Revenue management approach to stochastic capacity allocation problem

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European Journal of Operational Research 192 (2009) 442–459 www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Revenue management approach to stochastic capacity allocation problem Mohammad Modarres *, Mehdi Sharifyazdi Sharif University of Technology, Tehran, Iran Received 26 December 2006; accepted 11 September 2007 Available online 13 October 2007

Abstract To formulate stochastic capacity allocation problems in a manufacturing system, the concept and techniques of revenue management is applied in this research. It is assumed the production capacity is stochastic and hence its exact size cannot be forecasted in advance, at the time of planning. There are two classes of ‘‘frequent’’ and ‘‘occasional’’ customers demanding this capacity. The price rate as well as the penalty for order cancellation caused by overbooking is different for each class. The model is developed mathematically and we propose an analytical solution method. The properties of the optimal solution as well as the behavior of objective function are also analyzed. The objective function is not concave, in general. However, we prove it is a unimodal function and by taking advantage of this property, the optimal solution is determined.  2007 Elsevier B.V. All rights reserved. Keywords: Revenue management; Capacity allocation; Stochastic capacity; Optimization

1. Introduction Revenue management has considerable potential for planning of manufacturing operations in order to maximize the total profit, although it has not been applied in manufacturing systems as much as in service industry. Any revenue management problem contains the following common characteristics: • Capacity is perishable and limited. It cannot be enhanced easily in short term. • Demand is stochastic. • There are different customer classes. The available perishable asset can be sold at different prices, through different booking classes (usually at different periods). In the models developed in literature it is a common assumption that in each period the amount of available capacity is known and deterministic, although perishable. This assumption can be justified for airlines, hotels or service industries but not necessarily good enough for manufacturing system. In many real world manufacturing systems, the capacity is stochastic in nature which makes production planning more complicated, especially for make-to-order (MTO) or assemble-toorder (ATO) manufacturing systems. In a make-to-order (or assemble-to-order) manufacturing system, planning is on the basis of orders received in advance. Acceptance or rejection of an order depends on the availability of capacity as well as on its price rate. For each order, a

*

Corresponding author. E-mail addresses: [email protected] (M. Modarres), [email protected] (M. Sharifyazdi).

0377-2217/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.09.044

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delivery date is set at the time of arrival if it is accepted. However, if an order is accepted but not carried on, the system has to pay some penalty. This may happen when the required capacity for the accepted orders of one period exceeds the capacity of that period due to the stochastic nature of the capacity or by overestimating production capacity. On the other hand, rejecting an order means the loss of income. Therefore, the acceptance or rejection of an order results in a trade-off between income loss and penalty cost. Then, the objective is to develop an optimal policy of acceptance/rejection in order to earn the maximum expected total income (including the penalty cost). The complexity arises from the fact that the exact capacity is not known at the time of order arrivals, due to its stochastic nature. In reality, one cause of capacity uncertainty arises from machine failures, stops or breakdowns during the operation. As a result, since the maintenance duration is random, the system has stochastic working hours or actually a stochastic capacity of production. Fluctuation in availability of human force is another reason the exact capacity of some manufacturing systems cannot be forecasted, especially when there exists a high reliance on human experts. We assume there are two classes of customers, regular (or frequent) and occasional (or less frequent) customers. The first class customers have priority and their price rate is less than that of the customers of the second class. Since the price rate and penalty cost for each class is different, the acceptance/rejection policy for each class also differs appropriately. In fact, the optimal policy determines the maximum capacity which can be assigned to each customer class. To formulate the model as well as to obtain an optimal solution, we apply the concept of revenue management and modify some of its techniques to fit our problem. The paper is organized as follows. In the next section, the related literature is reviewed briefly and also a concise comparison between our model with the previous ones is presented. We define the problem in more detail in Section 3. A model is developed in Section 4 for capacity allocation problem with two customer classes and stochastic production capacity. The unimodality property of the objective function is proved in Section 5. An approach for determining the optimal solution is also introduced in that section. Furthermore, the capability of the proposed model to fit in the fixed-capacity case is also investigated. In Section 6, the sensitivity of optimal solution and objective function with respect to different parameters of the model is analyzed. In Section 7 the results of the previous section is studied through numerical examples. In the last section, the paper is summarized and some suggestions for future works are given. 2. Literature review Revenue management (also called yield management), which originated from airline industry is referred to a collection of methods, techniques and concepts aimed to assign some finite amount of a perishable asset to several classes of customers. Littlewood (1972) can be considered as the founder of revenue management. He presented an analytical model to assign seats to two price class of passengers in a flight. Mc Gill and Van Ryzin (1999) categorized the major areas of research in the field of revenue management as seat inventory control, overbooking, pricing and demand forecasting. A review of OR techniques in airline seat inventory control (both static and dynamic) carried out by Pak and Piersma (2002). There is a vast literature regarding revenue management, although to our knowledge there is no model similar to ours. The most familiar and oldest application of revenue management is in airline industry where a fixed capacity of seats must be sold (booked) before each flight departure. However, it also has been effectively applied to other areas such as car rental, broadcasting, cruise ships, internet service providers, railways, non-profit sector, lodging and hotels, restaurants, health care, tourism and holiday retail shopping (see Mc Gill and Van Ryzin, 1999; Harris and Pinder, 1995). Regarding production capacity as a perishable asset, many researchers applied revenue management techniques in production planning and capacity allocation. However, in the models which are available, the production capacity is assumed to be fixed. Not many studies regarding stochastic demand can be found in literature. In reality, due to many reasons such as breakdowns, maintenance operations, work force unreliability, manufacturing capacity usually has a stochastic nature. Thus, in this paper we propose a two-class revenue management (seat inventory control) model with probabilistic capacity. We review the application of review management in manufacturing, briefly. Many practitioners agree that revenue management is more applicable for environments with short-run fixed capacity, Smith et al. (1992), Cross (1988) and Harris and Pinder (1995). Stidham (1985) reviewed literature addressing customer admission policies in single-class make-to-order queues. Carr and Duenyas (2000) modeled decisions about both accept/reject and production control in a two-class M/M/ 1 production system. Carr and Lovejoy (2000) formulated a fixed-capacity problem to maximize profit in a structure like newsvendor problem by choosing market segments to be served. Ha (1997a,b, 2000) and Vericourt et al. (2002) studied the stock-rationing problem in a single-product make-to-stock system while all classes of customers request the same product, but backordering costs are different. Duenyas and Hopp (1995) proposed an admission control model in which it is the customer who makes the accept/reject decision regarding the quoted lead time. However, in most models this decision is made by the manufacturer. This system is modeled as a single-server queue. Duenyas (1995) extended the previous model

