Although air carriers derive revenue from both passengers and cargo, the majority of the literature on airline

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1 Vol. 43, No. 3, August 2009, pp issn eissn informs doi /trsc INFORMS Optimal Baggage-Limit Policy: Airline Passenger and Cargo Allocation Wai Hung Wong Department of Decision Sciences and Managerial Economics, Chinese University of Hong Kong, Shatin, N. T., Hong Kong, Anming Zhang Sauder School of Business, University of British Columbia, Vancouver, British Columbia V6T IZ2, Canada, Yer Van Hui Department of Management Sciences, City University of Hong Kong, Kowloon, Hong Kong, Lawrence C. Leung Department of Decision Sciences and Managerial Economics, Chinese University of Hong Kong, Shatin, N. T., Hong Kong, Although air carriers derive revenue from both passengers and cargo, the majority of the literature on airline management has focused on passengers. With the rapid growth in air freight, more studies are needed to examine the growing impact of air freight on air transportation. This paper addresses the optimal baggage-limit policy for airlines. Because much of the cargo is currently transported in the residual aircraft belly space after all of the passenger baggage has been enplaned, it is important for carriers to plan passenger and cargo levels together when setting passenger baggage limits. We formulate this problem as a variant of the price-dependent multi-item newsvendor model with weight-volume capacity constraints. The effects of baggage weight, prices, and costs on the number of passengers and amount of cargo carried are studied. Based on the model and carriers existing practice, we develop several illustrative cases. Our findings suggest that airlines may be able to increase profits with significant reductions in passenger baggage limits for large aircraft. Key words: combination airlines; baggage-limit policy; passenger-cargo allocation; optimization; aircraft sizes History: Received: June 2007; revisions received: February 2008, November 2008; accepted: January Published online in Articles in Advance August 6, Introduction In terms of weight, air freight accounts for less than 1% of the goods that are moved worldwide. In terms of value, it is about 39%, which highlights its importance in overall freight movement (Ammah-Tagoe 2004). With the growing importance of air cargo, many traditional airlines have changed from being pure passenger carriers to combination (passengercargo) carriers. Based on the cargo throughput of major airlines in the United States, Europe, and the Asia-Pacific region between 1995 and 2004 (as published on their respective websites), the airfreight volume has grown, on average, 50% faster than the air passenger volume in the past decade. 1 1 In addition, future growth is expected to be robust. According to Boeing s forecast in 2003, world airborne cargo will grow by 6.4% per year for the next 20 years, which is faster than the projected growth in air passenger traffic (5.1%) (Boeing 2003). A similar forecast was given by Airbus (Airbus 2002). Although passenger traffic is still the major source of revenue for combination carriers, air cargo carriage has become an increasingly important revenue source for these carriers, with some deriving 40% of their revenue from cargo operations. For example, among the top eight Asian airlines (Japan Air Lines, Korean Air, Asiana, Cathay Pacific, China Airlines, EVA Air, Singapore Airlines, and Thai Airways), the proportion of total revenue earned from cargo is on average 30%, with more than half of the cargo being carried in the belly space of passenger aircraft. Indeed, some carriers (e.g., British Airways and certain U.S. carriers) only use the belly-holds of their passenger aircraft to carry cargo. Cargo delivery for these carriers is therefore heavily influenced by passenger operations, such as flight scheduling, routing, and the amount of baggage that each passenger can bring (Kasilingam 1996; Zhu, Ludema, and van der Heijden 2000; Zhang and Zhang 2002; Billings, Diener, and Yuen 2003; LaDue 2004). 355

2 356 Transportation Science 43(3), pp , 2009 INFORMS In this paper, we investigate the baggage-limit policy of combination airlines i.e., the amount of checked baggage that each passenger is allowed to bring on board in the context of recent developments in air transportation. It is the current practice for combination airlines to only enplane cargo in the belly space that remains after all the passenger baggage has been loaded; hence, there is no guarantee that a cargo shipment will enplane a specific flight. Because the current baggage-limit policy of many airlines was set when cargo business was a secondary and, compared to passenger business, likely less profitable business, it will be interesting to determine whether the policy still results in profit maximization. In effect, several major carriers are changing, or reviewing, their passenger baggage allowance policy. For example, United Airlines released its new policy in December 2007, under which a passenger on a U.S. domestic flight (including flights to Canada) may check one bag for free, but a second for a (one-way) $25 fee. This change in policy effectively tightened checked baggage allowance. In September 2008, United increased the fee from $25 to $50 (see 2 In addition to air cargo gaining importance relative to passenger business, airlines have cited the sharp rise in fuel costs as a reason for their tightening up baggage allowance. 3 It would be interesting to see if these rationales are sensible. For example, the yield for scheduled airfreight declined 3.4% per year, after adjusting for inflation, during the period. In comparison, passenger yield declined 2.1% per year during the same period (Boeing 2003; LaDue 2004). Thus, relative to passenger price, cargo price became lower and so one might expect that less belly space should be allocated via a more generous baggage policy (rather than a more restrictive policy). To formalize this reasoning, two key questions need to be addressed: First, how should the belly space of a flight be allocated to passenger baggage and cargo? Second, how is this allocation affected by baggagelimit policies? We investigate these questions by determining baggage-limit policies when allowance is made for the interaction of baggage and cargo capacity utilization. The problem is formulated as a 2 These are in terms of US$, which are used throughout the paper. 