The newsboy problem with multiple demand classes.
We consider the single item newsboy problem, where the item can be sold to different demand classes at different prices. The demands are realized sequentially over time. That is, the newsboy purchases newspapers at the beginning of the day and sells them in the morning and in the afternoon with
1. Introduction
Many inventory systems operate in segmented markets represented by customer classes that are differentiated by price. For example, fashion and sporting goods may have two or more market segments: regular and discount, where excess supply in the regular market is subsequently disposed of in the discount market. Many perishable items in retailing also follow the pattern of multiple demand classes with non-increasing prices: the first demand class consists of customers who buy the product on the first day (or week) when the commodity is freshest; other customers may constitute the second and third day (or week) demand classes, who pay less for less fresh products. On the other hand, some non-conventional inventory items such as airline seats and special event tickets have demand classes that are increasing in price over time. For example, a 21-day advance purchase is generally less expensive than a 3-day advance purchase; an advance reservation ticket of a special event may cost less than a ticket purchased at t he door. Pfeifer [1] and Bodily and Weatherford [2] give extensive examples of items with time sensitive pricing.
In these contexts, there is often only one inventory replenishment opportunity which exists before the selling season; there is no second replenishment opportunity should the subsequent sales turn out to exceed the amount of the stock purchased. This is usually due to the long replenishment cycle relative to the sales season (as often exhibited in fashion goods systems) or due to the fixed replenishment capacity (such as in airline seats). In many situations, however, the replenishment quantity can be set as a decision variable. It would be interesting to evaluate the value of the ability to choose the initial capacity optimally, and hence the value of additional units of capacity.
In this paper, we study the problem of determining the optimal replenishment quantity to maximize the expected profit. We consider the demand classes as segmented by time and price; each demand class has a single price that is exogenously determined. The demand quantities are stochastic with known probability distributions. Thus, the proverbial newsboy orders his papers early in the morning and sells them at different prices in the morning and in the afternoon, where the demands are realized sequentially. We will first assume that demands are independent and later examine the cases with dependencies created by the returning of a fraction of the unsatisfied demand in a price class to another demand class. An example of this situation is that a fraction of the "low fare" buyers will join the "high fare" customer class (perhaps unwillingly) if their initial request is not met at the lower price. This occurs frequently when the prices are increasing over time. While the most general situation would be one in whi ch the prices fluctuate in either direction, we will examine two special cases with monotonically increasing or decreasing prices. As it turns out, the model structures between these two cases are drastically different.
We assume that demand quantities, often for two or more sub-periods within a fixed selling horizon, are realized sequentially over time. This simplification, which is utilized in most Perishable Asset Revenue Management (PARM) literature, could be restrictive in a general yield management scenario. In airlines, different demand classes might be viewed as different products (from the customer's perspective), each having its own restrictions, penalties and market characteristics; this may cause demands with different prices for essentially the same airline seat to co-exist simultaneously. However, we note that this is not very significant in airline applications, mainly due to the price sensitive travel and with the help of advance purchase restrictions [3].
This paper is organized as follows. In Section 2, we review the literature on both the single period stochastic inventory (newsboy) problem and the related models in PARM. In Section 3, we analyze the case with decreasing prices, and examine the effect of applying single demand newsboy model in multiple demands. In Section 4, we examine the case with increasing prices and consider a simple, "booking limit" type of policy structure. Through an example, we discuss the results and managerial implications. We give some concluding remarks in Section 5.
2. Literature
The newsboy problem has attracted considerable attention since the pioneering papers of Arrow et al. [4], and Morse and Kimball [5]. Some notable extensions follow the lines of considering alternative objectives other than maximizing the expected profit, and relaxing the requirements on the demand distribution. For example, Ismail and Louderback [6] and Lau [7] have considered the objective of satisfying the maximization of the probability of achieving a given profit level. Eeckhoudt et al. [7a] have considered the case of the risk averse newsboy whilst Reyniers [8] has discussed the situation in which there is a delayed observation of sales. Gallego and Moon [9] have studied the case in which the distribution of the demand is unknown.
Our decreasing price model extends the newsboy problem from a single demand to multiple demands with different selling prices. A few papers exist in the literature that are related to ours. On the supply side, Jucker and Rosenblatt [10] have studied the newsboy problem with a single demand with quantity discounts for purchasing costs. On the demand side, Khouja [11] has considerd the newsboy problem when it contains progressive discounts to sell off excess inventory. The amount of inventory sold (demand) at each discounted price is completely determined by (a fraction of) the amount of inventory sold at the original price; hence the multiple demands are perfectly correlated. Khouja presented a model formulation to maximize the expected profit and the probability of achieving a target profit. Khouja [12] has further extended his earlier model to include multiple discounts from the supplier side. The problem setting that we study differs from that of Khouja [11] in that the demands are independent across price classes in our decreasing price model, and dependent through residual demands in our increasing price model. Our treatment of demand independency and dependency tends to make our models more applicable. Lau and Lau [13] have examined the price-dependent demand distribution and presented efficient solution procedures for finding optimal order quantity and price under different optimization objectives. Although Lau and Lau optimize both the order quantity and the price, it is a single demand model where the demand distribution solely depends on price. We assume that prices are exogenous in each market segment. This allows us, without losing much realism, to obtain analytical results which are relatively simple and which allow for easy interpretations.
Other related models include the newsboy problem with multiple products. These products may share a given space or budget constraint as is discussed by Silver and Peterson [14], Li et al. [15] have, studied a two-product newsboy problem with independent exponential demands to maximize the probability of achieving a targeted (total) profit. However, products are not substitutable so that each of the products has one demand. Our decreasing price model assumes a single product which can be used to satisfy multiple demand classes. Kouvelis and Gutierrez [16] have further considerd the newsboy problem with a single product but in two markets (locations) and introduced differential ordering (production) costs, exchange risk and transportation cost. Our model differs from that of Kouvelis and Gutierrez as we assume onetime production for multiple markets and we focus on the determination of the optimal production quantity. Our model for the increasing price case also applies when the secondary market has a higher p rice so that it may be profitable to ship the product from the primary market to the secondary market although there may be sufficient demand in the primary market (the protection policy, as we will discuss in Section 4).
For the second case with increasing prices, our model is related to the PARM (Perishable Asset Revenue Management) models, which have been formulated mostly with reference to application in airline seat inventory control. Belobaba [17] and also Weatherford and Bodily [18] have presented a taxonomy and overview of research in PARM. Most of the published work utilizes probabilistic decision models to decide the optimal booking limit for the discount demand [3], while the initial capacity, equivalent to the order quantity in the newsboy problem, is assumed to be fixed. These models also differ by their treatment of dependencies of demands and diversion where a customer purchases a seat in a different fare class than he or she is originally willing to pay (upgrade or downgrade), as described by Belobaba and Weatherford [19]. Most of the models assume independent demands for the fare classes; one example is the two-class demand model by Pfeifer [1] who considered setting an allocation of a given initial seat capa city to the low fare class (i.e. choosing a booking limit) and who also highlighted the analogy to the newsboy problem.
Two papers that consider dependent demands are Brumelle et al. [20] who treated two-class dependent demands via a bivariate normal distribution, and Belobaba and Weatherford [19] who used a probability ([[beta].sub.2]) to represent the diversion effect ([beta.sub.2] is then equal to 1 -- the upgrade percent from low fare to high fare) and numerically compared decision rules incorporating diversion. We model the demand dependency through the diversion fraction, similar to Belobaba and Weatherford; however, we also consider the selection of initial seat capacity, and we derive analytical results. Gerchak et al. [21] have studied a two-class model similar to ours, although they use a discrete time, discrete demand unit formulation (ours is aggregated demand) with backward-recursive dynamic programming. They also considerd selecting the optimal initial order quantity. Later, Lee and Hersh [22] extended the work of Gerchak et al. to multiple classes and multiple seat bookings; however, in both papers, demand depend encies, likely created by upgrading or diversion, are not considered. Our model combines all three aspects: (1) fair allocation or booking limit decisions; (2) selection of initial seat capacity; and (3) explicit consideration of demand dependency through diversion. We are then able to examine the behavior of the booking limit. Our model enables us to derive economic interpretations for the optimal order quantity using marginal analysis, and extending the marginal rule in setting protection limits to incorporate the capacity decision.