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by relaxing ‘‘single-class’’ assumption. Chatterjee et al. (2002) modeled a manufacturer who updates offered lead times on the basis of status of operations. In Plambeck (2004) queue length and lead time distributions were approximated while the vendor was modeled as an exponential single-server queue. Regarding the subject of manufacturing capacity control in competitive markets which focuses on trade-off between lead time and holding inventory, one can find some papers in literature. Li (1992) studied an oligopoly n-firm market and showed that competition can raise tendency of a firm to produce in order to hold inventory. Li and Lee (1994) showed in a duopoly market, high processing rate can benefit the firm with a price premium. In Lederer and Li (1997) an n-firm oligopoly was modeled where production rates and costs of the firms were different. Armory and Haviv (2003) studied a system with two make-to-order producers serving two classes of customers with different waiting costs, and evaluated trade-off between price and delay. Dai et al. (2005) studied pricing strategies for multiple competing firms, using concepts of revenue management and game theory, while demand and price have a linear relation. Feng and Xiao (2006) proposed a model to make pricing and capacity allocation decision at the same time for perishable assets in a case that a supplier sells the same product to different micro-markets at various prices. There are also some models relating price, demand and promised lead time. Dobson and Yano (2002) developed an integer programming model to determine the price and production system at the same time by assuming deterministic constant rate of demand. Charnsirisakskul et al. (2004) investigated the relative benefits of due date and partial orders flexibility in ‘‘order selection problem’’ by applying a numerical approach and a mixed integer programming model. Ray and Jewkes (2004) formulated a model in which demand depends on price and lead time. Boyaci and Ray (2003) used a similar method for a firm that can offer two changeable products which are manufactured in distinct facilities. Keskinocak et al. (2001) presented a method to make quoted lead times completely reliable while maximizing revenue in both real-time and delayed systems. Kapuscinski and Tayur (2002) guaranteed lead times for two classes of customers in a make-to-order system by reserving capacity. Akkan (1997) stated that keeping different amounts of capacity reserved may lead to rejecting upcoming orders. Gupta and Wang (2004) proposed a model which specifies how much capacity should a manufacturer allocate to the production of contractual and transactional items and which transactional orders must be accepted, in order to maximize its long-run expected profit to maximize total revenue. There exits a considerable motivation to employ revenue management techniques and concepts for capacity rationing and pricing in make/assemble-to-order production systems, due to many common characteristics between order-driven manufacturing systems and classic revenue management problems in service industries. These similarities include perishability, limited capacity, high capacity change cost, segmentable demand, advance sales, stochastic demand and historical sales data and forecasting capabilities, see Harris and Pinder (1995). The problem of rationing limited inventory (capacity) between competing customer classes was mentioned by Haynsworth and Price (1989) and also by Ha (1997b). Harris and Pinder (1995) proposed two models for pricing and capacity reallocation in a two-class assemble-to-order environment using a revenue management approach. Sridharan (1998) and Ray and Jewkes (2004) used the idea of perishable asset revenue management (PARM) in make-to-order manufacturing environments. Bertrand and Sridharan (2001) modeled subcontracting policies (when demand overtakes capacity and all of the orders are accepted) as another class of the perishable asset (production capacity) in machine-tool industry. Balakrishnan et al. (1996) proposed a single-period heuristic capacity apportionment procedure for two different product classes in fashion apparel industry when demand is stochastic. A multiperiod extension of this model is presented by Sridharan and Balakrishnan (1996) while assuming tardiness in delivery of orders is permitted. Barut and Sridharan (2004) extended the former model for a short-term multi-product situation. Barut and Sridharan (2005) also generalized dynamic capacity apportionment procedure (DCAP) relaxing a couple of the assumptions. Patterson et al. (1997) also investigated rationing policy. Finally, we should mention a class of models, called DCAP in literature which may have some similarity with ours, see Barut and Sridharan (2004, 2005), Balakrishnan et al. (1996), Sridharan and Balakrishnan (1996), Sridharan (1998) and Bertrand and Sridharan (2001). They assumed production capacity is deterministic and fixed and the decision to make is regarding acceptation/rejection of orders through a dynamic process. However, what makes our model distinguished from theirs is that the capacity is stochastic and the decision maker has to partition it before the arrival of orders statically. Furthermore, this model finds the optimal solution analytically, while the latter models mostly apply heuristics approaches. Our model also have some similarity with two-class static revenue management models, like Smith et al. (1992), Cross (1988), Belobaba (1989) and Harris and Pinder (1995). Although the main difference of this model in comparison with its ancestors is stochastic capacity. Furthermore, the concepts of ‘‘Nesting’’ and ‘‘Protection Level’’ are not meaningful in this model, because due to the random nature of capacity, not any part of it can be ‘‘protected’’ for a special class of customers and there are only booking limits set for classes. 3. Capacity allocation problem In this section, we define the problem with more detail. In subsequent sections, this problem is formulated mathematically and then our approach to obtain the optimal solution is presented.

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3.1. Problem definition Consider a make-to-order (or assemble-to-order) manufacturing system with stochastic capacity. As mentioned before, the system has two types of customers, regular (or frequent) and occasional (or less frequent) customers. In case of limited capacity, the first class customers have priority over the second class customers as far as the acceptance of orders is concerned. Furthermore, they take advantage of some kind of discount and their price rate is usually less than that of the customers of the second class. Although the system is capable of producing different products or services, the size of each order is measured in terms of its capacity utilization, regardless of the type of product. It is also assumed each order is processed uninterrupted. Therefore, orders are processed one by one. Orders arrive in advance and may include different types of products. Demand (or orders) in each class is a non-negative random variable with a known continuous distribution function and independent of the demand of the other class. Each order is either accepted or rejected at the time of arrival, according to the adopted policy of the system. However, if an order is accepted but not delivered on time, the system has to pay some penalty. Therefore, it is important to determine how much order (in terms of capacity) can be accepted from each class to maximize the expected total income (including the penalty cost).The price rate depends on the class of customer. Similarly, the penalty of not delivering an order at the agreed delivery date differs, according to the customer class. The penalty rate of order cancellation for frequent customers is higher than that of the second class, because they are more reliable as well as the existence of long-term relationship. The model presented in this paper is applicable for some cases introduced by Harris and Pinder (1995), as potential applications of revenue management in production environments. They exampled an industrial transformer repairing facility, which has two groups of customers. The first group consists of customers who have long-term contracts and send their transformers to the facility through a prescheduled preventive maintenance plan. The second group consists of customers, who have emergent need of repair, because of accidental breakdowns. Normally, the first group takes advantage of lower fees, because they usually have long-term contracts and more monetary relationships with the repair facility. Furthermore, the second group pays more for the same service, because they usually have fewer choices and less time in urgency cases. In addition, the penalty to cancel the orders of regular customers (first group) is generally higher, because the company does not wish to lose these reliable customers and their future contracts. There are some other cases mentioned in Harris and Pinder (1995), all have two groups of regular (frequent) and occasional (emergent) customers with properties similar to the latter case. For example, a sport apparel producer has constant customers like sport clubs and sporting goods shops, as well as short-term customers as in special occasions, festivals and tournaments. Other cases which can be exampled are ceramic gift manufacturers, single-use dinnerware suppliers and flower packing and distributing companies. Orders are indexed by their period of due date. Without loss of generality, we can assume all orders with the same period of due date are processed within the same period. Then, the total required capacity for all orders of each period can not exceed the available capacity of that period. In other words, the problem is formulated in a static one-period model. However, due to the stochastic nature of capacity the exact capacity size cannot be forecasted at the time of planning. The decision maker’s problem is how to allocate the production capacity among the different booking classes. In other words, the optimal policy must determine the booking limit for each customer, while the capacity is not known at the time of order arrivals. A booking limit is the maximum number of daily capacity units, which can be assigned to a specific class (to the orders of that class arrived for processing in a specific time interval). In the previous two-class revenue management models in literature, a protection level for more desirable classes against a booking limit for the other classes is set. A protection level is a fraction of capacity kept only for orders of a special class. In this model, we do no set any protection level for a class, because the total capacity is not known at the time of planning and thus, protection level is not applicable in this case. In this model nesting and recapturing is not allowed. Nesting assumption implies that any unit of capacity protected for a class is also protected for its higher profitable classes. In a two-class case, it means there is no protection level for the lower profitable class and any order for the higher profitable class is accepted if nesting assumption is applied. By recapturing it is meant a low-value booking request is turned into a higher value booking class when the low-value booking class is not available. 3.2. Priority rule In each period if the outcome of capacity is less than the required planned capacity, then the customers who order frequently (first class) have priority over the other class. In other words, when the capacity is not sufficient for all orders of one period, then the orders of the second class are denied, first. The orders of the first class customers are canceled only if all orders of the second class have been cancelled.