3 In justifying its policy change to charge a second checked baggage for $25 on domestic and U.S. flights (from free service) in May 2008, Duncan Dee, Executive Vice President of Air Canada, indicated In an environment of record high and unrelenting fuel costs it is more critical than ever that the airline reviews its product offering to ensure it can continue to offer everyday low fares. This policy change is part of the ongoing review of our activities that allow us to keep pace with current industry standards and economic realities, while remaining competitive with our main North American competitors. ( variant of the price-dependent multi-item newsvendor model with capacity constraints. For passenger management, the newsvendor model has been used for single-leg revenue management problems (McGill and van Ryzin 1999). Because the present research problem considers passengers and cargo, it is a newsvendor model of multiple items with capacity constraints (Lau and Lau 1988, 1996; Erlebacher 2000). Moreover, two major issues that will also need to be addressed are multiple fare classes and price-dependent demand. This means that it is a newsvendor model that incorporates the relationship between pricing and stock quantity (Gallego and van Ryzin 1994; Weatherford 1997; Petruzzi and Dada 1999; Khouja 2000). More specifically, we first develop an analytical model on baggage allowance policy, which addresses profit maximization for an airline that faces random passenger and cargo demands. We find that as the cost of cargo falls, it is profit improving for the airline to allocate more belly space to cargo, for any given baggage allowance policy. This may not be very surprising given that falling cargo cost means, for any given price, more-profitable cargo business. On the other hand, when the cargo price (yield) falls, and hence cargo business becomes less profitable, should the airline allocate less belly space to cargo (recall the above discussion)? Our analysis shows, somewhat surprisingly, that the answer is in general ambiguous. Although a lower cargo price would induce an allocation of more passenger space, there is a countering force here: A price fall will stimulate the (random) demand. This demand-stimulating effect in turn will reinforce both the positive effect of the cargo capacity allocation on its expected sales and the effect of the cargo capacity allocation in reducing the penalty cost of turning away excess demand. Moreover, we identify both a direct effect of a change in baggage allowance on passenger-cargo allocation, and an indirect, countering effect. The indirect effect indicates that a more restrictive policy may actually increase passenger allocation, which in turn will reduce the space allocated to cargo. Air carriers need to be aware of this indirect effect when revising (tightening) their existing baggage-limit policies. We further specify conditions for one effect dominating the other. The analytical model is followed by an empirical study on current policies. We collect data from publicly available airline sources. For a given flight, we calculate the revenues and costs associated with different baggage-limit policies. The empirical results suggest that from the perspective of airline profitability, current baggage limits may be optimal for short-haul flights using small aircraft, but may not be optimal for long-haul flights using large aircraft. Our basecase numerical simulation suggests that large aircraft

3 Transportation Science 43(3), pp , 2009 INFORMS 357 might need to reduce their baggage limits by as much as 57% from the current policies. The empirical analysis also illustrates the demand-stimulating effect, as well as both the direct and indirect effects of a change in baggage allowance on passenger-cargo allocation, identified in our analytical modeling. Furthermore, our numerical examples show that when fuel costs rise, the allocation of passengers in general decreases, whereas the allocation of cargo increases. This result seems to be consistent with the airlines argument for their tightening up baggage allowance. Various sensitivity analyses are performed to investigate how the changes in some of the important parameters affect the simulation results, and a preliminary analysis of no-shows and overbooking is provided. We wish to emphasize that the present paper is a preliminary study that attempts to integrate both the passenger baggage policy and cargo policy. To avoid excessive complexity, our model has left out a number of aspects of real-world airline baggage and cargo management practices such as multiple fare classes, multidimensional air cargo demand, seasonality, route differences, and various pricing schemes. Our simulation findings should thus be considered suggestive rather than conclusive. The problem of air cargo booking, booking control, and revenue management has received considerable attention (e.g., Bartodziej and Derigs 2004; Pak and Dekker 2004; Prior, Slavens, and Trimarco 2004; Jonker 2006; Sandhu and Klabjan 2006; Amaruchkul, Cooper, and Gupta 2007; Becker and Dill 2007; Pilon 2007). In addition, there is a large body of literature on passenger revenue-management problems (e.g., Lee and Hersh 1993; Van Slyke and Young 2000; Kleywegt and Papastavrou 2001; Zhao and Zheng 2001; Karaesmen and van Ryzin 2004). Excellent general literature reviews on the subject include McGill and van Ryzin (1999), O Connor (2001), and Talluri and van Ryzin (2005). Nonetheless, the subject matter of integrating an airline s baggage-limit policy with cargo policy has not yet, to our knowledge, been examined in the literature. 2. Analysis 2.1. The Model Consider an airline s single-leg flight with an airplane of known capacities. Let W be the maximum allowable weight and V be the total belly volume available. If the airline allocates x p passengers to the flight (which is assumed not to exceed the maximum number of passengers allowable), then the allocation of x c units of cargo may be constrained by volume, weight, or both. The weight constraint may be expressed as: w pi + i w Li x pi + w cj x cj + b h u W (1) i j where x pi is the number of passengers allocated to fare class i, x cj is the number of cargo units allocated to class j, w pi denotes a passenger s weight (including carry-on baggage and in-flight facilities) in class i, w cj is the weight of a unit of class-j cargo, and b h denotes the flight s block hours with u as the amount of fuel per block hour. 4 In (1), w Li is the maximum baggage weight with which each passenger in class i is allowed. As passengers may bring less baggage weight than they are allowed, we use i w Li to represent a passenger s actual baggage weight, with baggage ratio i 0 < i 1 denoting the fraction of the maximum baggage weight. Similar to (1), the volume constraint may be expressed as: v pi x pi + v cj x cj V (2) i j where v pi (v cj, respectively) denotes the volume of the aircraft belly that is occupied by a class-i passenger s baggage (a unit of class-j cargo, respectively). For simplicity we assume that: v pi = v i w Li with v >0 (3) That is, v pi depends positively on the actual amount of baggage a passenger brings and, holding i constant, increases in w Li. 5 Given the capacity that is allocated to passengers x pi, the actual number of class-i passengers carried on the flight, denoted S pi, should be the smaller number of x pi and the number of passengers who show up for the flight: { xpi if x pi < pi D pi S pi = (4) pi D pi if x pi pi D pi where D pi is the market demand for class-i passengers and pi is the corresponding passenger show-up ratio for the flight, with pi = 1 being zero noshows. Similarly, the actual amount of cargo carried is given by: { xcj if x cj < cj D cj S cj = (5) cj D cj if x cj cj D cj 4 Block hour refers to the interval of time from the moment that a plane first moves under its own power for the purpose of flight at the originating airport to the moment when it comes to rest at the destination airport. 5 Note that further simplification of constraints (1) and (2) is not straightforward. If x cj unit is based on weight, then w cj = w ck, and hence (1) is simplified to i w pi + i w Li x pi + w c x c + b h f W with x c = j x cj. In this case, however, v cj may not equal v ck. In a similar fashion, if x cj unit is based on volume and hence v cj = v ck, then w cj may not equal w ck.

