A decomposition approach to determining fleet size and structure with network flow effects and demand uncertainty

Transcription

1 JOURNAL OF ADVANCED TRANSPORTATION J. Adv. Transp. 2016; 50: Published online 28 September 2016 in Wiley Online Library (wileyonlinelibrary.com) A decomposition approach to determining fleet size and structure with network flow effects and demand uncertainty Yu Wang 1,2 *, Jinfu Zhu 1 and Hong Sun 2 1 College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing , Jiangsu, China 2 School of Airport Engineering & Transportation Management, Civil Aviation Flight University of China, Guanghan , Sichuan, China SUMMARY This paper presents a new methodology to determine fleet size and structure for those airlines operating on hub-and-spoke networks. The methodology highlights the impact of stochastic traffic network flow effects on fleet planning process and is employed to construct an enhanced revenue model by incorporating the expected revenue optimization model into fleet planning process. The objective of the model is to find a feasible allocation of aircraft fleet types to route legs using minimum fleet purchasing cost, thus ensuring that the expected fleet profit is maximized subject to several critical resource constraints. By using a linear approximation to the total network revenue function, the fleet planning model with enhanced revenue modeling is decomposed into the nonlinear aspects of expected revenue optimization and the linear aspects of determining fleet size and structure by optimal allocation of aircraft fleet types to route legs. To illustrate this methodology and its economic benefits, an example consisting of 6 chosen aircraft fleet types, 12 route legs, and 57 path-specific origin-destination markets is presented and compared with the results found using revenue prorated fleet planning formulation. The results show that the fleet size and structure of the methodology proposed in this paper gain 211.4% improvement in fleet profit over the use of the revenue prorated fleet planning approach. In addition, comparison with the deterministic model reveals that the fleet size and structure of this proposed methodology are more adaptable to the fluctuations of passenger demands. Copyright 2016 John Wiley & Sons, Ltd. KEY WORDS: air transportation management; fleet planning; hub-and-spoke network; network flow effects; non-linear programming model; decomposition algorithm 1. INTRODUCTION A majority of large-scale airlines have set up hub-and-spoke network operation mode in response to the increasing pressure of high cost and low fare competition environment in the airline industry. In order to maintain a wide market scope, these airlines may have to provide a both large and flexible fleet composition to realize the collection, transfer, and distribution functions of passenger flow so that the airlines can achieve the economies of scale on hub-and-spoke networks. On one hand, a large fleet size has to be offered to numerous flights to maintain huge connecting opportunities at large airports and accomplish the transport task over the network. On the other hand, both heavily and densely traveled flights across the entire network require these airlines to establish a flexible fleet structure to adjust the big aircraft types with large seating capacities to heavily traveled flights and allocate the small aircraft types with small seating capacities to densely traveled flights. More intricately, these collection, transfer, and distribution functions on hub-and-spoke network enable various passenger flows to be flown on more than one flight. Among these flights, one travel passenger spilled from a flight suggests that the travel passenger is impossible of appearing in the connecting flights. These so-called network flow effects suggest that the supply-demand interactions between seating capacity and passenger flows on *Correspondence to: Dr. Yu Wang, College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing , Jiangsu, China. wangyu @163.com Copyright 2016 John Wiley & Sons, Ltd.