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3.3. Boundary assumption We assume the following boundary properties holds. These assumptions are very common and rational and not restrictive in real world problems. limxi !1 fi ðxi Þ ¼ 0;

i ¼ 1; 2;

limc!1 fc ðcÞ ¼ 0;

ð1Þ ð2Þ

where x1 ; x2 and c represent demand of classes 1 and 2 and the stochastic capacity, respectively and f1 ðx1 Þ; f2 ðx2 Þ and fc(c) are the probability density functions of x1 ; x2 and c, respectively. 3.4. Perishable asset and nesting in capacity allocation One major difference between airline and manufacturing revenue management roots in the nature of perishable asset. In classical revenue management, the only perishable item which is booked by passengers is airplane seats. However, in a manufacturing system the perishable asset is not its products which customers buy or order, since they can be stored and are not perishable by nature. What can be considered as perishable asset in a manufacturing system is its capacity. If this capacity is not used during a period, then it will be lost at the end of that period and can not be utilized later. Therefore, although the system can produce different products, we consider its capacity as the only output of this system. Furthermore, in production management problems, normally capacity is not fixed and varies dynamically due to some factors like preventive maintenance program and sudden breakdowns. However in airline RM problems, usually seat capacity is fixed. Other difference between RM and capacity allocation is the concept of nesting. It is not desirable to accept more shortterm orders with higher price but less reliable in long-term. This causes losing long-term and reliable customers, although they pay less for a unit of capacity. Otherwise it results in allocating less capacity for frequent customer and consequently loosing them. It is worth mentioning that in airline revenue management overbooking, researches focus more on no-shows. However, in capacity allocation it should be focused more on order cancellations and capacity changes, which has different nature. 4. The model 4.1. Notation Input data: r1 ; r2 : Price rate of orders (per one unit of capacity) for classes 1 and 2, respectively; p1 ; p2 : Penalty rate of cancelling orders for classes 1 and 2, respectively; p1 ; p2 : Lost profit rate of cancelling orders in classes 1 and 2, respectively; Note: (1) Logically r1 < r2 and we assume that p1 > p2. However, the objective function and the method is not directly influenced by this assumption. (2) p1 ; p2 are not independent parameters and it is assumed p1 > p2. They are derived from r1 ; r2 ; p1 ; p2 , as follows: pi ¼ ri þ pi ;

i ¼ 1; 2:

Random variables: x1 ; x2 : demand (in term of capacity size) for classes 1 and 2, respectively, (x1 ; x2 P 0Þ; c:stochastic capacity of production. (c P 0); f1 ðx1 Þ; f2 ðx2 Þ; fc ðcÞ : probability density functions of x1, x2 and c, respectively; F 1 ðx1 Þ; F 2 ðx2 Þ; F c ðcÞ : cumulative distribution function for x1, x2 and c, respectively; F 1 ð:Þ ¼ 1  F ð:Þ:

ð3Þ

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Decision variables: b1 ; b2 : a1 ; a2 : d 1; d 2: R:

booking limits (maximum capacity size) allocated for orders of classes 1 and 2, respectively; the size of capacity accepted (registered) for orders of classes 1 and 2, respectively; the size of capacity denied due to shortage for orders of classes 1 and 2, respectively; total revenue gained from the profit of order registration minus penalty of order cancellation.

Note: b1 ; b2 are independent variables. However, the others decision variables are derived from b1 ; b2 and the outcome of the random variables, i.e. Objective function: ai ¼ minfxi ; bi g;

i ¼ 1; 2:

ð4Þ

The size of capacity for denied orders of class 1 customers depends on the outcome of the capacity, only. It is independent of the orders of the second class, as explained by ‘‘priority rule’’.  a1  c; if 0 6 c 6 a1 ; d1 ¼ ð5Þ 0; if a1 6 c: However, the size of capacity for denied orders of class 2 customers depends on the outcome of the capacity as well as the orders of the first class, as explained by ‘‘priority rule’’. * a1  c; d2 ¼

if

0 6 c 6 a1 ;

a1 þ a2  c;

if

a 1 6 c 6 a1 þ a2 ;

0;

if

a1 þ a2 6 c:

ð6Þ

On the other hand, the objective function is R ¼ r1  a1 þ r2  a2  p1  d 1  p2  d 2 : Therefore, EðRÞ ¼ r1 Eða1 Þ þ r2 Eða2 Þ  p1 Eðd 1 Þ  p2 Eðd 2 Þ:

ð7Þ

The expected size of capacity for accepted orders, Eðai Þ; i ¼ 1; 2; is obtained from (4), as follows: Z bi Z 1 Z bi Eðai Þ ¼ xi  f i ðxi Þ  dxi þ bi  f i ðxi Þ  dxi ¼ ðxi  bi Þ  f i ðxi Þ  dxi þ bi : 0

bi

ð8Þ

0

The expected size of capacity for denied orders, Eðd i Þ; i ¼ 1; 2; are obtained from (5) and (6), as follows: Z b1 Z x1 Z 1 Z b1 Eðd 1 Þ ¼ ðx1  cÞ  f c ðcÞ  f 1 ðx1 Þ  dc  dx1 þ ðb1  cÞ  f c ðcÞ  f 1 ðx1 Þ  dc  dx1 0

0

b1

ð9Þ

0

and similarly, Eðd 2 Þ ¼

Z

Z

b1 0

þ þ þ þ þ

0

Z Z Z Z Z

b1 0

Z

b2

0

Z

1

b2

b1 0

Z

b2

0

1 b1 b1 0

Z Z

b2

0

b1

1

Z 0

b1

Z Z

Z

x1

b2  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1 þ

0

Z

1 b1

b2

Z

Z

b1

x2  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1 0

1

Z

b2

b1

b2  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1

0

x1 þx2

ðx1 þ x2  cÞ  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1

Z

b1 þx2

ðb1 þ x2  cÞ  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1

b1

1

Z

x2  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1 þ

x1

b2

1

x1

1

b2

Z

x1 þb2

ðx1 þ b2  cÞ  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1

x1

Z

b1 þb2

ðb1 þ b2  cÞ  fc ðcÞ  f2 ðx2 Þ  f1 ðx1 Þ  dc  dx2  dx1 :

ð10Þ

b1

Although the objective function in our model is to maximize the expected total revenue (minus penalty), it is possible to consider other objectives such as maximization of capacity utilization, maximization of average revenue per customer, minimization of lost customer good will, minimization of opportunity cost (see Mc Gill and Van Ryzin (1999) or Pak and Piersma (2002)).

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5. Optimality condition In general, the objective function E(R) is neither convex nor concave. This fact is verified through a counter example in Appendix B. However, it will be proved that E(R) is a unimodal function. Thus, this function has a maximum, which can be on the boundary of the feasible solutions region. As a result, the optimal solution is obtained by setting the partial derivatives of E(R) with respect to b1 and b2 equal to zero, provided this set of equations has a solution. Otherwise, the optimal solution is zero. 5.1. Unimodality property of objective function To prove E(R) is a unimodal function, first we present the following lemma. Lemma 1.

oEðRÞ ob1

is a product of a non-increasing function of b1 and a non-negative non-increasing function of b1.