3. The model with decreasing prices
We shall label the first demand to realize by class 1; and the last demand by class n. The demand in class j, [D.sub.j], has a pdf [f.sub.j](*). We assume that [D.sub.j], j = 1, 2, ..., n, are independent. The selling price in class j is [r.sub.j], with [r.sub.1] [greater than or equal to] [r.sub.2] [greater than or equal to] ... [greater than or equal to] [r.sub.n] [greater than] 0. We also assume [r.sub.1] [greater than] c, for otherwise the optimal order quantity will be zero. With the decreasing prices, clearly the optimal allocation rule for existing stock is of the "no reservation" type; i.e., the available stock is allocated, to the fullest extent and without any reservation for future demand classes, to [D.sub.1] first; the remaining stock, if there is any, is then fully allocated to [D.sub.2]; and so on. We wish to point out that this model also applies to a situation in which all n demands are realized simultaneously before the allocation decision is made, such as in a distribution system where the demand classes are divided geographically.
We also assume that beyond the n classes, any unsold units have zero salvage value. This is not a real limitation of our model, since any positive salvage value can be modeled as a last demand class with price [r.sub.n] and a very large mean demand. Let the unit purchase or production cost be c, and the order quantity be X (the decision variable). Then the expected profit maximization problem is
[max.sub.x] [pi](X) = [[[sigma].sup.n].sub.j=1][r.sub.j]E[[Q.sub.j]] - cX,
where [Q.sub.j] is the sales units in demand class j. Next, we obtain E[[Q.sub.j]]. We have,
[Q.sub.1] = min{[D.sub.1], X},
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hence,
E[[Q.sub.1]] = [[[integral of].sup.X].sub.0] [D.sub.1][f.sub.1]([D.sub.1])d[D.sub.1] + [[[integral of].sup.[infinity]].sub.0] X[f.sub.1]([D.sub.1])d[D.sub.1].
For j [greater than or equal to] 2, let [T.sub.j] [equivalent to] [[[sigma].sup.j-1].sub.i=1] [D.sub.i], and [g.sub.j](*) be the pdf, and [G.sub.j](*) the cdf, of [T.sub.j] and [D.sub.j], as shown in Fig. 1. By integrating over the three regions, we have,
E[[Q.sub.j]] = [[[integral of].sup.X].sub.0] [[[integral of].sup.[X-D.sub.j]].sub.0] [D.sub.j][g.sub.j]([T.sub.j])d[T.sub.j][f.sub.j]([D.sub.j])d[D.sub.j]
+ [[[integral of].sup.X].sub.0] [[[integral of].sup.X].sub.[X-D.sub.j]] ([X - T.sub.j])[g.sub.j]([T.sub.j])d[T.sub.j][f.sub.j]([D.sub.j])d[D.sub.j]
+ [[[integral of].sup.X].sub.0] [[[integral of].sup.[infinity]].sub.X] (X - [T.sub.j])[g.sub.j]([D.sub.j])d[D.sub.j][g.sub.j]([T.sub.j])d[T.sub.j ],
= [[[integral of].sup.X].sub.0] [D.sub.j][G.sub.j](X - [D.sub.j])[f.sub.j]([D.sub.j])d[D.sub.j]
+ [[[integral of].sup.X].sub.0] [[[integral of].sup.X].sub.[X-D.sub.j]] (X - [T.sub.j])[g.sub.j]([T.sub.j])d[T.sub.j][f.sub.j]([D.sub.j])d[D.sub.j ]
+ [[[integral of].sup.X].sub.0] (X - [T.sub.j])[1 - [F.sub.j](X)][g.sub.j]([T.sub.j])d[T.sub.j]. (1)
In Appendix 1, we show that [pi](X) is concave so that the optimal X is given by Proposition 2.
Proposition 1. The optimal X satisfies
[r.sub.1] Pr{[D.sub.1] [greater than or equal to] X} + [[[sigma].sup.n].sub.j=2] [r.sub.j] [Pr{[[[sigma].sup.j-1].sub.i=1][D.sub.i] [less than or equal to] X}
- Pr{[[[sigma].sup.j].sub.i=1] [D.sub.j] [less than or equal to] X}] = c, (2)
or equivalently,
[[[sigma].sup.n-1].sub.j=1]([r.sub.j] - [r.sub.j+1]) Pr{[[[sigma].sup.j].sub.i=1] [D.sub.i] [greater than] X}
+ [r.sub.n] Pr{[[[sigma].sup.n].sub.i=1][D.sub.i] [greater than] X} = c. (3)
Equations (2) and (3) allow for a straightforward marginal cost and revenue interpretation. The left-hand-side of both equations is the expected marginal revenue with an extra unit of stock which might be sold in one of the n demand classes; the right-hand-side is of course the unit marginal cost. Observe that the solution to equation (3) is the classical newsboy solution, with the total demand distribution, if [r.sub.j] = r, = 1,...,n. Also, setting [D.sub.n] = [infinity] (so Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] X} = 1) guarantees a unit salvage value [r.sub.n]; when n = 2, Equation (2) reduces to the standard newsboy formula
Pr{[D.sub.1] [less than or equal to] X} = [r.sub.1] - c/[r.sub.1] - [r.sub.2],
where [r.sub.2] is now the salvage value.
The next result shows that the optimal order quantity is bounded by the two newsboy quantities with the highest and lowest prices across all market segments.
Proposition 2. The optimal order quantity satisfies
[[G.sup.-1].sub.n+1] ([r.sub.n] - c/[r.sub.n]) [less than or equal to] [X.sup.*] [less than or equal to] [[G.sup.-1].sub.n+1] ([r.sub.1] - c/[r.sub.1]), (4)
where [G.sub.n+1](*) is the cdf of [[[sigma].sup.n].sub.j=1] [D.sub.j].
Proof. From (3), we have [r.sub.n] Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] [X.sup.*]} [less than or equal to] c, hence [X.sup.*] [greater than or equal to] [[G.sup.-1].sub.n+1]([r.sub.n] - c/[r.sub.n]). Replacing Pr{[[[sigma].sup.j].sub.i=1] [D.sub.i] [greater than] X} by a larger quantity Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] X} in (3), we obtain
c [less than or equal to] [[[sigma].sup.n-1].sub.j=1]([r.sub.j] - [r.sub.j+1]) Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] [X.sup.*]}
+ [r.sub.n] Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] [X.sup.*]} = [r.sub.1] Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] [X.sup.*]},
hence [X.sup.*] [less than or equal to] [[G.sup.-1].sub.n+1]([r.sub.1] - c/[r.sub.1]).
If we define [P.sub.s] as the probability of shortage (i.e. Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] X}), then Proposition 2 implies c/[r.sub.i] [less than or equal to] [P.sub.s] [less than to equal to] c/[r.sub.n]. This compares with the standard newsboy model with [P.sub.s] = c/r.
3.1. The effect of aggregating demand classes
We now consider the effect of aggregating the demand classes in order to use the single demand newsboy model. When the multiple demand class model is not available, a first heuristic approach might be to treat the market as an aggregated demand with some averaged price. Thus, the order quantity of this approach, [X.sup.c], is given by
Pr{[[[sigma].sup.n].sub.i=1] [D.sub.i] [greater than] [X.sup.c]} = c/r,
where the average price r can be weighted by the mean demands: r = [[[sigma].sup.n].sub.i=1] [[micro].sub.i][r.sub.i]/[[[sigma].sup.n].sub.i=1] [[micro].sub.i].
