub.t] [D.sup.2.sub.t] + [J.sub.t+1] ([R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t]), [[bar.V].sub.t] = [p.sup.1.sub.t] ([S.sub.t] - [R.sup.1.sub.t] + 1 + [B.sup.1.sub.t] + [B.sup.2.sub.t]) - [h.sub.t] ([R.sup.1.sub.t] - 1) - [l.sup.1.sub.t] ([D.sup.1.sub.t] - [S.sub.t] + [R.sup.1.sub.t] - 1 - [B.sup.1.sub.t] - [B.sup.2.sub.t]) - [[beta].sup.1.sub.t] ([B.sup.1.sub.t] + [B.sup.2.sub.t]) - [l.sup.2.sub.t] [D.sup.2.sub.t] + [J.sub.t+1] ([R.sup.1.sub.t - 1 - [B.sup.1.sub.t] - [B.sup.2.sub.t]) = [V.sub.t] + ([p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t]) - ([J.sub.t+1] ([R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t]) - [J.sub.t+1] ([R.sup.1.sub.t] - 1 [B.sup.1.sub.t] - [B.sup.2.sub.t])) > [V.sub.t].
The last inequality follows from the fact that the current net revenue from selling out of inventory is higher than the marginal expected profit from carrying one more unit of inventory forward in this market environment.
Case 2: Since the future marginal expected profit is better than the current net revenue from backlogging, promising one less item in the current period is the alternate policy.
So the alternate policy is {[[[bar.B].sup.1.sub.t] + [[bar.B].sup.2.sub.t]], [[bar.R].sup.1.sub.t]} = {[B.sup.1.sub.t] + [B.sup.2.sub.t] - 1, [R.sup.1.sub.t]}. In both policies, decisions for the second class ([R.sup.2.sub.t] and [B.sup.2.sub.t]) are the same. If we compare [V.sub.t] and [[bar.V].sub.t]:
When [D.sup.1.sub.t] - [S.sub.t] - [R.sup.1.sub.t] [greater than or equal to] [B.sup.1.sub.t] + [B.sup.2.sub.t], we will have:
[V.sub.t] = [P.sup.1.sub.t] ([S.sub.t] - [R.sup.1.sub.t] + [B.sup.1.sub.t] + [B.sup.2.sub.t]) - [h.sub.t] [R.sup.1.sub.t] - [l.sup.1.sub.t] ([D.sup.1.sub.t] - [S.sub.t] + [R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t]) - [l.sup.2.sub.t] [D.sup.2.sub.t] - [[beta].sup.1.sub.t] ([B.sup.1.sub.t] + [B.sup.2.sub.t]) + [J.sub.t+1] ([R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t]), [[bar.V].sub.t] = [p.sup.1.sub.t] ([S.sub.t] - [R.sup.1.sub.t] + [B.sup.1.sub.t] + [B.sup.2.sub.1] - 1) - [h.sub.t] [R.sup.1.sub.t] - [l.sup.1.sub.t] ([D.sup.1.sub.t] - [S.sub.t] + [R.sup.1.sub.t]
- [B.sup.1.sub.t] - [B.sup.2.sub.t] + 1) - [l.sup.2.sub.t] [D.sup.2.sub.t] - [[beta].sup.1.sub.t] ([B.sup.1.sub.t] + [B.sup.2.sub.t] - 1) + [J.sub.t+1] ([R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t] + 1) = [V.sub.t] + ([J.sub.t+1] ([R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t] + 1) - [J.sub.t+1] ([R.sup.1.sub.t] - [B.sup.1.sub.t] - [B.sup.2.sub.t])) - ([p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t]) > [V.sub.t].
The last inequality follows from the fact that the marginal future expected profit from one more unit of inventory is higher than the current net revenue from backlogging in this market environment.
For the second condition again assume that there exists an optimal policy in the form of {[B.sup.2.sub.t], ([R.sup.1.sub.t] + [R.sup.2.sub.t])} where [B.sup.2.sub.t] x ([R.sup.1.sub.t] + [R.sup.2.sub.t]) > 0. We will show that there exists an alternate policy, which is at least as good as and sometimes better than the original policy. We consider the same two main market environments as for the first condition.
Case 1: Since the current net revenue from selling out of inventory is better than the future marginal expected profit, the alternative policy is saving one less item in the current period.