Proof. As calculated in Appendix A,

oEðRÞ ob1

w1 ðb1 Þ ¼ r1  F c ðb1 Þðp1  p2 Þ  p2

Z

¼ F 1 ðb1 Þ  w1 ðb1 Þ, where b2

 F c ðb1 þ x2 Þ  f 2 ðx2 Þ  dx2 þ F c ðb1 þ b2 Þ  F 2 ðb2 Þ :

ð11Þ

0

F 1 ðb1 Þ is a non-negative non-increasing function of b1 and w1(b1) is a non-increasing function of b1. The reason is that in the following relation, Fc(b1), Fc(b1 + b2) and Fc(b1 + x) are non-decreasing functions of b1 (for fixed value of b2) and By our assumption p1 > p2 P 0. The same reasoning is applied for oEðRÞ ob2 ¼ F 2 ðb2 Þ  w2 ðb2 Þ, where Z b1  w2 ðb2 Þ ¼ r2  p2 F c ðb2 þ x1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ : ð12Þ 0

Now, unimodality property of E(R) will be proved through three mutually exclusive and comprehensive cases.

h

Theorem 1. For a fixed value of b2, let w1(b1) P 0. Then, no order of the first class customers is rejected. In other words, E(R) is unimodal and the maximum value for E(R) is attained when b1 tends to infinity. Fig. 1a shows the typical curve of this function with respect to b1, when b2 is fixed. Proof. Since both w1(b1) and F 1 ðb1 Þ are non-negative, the partial derivative is also non-negative and E(R) is non-decreasing. According to Appendix C, oEðRÞ tends to 0 at infinity. So, E(R) increases till it reaches (tends) to a fixed value (peak) ob1 when b1 tends to infinity. The structure of oEðRÞ in this case is illustrated in Fig. 1b. h oðb1 Þ Note: By our assumption, the first case never occurs. The reason is that limb1 !1 w1 ðb1 Þ ¼ r1  p1 ¼ p1 < 0 and limb2 !1 w2 ðb2 Þ ¼ r2  p2 ¼ p2 < 0. Theorem 2. For a fixed value of b2, let w1(b1) > 0 at b1 = 0 and w1(b1) < 0 for large values of b1. Then, E(R) is a unimodal function with respect to b1 and attains its maximum at the point where oEðRÞ oðb1 Þ ¼ 0. Fig. 2a shows the typical curve of this function with respect to b1. Proof. Since at 0 the partial derivative is positive, E(R) has an ascending start. It increases till it reaches to a peak when w1(b1) equals zero and consequently oEðRÞ ¼ 0. Then, it starts to fall down because F 1 ðb1 Þ is positive and w1(b1) is negative. oðb1 Þ Finally, it tends to a fixed value because F 1 ðb1 Þ tends to zero and as a result, the derivative tends to zero. As mentioned before, this is discussed in Appendix C. Clearly, in this case the optimal value of b1 is found by setting oEðRÞ equal to zero. oðb1 Þ Fig. 2b shows the typical structure of oEðRÞ in this case. h oðb1 Þ

b

a

∂E( R ) ∂ (b1)

E (R)

ψ 1 (b1)

F 1 (b1)

b1 Fig. 1. (a) First case for E(R) curve. (b) First case for partial derivative of

b1 oEðRÞ ob1

with a fixed value of b2.

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b

a E (R )

F 1 (b1 )

b1

ψ 1 (b1)

b1 Fig. 2. (a) Second case for E(R) curve. (b) Second case for

oEðRÞ oðb1 Þ

∂E( R ) ∂ (b1)

with a fixed value of b2.

Theorem 3. For a fixed value of b2, let w1(b1) 6 0. Then E(R) is a unimodal function with respect to b1 and attains its maximum at b1 = 0. Fig. 3a shows a typical curve of this function with respect to b1. Proof. Since at 0 the partial derivative is non-positive then E(R) has a descending start. It tends to a fixed value for large values of b1, because F 1 ðb1 Þ tends to zero (see Appendix C) and makes the partial derivative tend to zero, too. Clearly, in this case, b1 = 0 is the optimal value. For better illustration, the nature of oEðRÞ is shown in Fig. 3b. h oðb1 Þ Note: Theorems 1–3 also hold for

oEðRÞ oðb2 Þ

with respect to b2.

5.2. Optimal solution Let ðb1 ; b2 Þ be the optimal values of booking limit for classes 1, and 2 and also let (b1, b2) be the possible solution of the set of simultaneous equations of w1 ðb1 ; b2 Þ ¼ 0, w2 ðb2 ; b1 Þ ¼ 0. Then, by regarding the above cases and also by considering the fact that E(R) is a unimodal function, ðb1 ; b2 Þ can be obtained, by considering the following situations. Situation 1. If there are positive values for both b1 and b2, then ðb1 ; b2 Þ ¼ ðb1 ; b2 Þ. 1 ; 0Þ, while b 1 is the positive root of the Situation 2. If b1 > 0 but a positive value cannot be found for b2, thenðb1 ; b2 Þ ¼ ðb equation w1 ðb1 ; b2 Þ ¼ 0jb2 ¼0 , and ðb1 ; b2 Þ ¼ ð0; 0Þ when the mentioned equation has no positive root. Similarly, 2 Þ, if the equation w ðb2 ; b1 Þ ¼ 0j If b2 > 0 but a positive solution does not exist for b1, then ðb1 ; b2 Þ ¼ ð0; b 2 b1 ¼0 2 , and ðb ; b Þ ¼ ð0; 0Þ otherwise. has a positive root like b 1 2 Situation 3. If no positive value can be gained either for b1 or forb2, then ðb1 ; b2 Þ ¼ ð0; 0Þ: ^1 and b be the smallest root of F 1 ðb1 Þ ¼ 0 and the positive root of w ðb1 ; b2 Þj  ¼ 0, respectively. If Note: Let b 1 1 b2 ¼b2 ^1 P 0, then by Theorem 2, b can be all values of b1 P b ^1 . In this case, b ^1 is the maximum possible b1 P b demand of fre1 ^1 , then b ¼ b . Similar result also holds for the second class of customers. quent customers. However, if 0 6 b1 < b 1 1 Further studies about optimal solution and its properties need specification of probability distributions of demand and capacity. Example 1. A function of E(R) is depicted in Fig. 4 in terms of b1 and b2 to illustrate its structure. In this example, c and both x1 and x2 have truncated normal distributions from the following original distributions: c  N ½12; 2;

x1  N ½8; 1;

x2  N ½7; 3;

where N[l, r] represents a normal distribution with a mean of l and a standard deviation of r.

b

a E (R )

F 1 (b1)

b1

ψ 1 (b1)

b1 Fig. 3. (a) Third case for E(R) curve. (b) Third case for

oEðRÞ oðb1 Þ

∂E( R ) ∂ (b1)

with a fixed value of b2.

M. Modarres, M. Sharifyazdi / European Journal of Operational Research 192 (2009) 442–459

180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

18

6 21

12

b2

15

6

9

0

18 12 3

E (R )

Expected Revenue

450

b1

0

Fig. 4. An example of E(R) function.