A second heuristic that we consider is to treat each demand class in a separate newsboy model with price [r.sub.j], cost c and demand cdf [F.sub.j](*). A newsboy quantity is computed for each demand class; these individual newsboy quantities are then summed and used as the single order quantity X. Thus, the order quantity for this additive approach, [X.sup.s], is the sum of [X.sub.i] where [X.sub.i], i = 1,2,...,n, is given by
Pr{[D.sub.i] [greater than] [X.sub.i]} = c/[r.sub.i].
To compare the optimal solution with the above two heuristics, we consider the two-demand problem (n = 2). From the optimality condition (3), the optimal order quantity [X.sup.*] is given by the following equation:
([r.sub.1] - [r.sub.2]) Pr{[D.sub.1] [greater than] [X.sup.*]} + [r.sub.2] Pr{[D.sub.1] + [D.sub.2] [greater than] [X.sup.*]} = c. (5)
For a numerical comparison, we normalize c = 1 and [[micro].sub.1] = 1. The relative size of the second market, [[micro].sub.2] is expressed as a multiple of [[micro].sub.1]; ([[micro].sub.2]/[[micro].sub.1]) has three values 0.5, 1, and 2. The selling price in the first market [r.sub.1] has four values, 1.2, 2, 3 and 5; the selling price in the second market [r.sub.2] is expressed as a fraction of [r.sub.1], with [r.sub.2]/[r.sub.1] taking four values 0.2, 0.4, 0.6 and 0.8. We consider normally distributed demands; the coefficient of variation (cv), to be the same for [D.sub.1] and [D.sub.2], having five values 0.1, 0.2, 0.3, 0.4 and 0.5. In total, we solve 3 x 4 x 4 x 5 240 problems. The relative sub-optimality of each heuristic, computed as [(optimal profit) -- (heuristic profit)]/(optimal profit), is tabulated (Table 1). To reduce the size of the table, we only show the case with cv fixed at 0.5. Later we graphically demonstrate the impact of cv on the performance of the heuristics.
We first note that the "average price" heuristic would give a zero order quantity (and hence 100% sub-optimality) when the average price (r) is less than the cost. This happens, for example, when [r.sub.1] is greater than but close to c. The "separate newsboys" heuristic which approximates the optimal order quantity by treating each demand class with a separate newsboy order quantity is found to be more robust in a broad range of problems. We therefore focus on the separate newsboys heuristic in our subsequent discussions. Figure 2 shows the performance gap between the optimal and the separate newsboys heuristic as [r.sub.1] takes increasing values; as the ratio [r.sub.2]/[r.sub.1] increases from 0.2 to 1; and as the demand cv increases from 0.1 to 0.5, all with the [[micro].sub.1]/[[micro].sub.2] ratio fixed at 1. These results indicate that the separate newsboy heuristic performs poorly when [r.sub.1] [greater than] c [greater than] [r.sub.2] but all three take on close values. As seen from Table 1, when [r.sub.1] = 1.2, [r.sub.2]/[r.sub.1] = 0.8, c = 1, (and [[micro].sub.1] = [[micro].sub.2] = 1, cv = 0.5), the separate newsboys heuristic is 31.34% off from optimality. When [r.sub.1] is close to c, both heuristics perform poorly. These cases are not at all unrepresentative of practical situations; in these situations, the optimal order quantity given by our decreasing price model should be used over the heuristics. When the demand cv is small, the separate newsboys heuristic tends to perform well.
4. An increasing price model: two demand classes
Now we consider the case [r.sub.l] [less than or equal to] [r.sub.2] [less than or equal to] ... [less than or equal to] [r.sub.n] with [r.sub.n] [greater than] c. Again, these prices are pre-announced or fixed under exogenous market forces. As in Pfeifer [1] and many others, we will consider the simplest case with n = 2. We will also assume that the demands are positive with probability 1.
This type of demand realization with increasing prices does not allow for a trivial inventory allocation scheme across demand classes. Here, we adopt a static allocation policy of the following type: the maximum units sold at demand class 1 (low fare) is set to a pre-specified level P (the booking limit); any unsold units after the realization of class 1 demand will be made available to class 2 (high fare) demand. This corresponds to the closing of the discount fare class in airlines. Our model setting resembles that of Pfeifer [1] in that the low fare demand precedes the full fare demand and that once the low fare class is closed it is never re-opened (a common assumption in PARM). The order quantity (for both classes) is X. Under this scheme, a portion of the initial stock X is protected or reserved for the higher priced demand class, as the units available to the higher price demand will be at least X -- P. We term X -- P the protection level for the high fare class. Our problem here is to find the optima l X and P to maximize the expected profit. Our model differs from the previous research in this field in that the initial order quantity (or capacity), X, is a decision variable in our model.
A reasonable assumption in this environment is that some of the customers in the lower price class may be ready to pay the price of the next higher class (upgrading), if they are not able to buy the product at the price they have requested. This is described as diversion in its general sense by Belobaba and Weatherford [19], although the term diversion is often used to describe downgrading in fare classes. We model diversion by assuming that a fixed portion, s, of the unsatisfied lower price demand will join the higher price demand. For now, we assume 0 [less than] s [less than] 1. Thus, the proverbial newsboy sells papers at a higher price in the afternoon than in the morning; he then deliberately sets a limit to the amount of papers to sell in the morning and reserves some papers for the afternoon. Perhaps some of the customers who want to buy papers in the morning at the lower price will return in the afternoon.
Under these assumptions, the units sold at the low and high fare classes are given by
[Q.sub.1] = min{[D.sub.1],X,P},
[Q.sub.2] = min{X - [Q.sub.1], [D.sub.2] + s([D.sub.1] - [Q.sub.1])},
where s([D.sub.1] - [Q.sub.1]) is the diverted demand from class 1 (low fare) to class 2 (high fare). We assume that [D.sub.1] and [D.sub.2] are independent; however, with diversion, the effective demand (and sales) in class 2 is dependent on the class 1 realization through the policy parameter P. The problem to maximize the expected profit can now be expressed as
[max.sub.x,p] [pi](X,P) = [[[sigma].sup.2].sub.j=1] [r.sub.j]E[[Q.sub.j]] - cX.
Define F(x) [equivalent to] 1 - F(x). Obviously, X [greater than or equal to] P [greater than or equal to] 0, which leads us to rewrite [Q.sub.1] = min{[D.sub.1],P}. Therefore, for x [greater than] P,
[delta]E[[Q.sub.1]]/[delta]X = 0,
[delta]E[[Q.sub.1]]/[delta]P = [F.sub.1](P).
Our following derivations concern E[[Q.sub.2]]. We write
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The value of 22 is shown in Fig. 3.
Integrating over the disjoint regions of [Q.sub.2] in Fig. 3, we obtain
E[[Q.sub.2]] = [[[integral of].sup.P].sub.0] [[[integral of].sup.[infinity]].sub.[X-D.sub.1]] (X - [D.sub.1])[f.sub.2]([D.sub.2])d[D.sub.2][f.sub.1]([D.sub.1])d[D.sub.1 ]
+ [[[integral of].sup.P].sub.0] [[[integral of].sup.[X-D.sub.1]].sub.0] [D.sub.2][f.sub.2]([D.sub.2])d[D.sub.2][f.sub.1]([D.sub.1])d[D.sub.1]
+ [[[integral of].sup.X-P].sub.0] [[[integral of].sup.X-P+sP-[D.sub.2]/s].sub.P] ([D.sub.2] + s[D.sub.1] - sP)[f.sub.1]([D.sub.1])d[D.sub.1][f.sub.2]([D.sub.2])d[D.sub.2]
+ [[[integral of].sup.X-P].sub.0] [[[integral of].sup.[infinity]].sub.X-P+sP-[D.sub.2]/s] (X - P)[f.sub.1]([D.sub.1])d[D.sub.1][f.sub.2]([D.sub.2])d[D.sub.2]
+ [[[integral of].sup.[infinity]].sub.X-P] [[[integral of].sup.[infinity]].sub.P] (X - P)[f.sub.1]([D.sub.1])d[D.sub.1][f.sub.2]([D.sub.2])d[D.sub.2].