4 358 Transportation Science 43(3), pp , 2009 INFORMS where D cj is the market demand for class-j cargo, and cj is its cargo show-up ratio. These passenger and cargo demands are functions of price and a random variable: D pi = D pi p pi pi D cj = D cj p cj cj (6) where p pi and p cj are the passenger and cargo prices, respectively, and pi, cj are random variables. The demands satisfy the usual downward-sloping demand property: D pi p pi < 0 D cj p cj < 0 (7) Furthermore, D pi and D cj are assumed to be independent, and their density functions are denoted f i q pi p pi and g j q cj p cj. Let c pi and c cj be the airline s constant marginal costs of the passenger and cargo traffic, respectively. The (realized) profit of the flight may then be written as: = i p pi c pi S pi + j C f i O pi j p cj c cj S cj O cj (8) where C f is the flight s fixed cost, and O pi and O cj are, respectively, total penalty costs to the airline for turning away passengers and cargo in the case of excess demands. Specifically, the penalty cost for rejecting class-i passengers is given by: { opi pi D pi x pi if x pi < pi D pi O pi = (9) 0 if x pi pi D pi where o pi is the unit penalty cost. Similarly, the cargo penalty cost is given by: { ocj cj D cj x cj if x cj < cj D cj O cj = (10) 0 if x cj cj D cj with o cj being the corresponding unit penalty cost. To avoid allocating space to unprofitable products, only booking classes that have positive profit margins should be considered, i.e., p pi c pi > 0 and p cj c cj > 0. The baggage-limit policy is analyzed in the context of a two-stage setup. In stage 1 the airline decides on baggage allowance w L1 w L2 w LI for i = 1 2 I (i.e., there are I fare classes for passengers). In stage 2, the carrier determines the passenger and cargo allocation x p1 x pi x c1 x cj for i = 1 2 J. This sequential setup is justified by the observation that compared to the passengercargo allocation decision, the baggage-limit policy represents a longer-term decision that involves, for example, the airline s informing numerous (potential) passengers, as well as travel agents, far in advance of the actual flight time. Once decided, the policy cannot be changed as easily or frequently as the airline s internal passenger-cargo allocation decision. In both stages, however, the airline s objective remains the same: that is, to maximize the flight s expected profit, E = p pi c pi E S pi + p cj c cj E S cj i j C f E O pi E O cj (11) i j subject to constraints (1) and (2). 6 The expected sales and penalty costs can, by (4), (5), (9), and (10), be expressed as: xpi / pi S pi E S pi = pi q pi f i q pi p pi dq pi 0 + x pi f i q pi p pi dq pi (12) x pi / pi xcj / cj S cj E S cj = cj q cj g j q cj p cj dq cj 0 + x cj g j q cj p cj dq cj (13) x cj / cj O pi E O pi =o pi pi q pi x pi f i q pi p pi dq pi (14) x pi / pi O cj E O cj = o cj cj q cj x cj g j q cj p cj dq cj (15) x cj / cj Before proceeding to the analysis of this two-stage model, we discuss three more aspects of the model. First, in the above formulation we have, as is common in the revenue-management literature, assumed the perishability of a seat and a unit of cargo belly space on a particular flight. To deal with the problem of perishability, an airline might undertake overbooking such that the sum of wasted costs on seats and belly space, as well as the total penalty costs, are minimized. As a result, the sales functions for passenger and cargo (4) and (5), respectively need modifications. Although we continue to use (4) and (5) in the following analysis, our empirical study will address certain aspects of overbooking with the help of modified sales functions. Second, both constraints (1) and (2) take a linear functional form, implying that the opportunity costs of expanding cargo (in terms of passenger allocation) are constant. However, given our multiclass formulation, unit revenues and profit margins are usually 6 Note that constraints (1) and (2) are used to plan the allocation for respective passenger and cargo classes. If such an allocation is implemented without dynamic capacity adjustments between classes during operation, it may lead to capacity underutilization.