So the alternate policy is {[[bar.B].sup.2.sub.t], [[[bar.R].sup.1.sub.t] + [R.sup.2.sub.t]]} = {[B.sup.2.sub.t], [R.sup.1.sub.t] + [R.sup.2.sub.t] - 1}. In both policies, decisions for the first class are the same, namely, the items saved from first-class customers are [R.sup.1.sub.t], and the maximum amount of orders to backlog is [B.sup.1.sub.t] + [B.sup.2.sub.t]. Let us consider the following case:
When [S.sub.t] - [R.sup.1.sub.t] - [R.sup.2.sub.t] [greater than or equal to] [D.sup.1.sub.t], [D.sup.2.sub.t] > [S.sup.2.sub.t] - [R.sup.2.sub.t] + [B.sup.2.sub.t] => [[bar.B].sup.2, ef.sub.t] = [B.sup.2, ef.sub.t] = [B.sup.2.sub.t] and [[bar.S].sup.2.sub.t] = [S.sup.2.sub.t]:
[V.sub.t] = [P.sup.1.sub.t] [D.sup.1.sub.t] + [P.sup.2.sub.t] ([S.sup.2.sub.t] - [R.sup.2.sub.t] + [B.sup.2.sub.t] - [h.sub.t] ([R.sup.1.sub.t] + [R.sup.2.sub.t]) - [l.sup.2.sub.t] ([D.sup.2.sub.t] - [S.sup.2.sub.t] + [R.sup.2.sub.t] - [B.sup.2.sub.t]) - [[beta].sup.2.sub.t] [B.sup.2.sub.t] + [J.sub.t+1] ([R.sup.1.sub.t] + [R.sup.2.sub.t] - [B.sup.2.sub.t]), [[bar.V].sub.t] = [p.sup.1.sub.t] [D.sup.1.sub.t] + [p.sup.2.sub.t] ([S.sup.2.sub.t] - [R.sup.2.sub.t] + 1 + [B.sup.2.sub.t]) - [h.sub.t] ([R.sup.1.sub.t] + [R.sup.2.sub.t] - 1) - [l.sup.2.sub.t] ([D.sup.2.sub.t] - [S.sup.2.sub.t] + [R.sup.2.sub.t] - 1 - [B.sup.2.sub.t]) - [[beta].sup.2.sub.t] [B.sup.2.sub.t] + [J.sub.t+1] ([R.sup.1.sub.t] + [R.sup.2.sub.t] - 1 - [B.sup.2.sub.t]) = [V.sub.t] + ([p.sup.2.sub.t] + [h.sub.t] + [l.sup.2.sub.t]) - ([J.sub.t+1] ([R.sup.1.sub.t] + [R.sup.2.sub.t] - [B.sup.2.sub.t]) - [J.sub.t+1] ([R.sup.1.sub.t] + [R.sup.2.sub.t] - 1 - [B.sup.2.sub.t])) > [V.sub.t].
The last inequality follows from the fact that the current net revenue from selling out of inventory is higher than the marginal future expected profit from one more unit of inventory in this market environment.
Demand management of differentiated customers
Case 2: Since the future marginal expected profit is better than the current net revenue from backlogging, promising one less item in the current period is the alternate policy.
So the alternate policy is {[[bar.B].sup.2.sub.t], [[[bar.R].sup.1.sub.t] + [R.sup.2.sub.t]]} = {[B.sup.2.sub.t] - 1, [R.sup.1.sub.t] + [R.sup.2.sub.t]}. Again, in both policies, decisions for the first class are the same, namely, the items saved from first-class customers are [R.sup.1.sub.t], and the maximum amount of orders to backlog is [B.sup.1.sub.t] + [B.sup.2.sub.t]. Let us consider the following case.