2 1448 Y. WANG ET AL. one flight may significantly affect passenger flow volumes of subsequent flights. These characteristics of hub-and-spoke networks make the determination of fleet size and structure more complicated and challengeable than the decision-making process of fleet size and structure on point-to-point networks, in which a specific passenger is to be exactly flown on one flight. Optimizing fleet size and structure for a hub-and-spoke network to enhance the accuracy of match-ups between fleet capacities and market demands is a strategy falling into the scope of airline fleet planning, a strategic planning involving the optimal decision-making process to airline fleet size and structure. Through the acquisition and operation of different aircraft, airlines can provide a certain number of seats to acquire the expected market but meanwhile incur the undesired costs. Therefore, the motivation of airline fleet planning is to maximize the fleet profit by using the minimal quantity of aircraft as well as optimizing their corresponding structure to ensure that the passenger demand throughout the network could be satisfied. Considering that the acquisition of aircraft requires a huge investment, there is explicit evidence showing that a small saving of a few percent is not negligible for an airline. In order to achieve such a small savings, the adoption of an appropriate fleet planning approach is essential. The planning result for fleet size and structure with better adaptability to passenger market can improve an airline's profitability and significantly reduce the risk of fleet operations. As a result, more researchers and practitioners are focusing their studies on the fleet planning. The Top-Down macro fleet planning approach considered as one of the most useful approaches has been widely used in airline industry. The approach [1, 2] is intended to construct a simplifying supply-demand equation to ensure that the available seat miles (or sub-route network or even single route) needed throughout a future network could accommodate the growth of the forecast revenue passenger miles. This approach concerns more with the match-ups between aggregate demand and total seating capacity. Therefore, it only provides a rough estimate of the needed number of aircraft of different types for the planning horizon. The major problem of such an approach is that it could not provide more detailed information on routes (or flights) flown by various types of aircraft, for example, the number of not accepted passengers (spill) results from a type of aircraft with small seating capacity assigned to a big passenger flow market. In order to solve this drawback, the Bottom-Up micro fleet planning approach seems to be more applicable. This kind of approach conducts the fleet planning process with more concerns on detailed evaluations of route legs (or flight legs) and aircraft requirements. A route leg refers to a non-stop route with one take-off and landing. These evaluations mainly include the technical adaptabilities of different aircraft types flying on various route legs, as well as the market match-ups between passenger demand on a single route leg (or flight legs) and a specific aircraft type with a certain seating capacity. Fleet assignment technique is one of the typical approaches to be utilized to determine fleet size and structure. For a given flight schedule and aircraft of different types, the objective of such an approach (called as fleet assignment model-based micro approach for simplicity) is to maximize the fleet profit minus the acquisition cost of aircraft, subject to flight covering, and aircraft flow balance constraints. The advantage of FAM-based micro approach lies in the fact that the detailed aircraft routing schemes could be provided to guarantee the feasibility of fleet planning result. The major problem of such an approach is that the flight schedule must be exactly presented in advance. It is known to all that an airline future unknown operation environment can be hardly simulated and forecasted precisely, especially the future flight schedule used for a long-term strategic planning process. As a result, the FAM-based micro approach may easily make a flight schedule inconsistent with the airline's future realities and result in an unreliable fleet size and structure incapable of meeting the airline's future operation requirements. Using direct allocation of aircraft types to route legs is an alternative Bottom-Up micro fleet planning approach, in which the objective is still to maximize the fleet profit subject to the number of flights available on each route leg as well as the flying hours available for each aircraft fleet (called as route leg-based micro approach for simplicity). In this approach, no considerations need to be given to the operating feasibility of the planned fleet as in the FAM-based micro approach, but the advantage of such a route leg-based micro approach is still explicit, that is, the direct allocation between aircraft types and route legs for a series of given market demands needs no consideration to be given to the future flight schedule, while the necessary information on the assignment of a specific aircraft type to a route leg still can be explicitly presented as in the FAM-based micro approach.

3 A NEW MICRO AIRLINE FLEET PLANNING APPROACH 1449 However, route leg-based micro approach could not offer a direct and reliable solution to the fleet planning problem on hub-and-spoke networks because path-specific markets contain more than one route leg, and no consideration is given to the passenger network flow effects. A feasible way to solve this problem is the mileage-based (or flying hour-based) prorated fare allocation, in which each route leg in the path-specific market is assigned a fraction of the total fare, proportional to the ratio of the route leg's mileage to total mileage. Then, a point-to-point network without any connection through passengers is formed, and the corresponding problem could be solved by the route leg-based micro approach (called route leg-based approach with prorated revenue for simplicity). Explicitly, this approach is easy to over-estimate expected revenues on route legs because of the network flow effects. Therefore, this paper extends the route leg-based micro approach to make it adaptable to the characteristics of hub-and-spoke network, in which network flow effects are highlighted. In addition, the passenger demand with uncertainty is well dwelled on to explore the characteristics of its fluctuation. The main contributions of this paper therefore can be summarized as follows. (1) It provides a method of incorporating the network flow effects and demand uncertainty into the fleet planning process. (2) It devises a solution algorithm to isolate the nonlinear aspects of expected revenue optimization from airline fleet planning model using an approximation to the total network revenue function. (3) Through the case study of comparison with revenue prorated fleet planning methodology, we quantify the benefits of the methodology presented in this paper. (4) It provides a set of tests showing that our methodology is capable of producing better fleeting decisions when an airline operates in a more fluctuating network. The remainder of this paper is organized as follows. Section 2 reviews previous studies on fleet planning approaches and several related topics, including fleet assignment problem and the choice of aircraft size and flight frequency on a single route. In Section 3, we illustrate the drawbacks of the route leg-based micro model when applied to hub-and-spoke network, and then propose a fleet planning model with enhanced revenue modeling. In Section 4, we develop a solution algorithm using an approximation of the total network revenue function to solve the proposed model and provide case study results and discussions on simplified route networks. Finally, Section 5 formulates our documented conclusions. 2. LITERATURE REVIEW As mentioned previously, airlines can determine fleet size and structure from mainly two aspects, namely by fleet assignment and direct allocation of aircraft types to route legs. Fleet assignment problem is a well-researched topic, but explicit approaches to fleet size and structure problem are not yet found in large quantities in the literature. The representative contribution using fleet assignment to directly deal with airline fleet size and structure problem was the scenario aggregation-based approach presented by Listes and Dekker [3]. The approach assigned exactly one aircraft type to each flight leg to maximize fleet assignment contribution minus the fixed cost of aircraft fleet under plane flow balance and covering constraints for a given flight schedule and aircraft of several types. Moreover, an elaborated solution methodology oriented toward computational application was also presented by using both small and large-scale cases with consideration of stochastic nature of passenger demand. The numerical procedures included scenario-generated method, scenario aggregation algorithm, and solution rounding procedure, resulting in a so-called scenario aggregation-based solution algorithm. Recently, Wang and Sun [4] applied time sequence network model to determine fleet size and structure for those airlines operating in the single maintenance and crew base around the world, which is a representative operational mode of small and medium airlines mostly in Latin American and China [5]. To do so, they reduced the number of flights for a given flight schedule by forming flight pairings. A flight pairing is a combination of several ordered flights in which an airline can only swap aircraft until the last flight within flight pairing landing at the airline's single base. They [6] also concluded that an airline could determine a robust fleet composition by increasing the fleeting purity of base station, namely reducing the number of aircraft types deployed on the single maintenance and crew base. These previous approaches actually represent the operational