5 Transportation Science 43(3), pp , 2009 INFORMS 359 different among the classes. For example, full-fare passengers generally provide the airline higher yield than discount-fare passengers; hence, the displacement cost of cargo (in terms of passenger revenue) is increasing in cargo output (i.e., a convex function). The same applies to the displacement cost of passengers (in terms of cargo revenue) because different cargo shipments have different yields. Thus, the weight and volume constraints may be expressed, more generally, as: h x p1 x pi x c1 x cj 1 w L1 I w LI W (16) n x p1 x pi x c1 x cj 1 w L1 I w LI V respectively, where h and n are convex functions of passenger-cargo allocation variables. Note that w Li enters the second inequality in (16) via (3). Third, although the prices and costs are taken as given in the model, these prices and costs will be affected by certain parameters. For example, the cargo price is affected by cargo weight, cargo volume, or both, and the cargo cost is affected by the cargo weight. These relationships will be further specified in our empirical analysis. On the passenger side, a more generous baggage allowance represents a higher quality of service offered by the airline; as such, it may cost the carrier more to supply the service. On the other hand, the generous policy would, other things being equal, allow the airline to commend a higher price for the service, because a better service would positively affect customers demand. Mathematically, these observations are expressed as: p pi = p pi w Li c pi = c pi w Li with p pi >0 c pi >0 (17) Given these further specifications, the optimal passenger-cargo allocation and baggage-limit policy are investigated in the remainder of the section Optimal Passenger-Cargo Allocation As indicated in the introduction, air cargo has become an increasingly important revenue source for many combination carriers. Is this caused by an increase in airfreight yield (relative to passenger yield), a decline in airfreight cost, or both? To investigate this question, the model is solved starting with an analysis of the second stage. In this stage the airline determines, for a given baggage allowance policy w L1 w L2 w LI and given prices and costs, the optimal allocation xp1 x pi x c1 x cj that solves the following constrained optimization problem (recall, given by (11), is the expected profit): max x p1 x pi x c1 x cj = x p1 x pi x c1 x cj w L1 w LI (18) subject to the (general) constraints (16). To characterize the optimal passenger-cargo allocation, the Lagrangian function is formed: L = + W W h x p1 x pi x c1 x cj 1 w L1 I w LI + V V n x p1 x pi x c1 x cj 1 w L1 I w LI (19) where W and V are the (nonnegative) Lagrangian multipliers. The optimal allocation is then characterized by the first-order conditions and second-order conditions of (19). Its comparative-static effects with respect to various parameters are given below. Proposition 1. For a given baggage allowance policy, the optimal passenger-cargo allocation xp1 x pi xc1 x cj has the following properties: (a) x pi / c pi < 0, xcj / c cj < 0; (b) xpi / pi < 0, xcj / cj < 0; (c) xpi / o pi < 0, xcj / o cj < 0; and (d) the signs of xpi / p pi and xcj / p cj are in general undetermined. The proof of Proposition 1 is available on request. The proposition addresses four aspects: (a) as the cost of passenger operation falls, it is profit improving for the airline to allocate more space to passengers, for any given baggage allowance policy; (b) if passenger no-shows rise, then the airline should allocate more space to passengers, so as to minimize unfilled space when the flight departs; (c) as the penalty costs per passenger increase, the airline should allocate more space to passengers, so as to reduce the chance for excess demands and hence rejecting passengers. These three sets of results also apply to the cargo side: e.g., as the cargo cost falls, it is profit improving for the airline to allocate more belly space to cargo, for any given baggage allowance policy. Because falling cargo cost means, for a given price, more-profitable cargo business, it is not surprising that more space should in this case be allocated to cargo. Would the same logic apply to the price case? That is, as the cargo price rises, and hence cargo business becomes more profitable (for given cost), should the airline allocate more belly space to cargo? Aspect (d) of the proposition shows that the answer is generally ambiguous. A closer look at the proof reveals that two effects are at work here. Whereas higher cargo price would, holding the cargo demand constant, induce an allocation of more cargo space just as in the cost case in the price case there is an additional, countering force: A price rise will, by (7), dampen the (random) demand. This in turn will moderate both the positive effect of the cargo capacity allocation on its expected sales and the effect of the cargo capacity allocation in reducing the penalty cost. This is because both the sales effect and the penaltycost effect rely on the fact that more capacity, relative to the demand, results in less chance for excess

6 360 Transportation Science 43(3), pp , 2009 INFORMS demand to occur. The two opposite forces lead to the overall sign being ambiguous. Only if the demanddampening effect is sufficiently small would a rising cargo price (and hence more-profitable cargo business) lead to more space being allocated to cargo. These offsetting effects are empirically demonstrated later in Optimal Baggage-Limit Policy Given the optimal passenger-cargo allocations in stage 2, we can now investigate the optimal baggagelimit policy. This requires an analysis of stage 1 of the model. Before doing so, we examine the effects of the baggage-limit policy on the optimal passengercargo allocations. Because w Li enters both the objective function via functions (17) and the constraints, these allocation effects cannot be signed in general. Nevertheless, more can be said in the special singleclass case. Proposition 2. Consider a single fare class and linear weight and volume constraints. A more restrictive