When [S.sub.t] - [R.sup.1.sub.t] [greater than or equal to] [D.sup.1.sub.t] > [S.sub.t] - [R.sup.1.sub.t] - [R.sup.2.sub.t], [D.sup.2.sub.t] [greater than or equal to] [B.sup.2.sub.t] => [B.sup.2,ef.sub.t] = [B.sup.2.sub.t], [[bar.B].sup.2,ef.sub.t] = [B.sup.2.sub.t] - 1 and [[bar.S].sup.2.sub.t] = [S.sup.2.sub.t]:
[V.sub.t] = [p.sup.1.sub.t] [D.sup.1.sub.t] + [p.sup.2.sub.t] [B.sup.2.sub.t] - [h.sub.t] ([R.sup.1.sub.t] + [S.sup.2.sub.t]) - [l.sup.2.sub.t] ([D.sup.2.sub.t] - [B.sup.2.sub.t]) - [[beta].sup.2.sub.t] [B.sup.2.sub.t] + [J.sub.t+1] ([S.sup.2.sub.t] + [R.sup.1.sub.t] - [B.sup.2.sub.t]), [[bar.V].sub.t] = [p.sup.1.sub.t] [D.sup.1.sub.t] + [p.sup.2.sub.t] ([B.sup.2.sub.t] - 1) - [h.sub.t] ([R.sup.1.sub.t] + [S.sup.2.sub.t]) - [l.sup.2.sub.t] ([D.sup.2.sub.t] - [B.sup.2.sub.t] + 1) - [[beta].sup.2.sub.t] ([B.sup.2.sub.t] - 1) + [J.sub.t+1] ([S.sup.2.sub.t] + [R.sup.1.sub.t] - [B.sup.2.sub.t] + 1) = [V.sub.t] + ([J.sub.t+1] ([R.sup.1.sub.t] + [S.sup.2.sub.t] - [B.sup.2.sub.t] + 1) - [J.sub.t+1] ([R.sup.1.sub.t] + [S.sup.2.sub.t] - [B.sup.2.sub.t])) - ([p.sup.2.sub.t] + [l.sup.2.sub.t] - [[beta].sup.2.sub.t]) > [V.sub.t].
The last inequality follows from the fact that the marginal future expected profit from one more unit of inventory is higher than the current net revenue from backlogging in this market environment.
For both of the conditions, the expected profit under the alternative policies is higher than the policy we initially assumed to be optimal in both of the market environments defined at the beginning of the proof, and it can be shown easily that in all other cases in the two market environments, the alternate policies produce exactly the same or higher expected profit as the starting policy. Since alternate policies are at least as good as and sometimes better than the starting policy, a contradiction has been reached.
Proof of concavity for the Priority Differentiation Strategy (Theorem 1)
Let [j.sub.t]([I.sub.t], [S.sub.t]) = -[c.sub.t]([S.sub.t] - [I.sub.t]) + [G.sub.t]([S.sub.t]), so [J.sub.t]([I.sub.t]) = ma[x.sub.[S.sub.t]:[I.sub.t][less than or equal to][S.sub.t][less than or equal to][I.sub.t]+[q.sub.t]] [j.sub.t]([I.sub.t], [S.sub.t]). We prove by induction.
1. For the last period:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where v is the salvage value per item at the end of the horizon: [p.sup.1.sub.T] > [p.sup.2.sub.T] > v > 0.
It is clear that the first derivative of [G.sub.T] is equal to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Now we can check whether or not [G'.sub.T]([S.sub.T]) is non-increasing (superscript T denotes the total of demand classes 1 and 2):
[G.sup.".sub.T]([S.sub.T]) = [[??].sup.1.sub.T]([S.sub.T])([p.sup.2.sub.T] - [p.sup.1.sub.T]) + [[??].sup.T.sub.T]([S.sub.T]) x (v - [p.sup.2.sub.T]).
Since [p.sup.1.sub.T] > [p.sup.2.sub.T] > v > 0, it is easily seen that [G.sup.".sub.T]([S.sub.T]) [less than or equal to] 0, therefore [G.sub.T]([S.sub.T]) is concave.
2. Given t + 1 [less than or equal to] T, assume that [G.sub.t+1]([S.sub.T]) is concave in [S.sub.T], then we can prove that [j.sub.t+1]([I.sub.t+1], [S.sub.t+1]) is jointly concave in [I.sub.t+1] and [S.sub.t+1] by the following. For any ([I.sub.1], [S.sub.1]), ([I.sub.2], [S.sub.2]) [member of] [R.sup.2], let [I.sub.[lambda]] = [lambda][I.sub.1] + (1 - [lambda])[I.sub.2], [S.sub.[lambda]] = [lambda][S.sub.1] + (1 - [lambda])[S.sub.2]. Then,
[j.sub.t+1] ([I.sub.[lambda]], [S.sub.[lambda]]) = - [c.sub.t+1]([S.sub.[lambda]] - [I.sub.[lambda]) + [G.sub.t+1]([S.sub.[lambda]) = - [c.sub.t+1]([lambda][S.sub.1] + (1 - [lambda])[S.sub.2] - [lambda][I.sub.1] - (1 - [lambda])[I.sub.2]) + [G.sub.t+1]([lambda][S.sub.1] + (1 - [lambda])[S.sub.2]) [greater than or equal to] - [lambda][c.sub.t+1]([S.sub.1] - [I.sub.1]) - (1 - [lambda])[c.sub.t+1]([S.sub.2] - [I.sub.2]) + [lambda][G.sub.t+1]([S.sub.1]) + (1 - [lambda])[G.sub.t+1]([S.sub.2]) = [lambda][j.sub.t+1]([I.sub.1], [S.sub.1]) + (1 - [lambda])[j.sub.t+1]([I.sub.2], [S.sub.2]).