It is assumed that these random variables take only non-negative values, or truncated from zero. In this example, r1 ¼ 14; r2 ¼ 20; p1 ¼ 30 and p2 = 25. This chart is drawn through a simulation method. To do that, a random sample of 1000 items is generated for random variables (x1 ; x2 and c), using Microsoft Excel Random Number Generator. Then, the corresponding revenue R ¼ r1  a1 þ r2  a2  p1  d 1  p2  d 2 is calculated for each (b1, b2) 2 B as well as for each random sample generated of ðx1 ; x2 ; cÞ, and also by considering relations (4)–(6). (B is the Cartesian product of B1 and B2, where B1 ¼ B2 ¼ f0; 1; 2; . . . ; 23g.) Then, the resulting points of (b1, b2, E(R)) are interpolated by using Microsoft Excel. In this case, capacity is constrained for demand on average, because mean capacity is less than sum of mean demands of classes 1 and 2. oEðRÞ Here, for each b2 P 0, we have oEðRÞ oðb1 Þ jb1 ¼0 > 0 and similarly for each b1 P 0, oðb2 Þ jb2 ¼0 > 0. So, E(R) treats like the pattern shown in Theorem 2. Hence, the optimal solution lies within ðb1 ; b2 Þ (Situation 1). 5.3. Special case, deterministic capacity To check the result of our model, consider a special case in which the capacity is deterministic and equal to ^c > 0. In other words, c has a one-point probability distribution.Then, w1(b1) and w2(b2) are as follows: 8 if b1 6 ^c  b2 ; > < r1 > 0; w1 ðb1 Þ ¼ r1  p2  F 2 ð^c  b1 Þ; if ^c  b2 < b1 < ^c; ð13Þ > : r1  p1 < 0; if b1 P ^c and w2 ðb2 Þ ¼

8 > < > :

r2 > 0;

if

r2  p2  F 1 ð^c  b2 Þ; if r2  p2 < 0; if

b2 6 ^c  b1 ; ^c  b1 < b2 < ^c; b2 P ^c:

ð14Þ

Since both w1(b1) and w2(b2) are non-decreasing functions, then for a given b2, a root of w1(b1) = 0 lies within ½^c  b2 ; ^c. Similarly, for a given b1, a root of w2(b2) = 0 lies within ½^c  b1 ; ^c. Thus, the optimal solution of ðb1 ; b2 Þ is obtained by solving the set of simultaneous equations, w1 ðb1 ; b2 Þ ¼ 0 and w2 ðb2 ; b1 Þ ¼ 0, as follows: r1 r2 F 2 ð^c  b1 Þ ¼ ; F 1 ð^c  b2 Þ ¼ : ð15Þ p2 p2 By our assumptions, 0< pr12 , pr22 < 1, then the optimal solution can be found appropriately. Note: If the total capacity has to be completely divided between booking limits of classes 1 and 2, then (15) is as follows, by considering c = b1 + b2. r1 r2 F 2 ðb2 Þ ¼ ; F 1 ðb1 Þ ¼ : ð16Þ p2 p2

M. Modarres, M. Sharifyazdi / European Journal of Operational Research 192 (2009) 442–459

451

Consequently, r1 F 2 ðb2 Þ F 2 ð^c  b1 Þ ¼ : ¼ r2 F 1 ðb1 Þ F 1 ðb1 Þ

ð17Þ

This is exactly like the results of EMSRa model (Belobaba, 1987) for two classes. 6. Sensitivity analysis In this section, the sensitivity of the objective function as well as the optimal solution is studied with respect to expense parameters (r1 ; r2 ; p1 and p2) or with respect to the parameters of probability distributions of demand (x1 and x2) and capacity (c). From (3) and (7), EðRÞ ¼

2 X ½ri  ðEðai Þ  Eðd i ÞÞ  pi  Eðd i Þ:

ð18Þ

i¼1

Obviously, E(R) is an increasing function of r1 or r2 and also a decreasing function of p1 or p2, by considering the assumption of E(ai) P E(di) "i. However, we show that the optimal solution is sensitive to pr12 and pr22 , rather than to single parameters. From (11), for any non-negative value of b2: Z b2  F c ðx2 Þ:f2 ðx2 Þ  dx2 þ F c ðb2 Þ  F 2 ðb2 Þ : ð19Þ w1 ð0Þ ¼ r1  p2 0

Rb If pr12 is less than h2 ðb2 Þ ¼ 0 2 F c ðx2 Þ  f2 ðx2 Þ  dx2 þ F c ðb2 Þ  F 2 ðb2 Þ, then the best value for b1 is zero by Theorem 3. Conversely, if pr12 P h2 ðb2 Þ, then for any given non-negative b2, the optimal value of b1 is the positive root of w1(b1) = 0. Note that h2(b2) depends on the parameters of probability distributions of capacity and the demand of second class customers, but not on other parameters. From (11), w1 ðb1 Þ ¼ r1  F c ðb1 Þ  p1  F c ðb1 Þ Z b2  Prfb1 6 c 6 b1 þ x2 g  f 2 ðx2 Þ  dx2 þ Prfb1 6 c 6 b1 þ b2 g  F 2 ðb2 Þ :  ðr2 þ p2 Þ  0

ð20Þ

hR i b2 Since F c ðb1 Þ, Fc(b1) and Prfb 6 c 6 b þ x g  f ðx Þ  dx þ Prfb 6 c 6 b þ b g  F ðb Þ are non-negative 1 1 2 2 2 1 1 2 2 2 2 0 r1 phrases and w1(b1) is a non-increasing function, when p2 P h2 ðb2 Þ, as r1 grows or any of p1, p2 or r2 decrease, the optimum value of b1 (the value at which w1(b1) = 0) increases in a close neighborhood (to be more exact, we should say ‘‘does not We cannot extend this statement form ‘‘a close neighborhood’’ to the whole domain of b1, because hdecrease’’). i R b2 Prfb1 6 c 6 b1 þ x2 g  f 2 ðx2 Þ  dx2 þ Prfb1 6 c 6 b1 þ b2 g  F 2 ðb2 Þ is generally neither increasing nor decreasing. 0 In fact, if E(c) increases to E(c) + d while the variance as well as the shape of density function of c remains unchanged, then the optimal value of b1 rises exactly by d. 1Þ In this case, if pr1 ¼ FF c ðb for a b1 at which Pr{c 6 b1 + b2} = 0, then w1(b1) = 0 for the same b1 and this point returns the 1 c ðb1 Þ optimum value of b1. Similarly, for any b1, the sensitivity of b2 with respect to all parameters can be examined, by considering the properties of w2(b2). From (12) for a non-negative value of b1: Z b1  w2 ð0Þ ¼ r2  p2 F c ðx1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 Þ  F 1 ðb1 Þ : ð21Þ hR

0

i Thus, if pr22 6 F c ðx1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 Þ  F 1 ðb1 Þ , then the optimal value of b2 is 0. Fromp2 = r2 + p2, it is implied that: 2 3 p2 1 i5  1 ¼ h1 ðb1 Þ: P 4hR b 1 r2 F c ðx1 Þ  f ðx1 Þ  dx1 þ F c ðb1 Þ  F 1 ðb1 Þ b1 0

0

So, if

p2 r2

1

6 h1 ðb1 Þ, the optimum b2 lies at the point where w2(b2) = 0.