baggage allowance policy may or may not lead to an increase in the passenger allocation in general, but will increase passenger allocation if p p c p / w L 0, i.e., if it does not reduce the profit-cost margin for passenger operation. Furthermore, a more restrictive baggage allowance policy may or may not lead to an increase in the cargo allocation in general, but will increase cargo allocation if p p c p / w L > 0 and is sufficiently small. The intuition behind the cargo-allocation result is as follows. There is a direct effect of a change in baggage allowance, in that a more restrictive baggage allowance policy will, owing to less checked-in luggage, create space for cargo and, hence, more cargo allocation. However, there is also a secondary effect: A more restrictive policy may increase passenger allocation this will be the case, according to the first part of Proposition 2, if p p c p / w L 0 which in turn reduces the allocated cargo space. The presence of this indirect effect means that a more restrictive baggage allowance policy will not necessarily increase cargo space. Rather, cargo space will be increased only if the direct effect outweighs the indirect effect. We found that when p p c p / w L > 0, the indirect effect weakens, and so it is more likely to be dominated by the direct effect. A more generous policy represents a higher quality of service offered by the airline; as such, it would likely allow the carrier to commend not only a higher price for the service, but a higher price-cost margin as well (i.e., p p c p / w L > 0. Itis noted that our subsequent empirical study shows that although both scenarios are possible, a more restrictive baggage policy would in general induce more cargo allocation. We now try to determine the optimal baggage allowance policy, taking into account the stage-2 optimal passenger-cargo allocations, x pi w L1 w LI and x cj w L1 w LI for i = 1 I, j = 1 J. We define x p1 x pi x c1 x cj w L1 w LI = w L1 w LI (20) Here, w L1 w LI is the maximum value of expected profit for any given allowance policy w L1 w LI, and for allocations xs that satisfy constraints (16). Then, the optimal baggage allowance policy wl1 w LI can be obtained from the following first-order conditions: / w Li = L/ w Li = 0 i= 1 I (21) where L/ w Li is the partial derivative of the Lagrangian function, given in (19), with respect to w Li. At this level of generality, however, the explicit solution wl1 w LI is hard to obtain. In our subsequent empirical study, with more restrictions imposed on both the objective function and constraints, we will solve the problem for explicit solution. 3. Empirical Study: Is the Current Baggage Policy Optimal? In the remainder of this paper, we use real-life data to examine the proposed model and illustrate the relationship between passenger and cargo allocations under alternative baggage-limit policies. To avoid excessive complexity, our empirical study will leave out a number of aspects of real-world airline baggage and cargo management practices. These include: (a) multiple fare classes for each passenger compartment (e.g., economy-class compartment); (b) multiple (or even continuous) fare classes for cargo, in terms of a unique weight/volume/fare combination; (c) dynamic booking control and allocation limits on each passenger compartment; (d) routing differences such as baggage densities, weight ratios, and cargo densities; (e) multiple methods for demand forecasting; and (f) various pricing schemes for different airlines and for different seasons. The findings in the following empirical results should thus be considered as suggestive rather than conclusive Explicit Forms of Demand, Cost, Price, and Weight-Volume Relationship As an initial, illustrative study, we consider only one cargo class and two fare classes for passenger, with i = 1 being the full-fare (business) class and i = 2 the discount-fare (economy) class, assuming that the business-class passengers are assigned to the business compartment, whereas the economy class passengers

7 Transportation Science 43(3), pp , 2009 INFORMS 361 are assigned to the economy compartment and that there are no subclasses within each compartment. Next, we operationalize demand functions (6). Following Brumelle and McGill (1993), Queenan et al. (2007), Li, Zhang, and Zhang (2008), and others who have assumed normal distribution for modeling airline demand, we use normal distributions for the random variables in (6). Furthermore, we use the following log-linear relationship for the mean passenger and cargo demands: E D pi = k 1pi k 2pi ln p pi /k 3pi E D c = k 1c k 2c ln p c /k 3c i = 1 2 (22) where k 1pi, k 2pi, k 3pi, k 1c, k 2c, k 3c are positive (constant) parameters. Log-linear relationship is a common approach in empirical demand modeling in the airline industry (e.g., Anderson and Kraus 1981; Oum, Zhang, and Zhang 1993; Ghobrial and Kanafani 1995; Jorge-Calderon 1997). Equations (22) show that the mean passenger demand and cargo demand depend negatively on prices. We also assume an 80% service level for both demands D pi and D c, which is a general practice that limits the fraction of demand satisfied. 7 We now specify the empirical relationships presented in Equation (17), where passenger price p pi and passenger cost c pi are related to baggage allowance w Li. In such specification we try to incorporate existing practices and data so as to make the analysis both relevant and manageable. Consider first the specification of costs c pi, c c, and C f. There are very few studies in the literature that analyze airline flight costs, three notable exceptions being O Connor (2001), who discusses the common categories of airline service cost; Gaier et al. (1999), who proposes an air carrier cost-benefit model that gives a clear description of airline operating costs and their calculation; and Tsai and Kuo (2004), who suggests the use of an activity-based cost model to calculate a carrier s costs based on its individual flights. Using these three studies as a basis, we define the costs as follows. The marginal cost of passenger c pi is decomposed into two parts: the weight-dependent unit cost pi and the passenger-related unit cost pi. pi is the sum of costs such as fuel and airport landing charges per kilogram (kg) per block hour, 8 7 More specifically, the 80% service level is set as a constraint to bind x pi and x c in case the demands for passengers and cargo are very low, owing to high passenger and cargo prices. For the cases we studied, most of the time this constraint is not binding, and removing the constraint will not change the results materially. 