So by Lemma A1, [J.sub.t+1]([I.sub.t]) is concave in [I.sub.t], and as a result [J'.sub.t+1]([I.sub.t]) is non-increasing in [I.sub.t].
3. Next let us prove that [g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]) is quasi-concave in [R.sup.1.sub.t] and [R.sup.2.sub.t].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let us define [R.sup.1*.sub.t] and [R.sup.2*.sub.t] as
[R.sup.1* t] = max {I : [p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t] [less than or equal to] [J'.sub.t+1](I)} if [p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t] < [J'.sub.t+1](0) (= 0 o.w.), [R.sup.1*.sub.t] + [R.sup.2.sub.t] = max {I: [p.sup.2.sub.t] + [l.sup.2.sub.t] + [h.sub.t] [less than or equal to] [J'.sub.t+1](I)} if [p.sup.2.sub.t] + [l.sup.2.sub.t] + [h.sub.t] < [J'.sub.t+1](0) (= 0 o.w.).
Thus, we have [nabla][g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]) [greater than or equal to] [0, 0[].sup.T]] when 0 [less than or equal to] [R.sup.1.sub.t] [less than or equal to] [R.sup.1*.sub.t] and 0 [less than or equal to] [R.sup.2.sub.t] [less than or equal to] [R.sup.2*.sub.t], and [nabla][g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]) [less than or equal to] [0, 0[].sup.T] when [R.sup.1.sub.t] > [R.sup.1*.sub.t] and [R.sup.2.sub.t] > [R.sup.2*.sub.t]; thus, [g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]) is quasi-concave with respect to [R.sup.1.sub.t] and [R.sup.2.sub.t]. ([R.sup.1*.sub.t], [R.sup.2*.sub.t]) is the unique unconstrained optimizer of [g.sup.I.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]), and it is independent of inventory level [S.sub.t]. ([R.sup.1,c.sub.t], [R.sup.2,c.sub.t]) = (min([R.sup.1*.sub.t], ([S.sub.t][).sup.+]), min([R.sup.2*.sub.t], ([S.sub.t][).sup.+])) maximizes [g.sup.1.sub.t]([S.sub.t], [R.sup.1.sub.t], [R.sup.2.sub.t]), for 0 [less than or equal to] [R.sup.1.sub.t] [less than or equal to] ([S.sub.t][).sup.+] and 0 [less than or equal to] [R.sup.2.sub.t] [less than or equal to] ([S.sub.t][).sup.+].
4. Next let us prove that [g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) is quasi-concave in [B.sup.1.sub.t] and [B.sup.2.sub.t]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let us define [B.sup.1*.sub.t] and [B.sup.2*.sub.t] as
[B.sup.1*.sub.t] + [B.sup.2*.sub.t] = min {I : [J'.sub.t+1](-I) [greater than or equal to] [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t]} if [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t] > [J'.sub.t+1](0) (= 0 o.w.), [B.sup.2*.sub.t] = min {I : [J'.sub.t+1](-I) [greater than or equal to] [p.sup.2.sub.t] + [l.sup.2.sub.t] - [[beta].sup.2.sub.t]} if [p.sup.2.sub.t] + [l.sup.2.sub.t] - [[beta].sup.2.sub.t] > [J'.sub.t+1](0) (= 0 o.w.).