Taking the derivative w.r.t. X and after considerable algebraic simplification,
[delta]E[[Q.sub.2]]/[delta]X = [[[integral of].sup.P].sub.0] [F.sub.2](X - [D.sub.1])[f.sub.1]([D.sub.1])d[D.sub.1] + [F.sub.1](P)[F.sub.2](X - P) + [[[integral of].sup.X-P].sub.0] [F.sub.1](X - P + sP - p[D.sub.2]/s)[f.sub.2]([D.sub.2])d[D.sub.2] (6)
Similarly, taking the derivative w.r.t. P and simplifying,
[delta]E[[Q.sub.2]]/[delta]P = -s[F.sub.1](P) - (1 - s)[F.sub.1](P)[F.sub.2](X - P) - (1 - s) [[[integral of].sup.X-P].sub.0] [F.sub.1] (X - P + sP - [D.sub.2]/s)[f.sub.2]([D.sub.2])d[D.sub.2]. (7)
It can be shown that the second partial derivative of [pi](X, P) w.r.t. X is negative, hence [pi](X, P) is concave in X for any given P. However, [pi](X,P) is not necessarily concave in P (and hence is not jointly concave in (X, P)); this causes a slightly more complicated solution procedure. We discuss the optimal solution in three cases.
Case 1:0 [less than] P [less than] X
The first-order necessary conditions for an interiorly optimal (X, P) are
[r.sub.1] [delta]E[[Q.sub.1]]/[delta]X + [r.sub.2] [delta]E[[Q.sub.2]]/[delta]X = c,
[r.sub.1] [delta]E[[Q.sub.1]]/[delta]P + [r.sub.2] [delta]E[[Q.sub.2]]/[delta]P = 0,
which, using (6) and (7), can be written as
[r.sub.2] [[[[integral of].sup.p].sub.0] [F.sub.2](X - [D.sub.1])[f.sub.1] ([D.sub.1])d[D.sub.1] + [F.sub.1](P)[F.sub.2](X - P)
+ [[[integral of].sup.X-P].sub.0] [F.sub.1](X - P + sP - [D.sub.2]/s)[f.sub.2]([D.sub.2])d[D.sub.2]] = c, (8)
([r.sub.1] - [sr.sub.2])[F.sub.1](P) - [r.sub.2](l - s)[F.sub.1](P)[F.sub.2](X - P)
- [r.sub.2](1 - s) [[[integral of].sup.X-P].sub.0] [F.sub.1](X - P + sP - [D.sub.2]/s) [f.sub.2]([D.sub.2])d[D.sub.2] = 0, (9)
or equivalently in probability expressions, which we will use later to facilitate a marginal revenue discussion,
[r.sub.2] [Pr{[D.sub.1] + [D.sub.2] [greater than] X, [D.sub.1] [less than or equal to]P} + Pr{[D.sub.1] [greater than]P} x Pr{[D.sub.2] [greater than] X - P} + Pr{s([D.sub.1] - P) + [D.sub.2] [greater than] X - P, [D.sub.2] [less than or equal to] X - P}] = c, (10)
([r.sub.1] - [sr.sub.2]) Pr{[D.sub.1] [greater than] P} - [r.sub.2](1 - s) Pr{[D.sub.1] [greater than] P} x Pr{[D.sub.2] [greater than] X - P} - [r.sub.2](1 - s) Pr{s([D.sub.1] - P) + [D.sub.2] [greater than or equal to] X - P, [D.sub.2] [less than or equal to] X - P} = 0. (11)
For given [f.sub.1](*), [f.sub.2](*) and the parameters [r.sub.1], [r.sub.2], c and s, simultaneous Equations (8) and (9) can now be solved in a standard numerical procedure for X and P. Note that [r.sub.1] is missing from (8) and (10) (partial derivative w.r.t. X); this implies that once the booking limit P is fixed, the optimal X does not depend on the selling price [r.sub.1] This is simply because an additional order quantity (or capacity) beyond P will affect only the sales in class 2. However, the optimal X given P still depends on the demand distribution in class 1, as the amount of stock available to meet class 2 demand depends on the realization of class 1 demand.
It is easy to see that the solution X from Equations (10) and (11) will be non-negative, as the left-hand-side of Equation (10) will be equal to [r.sub.2] [greater than] c at X = 0 (for any P [greater than or equal to] 0). The next proposition can be used to determine if a solution from Equations (8) and (9) is a local maximum.
Proposition 3. If (X,P) satisfies s + (1 - s)[F.sub.2](X - P)[less than or equal to] [r.sub.1]/[r.sub.2], then [pi](X,P) is concave at (X,P).
Proof. See Appendix 2.
Case 2: P = 0
In this case, the optimal [X.sup.*] is solved from Equation (10), which reduces to
[r.sub.2] Pr{[sD.sub.1] + [D.sub.2] [greater than] [X.sup.*]} = c.
That is, [X.sup.*] is the newsboy quantity with price [r.sub.2], cost c, and demand s[D.sub.1] + [D.sub.2].
Case 3: P = X
This is the "unprotected" inventory allocation policy (all available units are open to class 1). We then have
[Q.sub.1] = min {[D.sub.1], X},
[Q.sub.2] = {min {X - [D.sub.1], [D.sub.2]}, if [D.sub.1] [less than or equal to]X, 0, if [D.sub.1] [greater than] X.
The above is identical to our previous decreasing price model described in Section 3, except here [r.sub.1] [less than] [r.sub.2]. The diversion fraction s does not affect the model in any way. The first-order necessary condition for optimal X is therefore given by (5). It is easy to see that the optimal X [greater than] 0 when [r.sub.1] [greater than] c, since when X = 0, the left-hand-side of Equation (5) is equal to [r.sub.1]. We note, however, that in this case Equation (5) is not sufficient to guarantee an optimal X, since [pi](X) may not be concave. The second order sufficient condition needs to be checked to ensure a local maximum.
Now for given problem parameters, we first consider case (1) and solve Equations (8) and (9) numerically to obtain (X,P). Proposition 3 is then used to determine if (X,P) is a local maximum. Any solution P [less than] 0 or P [greater than] X is discarded. Next we consider the cases (2) and (3) and solve for optimal X respectively. Finally the best solution from the three cases is chosen as the global optimum. In all three cases, the solution X will be non-negative if [r.sub.2] [greater than] [r.sub.1] [greater than] c.
We next give upper bounds on the optimal initial capacity X and the optimal protection level X - P. The proof of Proposition 4 is given in Appendix 3.
Proposition 4. (a) The optimal [X.sup.*] satisfies [X.sup.*] [less than or equal to] [[G.sup.-1].sub.3](1 - c/[r.sub.2]), where ([G.sub.3](*) is the cdf of [D.sub.1] + [D.sub.2].
(b) The optimal [X.sup.*] and [P.sup.*] satisfies [X.sup.*] - [P.sup.*] [less than or equal to] [[G.sup.-1].sub.s](1 - c/[r.sub.2]), where ([G.sub.s](*) is the cdf of s[D.sub.1] + [D.sub.2].
The condition under which it is optimal to close the low fare class (Case 2) deserves a closer look. We next give two conditions for the optimal P to be zero.
Lemma 1. If, for a given X, [delta][pi](X,P)/[delta]P [less than or equal to] 0 at P + 0, then [delta][pi](X, P)/[delta]P [less than or equal to] 0 for any P [greater than] 0.
Proof. See Appendix 4.
Proposition 5. For a fixed X, the corresponding optimal P = 0 if and only if [r.sub.2] (1 - s)Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] X} [less than or equal to] [r.sub.2] - [r.sub.1].