452

M. Modarres, M. Sharifyazdi / European Journal of Operational Research 192 (2009) 442–459

Regarding p2 = r2 + p2 and from (12) following conclusion can be carried out:   Z b1 w2 ðb2 Þ ¼ r2 1  F c ðb2 þ x1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ  p2

0

Z

b1

 F c ðb2 þ x1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ ;

ð22Þ

0

hR b1

i Rb F ðb þ x Þ  f ðx Þ  dx þ F ðb þ b Þ  F ðb Þ always lays between 0 and 1, because 0 1 F c ðb2 þ x1 Þ  f 1 ðx1 Þ  c 2 1 1 1 c 1 2 1 1 1 0 hR b dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ 6 F c ðb1 þ b2 Þ 6 1. So, in the case that pr22 6 h1 ðb1 Þ, considering (12), since 0 1 F c ðb2 þ x1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ is a non-negative non-decreasing function of b2, as r2 is increased or p2 is reduced,

the optimal value of b2 grows up. By the same reasoning as before, while pr22 6 h1 ðb1 Þ, when probability density function (PDF) of c moves forward or backward by d, optimal value of b2 shifts in the same direction by the same amount (d). To analyze sensitivity of optimal b2 with respect to PDF of x1, we see that: Z b1 Z b1 F c ðb2 þ x1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ ¼ ½F c ðb2 þ b1 Þ  Prfb2 þ x1 6 c 0

0

6 b1 þ b2 g  f 1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ Z b1 ¼ F c ðb1 þ b2 Þ  Prfb2 þ x1 6 c 0

6 b1 þ b2 g  f 1 ðx1 Þ  dx1 :

ð23Þ

Rb when PDF of x1 is shifted ahead, 0 1 Prfb2 þ x1i 6 c 6 b1 þ b2 g  f 1 ðx1 Þ  dx1 decreases (does not increase) and so, hHere, R b1 F c ðb2 þ x1 Þ  f 1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ grows. Hence, the point at which w2(b2) = 0 (optimal b2), is shifted 0 backward (does not move forward). It can be summarized as follows. In optimal solution, a positive booking limit is set for the first class (frequent customers) if the ratio of its price rate to expense paid back to a second class customer (price rate plus penalty in the case of cancellation), is high enough. As long as price rate for frequent customers grows up or price rate of occasional customers or penalty rate for the customers of either first or second class reduces, the optimal booking limit of regular customers tends to rise and does not decline. Extension of capacity results in higher booking limit of frequent customers. If proportion of price rate to penalty rate for occasional customers is too low, then no capacity is assigned to them. Growth of capacity has an increasing effect on booking limit of occasional customers, while rise of demand of frequent customers has a decreasing influence on it. More detailed sensitivity analysis is possible, only if the PDFs of demand and capacity are known. The results of some numerical examples of sensitivity analysis for the case where PDFs of both demand and capacity, are assumed to be uniformly distributed, are available in Section 7. 7. Numerical sensitivity analysis To check the results of sensitivity analysis in Section 6, we investigate effects of change in the various parameters by studying a basic problem, using numerical methods. To define this basic problem, let: c  U ½8; 14;

x1  U ½5; 9;

x2  U ½6; 10:

While U[a, b] represents a continuous uniform distribution between a and b. Also let: r1 ¼ 14;

r2 ¼ 20;

p1 ¼ 30;

p2 ¼ 25:

Such as in other real cases, in the above basic problem, mean capacity is less than the expected total demand. Fig. 5 demonstrates the expected revenue function, by applying the same approach as in Section 5.2. The optimal solu tion is approximated at b1 ¼ 4; b2 P 9; EðRÞ ffi 176:96. Then two other cases are analyzed. In the first one, mean capacity is equal to mean total demand (c  U ½12; 18Þ and in the other one, it is higher than mean demand (c  U ½16; 22Þ. Other parameters are assumed to remain unchanged. Resulting E(R) functions of the first and the second case are shown in Figs. 6 and 7 respectively. For the first case,   b1 ¼ 12; b2 P 9; EðRÞ ffi 226:73 and for the second one b1 P 14; b2 P 9; EðRÞ ffi 248:76.

24

20

16

12

8

453

170-180 160-170 150-160 140-150 130-140 120-130 110-120 100-110 90-100 80-90 70-80 60-70 50-60 40-50 30-40 20-30 10-20 0-10

b2

4

24 0

18

b1

21

12

15

6

9

180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

0 3

Expected Revenue

M. Modarres, M. Sharifyazdi / European Journal of Operational Research 192 (2009) 442–459

Fig. 5. E(R) function in the basic instance problem of numerical sensitivity analysis.

225-230 220-225

210-215 205-210 200-205 195-200 190-195 185-190 180-185 175-180 170-175 165-170 160-165 155-160

S25

S17

S13

S9

S5

25 S1

19

22

13

16

7

10

b1

S21

150-155

1 4

Expected Revenue

215-220 230 225 220 215 210 205 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

145-150 140-145 135-140

b2

130-135 125-130 120-125

Fig. 6. E(R) function in the case where mean capacity is equal to mean demand.

240-250 230-240 210-220 200-210 190-200 180-190 170-180 160-170 150-160 140-150 130-140 120-130 110-120 100-110

b2

S25

S21

S17

S13

S9

S5

25 S1

19

22

13

b1

16

7

10

1

90-100

4

Expected Revenue

220-230 250 240 230 220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

80-90 70-80 60-70 50-60 40-50 30-40

Fig. 7. E(R) function in the case where mean capacity is more than mean demand.

The growth of capacity makes the optimal value of booking limit of frequent customers and also the maximum value of expected revenue to be enhanced. However, it does not influence the optimal booking limit of occasional customers. Furthermore, second class booking limit has no upper bound when capacity is not tight for demand of occasional customers.

454

M. Modarres, M. Sharifyazdi / European Journal of Operational Research 192 (2009) 442–459 E(R)* 10

b* 1 b* 2

8

205

E(R)*

6

185

4 165

2

145

0 9

10 11 12 13 14 15 16 17 18 19 20 r1

9 10 11 12 13 14 15 16 17 18 19 20 r1

Fig. 8. Sensitivity of optimal solution (left) and optimal expected revenue (right) with respect tor1.

Similarly, the booking limit of frequent customers has no upper bound, if the average remaining capacity (after subtraction of occasional demand from available capacity) is not limiting for demand of the first class. To study the effects of changes in price rate of frequent customers (r1) on optimal solution and optimal revenue, we have specified variation range of r1, so that r1 6 r2 and p1 P p2. Therefore, it is deemed that r1 takes the integer values between 9 and 20. For each r1 in the mentioned range, a random sample of 100 items is generated for all of the random variables (x1, x2 and c) using Microsoft Excel Random Number Generator, regarding their PDFs. Then for each r1 and each ðb1 ; b2 Þ 2 B, when B is the Cartesian product of B1 and B2, while B1 ¼ B2 ¼ f0; 1; 2; . . . ; 23g, and each ðx1 ; x2 ; cÞ within the random sample generated before, the corresponding revenue (R) is calculated. Afterwards, sample mean of R is computed for each r1 and each ðb1 ; b2 Þ 2 B as estimation for E(R). Then, maximum E(R) and the optimal (b1, b2) are specified for each r1. After approximation of optimal solution for each r1, a second-degree-polynomial interpolation curve is drawn among optimal values of b1 and b2 to make it easier to study the pattern of changes in optimal solution with respect to changes in r1. Results are shown in Fig. 8. It is observed that when price rate of frequent customers grows up, booking limit of them increases, while the booking limit of the other customers (occasional) reduces by the same amount. As much as first class price rate increases, increase rate of its booking limit and expected revenue will go higher. Similar method is used to analyze sensitivity of b1 , b2 and E(R)* with respect to the other expense parameters. Only integer values are examined for these parameters (r2, p1 and p2). The range of values for each parameter is determined such that the assumptions made about price rates and penalties in Section 4.1 are regarded. So, r2, p1 and p2 are varied in the intervals [14, 25], [11, 30] and [0, 10], respectively. Results are depicted through Figs. 9–11. The same conclusions as for r1 can be made for r2 by replacing r2 instead of r1 in the conclusion paragraph after Fig. 8, except that as much as r2 rises, increase rate of expected revenue will reduce. Here, variations of penalty rate of frequent customers in the specified range, has no sizeable influence on either booking limits or optimal expected revenue. This is because first class’s penalty rate takes only such values that keep cancellation repayment rate of frequent customers higher than that of occasional customers (p1 P p2). So, the model tries to find the optimal solution such that avoid paying back repayments to first class customers rather than second class ones. Hence, at ðb1 ; b2 Þ usually there is no cancellation for frequent customers and variation of their penalty rate inside the acceptable range do not influence booking limits and expected revenue.