Thus, we have [nabla][g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) [greater than or equal to] [0, 0[].sup.T] when 0 [less than or equal to] [B.sup.1.sub.t] [less than or equal to] [B.sup.1*.sub.t] and 0 [less than or equal to] [B.sup.2.sub.t] [less than or equal to] [B.sup.2*.sub.t], and [nabla][g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) [less than or equal to] [0, 0[].sup.T] when [B.sup.1.sub.t] > [B.sup.1*.sub.t] and [B.sup.2.sub.t] > [B.sup.2*.sub.t]; thus, [g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]) is quasi-concave with respect to [B.sup.1.sub.t] and [B.sup.2.sub.t]. ([B.sup.1*.sub.t], [B.sup.2*.sub.t]) is the unique unconstrained optimizer of [g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]), and it is independent of inventory level [S.sub.t]. ([B.sup.1,c.sub.t], [B.sup.2,c.sub.t]) = (min([B.sup.1*.sub.t], [q.sub.t+1]), min([B.sup.2*.sub.t], [q.sub.t+1])) maximizes [g.sup.2.sub.t]([S.sub.t], [B.sup.1.sub.t], [B.sup.2.sub.t]), for 0 [less than or equal to] [B.sup.1.sub.t] [less than or equal to] [q.sub.t+1] and 0 [less than or equal to] [B.sup.2.sub.t] [less than or equal to] [q.sub.t+1].
5. Next let us prove that [g.sup.3.sub.t]([S.sub.t], [R.sup.2.sub.t], [B.sup.1.sub.t]) is quasi-concave in [B.sup.1.sub.t] and [R.sup.2.sub.t]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Let us define [R.sup.2*.sub.t] and [B.sup.1*.sub.t] as
[R.sup.2*.sub.t] = max {I : [p.sup.2.sub.t] + [l.sup.2.sub.t] + [h.sub.t] [less than or equal to] [J'.sub.t+1](I)} if [p.sup.2.sub.t] + [l.sup.2.sub.t] + [h.sub.t] < [J'.sub.t+1](0) (= O o.w.), [B.sup.1*.sub.t] = min {I : [J'.sub.t+1](-I) [greater than or equal to] [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t]} if [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t] > [J'.sub.t+1](0) (= 0 o.w.).
Thus, we have [nabla][g.sup.3.sub.t]([S.sub.t], [R.sup.2.sub.t], [B.sup.1.sub.t]) [greater than or equal to] [0, 0[].sup.T] when 0 [less than or equal to] [R.sup.2.sub.t] [less than or equal to] [R.sup.2*.sub.t] and 0 [less than or equal to] [B.sup.1.sub.t] [less than or equal to] [B.sup.1*.sub.t], and [nabla][g.sup.3.sub.t]([S.sub.t], [R.sup.2.sub.t], [B.sup.1.sub.t]) [less than or equal to] [0, 0[].sup.T] when [R.sup.2.sub.t] > [R.sup.2*.sub.t] and [B.sup.1.sub.t] > [B.sup.1*.sub.t]; thus, [g.sup.3.sub.t]([S.sub.t], [R.sup.2.sub.t], [B.sup.1.sub.t]) is quasi-concave with respect to [R.sup.2.sub.t] and [B.sup.1.sub.t]. ([R.sup.2*.sub.t], [B.sup.1*.sub.t]) is the unique unconstrained optimizer of [g.sup.3.sub.t]([S.sub.t], [R.sup.2.sub.t], [B.sup.1.sub.t]), and it is independent of inventory level [S.sub.t]. ([R.sup.2,c.sub.t], [B.sup.1,c.sub.t]) = (min([R.sup.2*.sub.t], [S.sub.t]), min([B.sup.1*.sub.t], [q.sub.t+1])) maximizes [g.sup.3.sub.t]([S.sub.t], [R.sup.2.sub.t], [B.sup.1.sub.t]), for 0 [less than or equal to] [R.sup.2.sub.t] [less than or equal to] ([S.sub.t][).sup.+] and 0 [less than or equal to] [B.sup.1.sub.t] [less than or equal to] [q.sub.t+1].
6. Let us prove the concavity of [G.sup.1.sub.t]([S.sub.t]) with respect to [S.sub.t], where [G.sup.1.sub.t]([S.sub.t]) = [g.sup.1.sub.t]([S.sub.t], [R.sup.1,c.sub.t], s[R.sup.2,c.sub.t]). We will consider [G.sup.1.sub.t]([S.sub.t]) in five cases:
Case I: [S.sub.t] [less than or equal to] [R.sup.1*.sub.t]
The profit-to-go after production in this case is: [G.sup.1.sub.t]([S.sub.t]) = -[h.sub.t][S.sub.t] + [J.sub.t+1]([S.sub.t]) - [[integral].sup.[infinity].sub.0] [l.sup.1.sub.t]kd[[PHI].sup.1.sub.t](k) - [[integral].sup.[infinity].sub.0] [l.sup.2.sub.t]kd[[PHI].sup.2.sub.t](k), so its first derivative is [G.sup.1'.sub.t]([S.sub.t]) = -[h.sub.t] + [J'.sub.t+1]([S.sub.t]). Thus, it is clear that [G.sup.1'.sub.t]([S.sub.t]) is non-increasing since: [G.sup.1".sub.t]([S.sub.t]) = [J.sup.".sub.t+1]([S.sub.t]) and [J.sup.".sub.t+1]([S.sub.t]) [less than or equal to] 0.