Proof. The partial derivative of [pi](X, P) w.r.t. P at P = 0 is given by [r.sub.2](1 - s)Pr{[s[D.sub.1] + [D.sub.2] [less than or equal to] X} - ([r.sub.2] - [r.sub.1]) (see (A1) in Appendix 4). By Lemma 1, if [r.sub.2] (1 - s)Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] X} [less than or equal to] [r.sub.2] - [r.sub.1], then the partial derivative of [pi](X, P) w.r.t. P is non-positive for all P [greater than] 0. Hence the optimal P = 0, then the partial derivative of [pi](X,P) w.r.t. P should be non-positive at P = 0.
Proposition 6. If [r.sub.1]/[r.sub.2][less than or equal to]s, then optimal P = 0.
Proof. The partial derivative of [pi](X,P) w.r.t. P is given by the LHS of Equation (9) or (11). If [r.sub.1]/[r.sub.2][less than or equal to] s, then [delta][pi](X,P)/[delta]P [less than] 0 for all X and P [greater than or equal to] 0. Hence [P.sup.*] = 0.
We note that for the special case s = 0, it can be shown that the last term in the left-hand-sides of Equations (8), (9), (10) and (11) will disappear. Then, (11) can be rewritten simply as
Pr{[D.sub.2] [greater than or equal to] X - P} = [r.sub.1]/[r.sub.2] (12)
It can be shown that Proposition 3 will be satisfied from Equation (12) so that any solution satisfying Equations (8) and (9) will be a local maximum (in Case 1). Once the capacity, X, is fixed, Equation (12) is known to be Littlewood's [23] rule. This is generalized for more than two classes as the Expected Marginal Seat Revenue (EMSR) model by Belobaba [3].
When s = 1, clearly an optimal solution is P = 0 and X = the newsboy quantity with price [r.sub.2], cost c and demand [D.sub.1] + [D.sub.2] (Case 2). In this case, class 1 is closed to all customers so that all demands are diverted to class 2.
Example. Consider demands [D.sub.1] [D.sub.2] that are uniformly distributed between 0 and 20. Let [r.sub.1] = 2, [r.sub.2] = 3, c = 1. For s values ranging from 0 to 1, we compute the optimal policy values in Table 2. A dash line (--) in the table indicates that the solution from the first-order conditions (8) and (9) does not satisfy 0 [less than] P [less than] X or that the solution is not a local maximum.
Next, we arbitrarily fix s = 0.3 and vary [r.sub.2]. The results are shown Table 3.
We note that, by comparing Case (1) and Case (3) (the "unprotected case"), the policy that limits sales in the lower priced class, thus allowing "reserved" units for the future, higher priced demands, can generate higher expected profit. This is true even if no low fare demand is diverted to the high fare class (s = 0).
To evaluate the value of the ability to choose the initial capacity optimally, we compare the optimal [[pi].sup.*] with the resulting maximum [pi](X,P(X)) where X is given and P(X) is optimized for the given X. We set the problem parameters as [r.sub.1] = 2, [r.sub.2] = 3, s = 0.3. Then [X.sup.*] = 22.12,
[P.sub.*] = 11.29 and [[pi].sup.*] = 21.228. Figure 4 shows the function [pi](X, P(X)) when X is given in a range from 0 to 30. The difference between a point on the curve and the peak point is the value of choosing the capacity optimally.
Finally, it would be interesting to compare the sensitivity of profit w.r.t. P when X is chosen optimally ([X.sup.*] = 22.12). In Fig. 5, we plot the function [pi]([X.sup.*],P). The profit curve here w.r.t P is flatter than the curve w.r.t. X (in Fig. 4). Comparing Figs. 4 and 5 suggests that the profit is more sensitive to the choice of X than to P; i.e., the gain in profit with a well chosen initial capacity is greater than the gain in profit with an optimized booking limit.
4.1. Discussion of results and managerial implications
The behavior of the optimal booking limit P in the increasing price model has clear implications. P = 0 corresponds to the closing of the lower fare class. The example indicates that the optimal P value is sensitive to the diversion fraction s. The s value at which the low fare class is closed for all customers can indeed be small. When the fare difference between the two classes increases, the critical s value to close the low fare class is further reduced. From Proposition 6, the critical value of s to close the low fare class is no greater than the fare ratio [r.sub.1]/[r.sub.2]. This result has an intuitive expected marginal cost interpretation: by closing the low fare class, we forego a revenue of [r.sub.1] for each customer we reject, but gain [sr.sub.2] from each customer diverted to the high fare class. The example also shows that the optimal P value decreases when the diversion fraction s increases, thus forcing more demand units into the higher fare class. Also, for a given s, the optimal P decrease s when the fare difference [r.sub.2] - [r.sub.1] increases.
The protection level X - P for the high fare class is also sensitive to the diversion fraction s and the fare difference [r.sub.2] - [r.sub.1]. The optimal protection level increases in s and in [r.sub.2] - [r.sub.1], as shown in our example. If there were a second replenishment opportunity for the high fare class, the "protection" level X - P would be equal to the newsboy quantity with a unit revenue [r.sub.2] and the effective demand s max {[D.sub.1] - P, 0} + [D.sub.2]. As formally stated in Proposition 4b, the protection level X - P never exceeds the newsboy quantity with revenue [r.sub.2] and the maximum effective Class 2 demand [sD.sub.1] + [D.sub.2].
So far, our model has focused on the static decision problem where permanent allocation of the seat inventory is determined for the two fare classes. In many applications, P can be recomputed based on continuously updated (remaining) capacity X and demand. In these cases, Proposition 5 gives the optimal closing limit for the low fare class, when [D.sub.1] and [D.sub.2] are interpreted as the remaining demands during the two sales intervals (and X as the remaining capacity). The value of P determines if the next arriving customer should be granted a low fare. Proposition 5 implies that there exists an [X.sub.0] such that [P.sup.*] = 0 if (the remaining) X [less than or equal to] [X.sub.0], and [P.sup.*] [greater than] 0 if X [greater than] [X.sub.0]. Again, Proposition 5 allows for a simple Expected Marginal Revenue (EMR) interpretation: rewriting the condition in Proposition 5, we have
[r.sub.2][P.sub.r]{s[D.sub.1] + [D.sub.2] [greater than] X} + [r.sub.2]s [P.sub.r] {s[D.sub.1] + [D.sub.2] [less than or equal to] X} [greater than or equal to [r.sub.1]. (13)
The right-hand-side of the above is the marginal revenue to increase P from 0 to 1 unit (which will be sold since we assume [D.sub.1] [greater than] 0 with probability 1). The left-hand-side of the above is the expected marginal revenue lost when the total effective high-fare demand is greater than X (in which case we would lose a revenue of [r.sub.2]) or when the total effective high-fare demand is less than X (in which case we would lose only the diversion revenue [r.sub.2]s]). We also note that (13) is a more general rule than a rule shown as Equation (3) in Belobaba and Weatherford [19].
5. Concluding remarks
In both the cases of decreasing and increasing prices, our results show that the optimal X never exceeds the single newsboy quantity with the best possible unit revenue and the total demand (Proposition 2 and Proposition 4a). Moreover, all the optimality conditions allow for an interpretation in expected marginal cost and revenue when the revenues are properly weighted by the probabilities. These interpretations are consistent with those found in the PARM literature. Our numerical examples show that approximating the optimal order quantity by applying a single newsboy model on an aggregated demand (the "average price" heuristic) or summing separate newsboy order quantities for each demand class (the "separate newsboys" heuristic) may lead to poor performances.
We wish to point out that the alternative objective of maximizing the probability of achieving a targeted profit T [7,11,15] results in [X.sup.*] [greater than or equal to] T/([r.sub.1] - c) in the decreasing price model with [X.sup.*] T/ ([r.sub.l] - c) if [r.sub.1] [greater than] c [greater than] [r.sub.2] [greater than] ... [greater than] [r.sub.n].