E(R)*

10 8

b*1 b*2

6 4 2 0

190 170 150 130

14 15 16 17 18 19 20 21 22 23 24 25 r2

E(R)*

14 15 16 17 18 19 20 21 22 23 24 25 r2

Fig. 9. Sensitivity of optimal solution (left) and optimal expected revenue (right) with respect to r2.

E(R)*

8 170.6

6

170.4

4

b*1 b*2

2 0

170.2

E(R)*

170 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

p1

p1

Fig. 10. Sensitivity of optimal solution (left) and optimal expected revenue (right) with respect to p1.

M. Modarres, M. Sharifyazdi / European Journal of Operational Research 192 (2009) 442–459

455

E(R)*

10 8 6 4 2 0

180 175 170 165 160

b*1 b*2 0

1

2

3

4

5 p2

6

7

8

9

10

E(R)*

0

1

2

3

4

5 p2

6

7

8

9

10

Fig. 11. Sensitivity of optimal solution (left) and optimal expected revenue (right) with respect to p2.

In the latter figure, two third-degree polynomials are used to draw the trend curves in the left side of the figure. When penalty rate of occasional customers increases, booking limit of them cuts down a little to lessen overbooking. But, since it is assumed that p2 holds only values which make p1 P p2, and model tends to take advantage of the unused capacity by accepting the risk of overbooking and missed customers, there is always some denied orders for occasional customers. Because they pay more than the other customers and their corresponding repayment rate is less than the other group. So, overbooking always holds in second class. Therefore, as long as penalty rate of the second class rises, expected revenue reduces until it does not become too high in comparison with the price rate. Afterward, sensitivity of the optimal solution and expected revenue with respect to distribution parameters is investigated. To study about mean demand of frequent customers, we assume that variance of demand does not change. So, the minimum level that mean x1 can take, is 2. We set the maximum value of mean x1 on 18 and study all of the integer values between 2 and 18 for sensitivity analysis. The range determined for mean x1 has the property that we can see the case where average capacity is lower than average demand, as well as the case where it is higher than average demand. Even at the upper bound of the range (mean x1 = 18), we see the case where minimum value of frequent customers demand (that is 16) is higher than maximum possible value of capacity (that is 14). For each integer value of mean x1 in the specified interval, a random sample of 50 items is generated for x1 and another random sample of 100 items for x2 and c. Then for each x1 generated corresponding to each mean x1, each (b1, b2) 2 B, and each item of (x2, c) within the random sample generated before, corresponding revenue (R) is calculated. Afterwards, sample mean of R is computed for each mean x1 and each (b1, b2) 2 B as estimation for E(R). Then, maximum E(R) and the optimal (b1, b2) are specified for each mean x1. Then, a second-degree-polynomial interpolation curve is drawn among optimal values of b1 and b2 as a trend curve. Results are shown in Fig. 12. Rise of mean demand of frequent customers increases expected revenue at first, because, there is enough capacity inside the booking limit to respond demand. But, after a while, expected revenue will not increase, since the booking limit is constraining for demand and no more orders can be accepted. Sensitivity of optimal solution and expected revenue with respect to mean x2 and mean c is analyzed through a process which is very similar to that of mean x1. Results are depicted in Figs. 13 and 14. Pattern of changes in optimal solution and expected revenue with respect to mean x2 is similar to that of x1. That is, when mean demand of occasional customers increases, firstly, booking limit of them and expected revenue rise and booking limit of frequent customers reduces by the same amount as increase in booking limit of occasional ones. But, after a while

E(R)*

9 8 7 6 5 4 3 2

175

b*1 b*2

170 165

E(R)*

160

18

16

14

12

10

8

6

4

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Mean x1

2

155

Mean x1

Fig. 12. Sensitivity of optimal solution (left) and optimal expected revenue (right) with respect to mean x1.

E(R)* 205

b*1 b*2

185 165

E(R)*

145

Mean x2

18

16

14

12

10

8

6

4

2

18

16

14

12

10

8

6

125

4

2

14 12 10 8 6 4 2 0

Mean x2

Fig. 13. Sensitivity of optimal solution (left) and optimal expected revenue (right) with respect to mean x2.

M. Modarres, M. Sharifyazdi / European Journal of Operational Research 192 (2009) 442–459 E(R)*

Mean c

300 250 200 150 100 50 0

22

20

18

16

14

12

8

10

6

E(R)*

4

2

20

18

16

14

12

8

10

6

b* 1 b* 2

4

2

12 10 8 6 4 2 0

22

456

Mean c

Fig. 14. Sensitivity of optimal solution (left) and optimal expected revenue (right) with respect to mean c.

all of them stop changing and tend to a fixed value. That is because there will not be enough capacity to fulfill orders and rise of second class booking limit will lead to overbooking and penalty payment. Trend curves in the left part of Fig. 14 are approximated by a sixth-degree polynomial. When mean capacity is very low, only occasional customers have a booking limit that rises by increase of mean capacity, because, they have higher price rate and if a booking limit is assigned to frequent customers, capacity shortage occurs and penalty payment will be necessary. After that, booking limit of second class customers, stops rising and first class booking limit grows. That is because second class demand cannot fill the capacity singly. At last, both booking limits will tend to a fixed value, because, despite there is adequate capacity to fulfill all of the arrived orders, maximum possible accepted orders cannot fill the capacity. As expected, all of the results concluded in this section are in compatibility with Section 6. 8. Conclusion In this paper, we formulated a capacity allocation problem for two groups of frequent and occasional customers, who pay different prices for the same product. We developed a method for a stochastic capacity allocation problem by applying the concept of revenue management and introduced a new approach. The objective function has a unimodal shape and the optimal solution can be found analytically. Four possible cases may happen, depending on such factors as demand and capacity distribution function and product price and order cancellation penalty rate for each class. A method for finding optimal solution in each case is also presented. The model is checked against previous models with deterministic capacity and showed compatibility. Sensitivity of the optimal solution with respect to various expense and distribution parameters is also investigated using analytical and numerical methods. The model presented in the paper can be applied for make-to-order random-capacity production environments which have long-term frequent regular low-price high-penalty customers, as well as short-term occasional high-priced low-penalty ones. This model is better used when capacity is tight for total demand but cannot be totally utilized dealing with any single group of customers. For further research it is recommended to investigate the case in which only a booking limit is determined for the less desirable class (class 2). In such case, all orders of more desirable class are accepted but they have priority to be cancelled. One more open field of research is to generalize this model for the case where total repayment rate for the frequent customers (p1) is not necessarily more than that of occasional customers (p2). Furthermore, it is good to study the backorder demand case, where in the case of order cancellation, instead of paying back the whole price paid by the customer in addition to cancellation penalty (pi), only a cancellation penalty (pi) will be paid to the customer and the order will be only postponed (backordered) not totally canceled. It is also desirable to find exact formulas for optimal solution and sensitivity analysis, corresponding to some specific probability distribution functions for both demand and capacity. Another useful way of extending this model is to increase the number of classes. Developing a dynamic order acceptation/rejection policy for the stochastic capacity case will also be a good idea. The multi-period random capacity allocation with revenue management approach can attract further studies. Acknowledgement The authors would like to thank anonymous referees for their valuable comments that improved the quality of this paper. Appendix A. Derivatives of E(R) To have partial derivatives of the objective function, first we have to find partial derivatives of expected values of a1, a2, d1 and d2.