4 1450 Y. WANG ET AL. use of airline fleet assignment models in nature, but they provide a fundamental direction for further investigations into an appropriate fleet size and structure. Barnhart et al., [7] proposed a new deterministic formulation and solution algorithm to capture hub-and-spoke network flow effects in response to primary drawbacks of the basic leg-based fleet assignment models [8 10]. Jacobs et al., also intended to [11] incorporate hub-and-spoke network flow effects into airline fleet assignment process through combining network flow aspects of probabilistic origin-destination yield management with a leg-based fleet assignment model, in which stochastic nature of passenger demand was sufficiently taken into account. Recently, Barnhart et al., [12], once again, constructed a sub-network fleet assignment model to better model the revenue side of the objective function by using composite decision variables representing simultaneous assignment of fleet types to sub-networks. Sherali et al., [13, 14] constructed a coupled optimization model of flight schedule design and fleet assignment and developed a BENDERS decomposition to solve the model. Cacchiani and Salazar-Gonzalez [15] incorporated fleet assignment, aircraft-routing and crew-pairing problem into one integrated model and described a column-generation algorithm for obtaining a heuristic solutions. Liang and Chaovalitwongse [16] combined airline fleet assignment with weekly aircraft maintenance routing to construct a network-based model. It provided a tight linear programming relaxation in nature and could be solved by a diving heuristic. Gao et al., [17] developed a robust model integrating airline fleet and crew assignment through limiting the number of aircraft types and crew-bases. Pita et al., [18] explored a coupled optimization model of flight scheduling and fleet assignment for those slotcongested airports. Cadarso and Marin [19] presented a robust optimization model to optimize the airline flight schedule, fleet assignment, and passengers. Although these studies are direct contributions to airline fleet assignment problem, they show an explicit and interesting direction for airlines applying fleet assignment techniques to determine fleet size and structure. It is apparent that FAM-based micro fleet planning approach is only a short-term solution pathway to determine fleet size and structure because of the inaccurate flight schedule typically estimated from a long-term fleet planning horizon. To solve this drawback, Wang et al., [20, 21] attempted to construct a fleet planning approach using an optimal allocation between route legs and seating capacities of different aircraft types within artificial divided time intervals. It was concluded that an airline's fleet resulting from the route leg-based approach performed better than the use of the FAM-based approach because of the fluctuation of the future flight schedule. In route leg-based micro approach, Schick and Stroup [22] developed a multi-year fleet planning model using the concept of route leg-based approach with considerations of fleet operation and investment constraints. Du [23] and Sun et al., [24] constructed similar models for those airlines in China from the perspective of airline operational level. Dozic and Kalic [25] presented a three-stage fleet planning model, in which aircraft types were only divided into small or medium category. In the first stage, they constructed an approach for the choice of aircraft types on route legs using fuzzy logic. Based on the results from the first stage, the airline's flights were divided into flight subsets by the type of aircraft assigned to each route, and then a heuristic and analytic approach was developed to determine the minimal needed number of aircraft for each flight subset. Finally, a multi-criteria decision-making approach was established using a swapping method to ensure that the optimal number and type of aircraft are achieved. In fact, all these approaches focus their discussion on fleet planning problems with an assumption that the market share for the planned airline is known in advance. Takebayashi [26] released this assumption and explored the impacts of runway capacity on airline fleet planning and network design from the perspective of passenger routing choice and airlines' profit maximization. It is concluded that the choice of a downsizing aircraft strategy for a specific route leg is not always adopted when runway capacity is expanded. Wang et al., [20, 21] also attempted to extend the previous model [24] to construct a multi-airline fleet planning model, in which the interaction between flight frequency and market share was sufficiently considered. They designed a heuristic algorithm based on the concept of equilibrium optimum. It concluded that the fleet size and structure deprived from the proposed approach could make a significant improvement in fleet profit for those airlines in a competitive environment. Very recently, more studies have been made to incorporate environmental factors (e.g., noise and carbon dioxide emission) into airline fleet planning process. For instance, Tsai et al., [27] combined European Union Emissions Trading Scheme with airline fleet planning problem using the mixed activity-based costing decision model. The study analyzed the choice and introduction mode of