Finally, a possible direction for future research might be to extend the two-class model with increasing prices to an n-class model. We expect the same managerial implications from the two-class model; however, the mathematical expressions will be much more complex, since the sales in class j now depends recursively on the sales in all lower numbered classes.
Biographies
Alper Sen is currently a doctoral student in Operations Management in the Marshall School of Business at the University of Southern California. He holds B.S. and an MS. degrees in Industrial Engineering from Bilkent University, Ankara, Turkey. His current research interests are revenue management models under competition and with forecasting updates. His previous work in machine scheduling has been published in EJOR and OR Letters.
Alex X. Zhang is an Assistant Professor of Operations Management in the Marshall School of Business at the University of Southern California. His current research interests are multi-item inventory systems modeling and models in revenue management. His previous work has been published in IIE Transactions and other journals. He holds a Ph.D. in Industrial Engineering from Stanford University.
References
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Optimal and heuristic order quantities c = 1;
[[micro].sub.1] = 1; coefficient of
variation cv = 0.5
[[micro].sub.2]/ [r.sub.1] [r.sub.2]/[r.sub.1] Optimal
[[micro].sub.1] [X.sup.*] [[pi].sup.*]
0.5 1.2 0.2 0.5724 0.0642
0.4 0.6409 0.0755
0.6 0.7262 0.0900
0.8 0.8323 0.1095
2 0.2 1.0853 0.6717
0.4 1.1833 0.7464
0.6 1.2904 0.8359
0.8 1.3988 0.9414
3 0.2 1.3212 1.6082
0.4 1.4358 1.7735
0.6 1.5493 1.9631
0.8 1.6523 2.1743
5 0.2 1.5529 3.6349
0.4 1.6836 3.9920
0.6 1.7986 4.3846
0.8 1.8935 4.8028
1 1.2 0.2 0.5480 0.0653
0.4 0.6745 0.0788
0.6 0.8051 0.0988
0.8 1.0114 0.1316
2 0.2 1.1242 0.6892
0.4 1.3002 0.7986
0.6 1.5432 0.9558
0.8 1.8001 1.1744
3 0.2 1.3936 1.6614
0.4 1.6515 1.9323
0.6 1.9374 2.3020
0.8 2.1536 2.7494
5 0.2 1.6927 3.7906
0.4 2.0480 4.4459
0.6 2.3103 5.2485
0.8 2.4778 6.1222
2 1.2 0.2 0.5871 0.0648
0.4 0.6859 0.0785
0.6 0.8426 0.1003
0.8 1.1673 0.1436
2 0.2 1.1434 0.6953
0.4 1.3916 0.8251
0.6 1.9881 1.0689
0.8 2.6441 1.5301
3 0.2 1.4442 1.6865
0.4 2.0388 2.0693
0.6 2.8440 2.8177
0.8 3.2353 3.7657
5 0.2 1.8784 3.8967
0.4 3.0001 5.1292
0.6 3.4816 6.8017
0.8 3.7541 8.6001
[[micro].sub.2]/ Average price heuristic
[[micro].sub.1] [X.sub.1] [[pi].sub.1] [[delta].sub.1] (%)
0.5 0.0000 0.0000 100.00
0.0000 0.0000 100.00
0.5112 0.0791 12.19
0.8058 0.1093 0.20
1.2357 0.6553 2.44
1.3219 0.7334 1.74
1.3915 0.8293 0.80
1.4499 0.9397 0.18
1.5638 1.5553 3.29
1.6176 1.7450 1.60
1.6640 1.9518 0.58
1.7047 2.1719 0.11
1.8380 3.5508 2.31
1.8771 3.9533 0.97
1.9116 4.3708 0.32
1.9425 4.8000 0.06
1 0.0000 0.0000 100.00
0.0000 0.0000 100.00
0.0000 0.0000 100.00
0.9775 0.1313 0.21
1.3159 0.6655 3.45
1.5998 0.7358 5.61
1.7747 0.9324 2.44
1.9012 1.1695 0.41
1.9012 1.4939 10.08
2.0422 1.8506 4.23
2.1488 2.2766 1.10
2.2340 2.7451 0.16
2.3046 3.5582 6.13
2.4002 4.3679 1.75
2.4769 5.2274 0.40
2.5407 6.1187 0.06
2 0.0000 0.0000 100.00
0.0000 0.0000 100.00
0.0000 0.0000 100.00
1.0224 0.1400 2.55
0.0000 0.0000 100.00
1.9184 0.7377 10.60
2.4714 1.0258 4.04
2.7813 1.5248 0.35
2.3673 1.3452 20.24
2.8438 1.9374 6.50
3.1277 2.7919 0.91
3.3280 3.7621 0.10
3.2013 3.4674 11.02
3.4816 5.0477 1.59
3.6760 6.7837 0.26
3.8232 8.5974 0.03
[[micro].sub.2]/ Separate newsboys heuristic
[[micro].sub.1] [X.sub.2] [[pi].sub.2]
0.5 0.5163 0.0633
0.5163 0.0714
0.5163 0.0795
0.5163 0.0876
1.0000 0.6663
1.0000 0.7230
1.2581 0.8352
1.4203 0.9411
1.2154 1.5968
1.4735 1.7722
1.6804 1.9483
1.7680 2.1624
1.4208 3.6120
1.9208 3.9351
2.0285 4.3305
2.0894 4.7613
1 0.5163 0.0640
0.5163 0.0728
0.5163 0.0816
0.5163 0.0903
1.0000 0.6786
1.0000 0.7478
1.5163 0.9554
1.8407 1.1736
1.2154 1.6334
1.7317 1.9286
2.1455 2.2774
2.3206 2.7310
1.4208 3.7152
2.4208 4.3590
2.6362 5.1717
2.7581 6.0581
2 0.5163 0.0633
0.5163 0.0715
0.5163 0.0796
0.5163 0.0878
1.0000 0.6818
1.0000 0.7540
2.0326 1.0686
2.6814 1.5298
1.2154 1.6460
2.2479 2.0598
3.0757 2.8005
3.4258 3.7507
1.4208 3.7573
3.4208 5.0667
3.8515 6.7385
4.0953 8.5387
[[micro].sub.2]/
[[micro].sub.1] [[delta].sub.2] (%)
0.5 1.39
5.36
11.66
19.98
0.81
3.13
0.09
0.03
0.71
0.07
0.75
0.55
0.63
1.42
1.23
0.86
1 1.96
7.70
17.48
31.34
1.54
6.37
0.04
0.06
1.68
0.19
1.07
0.67
1.99
1.95
1.46
1.05
2 2.21
8.92
20.61
38.88
1.94
8.61
0.03
0.02
2.40
0.46
0.61
0.40
3.58
1.22
0.93
0.71
Optimal X and P as a function of s, Parameters:
[r.sub.1] = 2, [r.sub.2] = 3, c = 1
s Case 1 (0 [less than] P Case 2 (P = 0)
[less than] X
X P [pi](X, P) X [pi](X, P)
0 23.33 16.67 20.927 13.33 13.333
0.1 23.12 15.48 20.968 14.33 15.309
0.2 22.75 13.80 21.051 15.33 17.233
0.3 22.12 11.29 21.228 16.33 19.109
0.4 20.92 7.26 21.640 17.33 20.934
0.5 - - - 18.33 22.709
0.6 - - - 19.33 24.435
0.7 - - - 20.34 26.106
0.8 - - - 21.39 27.765
0.9 - - - 22.51 29.312
1.0 - - - 23.67 30.899
s Case 3 (P = X) Optimal