Case II: [R.sup.1*.sub.t] + [member of] < [S.sub.t] [less than or equal to] [R.sup.1*.sub.t] + [R.sup.2*.sub.t]
The first derivative of the profit-to-go after production function is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus, it is clear that [G.sup.1'.sub.t]([S.sub.t]) is non-increasing since:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Case III: [S.sub.t] = [R.sup.1*.sub.t] + [member of]
It is clear that [G.sup.1'.sub.t]([S.sub.t]) is non-increasing in this case since:
[G.sup.1".sub.t]([R.sup.1*.sub.t] + [member of]) = (1 - [[??].sup.1.sub.t](0))[p.sup.1.sub.t] + [h.sub.t] + [l.sup.1.sub.t] - [J'.sub.t+1]([R.sup.1*.sub.t])) + [[??].sup.1.sub.t](0)([J'.sub.t+1]([R.sup.1*.sub.t] + [member of]) - [J'.sub.t+1]([R.sup.1*.sub.t])); also, [J'.sub.t+1]([S.sub.t]) is non-increasing and [p.sup.1.sub.t] + [h.sub.t] + [l.sup.1.sub.t] = [J'.sub.t+1]([R.sup.1*.sub.t]).
Case IV: [R.sup.1*.sub.t] + [R.sup.2*.sub.t] + [member of] < [S.sub.t]
In this case the first derivative of the profit-to-go after production is equal to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The second derivative is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus, it is clear that [G.sup.1'.sub.t] ([S.sub.t]) is non-increasing since [J.sup.".sub.t+1]([S.sub.t]) [less than or equal to] 0, [p.sup.1.sub.t] + [h.sub.t] + [l.sup.1.sub.t] = [J'.sub.t+1]([R.sup.1*.sub.t]) due to the [R.sup.1*.sub.t] decision, and [J'.sub.t+1]([R.sup.1*.sub.t] + [R.sup.2*.sub.t]) = [p.sup.2.sub.t] + [h.sub.t] + [l.sup.2.sub.t] due to the [R.sup.1*.sub.t] + [R.sup.2*.sub.t] decision.
Case V: [R.sup.1*.sub.t] + [R.sup.2*.sub.t] + [member of] = [S.sub.t]
In this case, the second derivative is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It is clear that [G.sup.1'.sub.t]([S.sub.t]) is non-increasing in this case since [J'.sub.t+1]([S.sub.t]) is non-increasing, [p.sup.1.sub.t] + [h.sub.t] + [l.sup.1.sub.t] = [J'.sub.t+1]([R.sup.1*.sub.t]) due to the [R.sup.1*.sub.t] decision, and [J.sub.t+1]([R.sup.1*.sub.t] + [R.sup.2*.sub.t]) = [p.sup.2.sub.t] + [h.sub.t] + [l.sup.2.sub.t] due to the [R.sup.1*.sub.t] + [R.sup.2*.sub.t] decision.
7. Let us prove the concavity of [G.sup.2.sub.t]([S.sub.t]) with respect to [S.sub.t], where [G.sup.2.sub.t]([S.sub.t]) = [g.sup.2.sub.t]([S.sub.t], [B.sup.1,c.sub.t], [B.sup.2,c.sub.t]).
The first derivative of the profit-to-go after production is equal to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The second derivative is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus, it is clear that [G.sup.2'.sub.t]([S.sub.t]) is non-increasing since [J.sup.".sub.t+1]([S.sub.t]) [less than or equal to] 0; [p.sup.2.sub.t] + [l.sup.2.sub.t] - [[beta].sup.2.sub.t] = [J'.sub.t+1](-[B.sup.2*.sub.t]) due to the [B.sup.2*.sub.t] decision; [J.sub.t+1](-[B.sup.1*.sub.t] + [B.sup.2*.sub.t])) = [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t] due to the [B.sup.1*.sub.t] + [B.sup.2*.sub.t] decision, and [[beta].sup.1.sub.t] [greater than or equal to] [[beta].sup.2.sub.t].