8 The existing airport landing fees depend on aircraft weight. In the United States, for example, the fee rates are generally equal to the annual residual costs those remaining after all other nonaeronautical revenue sources are fully exploited divided by the weight of all aircraft landing during the year (Daniel 2001). whereas pi is the sum of costs such as of cabin crew, loss, and damage per passenger per block hour. Therefore, c pi is expressed as: c pi = b h w pi + i w Li pi + b h pi (23) Thus, c pi w Li = b h i pi > 0 reflecting, as indicated in (17), that more generous baggage allowance represents a higher quality of service, and hence would be more costly for the carrier. The marginal cost of cargo c c also consists of the weight-dependent unit cost c and the cargo-related unit cost c, and is expressed as: c c = b h w c c + b h c (24) For fixed cost C f, certain costs are considered to be proportional to block hours b h and are aggregated into (Types 1 6, Table 1), whereas other costs are considered to be proportional to an aircraft s gross weight G w and are aggregated into (Types 7 11, Table 1). That is, C f = b h + G w (25) According to the airline data we collected, the fixed cost C f is, on average, about 57.6% of a flight s total cost (Table 1). Before specifying the passenger price p pi we determine cargo price p c. Following the International Air Transportation Association (IATA) Tact Rules (IATA 2005), p c is based on the chargeable weight: the gross weight or the volumetric weight, whichever is higher. Consequently, p c may be expressed as: p c = p cu b h max w c v c (26) where p cu is the unit (i.e., per block hour) charge price per kg and = kg, the IATA volumetric weight conversion. Note that in general we have p cu w c <0: The unit price per kg falls as the cargo weight increases (i.e., the heavier a cargo shipment, the lower its unit price). Now we turn to p pi or the first part of (17): the relationship between passenger price p pi and baggage allowance w Li. This relationship is approximated as follows: p pi = b h w pi p pui + b h p L w Li (27) where p pui is a fixed unit charge price per passenger weight w pi, including the carry-on luggage and inflight facilities, and p L is the unit charge for baggage weight w Li. IATA has no guidelines for the relationship between p L and w Li. Here, we follow Leung and Cheung (2000) by assuming the relationship to be the following decreasing function: p L = m/ 1 + wli 0 5, with m being a positive constant that varies with route distance. This function, together with (27), ensures

8 362 Transportation Science 43(3), pp , 2009 INFORMS Table 1 Components of the Fixed Operating Costs of a Flight Table 2 B and B Specifications Cost type Percent of a flight s total cost Characteristics B B Flight crew costs Hull insurance Part and component costs Aircraft and traffic handling personnel Amortization Fixed fuel Air traffic control and aircraft servicing Interest Hangar costs Aircraft rental Fixed landing charge 1 8 Total 57 6% Source: Based on the annual reports of U.S., European, and Asia-Pacific carriers, p pi w Li >0, a relationship specified in (17): i.e., a more generous baggage allowance policy would allow the carrier to commend a higher price. Parameter values for the power of w Li in the range of (as opposed to value 0.5 in the above base-case function) were also used in the empirical analysis, but the differences in the results were found to be insignificant. Finally, the approximation (27) also attempts to reflect airline practice under which baggage is measured and charged mainly by weight. Like the case for cargo, we assume that the baggage weight-volume function (3) follows the IATA chargeable weight conversion: 1 m 3 = kg. Our empirical specification in this section indicates that passenger price p pi and cost c pi are determined by baggage allowance w Li, whereas cargo price p c is related to cargo weight w c and cargo volume v c via Equation (26) and cargo cost c c is related to w c. Furthermore, both the (mean) passenger and cost demands are determined by prices. Therefore, once w Li, w c, and v c are established, other parameters can be calculated accordingly. In what follows, we determine the optimal allocations x p1 x p2 x c for different flying distances (aircraft sizes) to reflect a major practical aspect of air transportation Scenarios: Long-Haul and Short-Haul Flights The existing airline practice is to set one baggage-limit policy for long-haul flights, which usually use large aircraft, and another policy for short-haul flights, which usually use small aircraft. In our study, the former case is represented by a wide-body B aircraft and the latter case by a narrow-body B aircraft. The specifications of these two aircraft types are given in Table 2. We use these specifications to define the weight and volume constraints (1) and (2) that will be used in the empirical analysis. Referring to Kasilingam (1996), we assume a 10% stacking loss for the volume capacity. (The total volume Max design takeoff weight (note 1) (kg) Spec operating empty weight (note 2) (kg) Max payload (kg) Max belly volume (m 3 ) Max seat 59 business-class, 12 business-class, 324 economy-class 150 economy-class Usable fuel capacity (note 3) (kg) Notes. 1. Maximum weight for takeoff as limited by aircraft strength and airworthiness requirements. 2. Weight of structure, power plant, furnishing systems, unusable fuel and other unusable propulsion agents, and other items of equipment that are considered to be an integral part of the configuration of a particular airplane. 