X [pi](X,P) [X.sup.*] [P.sup.*] [[pi].sup.*]
0 23.67 20.896 23.33 16.67 20.927
0.1 23.67 20.896 23.12 15.48 20.968
0.2 23.67 20.896 22.75 13.80 21.051
0.3 23.67 20.896 22.12 11.29 21.228
0.4 23.67 20.896 20.92 7.26 21.640
0.5 23.67 20.896 18.33 0 22.709
0.6 23.67 20.896 19.33 0 24.435
0.7 23.67 20.896 20.34 0 26.106
0.8 23.67 20.896 21.39 0 27.765
0.9 23.67 20.896 22.51 0 29.312
1.0 23.67 20.896 23.67 0 30.899
Optimal X and P as a function of [r.sub.2],
Parameters: [r.sub.1] = 2, c = 1, s = 0.3
[r.sub.2] Case 1(0 [less than] Case 2 (P = 0)
P [less than] X)
X P [pi](X,P) X [pi](X,P)
2.0 20.00 20.00 13.333 13.00 7.850
2.2 20.85 17.95 14.719 13.91 9.981
2.5 21.67 15.25 16.983 15.00 13.313
3.0 22.12 11.29 21.228 16.33 19.109
3.5 21.61 7.49 26.021 17.29 25.095
4.0 20.18 3.41 31.379 18.00 31.200
5.0 - - - 19.00 43.625
6.0 - - - 19.67 56.217
8.0 - - - 20.52 81.674
[r.sub.2] Case 3 (P = X) Optimal
X [pi](X,P) [X.sup.*] [P.sup.*] [[pi].sup.*]
2.0 20.00 13.333 20.00 20.00 13.333
2.2 20.93 14.717 20.85 17.95 14.719
2.5 22.11 16.947 21.67 15.25 16.983
3.0 23.67 20.896 22.12 11.29 21.228
3.5 24.88 25.095 21.61 7.49 26.021
4.0 25.86 29.429 20.18 3.41 31.379
5.0 27.35 38.413 19.00 0 43.625
6.0 28.45 47.699 19.67 0 56.217
8.0 30.00 66.625 20.52 0 81.674
Appendix 1
Proof of Proposition 1
Taking derivative w.r.t. X in (1) and simplifying,
dE[[Q.sub.j]]/dX = [[[integral of].sup.X].sub.0] [D.sub.j][g.sub.j](X - [D.sub.j]) [f.sub.j]([D.sub.j])d[D.sub.j] + 0
+ [[[integral of].sup.X].sub.0] [[[[integral of].sup.X].sub.[X-D.sub.j]] [g.sub.j]([T.sub.j])d[T.sub.j]
+ 0 - [D.sub.j][g.sub.j](X - [D.sub.j])][f.sub.j]([D.sub.j])d[D.sub.j]
+ [[[integral of].sup.X].sub.0] (X - [T.sub.j])[g.sub.j]([T.sub.j])d[T.sub.j][f.sub.j](X)
+ [[[integral of].sup.X].sub.0] [1 - [F.sub.j](X)][g.sub.j]([T.sub.j])d[T.sub.j]
- [[[integral of].sup.X].sub.0] (X - [T.sub.j])[f.sub.j](X)[g.sub.j]([T.sub.j])d[T.sub.j] + 0,
= [[[integral of].sup.X].sub.0] [[[integral of].sup.X].sub.[X-D.sub.j]] [g.sub.j]([T.sub.j])d[T.sub.j][f.sub.j]([D.sub.j])d[D.sub.j]
+ [[[integral of].sup.X].sub.0] [1 - [F.sub.j](X)][g.sub.j]([T.sub.j])d[T.sub.j],
= Pr{[T.sub.j] + [D.sub.j] [greater than or equal to] X, [T.sub.j] [less than or equal to] X, [D.sub.j] [less than or equal to] X}
+ Pr{[T.sub.j] [less than or equal to] X, [D.sub.j] [greater than or equal to] X},
= Pr{[T.sub.j] [less than or equal to] X} - Pr{[T.sub.j] + [D.sub.j] [less than or equal to] X},
= Pr{[T.sub.j] + [D.sub.j] [greater than] X} - Pr{[T.sub.j] [greater than] X}.
Together with
dE[[Q.sub.1]]/dX = 1 - [F.sub.1](X) = Pr{[D.sub.1] [greater than] X},
We obtain
d[pi](X)/dX = [r.sub.1] Pr{[D.sub.1] [greater than or equal to] X} + [[[sigma].sup.n].sub.j=2] [r.sub.j] [Pr{[[[sigma].sup.j-1].sub.i=1] [D.sub.i] [less than or equal to] X}
- Pr{[[[sigma].sup.j].sub.i=1] [D.sub.i] [less than or equal to] X}] - c,
= [[[sigma].sup.n-1].sub.j=1] ([r.sub.j] - [r.sub.j+1]) Pr{[[[sigma].sup.j].sub.i=1] [D.sub.i] [greater than] X}
+ [r.sub.n] Pr{[[[sigma].sup.n].sub.i=1]) [D.sub.i] [greater than] X} - c.
Taking the second derivative, we obtain
[d.sup.2][pi](X)/d[X.sup.2] = - [[[sigma].sup.n-1].sub.j=1] ([r.sub.j] - [r.sub.j+1])[g.sub.j](X) - [r.sub.n][g.sub.n](X),
which is always non-positive since [r.sub.j] [greater than or equal to] [r.sub.j+1] and [r.sub.n] [greater than or equal to] 0. This proves that expected profit is concave in X so that the solution to Equation (3) is the global maximum.
Appendix 2
Proof of Proposition 3
We need to show
[[delta].sup.2][pi](X, P)/[delta][X.sup.2] [less than or equal to] 0, [[delta].sup.2][pi](X, P)/[delta][P.sup.2] [less than or equal to] 0, and
[[[[delta].sup.2][pi](X, P)/[delta]X[delta]P].sup.2] [less than or equal to] [[delta].sup.2][pi](X, P)/[delta][X.sup.2]. [[delta].sup.2][pi](X, P)/[delta][P.sup.2].
From (6), we have
[[delta].sup.2]E[[Q.sup.2]]/[delta][X.sup.2] = [[[integral of].sup.P].sub.0] [-[f.sub.2](X - [D.sub.1])][f.sub.1]([D.sub.1])d[D.sub.1]
+ [F.sub.1] (P) [-[f.sub.2](X - P)]
+ [[[integral of].sup.X-P].sub.0] [-[f.sub.1] (X - P + sP - [D.sub.2]/s)]
x (1/s)[f.sub.2]([D.sub.2]d[D.sub.2] + [F.sub.1](P)[f.sub.2](X - P)
= - [[[integral of].sup.P].sub.0] [f.sub.2](X - [D.sub.1][f.sub.1]([D.sub.1])d[D.sub.1] - (1/s)
x [[[integral of].sup.X-P].sub.0] [f.sub.1](X - P + sP - [D.sub.2])/s[f.sub.2]([D.sub.2])d[D.sub.2],
hence
[delta][pi](X, P)/[delta][X.sup.2] = [r.sub.2] [[delta].sup.2]E[[Q.sub.2]]/[delta][X.sup.2] [less than or equal to] 0.
From (7), we have
[[delta].sup.2]E[[Q.sub.2]]/[delta][P.sub.2] = s[f.sub.1] (P) - (1 - s) [-[f.sub.1] (P)][F.sub.2](X - P) - (1 - s) x [F.sub.1](P)[-1[f.sub.2](X - P)](-1) (1 - s) x [[[integral of].sup.X-P].sub.0] [ - [f.sub.1] (X - P + sP - [D.sub.2]/s)] (-1 + s/s) x [f.sub.2]([D.sub.2])d[D.sub.2] - (1 - s)[F.sub.1](P)[f.sub.2](X - P) - (-1), = s[f.sub.1](P) + (1 - s)[f.sub.1](P)[F.sub.2](X - P) - [(1 - s).sup.2]/s x [[[integral of].sup.X-P].sub.0] [f.sub.1] (X - P + sP - [D.sub.2]/s) [f.sub.2]([D.sub.2])d[D.sub.2].