5 A NEW MICRO AIRLINE FLEET PLANNING APPROACH 1451 aircraft. Khoo and Teoh [28] constructed a bi-objective dynamic programming approach by incorporating the concept of green fleet as an objective into fleet planning problem. In addition to these studies, some researchers showed more interests to the choice of aircraft type on a single route. Wei and Hansen [29, 30] used game models to discuss the airlines' choice of aircraft size and service frequency on both individual long-haul and short-haul routes for the duopoly markets. They pointed out that the extra landing fees could force airlines to use larger aircraft to reduce airport congestion and flight delay. Through econometrical and empirical analysis, other related studies [31 35] focused on how the choice of aircraft size and flight frequency to affect the related factors (e.g., environment, demand, and market share) that both society and airlines are concerned with. 3. MODELING In this section, a simple example is presented to demonstrate the drawbacks of applying the basic route leg-based micro approach to hub-and-spoke network. Thereafter, a model with enhanced revenue modeling is presented based on path-specific market. To facilitate the description of the problem, we define the following term. A path-specific market is a specific path that passengers wish to travel on. It should be noted that a transportation market of an airline may include different path-specific markets so along as the path that the passenger travels on is not exactly the same. For instance, a passenger traveling from origin A to destination B could be called as a transportation market, but completing this trip A B, traveling passenger can take different journeys (such as A C B and A D B). Each of these journeys is such a market that is called as a path-specific market Problem description As analyzed previously, the core idea of the basic route leg-based approach is to allocate seating capacities of aircraft types onto route legs to determine fleet size and structure, in which no considerations are given to the effects of traffic flows between all route legs. In fact, in hub-and-spoke network, the majority of travel passengers need to complete their journey by means of one or more transfers at hub airports. As a result, one travel passenger spilled from a route leg suggests that the travel passenger is impossible of appearing in the connecting route legs. Different types of aircraft with various seating capacities assigned to a specific route leg may yield different traffic flows on the route leg and results in different passenger flows on other connecting route legs. These supply-demand interactions between all route legs are defined as the so-called network flow effects, under which the passenger revenues resulting from any connecting passenger flows must be prorated onto the related flown route legs before using the basic route leg-based approach. This pre-process may lead to a poor estimate in passenger revenue and affects the accuracy of fleet planning process. Note particularly that the stochastic nature of passenger flows across an entire network makes the fleet planning problem become even more complicated. Figure 1 is the simplest hub-and-spoke network with hub airport H, which is used to illustrate the previous problem. Explicitly, spilling a passenger from market A B means both reductions of one passenger from route leg A H and H B. Using the basic route leg-based approach to determine the optimal allocation between aircraft types and route legs, the revenue resulting from pathspecific market A B have to be prorated on route leg A H and H B, respectively. Each route leg in the path-specific market A B is assigned a fraction of the path-specific market fare, proportional to the ratio of the route leg's mileage to total mileage in the path-specific market. Consider an example detailed in Tables I and II for the network (Figure 1). Table I shows seating capacity, cost of acquisition, and unit operating cost of each chosen aircraft type. It should be noted that chosen aircraft type refers to a series of types that an airline may select to operate, which is determined by many factors, for example, aircraft related costs, aircraft deliver/useful time, aircraft Figure 1. Illustrative example of hub-and-spoke route network.

6 1452 Y. WANG ET AL. Table I. Seating capacity and related costs. Aircraft type Number of seats Cost of acquisition Cost (unit/mile) A , A , Table II. Passenger information. Path Origin Destination Travel mile Demand Average fare 1 A H 1200 (100, 50) H B 1600 (60,40) A B 2800 (60,30) 2000 Note: values in ( ) denote mean and deviation, respectively. physical attributes, and other factors. [36, 37]. The acquisition cost of aircraft is defined as the prorated cost by airline planning horizon. Table II shows demand and average fare data in three path-specific markets A H, H B, and A B (connecting through H). For simplicity, we assume that the airline provides only one flight on route leg A H and H B, respectively. Table III presents the different expected profit for the different schemes of introduced aircraft and aircraft assigning schemes. Column 5 of Table III is a weighted average fare after the fare of path-specific market 3 prorated on route leg A H and H B, respectively. Column 6 of Table III is the aggregated demand including the demand on path-specific market 3. The profit contribution listed in column 7 of Table III refers to the expected revenue minus the aircraft operating costs. The expected profit listed in column 8 of Table III is defined as the profit contribution minus the acquisition cost of aircraft. Column 8 of Table III shows that the expected profit in introduced Scheme 4 amounts to 20, 072 units, and it is the best one out of four schemes in Table III. Table IV also brings forth the expected profit for the comparisons of different introduced aircraft and aircraft assigning schemes. The results are deprived from the global distributing the revenues for each path-specific markets 1, 2, and 3 (method detailed in Section 3.2). The expected profit listed in column 6 of Table IV shows some remarkable differences compared with the use of revenue-prorated approach. Furthermore, Scheme 3, rather than previous Scheme 4 in use of revenue-prorated approach, is served as the optimal solution of aircraft introduced scheme. Through this example, the basic route leg-based approach with prorated revenue in advance is possible to be incapable of capturing the effects of passenger flows over hub-and-spoke network. The resulting fleet size and structure may be therefore overestimated as well. It should be noted that, in this example, it is possible to enumerate possible combinations of aircraft types and compute the maximum expected profit accordingly. However, with a network of many chosen aircraft types and path-specific markets, enumeration is computationally expensive, if not entirely impossible. In addition, it is also typically impossible for an airline to provide exactly one flight Table III. Results of illustrative example using prorating approach. Introduced Scheme (1) Route leg (2) Aircraft type (3) Operating cost (4) Average fare (5) Demand (6) Profit contribution (7) Expected profit (8) A321/1 A H A321 55, (160, 58) 79, , 167 A319/1 H B A319 66, 000 1, 122 (120, 50) A321/1 A H A319 49, (160, 58) 87, , 995 A319/1 H B A321 74, 000 1, 122 (120, 50) A321/0 A H A319 49, (160, 58) 56, , 766 A319/1 H B A319 66, 000 1, 122 (120, 50) A321/1 A H A321 55, (160, 58) 110, , 072 A319/0 H B A321 74, 000 1, 122 (120, 50)