8. Let us prove the concavity of [G.sup.3.sub.t]([S.sub.t]) with respect to [S.sub.t], where [G.sup.3.sub.t]([S.sub.t]) = [g.sup.2.sub.t]([S.sub.t], [R.sup.2,c.sub.t], [B.sup.1,c.sub.t]).
We will consider [G.sup.3.sub.t]([S.sub.t]) in three cases:
Case I: [S.sub.t] [less than or equal to] [R.sup.2*.sub.t]
So the first and second derivatives of the profit-to-go after production in this case are equal to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus, it is clear that [G.sup.3'.sub.t]([S.sub.t]) is non-increasing since [J.sup.".sub.t+1]([S.sub.t]) [less than or equal to] 0 and [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t] = [J'.sub.t+1](-[B.sup.1*.sub.t]) due to the [B.sup.1*.sub.t] decisions.
Case II: [R.sup.2*.sub.t] + [member of] < [S.sub.t]
Then the first and second derivatives of the profit-to-go after production are equal to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Thus, it is clear that [G.sup.3'.sub.t]([S.sub.t]) is non-increasing since: [J.sup.".sub.t+1]([S.sub.t]) [less than or equal to] 0; [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t] = [J'.sub.t+1](-[B.sup.1*.sub.t]) due to the [B.sup.1*.sub.t] decision, and [p.sup.2.sub.t] + [h.sub.t] + [l.sup.2.sub.t] = [J'.sub.t+1]([R.sup.2*.sub.t]) due to the [R.sup.2*.sub.t] decision.
Case III: [R.sup.2*.sub.t] + [member of] = [S.sub.t] and [B.sup.1*.sub.t] [less than or equal to] [q.sub.t+1]
The second derivative of [G.sup.3.sub.t]([S.sub.t]) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
We know that [G.sup.3'.sub.t]([S.sub.t]) is non-increasing in this case since [J'.sub.t+1]([S.sub.t]) is non-increasing; [p.sup.1.sub.t] + [l.sup.1.sub.t] - [[beta].sup.1.sub.t] = [J'.sub.t+1](-[B.sup.1*.sub.t]) due to the [B.sup.1*.sub.t] decision, and [p.sup.2.sub.t] + [h.sub.t] + [l.sup.2.sub.t] = [J'.sub.t+1]([R.sup.2*.sub.t]) due to the [R.sup.2*.sub.t] decision.
9. Let us prove the concavity of [G.sub.t]([S.sub.t]).
In each period, we must be in one of the following cases, which are independent of the [S.sub.t] values:
(i) If [J'.sub.t+1] (0) > [p.sup.1.sub.t] + [l.sup.1.sub.t] + [h.sub.t], we have [R.sup.1*.sub.t] [greater than or equal to] 0, [R.sup.2*.sub.t] [greater than or equal to] 0, [B.sup.1*.sub.t] = 0, and [B.sup.2*.sub.t] = 0, therefore [R.sup.1,c.sub.t] [greater than or equal to] 0, [R.sup.2,c.sub.t] [greater than or equal to] 0, [B.sup.1,c.sub.t] = 0, and [B.sup.2,c.sub.t] = 0; thus, we have [G.sub.t]([S.sub.t]) = [G.sup.1.sub.t]([S.sub.t]).
(ii) If [p.sup.2.sub.t] + [l.sup.2.sub.t] - [[beta].sup.2.sub.t] > [J'.sub.t+1](0), we have [B.sup.1*.sub.t] [greater than or equal to] 0, [B.sup.2*.sub.t] [greater than or equal to] 0, [R.sup.1*.sub.t] = 0, and [R.sup.2*.sub.t] = 0, therefore [B.sup.1,c.sub.t] [greater than or equal to] 0, [B.sup.2,c.sub.t] [greater than or equal to] 0, [R.sup.1,c.sub.t] = 0, and [R.sup.2,c.sub.t] = 0; thus, we have [G.sub.t]([S.sub.t]) = [G.sup.2.sub.t]([S.sub.t]).