Since [[delta].sup.2]E[[Q.sub.1]]/[[delta]P.sup.2] = - [f.sub.1](P), we have
[delta][pi](X, P)/[delta][P.sup.2]
= [r.sub.1] [[delta].sup.2]E[[Q.sub.1]]/[delta][P.sup.2] + [r.sub.2] [[delta].sup.2]E[[Q.sub.2]]/[delta][P.sup.2], = -[r.sub.1] - [sr.sub.2] - (1 - s)[r.sub.2][F.sub.2](X - P)][f.sub.1] (P) -[r.sub.2] [(1 - S).sup.2]/s [[[integral of].sup.X-P].sub.0] [f.sub.1] (X - P + sP - [D.sub.2]/s)[f.sub.2]([D.sub.2])d[D.sub.2].
When s + (1 - s)[F.sub.2](X - P) [less than or equal to] [r.sub.1]/[r.sub.1], the first term or the above is negative; hence the whole expression is negative.
Next, consider [[delta].sup.2][pi](X, P)/[delta]X[delta]P. From (6), we have
[[delta].sup.2]E[[Q.sub.2]]/[delta]X[delta]P
= [F.sub.2](X - P)[f.sub.1](P) + [-[f.sub.1](P)][F.sub.2](X - P) + [F.sub.1](P)[-[f.sub.2](X - P)](-1) + [[[integral of].sup.X-P].sub.0] [-[f.sub.1](X - P + sP - [D.sub.2]/s)] (-1 + s/s)[f.sub.2]([D.sub.2])d[D.sub.2] + [F.sub.1](P)[f.sub.2](X - P)(-1), = (1 - s/s) [[[integral of].sup.X-P].sub.0] [f.sub.1] (X - P + sP - [D.sub.2]/s)[f.sub.2]([D.sub.2])d[D.sub.2].
Let
A [equivalent to] [[[integral of].sup.X-P].sub.0] [f.sub.1] (X - P + sP - [D.sub.2]/s)[f.sub.2]([D.sub.2])d[D.sub.2].
Then we have
[[delta].sup.2][pi](X, P)/[delta]X[delta]P = [r.sub.1] [[delta].sup.2]E[[Q.sup.1]]/[delta]X[delta]P + [r.sub.2] [[delta].sup.2]E[[Q.sup.2]]/[delta]X[delta]P = [r.sub.2] (1 - s/s) A.
It remains to prove
[[[[delta].sup.2][pi](X, P)/[delta]X[delta]P].sup.2] [less than or equal to] [[delta].sup.2][pi](X, P)/[[delta]X.sup.2] x [[delta].sup.2][pi](X, P)/[[delta]P.sup.2].
Notice that the integral term A appears in both [[delta].sup.2][pi](X, P)/[[delta]X.sup.2] and [[delta].sup.2][pi](X, P)/[[delta]P.sup.2] (with a negative sign); all other terms in both expressions are negative. Hence
[[delta].sup.2][pi](X, P)/[[delta]X.sup.2] x [[delta].sup.2][pi](X, P)/[[delta]P.sup.2] [greater then or equal to] [r.sub.2] (1/s)A x [r.sub.2] [(1 - s).sup.2]/s A, = [[r.sup.2].sub.2] [(1 - s).sup.2]/[s.sup.2] [A.sup.2], = [[[[delta].sup.2][pi](X, P)/[delta]X[delta]P].sup.2].
This proves that the function [pi](X, P) is jointly concave in X and P.
X and P.
Appendix 3
Proof of Proposition 4
For Part (a), it is necessary and sufficient to prove [r.sub.2]Pr{[D.sub.1] + [D.sub.2] [X.sup.*]} [greater than or equal to] c. We consider three cases. In Case 1 (0 [less than] P [less than] X), the optimal [X.sup.*] satisfies either Equation (8) or (10). As shown in Fig. A1, a key observation is that the three probability terms on the left-hand-side of Equation (10) correspond to regions I, II and III; whereas the probability term Pr{[D.sub.1] + [D.sub.2] [greater than] X} corresponds to a region which includes regions I, II and III as sub-regions. Therefore,
Pr{[D.sub.1] + [D.sub.2] [greater than] X, [D.sub.1] [less than or equal to] P} + Pr {[D.sub.1] [greater than] X - P} + Pr{s[D.sub.1] - P) + [D.sub.2] [greater than] X - P, [D.sub.2] [less than or equal to] X - P} [less than or equal to] Pr{[D.sub.1] + [D.sub.2] [greater than] X}.
From (10), we have [r.sub.2] Pr{[D.sub.1] + [D.sub.2] [greater than] [X.sup.*]} [greater than or equal to] c.
In Case 2 (P = 0), the optimal [X.sup.*] satisfies [r.sub.2] Pr{s[D.sub.1] + [D.sub.2] [greater than] [X.sup.*]} = c. Since 0 [less than or equal to] s [less than or equal to] 1, [r.sub.2] Pr{[D.sub.1] + [D.sub.2] [greater than] [X.sup.*]} [greater than or equal to] [r.sub.2] Pr{s[D.sub.1] + [D.sub.2] [greater than] [X.sup.*]} = c.
In Case 3 (P = X), the optimal [X.sup.*] satisfies Equation (5), from which we write
[r.sub.2] Pr{[D.sub.1] + [D.sub.2] [greater than] [X.sup.*]} = c + ([r.sub.2] - [r.sub.1]) Pr{[D.sub.1] [greater than] [X.sub.*]} c.
For Part (b), we need to prove [r.sub.2] Pr{s[D.sub.1] + [D.sub.2] [greater than] [X.sup.*] - [P.sup.*]} [greater than or equal to] c. In Case 1 (0 [less than] P [less than] X), we observe from Figure Al that regions I, II and III are sub-regions of the region represented by the probability term Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] [X - P}. In Case 2 and Case 3, the proposition holds trivially.
Appendix 4
Proof of Lemma 1
Again referring to Fig. Al, we can rewrite [delta][pi](X,P)/[delta]P (the left-hand-side of Equation (11)) as
[delta][pi](X,P)/[delta]P = [r.sub.2](l - s)
x Pr{s([D.sub.1] - P) + [D.sub.2] [less than or equal to] X - P, [D.sub.1] [greater than] P} - ([r.sub.2] - [r.sub.1]) Pr{[D.sub.1] [greater than] P}. (Al)
If [delta][pi](X,P)/[delta]P [less than or equal to] 0 at P = 0, then from the above (setting P = 0), we have
[r.sub.2](1 - s) Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] X} [less than or equal to] [r.sub.2] - [r.sub.1]. (A2)
It can be easily shown that for two independent random variables A and B, if Pr{B [greater than] b} [greater than] 0, then Pr{A + B [less than or equal to] a \ B [greater than] b} [less than or equal to] Pr{A + B [less than or equal to] a} for all a. Hence Pr{s([D.sub.1] - P) + [D.sub.2] [less than or equal to] X - P, [D.sub.1] [less than P} = Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] [X - P + sP [D.sub.1] [greater than] P} Pr{[D.sub.1] [greater than] P} [less than or equal to] Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] X - P + sP} Pr{[D.sub.1] [greater than] P}.
Substituting inequality (A2) into (Al), we have, for any P [greater than] 0,
[delta][pi](X,P)/[delta]P [less than or equal to] [r.sub.2](l - s)Pr{s([D.sub.1] - P) + [D.sub.2] [less than or equal to] X - P, [D.sub.1] [greater than] P} - [r.sub.2](l - s) Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] X} Pr{[D.sub.1] [greater than] P} [less than or equal to] [r.sub.2](l - s) Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] X - P + sP}Pr{[D.sub.1] [greater than] P} - [r.sub.2](l - s)Pr{s[D.sub.1] + [D.sub.2] [less than or equal to] X}Pr{[D.sub.1] [greater than]P} [less than or equal to] 0.
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