7 A NEW MICRO AIRLINE FLEET PLANNING APPROACH 1453 Table IV. Results of an illustrative example using global distributing revenue approach. Introduced Scheme (1) Route leg (2) Aircraft type (3) Operating cost (4) Profit contribution (5) Expected profit (6) A321/1 A H A321 55, , , 736 A319/1 H B A319 66, 000 A321/1 A H A319 49, , , 760 A319/1 H B A321 74, 000 A321/0 A H A319 49, , 009 7, 009 A319/1 H B A319 66, 000 A321/1 A H A321 55, , 283 6, 283 A319/0 H B A321 74, 000 to each route leg. Thus, in this paper, we describe a new route leg-based fleet planning model with an enhanced revenue modeling, as well as a decomposition algorithm for the model across an entire airline hub-and-spoke network Notations Sets I: the set of route legs indexed by i. J: the set of path-specific markets indexed by j. K: the set of candidate aircraft types indexed by k Parameters p j : the average fare for path-specific market j. a k : the fixed cost of aircraft type k. c ik : the operating costs for route leg i flown by aircraft type k. r j : the passenger demand for path-specific market j that yields to normal distribution. b ik : the number of flying hours for route leg i flown by aircraft type k. F min i : the minimal number of flights on route leg j. F max i : the maximal available number of flights on route leg j. Cap ik : the number of seats offered to route leg i that flown by aircraft type k. u k : the expected flying time in hours of aircraft type k. 1 if route legiði IÞis flownbypath specific market j j J δ ij ¼ ð Þ; 0 otherwise: Decision variables x ik : the flight frequency assigned to route leg i flown by aircraft type k. s j : the number of seats offered to path-specific market j. l j : the average load factor for path-specific market j. z k : the number of aircraft of type k. R Total : the total revenue in a network. seat i : the number of seats offered to route leg i Functions f (r j ): the probabilistic density function with respect to r j. P Total (s j, x ik, z k ): the total fleet operation profit function with respect to s j, x ik, and z k. R j (s j ): the expected revenue function for path-specific market j with respect to s j. H i (seat i ): the expected revenue function for route leg i with respect to seat i.

8 1454 Y. WANG ET AL Mathematical formulation Airline fleet operation profit includes the passenger revenue, the operating costs, and the acquisition cost of aircraft fleet. Mathematically, the objective of the fleet planning model presented in this paper can be illustrated as follows: max P total s j ; x ik ; z k ¼ j J p j min l j s j ; r j c ik x ik a k z k (1) i I Objective (1) is to maximize the total fleet operation profit. The first term of objective (1) on the right hand is the total passenger revenue, the second term is the total operating costs, including the cost of fuel-consumptions, gate rental cost, aircraft maintenance cost and take-off and landing costs, and so forth, and the third term is the acquisition cost of aircraft fleet under the assumption of each chosen aircraft type with its self-purchased or leasing mode. It should be noted that the actual captured demand on each path-specific market is determined by the minimum value between the average captured demand volume (l j s j ) and the market demand volume on the path-specific market (r j ). Mathematically, each of the term for a path-specific market can be further rewritten as follows: ( p j l j s j ; l j s j r j p j min l j s j ; r j ¼ p j r j ; l j s j > r j (2) Because objective (1) is a nonlinear formulation that is hardly depicted and solved in a mathematical programming model, a conversion of objective (1) has to be developed and mathematically can be written as follows: max P total s j ; x ik ; z k ¼ j J p j l j s j i I c ik x ik a k z k (3) s:t: l j s j r j ; j J (4) Constraints (4) are maximum passenger number constraints, ensuring that the average captured passengers on each path-specific market must not exceed the total number of passengers on that path-specific market. In order to achieve the maximization of the fleet operation profit, if the average captured demand (l j s j ) on one path-specific market is no more than the corresponding market demand (r j ), the actual captured revenue could be written as the first term (p j l j s j ) on the right hand in objective (3) for that path-specific market according to formulas (2). Otherwise, the actual captured demand equals to the market demand on that path-specific market (r j ). In the latter case, the demand gap equals to the term (l j s j -r j ), from which no real revenue can be actually generated. This means that the first term on the right hand in objective (3) over-calculates (l j s j -r j ) passengers for the actual captured demand and the passenger revenue is overestimated. In this situation, constraints (4) can restrict the average captured demand (l j s j ) to the volume of market demand (r j ), thus ensuring the corresponding revenue is corrected. Therefore, objective (3) and constraints (4) are the equivalent conversion for objective (1). In addition, consider that too low flight frequency on any route leg leads to a large planned delay for an airline's passengers originally intending to board on the airline's flights. This means the airline's decease of market share in a competitive route leg. Therefore, airline generally has to set to a minimum flight frequency for each route leg to assure the operational feasibility of the route leg. On the other hand, the number of flight slots for an airline is limited during the planning period because of the runway capacity of airport as well as the slot allocation policies (such as grandfather law). Mathematically, these limitations can be expressed as follows: x ik F min ; i I (5) i