(iii) If [J'.sub.t+1](0) > [p.sup.2.sub.t] + [l.sup.2.sub.t] + [h.sub.t], and [J'.sub.t+1](0) < [p.sup.1.sub.t] - [[beta].sup.1.sub.t] + [l.sup.1.sub.t], we have [R.sup.2*.sub.t] [greater than or equal to] 0, [R.sup.1*.sub.t] = 0, [B.sup.1*.sub.t] [greater than or equal to] 0, and [B.sup.2*.sub.t] = 0, therefore [R.sup.1,c.sub.t] = 0, [R.sup.2,c.sub.t] [greater than or equal to] 0, [B.sup.1,c.sub.t] [greater than or equal to] 0, and [B.sup.2,c.sub.t] = 0, thus, we have [G.sub.t]([S.sub.t]) = [G.sup.3.sub.t]([S.sub.t]).
So in each period, [G.sub.t]([S.sub.t]) reduces to some function that is concave. Therefore [G.sub.t]([S.sub.t]) is concave.
Acknowledgements
The research was supported in part by NSF grants DMI-0245352 and DMI-0348532, Lucent Technologies through The Logistics Institute, The Logistics Institute-Asia Pacific, and The Center for Engineering Logistics and Distribution. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or any other sponsor. The authors would also like to thank the reviewers, Associate Editor, and Editor-in-Chief for their contributions, which greatly improved the paper.
Received August 2005 and accepted August 2006
(1) H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA E-mail: jswann@isye.gatech.edu
(2) School of Industrial Engineering and Management, Oklahoma State University, Stillwater, OK 74078, USA
(3) Department of Civil and Environmental Engineering and the Engineering Systems Division, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
* Corresponding author
(1) Each decision is equal to zero if the condition is never satisfied.
(2) If / indicates that orders must be filled by period t + l, and previously accepted orders must be filled before new orders are accepted, then the results in this paper hold as described for planned backlogs. However, for the version of the problem with specific and varying [l.sub.t], the assumption that previous orders are filled first may be too restrictive.
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SERHAN DURAN (1), TIEMING LIU (2), DAVID SIMCHI-LEVI (3) and JULIE L. SWANN (1), *
Biographies
Serhan Duran is a Ph.D. candidate in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology. He received an M.S. in Operations Research and an M.S. in Industrial Engineering from the Georgia Institute of Technology. His research interests include using stochastic models for the improvement of manufacturing, service and distribution operations and implementing revenue management techniques in manufacturing environments to improve flexibility.
Tieming Liu is an Assistant Professor at the School of Industrial Engineering and Management, Oklahoma State University. He received a doctoral degree in Transportation and Logistics from the Massachusetts Institute of Technology in 2005, a Master's degree in Industrial Engineering and Management Science from Northwestern University in 2001, and Master's and Bachelor's degrees in Control Theory and Control Engineering from Tsinghua University in 2000 and 1997, respectively. His research interests include supply chain management, revenue management, manufacturing flexibility and disruption management.
David Simchi-Levi holds a Ph.D. from Tel Aviv University and is currently a Professor at MIT in the Department of Civil and Environmental Engineering and the Engineering Systems Division. His research currently focuses on developing and implementing robust and efficient techniques for supply chains, logistics and manufacturing systems. He has published widely in professional journals on both practical and theoretical aspects of logistics and supply chain management. He has been the principal investigator for more than $5000 000 in funded academic research. He is the Editor-in-Chief of Operations Research, the flagship journal of INFORMS, the former Editor-in-Chief of Naval Research Logistics and a member of the board for several scientific journals. He is a co-author of the books: The Logic of Logistics, Managing the Supply Chain and Designing and Managing the Supply Chain, the latter of which received the book-of-the-year award and the Outstanding IIE Publication award given in 2000 by the Institute of Industrial Engineers. He is the founder and chairman of LogicTools (www.logic-tools.com), which provides software solutions and professional services for supply chain planning.
Julie Swann is an Assistant Professor in the School of ISyE at Georgia Tech. She received her B.S. in Industrial Engineering from the Georgia Institute of Technology in 1996 and her M.S. and Ph.D. in Industrial Engineering and Management Sciences from Northwestern University in 1998 and 2001, respectively. She is currently focused on the modeling and analysis of problems and algorithms in logistics and supply chain management. She has particular interests in developing and analyzing tools to manage demand, such as pricing, revenue management, or lead time quotation, to increase the flexibility in the system. Other research interests include applications of economics and optimization to healthcare policy. She was awarded an NSF CAREER grant in 2004, and in 2002, she received the Doctoral Dissertation Award from the Council of Logistics Management.
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