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457

Z b1 Z 1 oEða1 Þ ¼ 0dx1 þ b1 f1  ðb1 Þ þ f1 ðx1 Þ  dx1  b1 f1  ðb1 Þ ¼ F 1 ðb1 Þ; ob1 b1 0 oEða1 Þ oEða2 Þ ¼ 0; ¼ 0; ob2 ob1 Z b2 Z 1 oEða2 Þ ¼ 0dx2 þ b2 f2  ðb2 Þ þ f2 ðx2 Þ  dx2  b2 f2  ðb2 Þ ¼ F 2 ðb2 Þ; ob2 0 b2 Z b1  Z b1 Z b1 Z 1 oEðd 1 Þ o ¼ 0dx1 þ ðb1  cÞ  fc ðcÞ  f1 ðb1 Þ  dc þ ðb1  cÞ  fc ðcÞ  f1 ðx1 Þ  dc dx1 ob1 0 0 0 b1 ob1 Z b1  ðb1  cÞ  fc ðcÞ  f1 ðb1 Þ  dc ¼ F c ðb1 Þ  F 1 ðb1 Þ; 0 Z b2  oEðd 1 Þ oEðd 2 Þ ¼ 0; ¼ F 1 ðb1 Þ F c ðb1 þ x2 Þ  f2 ðx2 Þ  dx2  F c ðb1 Þ þ F c ðb1 þ b2 Þ  F 2 ðb2 Þ ; ob2 ob1 0 Z b1  oEðd 2 Þ ¼ F 2 ðb2 Þ F c ðx1 þ b2 Þ  f1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ : ob2 0 Now, it can be concluded that: oEðRÞ ¼ r1  F 1 ðb1 Þ  p1  F c ðb1 Þ  F 1 ðb1 Þ ob1 Z b2  F c ðb1 þ x2 Þ  f2 ðx2 Þ  dx2  F c ðb1 Þ þ F c ðb1 þ b2 Þ  F 2 ðb2 Þ :  p2  F 1 ðb1 Þ

ð24Þ

0

And similarly, oEðRÞ ¼ r2  F 2 ðb2 Þ  p2  F 2 ðb2 Þ ob2

Z



b1

F c ðx1 þ b2 Þ  f1 ðx1 Þ  dx1 þ F c ðb1 þ b2 Þ  F 1 ðb1 Þ :

ð25Þ

0

Appendix B. Studying convexity/concavity of E(R) The maximal value of E(R) will be obtained by setting (24) and (25) equal to zero, provided that E(R) is a concave function, or the Hessian matrix of E(R) is negative semi-definite. This property is checked as follows: o2 EðRÞ ¼ r1  f1 ðb1 Þ  p1 ½fc ðb1 Þ  F 1 ðb1 Þ  F c ðb1 Þ  f1 ðb1 Þ ob21 Z b2  þ p2  f1 ðb1 Þ F c ðb1 þ x2 Þ  f2 ðx2 Þ  dx2  F c ðb1 Þ þ F c ðb1 þ b2 Þ  F 2 ðb2 Þ 0 Z b 2   p2  F 1 ðb1 Þ fc ðb1 þ x2 Þ  f2 ðx2 Þ  dx2  fc ðb1 Þ þ fc ðb1 þ b2 Þ  F 2 ðb2 Þ ; 0

Z b1  o EðRÞ ¼ r  f ðb Þ þ p  f ðb Þ F ðx þ b Þ  f ðx Þ  dx þ F ðb þ b Þ  F ðb Þ 2 2 2 2 2 2 c 1 2 1 1 1 c 1 2 1 1 ob22 0 Z b 1   p2  F 2 ðb2 Þ fc ðx1 þ b2 Þ  f1 ðx1 Þ  dx1 þ fc ðb1 þ b2 Þ  F 1 ðb1 Þ ; 2

0

o2 EðRÞ o2 EðRÞ ¼ ¼ p2  F 1 ðb1 Þ  F 2 ðb2 Þ  fc ðb1 þ b2 Þ: ob1 ob2 ob2 ob1 The following counter example shows E(R) is neither convex nor concave. Consider the basic problem defined at the beginning of Section 7. Then, consider two solutions of this problem, (b1 = 13, b2 = 10) and (b1 = 4, b2 = 7). Now, the Hessian matrix is positive definite at first solution and negative definite at the second one. Consequently, the optimal solution cannot be obtained by setting the gradient vector of E(R) equal to zero vector. Appendix C. Boundary properties of E(R) The following boundary properties of the objective function and its derivatives are independent of probability distribution functions. By considering boundary assumptions (1) and (2),

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(1) When both booking limits are set zero, the expected revenue equals zero and first-degree derivatives of expected revenue are positive. b1 ¼ b2 ¼ 0 ) EðRÞ ¼ 0;

oEðRÞ ¼ r1 > 0; ob1

oEðRÞ ¼ r2 > 0: ob2

(2) When the value of a booking limit is very large, then the expected revenue is not sensitive to small changes in that booking limit.     oEðRÞ oEðRÞ limb1 !1 ¼ 0; limb2 !1 ¼ 0: ob1 ob2 (3) When both booking limits take large amounts, then the objective function will be flat. That is, it is either convex or concave. In other words, when H represents the Hessian matrix, we have:   0 0 limb1 ;b2 !1 H ðb1 ; b2 Þ ¼ : 0 0 (4) When just one of the booking limits has a large value, the objective function is flat in the direction of that variable (booking limit), but convexity direction of it in the other direction depends on the amount of the other decision variable.   0 0 limb1 !1 H ðb1 ; b2 Þ ¼ : 0 kðb2 Þ While kðb2 Þ ¼ r2  f2 ðb2 Þ þ p2  f2 ðb2 Þ  E½F c ðx1 þ b2 Þ  p2  F 2 ðb2 Þ  E½fc ðx1 þ b2 Þ.   cðb1 Þ 0 And limb2 !1 H ðb1 ; b2 Þ ¼ . While 0 0 cðb1 Þ ¼ r1  f1 ðb1 Þ þ p1  f1 ðb1 Þ  ½E½F c ðx2 þ b1 Þ  F c ðb1 Þ  p1  F 1 ðb1 Þ  ½E½fc ðx2 þ b1 Þ  fc ðb1 Þ  p1  ½fc ðb1 Þ  F 1 ðb1 Þ  F c ðb1 Þ  f1 ðb1 Þ:

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