9 A NEW MICRO AIRLINE FLEET PLANNING APPROACH 1455 x ik F max ; i I (6) i Constraints (5) (6) are minimum and maximum flight frequency constraints, respectively, ensuring that the flight frequency assigned to each route leg cannot be less than the minimum flight frequency, and the flight frequency assigned to each route leg cannot exceed the maximum flight frequency on the route leg. The number of flying hours for each aircraft fleet type is also strictly limited by airline's maintenance regulations. These limitations make some types of aircraft fleets incapable of being used during a period of time (e.g., 1 year). The mathematically constraints can be stated as follows: b ik x ik u k z k ; k K (7) i I Constraints (7) are fleet flying hour constraints, ensuring that the flying hours for fleet of each aircraft type can not exceed the maximum flying hours available. Constraints (8) are maximum seat capacity constraints, ensuring that seats offered to each route leg must not exceed the seating capacity provided by the choice of aircraft type mix and flight frequencies. Constraints (9) (11) are types and value ranges of variables. δ ij s j Cap ik x ik ; i I (8) j J x ik 0; int; i I; k K (9) z k 0; int; k K (10) s j 0; j J (11) The seating capacity variable s j in models (3) (11) is based on the path-specific market so that the revenue optimization level is enhanced in the revenue side of airline fleet planning model. Through constraints (8), variable s j is linked with the choice of aircraft type and flight frequency, as well as variable z k. However, this model only presents a deterministic fleet planning model that captures network effects for hub-and-spoke networks, in which uncertain characteristic of market demand is not sufficiently taken into account yet. The variability of airline's market demand fluctuates with the seasonality of passenger market and the competitions among airlines that operate on the same market. More importantly, these fluctuations show significantly stochastic feather. This paper therefore regards market demand on path-specific market as a stochastic parameter to represent the uncertain nature of market demand. If any possible mix of market demands on all path-specific markets resulting from stochastic demands is considered as a demand scenario, different fleet sizes and structures could be obtained from all possible scenarios using deterministic models (3) (11). Furthermore, each kind of fleet sizes and structures resulting from different scenarios may be significantly different. It is impossible for airlines to change their fleet size and structure instantaneously in response to the change of market demands. Therefore, the objective of this stochastic model is to ensure exactly one optimal fleet size and structure can make the maximum fleet operation profit on average for all possible demand scenarios. This means that such fleet operation profit and the corresponding fleet size and structure are actually two sets of expected values in nature. Mathematically, this objective can be rewritten as follows: max P total s j ; x ik ; z k ¼ R j s j c ik x ik a k z k (12) j J i I Where R j is a function with respect to variable s j. According to formulations (1) and (2), such expected revenue function can be written as follows:

10 1456 Y. WANG ET AL. " # R j s j ¼ pj l js j 0 r jf r j drj þ þ l j s j l j s j f r j drj (13) Other definitions of the formulation remain unchanged. The fleet planning's expected model with stochastic demand can be stated as follows. max P total s j ; x ik ; z k ¼ R j s j c ik x ik a k z k (12) j J i I p j s:t: x ik F min ; i I (5) i x ik F max ; i I (6) i b ik x ik u k z k ; k K (7) i I " # l js j 0 r jf r j drj þ þ l j s j l j s j f r j drj ¼ R j s j (13) δ ij s j Cap ik x ik ; i I (8) j J x ik 0; int; i I; k K (9) z k 0; int; k K (10) s j 0; j J (11) The main difference between basic route leg-based model and path-specific market-based model presented previously lies in the facts as follows: (1) before using the basic route leg-based model, the revenues resulting from through connecting market must be prorated onto the corresponding flowing route legs, and then the model can be used to optimize fleet size and structure by means of match-ups between aircraft types and route leg-based demands, while path-specific market-based model can directly optimize fleet size and structure by allocating seats for each path-specific market across an entire hub-and-spoke airline network; (2) no network flow effects are directly incorporated into the basic route leg-based model because none of interactions between all route legs are taken into account. This leads to errors in estimating the expected traffic and revenue for each route leg in a network, while the path-specific marketbased model succeeds in capturing the effects of traffic flow by incorporating the network revenue optimization problem into fleet planning process using either formulas (3) (11) for deterministic passenger demand or (5) (13) for stochastic passenger demand. Consider that the deterministic path-specific market-based models (3) (11) is actually a point estimate to passenger demand, which is typically inconsistent with the characteristics of passenger demand in nature. This paper therefore mainly focuses on the expected models (5) (13). 4. SOLUTION ALGORITHM Expected models (5) (13) is a nonlinear mixed integer programming model with integral sign. To solve this model, gradient method can be firstly used to converse the nonlinear model to a linear model with a variable step size, then interior point method can be applied to this transformed model, and finally, the model can be solved by branch and bound algorithm. However, the solving performance of interior point method depends largely on initial solution. Moreover, branch and bound is an