3. Fuel available for aircraft propulsion. Source: Boeing Technical Specification for B and B (Boeing 2007). for B is m 3, whereas the total volume for B is 40.6 m 3.) We build these two cases based on the data from the Cathay Pacific 2004 annual report, which states that the airline has flown 285 million kilometers in 386,000 block hours with 77,000 aircraft departures. The total expense of these flights is $4,197 million, and thus the average cost per block hour is $10,873. For our simulations, we need to specify the values for parameters i and pi. Although a fullfare passenger is likely to enjoy a more generous baggage allowance than a discount-fare passenger, i.e., w L1 >w L2, the passenger may bring less baggage relative to the allowable amount than his/her discount-fare counterpart, implying 1 < 2. This may especially be the case for the flights connecting with a large number of business travelers, many of whom having a return ticket for the same day and no checked baggage at all. In effect, a full-fare passenger might actually bring less baggage in terms of an absolute amount than a discount-fare passenger; that is, it is even possible that 1 w L1 < 2 w L2. In both long- and short-haul cases, we set 1 = 0 4 for business class and 2 = 0 7 for economy class. We assume, in this section, zero no-shows, i.e., pi = 1, and no overbooking for both passenger and cargo. The issues of overbooking and no-shows are discussed in 4. We approximate a long-haul flight as a flight from Hong Kong to Los Angeles, using a B airplane. Based on our Cathay Pacific data, the cost per block hour is $12,178, and its fixed cost is $116,920, which is about 55% of the total flight cost. The maximum checked baggage weight per passenger for the long-haul flight is 64 kg over two bags, and here we consider the business and economy classes having the same weight limits (i.e., w L1 = w L2 ). The charging price and cost for the business class are

9 Transportation Science 43(3), pp , 2009 INFORMS 363 Table 3 Parameter Specifications for the Long-Haul and Short-Haul Flights Aircraft b h w p p pu p cu m p p c c u k 1p1 k 1p2 k 2p1 k 2p2 k 3p k 1c k 2c k 3c B B Notes. If subscript i is dropped in a passenger parameter, this means that the parameter takes the same value between the business and economy classes (e.g., p pu1 = p pu2 = p pu ). assumed to be three times those of the economy class. A short-haul flight is approximated as a flight from Hong Kong to Singapore using a B airplane. The fixed cost is $15,036, about 55% of the total cost. For a short-haul flight the maximum checked baggage weight over two bags is, in practice, different between the business class (30 kg) and the economy class (20 kg), and so we assume that w L1 = 1 5w L2. Furthermore, the charging price and cost for the business class are taken to be two times those of the economy class in the short-haul case. In both the long- and short-haul cases, we consider cargo unit with the standard density, i.e., w c = 50 kg and v c = 0 3 m 3, which is consistent with the IATA volumetric weight conversion indicated earlier. The particulars of the other parameters are shown in Table 3. As can be seen from the table, the parameters are slightly modified for the short-haul flight to reflect its pricing and cost characteristics Simulation Results For each case, we find the optimal x pi x c by setting a specific set of w Li and v pi (via relationship (3)). The empirical analysis becomes a linear programming model, and we solve it using the Microsoft Solver. Table 4 shows the results for long-haul flights (with large aircraft) under different baggage weights and volumes. (Recall that in the long-haul case, w L1 = w L2 = w L.) We find that the largest total profit margin that is, the total passenger and cargo profits divided by total costs occurs when w L = 30 kg, indicating that the current allowance of 64 kg is nonoptimal. Our illustrative analysis suggests that to maximize profit, large aircraft might need to reduce their baggage limits by as much as 57% from the current policy. Table 4 also shows that a more restrictive baggage allowance policy may or may not lead to an increase in the cargo allocation. Specifically, as w L declines from 70 kg, initially the allocations for both passenger classes x p1 x p2 increase but the cargo allocation x c falls. When w L declines further from 40 to 0 kg, however, x c starts to rise, indicating that a more restrictive policy induces more cargo allocation. This result is consistent with Proposition 2, which shows the undetermined impact of offsetting direct and indirect effects. Furthermore, when w L moves toward 0 kg, x c rises, whereas x p2 falls, significantly. This suggests that cargo may become particularly profitable relative to economy-class passengers, when little, or even no, checked baggage is allowed. For all the baggage policies considered in Table 4 there are excess volume capacities, suggesting that carriers could have a more lenient policy on the baggage volume. Table 5 shows the results for short-haul flights using small aircraft. The results are reported for different baggage weights in terms of w L2 (recall that w L1 = 1 5w L2 in the short-haul case). The largest total profit margin occurs when w L2 = 20 kg, implying that the current baggage limits for short-haul aircraft (i.e., 20 kg for the economy class and 30 kg for the business class) are optimal. We also note that in contrast to the situation with large aircraft, the volume constraint is binding for short-haul flights when w L2 is greater than 40 kg. With small aircraft, there is very limited belly space left for enplaning cargo when w L2 is large. Table 4 Simulation Results for B Long-Haul Flight (w L = 70, (w L = 64, (w L = 60, (w L = 40, (w L = 30, (w L = 20, (w L = 10, (w L = 0, v p = 0 29) v p = 0 27) v p = 0 25) v p = 0 17) v p = 0 13) v p = 0 08) v p = 0 04) v p = 0) Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Cost Price Optimal X Surplus volume Surplus weight Revenue ( 000) Total profit margin (%) Notes. Pb: Business class; Pe: Economy class; C: Cargo. Cost, Price, and Revenue: in terms of US$.