11 A NEW MICRO AIRLINE FLEET PLANNING APPROACH 1457 exponential algorithm. Therefore, this solution algorithm is a time-consuming approach. Alternatively, neighborhood search-based algorithm (such as genetic algorithm, and simulated annealing) is a high-efficient algorithm in solving fleet planning problem, but it is dependent largely on the problem itself and easy to be trapped into suboptimal solution. In fact, the nonlinear aspects of models (5) (13) are only contained in constraints (13). If these nonlinear aspects could be isolated from the programming model, the scale of the remaining aspects of such programming model can be reduced. More importantly, such remaining model is a pure linear integer programming model and can be easily solved by classic integer programming algorithm. In addition, through such isolation the nonlinear aspects may become a nonlinear programming model that only includes one real type of variable s j and can be also easily solved by interior point method. Based on this viewpoint, the kernel step of this section is to present a linear approximation function to the total expected network revenue with respect to seats provided to a route leg, such that the fleet planning model is allowed to decompose into two separate but related problems: (i) a linear integer fleet planning model; and (ii) a nonlinear expected network revenue optimization model. Separately, each of these models can be solved by using conventional integer programming (IP) or non-linear programming (NLP) methods (interior point method). Through using this kind of an approximation function, any given solution to the pure linear integer programming model can yields a set of total seat numbers on each route leg. These seat numbers on each route leg could conversely form an approximation function by using the solution to the nonlinear programming model. Adding this new constructed approximation function in the linear model can yield a new solution, which is can again form another new approximation function. The whole iteration is terminated (or optimal solution could be found) until the objective values deprived from both side of the entire model equal each other Approximation function Considering the stochastic nature of passenger demand, the total expected revenue for a network can be defined as " # R Total ¼ p j l js j 0 r jf r j drj þ þ l j s j l j s j f r j drj j J Formula (14) is the sum of the expected revenues increased on each path-specific market. It is a function with respect to seats offered to path-specific market subject to the maximum number of seats offered to each route leg, or mathematically can be written as (14) δ ij s j Cap ik x ik ; i I (8) j J s j 0; j J (11) Through maximizing objective (14) under constraints (8) and (11), the previous programming model is a so-called expected revenue optimization model in nature using for enhancing the passenger revenues across the entire hub-and-spoke network. If the expected revenue on each path-specific market is an increasing concave function with respect to seats offered to the pathspecific market, all of inequalities shown in constraints (8) must take strict equal sign since the remaining seats on any route leg can be at least assigned onto the corresponding route leg, thus ensuring that the total expected revenue is maximized for the network. Therefore, the expected revenue H(seat i ) on route leg i is also a concave function with respect to seats offered to route leg i, and seat i resulting from traffic network flows within the network system. This suggests that the expected revenue H(seat i ) in the network system could account for the effect of all path-specific passenger flows over route leg i. Figure 2 shows an illustrative example of the revenue function for a single route leg in a network. To maximize the total expected fleet operation profit across the entire airline network, the total expected revenue is necessarily expected to be maximum. Therefore, a expected network revenue

12 1458 Y. WANG ET AL. Figure 2. Expected revenue as a function of seats offered to route leg. optimization model shown in formulas (8), (11), and (14) is actually included in the path-specific market based model, in which the solution yields a bid price for each route leg that represents the dual value of seats offered to the route leg and equals the slope of the revenue function at a given seat number offered to a route leg, Seat i. Figure 2 shows that this slope is constantly in the upper of the revenue curve at any given point. Therefore, it can be used to define a linear approximation and upper bound to the revenue function. Mathematically, this upper bound can be expressed as H 0i þ μ i seat i H i ðseat i Þ; i I (15) seat i ¼ Cap ik x ik ; i I (16) where H 0i represents the right-hand side of the linear approximation to the revenue function. μ i is defined as the marginal value of an extra seat given to route leg i resulting from the expected network optimization model. Mathematically, this bid price can be defined as μ i ¼ H iðseat i Þ ; i I (17) seat i Note that the value of Equation (17) can be deprived from the expected network revenue optimization for all path-specific markets in the network. Therefore, formula (15) accounts for the cumulative effect of all path-specific markets flowing over route leg i in nature and incorporates the interactions between all route legs in the network. Summing over the route legs in the network, the relationship between total expected revenue and these bid prices can be written as H 0i þ i I i I μ i Cap ik x ik H i R Total (18) i I Constraint (18) is defined as an approximation to the total expected revenue for the network Solving step Constraint (18) actually links the choice of aircraft types and flight frequencies on all route legs with the expected network revenue optimization model. The relationship allows for the decomposition of the fleet planning model presented in this paper into two separate but related problems: (1) a linear fleet planning model, or mathematically rewritten as max P total ¼ R Total i I c ik x ik a k z k (19)

13 A NEW MICRO AIRLINE FLEET PLANNING APPROACH 1459 s:t: x ik F min ; i I (5) i x ik F max ; i I (6) i b ik x ik u k z k ; k K (7) i I H 0i þ μ i Cap ik x ik R Total (18) i I i I x ik 0; int; i I; k K (6) z k 0; int; k K (7) (2) a nonlinear expected network revenue optimization model, or mathematically rewritten as " # max W Total ¼ p j l js j 0 r jf r j drj þ þ l j s j l j s j f r j drj j J (20) s:t: δ ij s j Cap ik x ik ; i I (8) j J s j 0; j J (11) where W Total represents the total expected revenue resulting from the nonlinear model and used to distinguish the total expected revenue R total resulting from the linear model. To achieve the maximum fleet profit, the optimal solution to the fleet planning model refers to the total expected revenues resulting from linear and nonlinear models that exactly equal to each other. The kernel of finding the optimal solution can be summarized in the following steps: Step 1 For an initialized choice of aircraft types and flight frequencies for each route leg within a network, the nonlinear model yields a set of bid prices for all route legs in the network. Step 2 These bid prices produce an approximation function to total expected revenue that adds into the linear model. This approximation function guarantees the maximum fleet profit in the linear model to be constantly declined because the feasible space of total expected revenue is typically reduced. Step 3 The linear model can yield a new solution to the choice of aircraft types and flight frequencies for each route leg within the network. This solution generates a set of seat allocating schemes to all route legs for the network. Step 4 Through adding this seat allocating schemes in the nonlinear model, the total expected revenue is generated and compared with the previous revenue resulting from the linear model. Step 5 If the revenue gap is not zero, the new set of bid prices for all route legs is again generated to produce another approximation function to total expected revenue for the linear model. Therefore, the entire iteration could be described as Figure 3. To begin the iterative algorithm, a choice of aircraft type and flight frequency for each route leg is needed to present the initial bid prices for each route leg. This paper obtains such starting value assuming that none of the seats offered to any route legs. More detailed solving steps are summarized as follows.

14 1460 Y. WANG ET AL. Figure 3. Description of decomposition algorithm. Algorithm 1 Decomposition Algorithm: INPUT All parameters. Initialize t =0, x ik (t)=0, z k (t)=0; seat i (t)=0; Do { - Use the nonlinear model to compute current optimal solution W total (t), decision variables s j (t), and bid price μ i (t) attth iteration; - Nonlinear model revenue Nmr_P total (t)=w total (t) - c ik x ik ðþ t a k z k ðþ t i I - For i < = the number of elements in set I - seat i ðþ¼ t Cap ik x ik ðþ t - End For; - For j < = the number of elements in set J " # - R j ðþ¼ t p j l js j ðþ t 0 r j f r j drj þ þ l j s j ðþ t l js j ðþf t r j drj - End For; - For i < = the number of elements in set I - Revenue i (t)=0; - For j < = the number of path-specific markets flown on leg i - Revenue i (t)=revenue i (t)+r j (t); - End For; - Revenue 0i (t)=revenue i (t) -μ i (t) seat i (t); - End For; - Add a new approximation function shown in constraints (15) into the linear model to compute current optimal solution Lmr_P total (t) and decision variables x ik (t), z k (t)attth iteration; - For i < = the number of elements in set I - seat i (t)=0; - For k < = the number of elements in set K - seat i (t)=seat i (t)+cap ik x ik (t); - End For; - End For } t = t + 1, until Nmr_P total (t) Lmr_P total (t); Output Optimal solution and objective.

15 A NEW MICRO AIRLINE FLEET PLANNING APPROACH NUMERICAL EXAMPLE To illustrate the fleet planning approach presented in this paper, we use the simplified 11-city example shown in Figure 4. In Figure 4, the non-strict hub-and-spoke network consists of 2 hub airports, 57 path-specific markets, 12 route legs, and 6 candidate aircraft types (2H + 57P + 12R + 6 T), in which passengers in either LXA LHW or WUH SHA market can directly travel on route leg LXA LHW or WUH SHA, respectively without any connecting through at either hub CTU or PEK. The first value on each link in the network represents average flight hours between two airports. The second value on each link denotes minimum flight frequency available between two airports. The third value on each link denotes maximum flight frequency available between two airports. These are all statistical data from a major Chinese airline. Typical 57 path-specific market demands are assumed to be normal distributions, which is one of the most conventional distribution modes, ensuring that the expected revenue yields to an increasing concave function. Their values and corresponding coefficients of variation are all assumed data. The average load factor is set to be 85% for each path-specific market demands. For this illustrative example, the model was formulated and solved using YALMIP with the commercially available solver GUROBI, a toolbox for modeling and optimization in MATLAB. In addition, Table V indicates the detailed operating information on candidate aircraft types, including nominal seating capacity, related-costs, and expected weekly flying time in hours for each aircraft type. All costs, revenues and profits in this example are in ten thousand Yuan as the unit. Figure 4. The non-strict hub-and-spoke network. Table V. Operation information for aircraft types under hub-and-spoke network. Index A319 A320 A321 B B A Capacity Operating cost in hours Purchasing Cost Expected weekly flying time in hours