10 364 Transportation Science 43(3), pp , 2009 INFORMS Table 5 Simulation Results for the B Short-Haul Flight w L2 = 70, w L2 = 60, w L2 = 50, w L2 = 40, w L2 = 30, w L2 = 20, w L2 = 10, w L2 = 0, v p = 0 29 v p = 0 25 v p = 0 21 v p = 0 17 v p = 0 13 v p = 0 08 v p = 0 04 v p = 0 Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Pb Pe C Cost Price Optimal X Surplus volume Surplus weight 1, Revenue ( 000) Total profit margin (%) Notes. Pb: Business class; Pe: Economy class; C: Cargo. Cost, Price, and Revenue: in terms of US$. Using the results of Tables 4 and 5, Figure 1 depicts the relationship between the total profit margin and baggage-limit policy (in terms of w L2 for both the long- and short-haul cases. As can be seen from the figure, in terms of this relationship the large and small aircraft behave quite similarly. Both exhibit a bell-shape curve, with the profit margin being quite sensitive to the changes in baggage-limit policy. Both the overgenerous and overstringent baggage policies would reduce the profit margins, and the large aircraft s profit margin appears to be more sensitive to a small deviation from the optimum than the small aircraft s. In particular, not allowing passengers to bring any checked baggage does not appear to be a profitable strategy for carriers. Furthermore, note that for a given w L2, the total profit margin for large aircraft is greater than that for narrow aircraft. We have also examined the special cases of single fare class for both the short-and long-haul flights and found that the results are quite similar to those of the multiclass case. The above empirical analysis illustrates that as far as the baggage-limit policy is concerned, the current industry practice may be optimal for small aircraft, but it may not be optimal for large aircraft. Furthermore, note that the optimal levels found from our Profit margin (%) Figure Economy class baggage weight (kg) B Total Profit Margin and Baggage-Limit Policy B analysis are lower than the weight levels suggested by the U.S. Federal Aviation Administration (FAA). In 2003, the FAA required domestic air carriers to use the federal weight standard of an estimated economyclass checked baggage weight of 30 lbs. This means that the carriers need to estimate the checked baggage weight accordingly, about 30 kg for two bags. Given our assumed baggage ratio of 2 = 0 7 (i.e., the fraction of the maximum allowable baggage weight) for both the long- and short-haul cases, the FAA weight levels are higher than the average baggage weight levels under our optimal policies: 21 kg for large aircraft and 14 kg for small aircraft. Finally, we note that rising passenger weight (including carry-on baggage) appears to be an industry trend. For example, the FAA increased its estimate of the average passenger weight from 180 lbs, estimated in 1993, to 190 lbs in This is because an increase in both average passenger body weight and carry-on baggage, such as laptop computers. On the other hand, owing largely to competition among carriers, passenger yield may not increase in proportion with this change. (In effect, passenger yield has been declining since the mid-1980s; see, e.g., Boeing 2003.) From our calculations of the two cases, if passenger weight is increased by 5 lbs, the total profit margin will decrease by 0.7% under the optimal baggage policy. As a result, it is important for carriers, in view of this industry trend, to consider reducing the baggage allowance and enplaning more cargo for a better profit margin. 4. Sensitivity Analysis To check the robustness of our base-case simulation results reported in 3, we have conducted a number of sensitivity analyses with respect to the formulations and parameter values assumed, some of which have already been mentioned in the previous sections. In this section, we conduct four sets of sensitivity analyses. For all the analyses, we only present the case

11 Transportation Science 43(3), pp , 2009 INFORMS 365 of large aircraft. The small-aircraft case and other baggage policies are also examined, and similar results are obtained Baggage Ratio Recall that baggage ratio i 0 < i 1 refers to the fraction of the maximum baggage weight for an average class-i passenger. In our base-case simulation, we have used 1 = 0 4 for business class and 2 = 0 7 for economy class. To see the effects of variation in baggage ratio, we vary the values of 1 between 0.2 and 0.8 and the values of 2 between 0.4 and 1. Figure 2 shows the profit effect of such variations. When both baggage ratios increase, the total profit margin decreases correspondingly. It is observed that increasing 1 by 0.1 reduces the total profit margin by 0.4%, whereas increasing 2 by 0.1 reduces the total profit margin by 1.5%. This is largely because the number of economy-class passengers is more than the number of business-class passengers. Consequently, it seems sensible to control the baggage weight allowance more tightly for the economy class than for the business class Cargo and Passenger Prices In our empirical formulation, cargo and passenger prices are determined via Equations (26) and (27), respectively by their respective unit prices p cu and p pui. In our base-case simulation, we have used p cu = 0 24 and p pu1 = p pu2 = p pu = To see the effects of variation in these unit prices, we vary the values of p cu between 0.22 and 0.28 and the values of p pu between 0.24 and We find that the effects on the optimal baggage policies are small. Furthermore, when both unit prices rise, the total profit Profit margin (%) Economy class baggage ratio Business class baggage ratio Allocation quantity Figure Unit price x p2 Passenger and Cargo Allocation for Various Unit Prices margin increases, as expected. However, the rates of increase may be different for the two unit prices. For example, increasing p cu by 0.01 improves the total profit margin by 0.2% 0.9%, whereas increasing p pu by 0.01 improves the total profit margin by about 2.5% 3.9%. The trade-off between increasing p pu by 0.01 is roughly equivalent to decreasing p cu by Moreover, the sensitivity analysis with respect to cargo/passenger prices can be used to illustrate Proposition 1. Although it is expected that passenger allocation x pi should increase when the passenger price rises, it is not always the case, as shown by part (d) of Proposition 1, owing to a demanddampening effect associated with a price rise. Only if the demand-dampening effect is sufficiently small would a rise in price induce more passenger allocation. Figure 3 shows that for a fixed cargo price, the economy-class passenger allocation x p2 first rises, then falls as p pu2 increases. Likewise, by holding passenger prices fixed, similar changes in x c are found as the cargo price increases. These observations suggest that carriers may need to be aware of these offsetting effects when facing price changes Passenger and Cargo Costs The sensitivity analysis with respect to passenger and cargo costs can be done similarly to that with respect to passenger and cargo prices. To further see how sensitive the total profit margin is to the changes in passenger/cargo costs, we modify the values of c pi and c c by multiplying them with certain fractions (0.8 to 1.2, and 0.6 to 1.4, respectively). We observe that the profit margin is dependent heavily on the passenger unit cost c pi. For example, increasing c pi by 5% reduces the profit margin by 2%, whereas increasing c c by 10% reduces the profit margin by only 0.5%. x c Figure 2 Total Profit Margin and Baggage Ratios The analysis behind these results is available on request.