The Planning of Aircraft Routes and Flight Frequencies in an Airline Network Operations

Transcription

1 Journal of Advanced Transportation, Vol. 3.5, No. I, pp www. advan ced-transport. corn The Planning of Aircraft Routes and Flight Frequencies in an Airline Network Operations Shungyao Yun Chung-Rey Wang This research is aimed at developing a model that maximizes system profit when determining the aircratt routes and flight frequencies in a network. The model employs network flow techniques to effectively collect or deliver passenger flows from all origins to all destinations using non-stop and multi-stop flights in multi-fleet operations. The model was formulated as a multicommodity network flow problem. A Lagrangian-based algorithm was developed to solve the problem. To test the model in practice, a case study is presented. Introduction Since aircraft routes and flight frequencies are essential for making airline timetables, it is important to plan them effectively in order to achieve a profitable timetable [Teodorovic, The traditional approach to such planning in Taiwan was not based on a systematic analysis, so was ineffective and inefficient, especially when the air transportation is growing today. Moreover, the government plans to develop Taiwan as an air hub in the Asia-Pacific areas. It is expected that the demand for air transportation will grow and that carrier operations in Taiwan will become more complicated. Thus, the traditional approaches for determining aircraft routes and flight frequencies may result in profit loss or decreased levels of service. From this, we see that a systematically optimised model, to assist carriers in planning aircraft routes and flight frequencies in Taiwan, would be useful for carriers. Shangyao Yan and Chung-Rey Wang are in the Department of Civil Engineering, National Central University, Chungli, Taiwan, Republic of China. Received: November 1996; Accepted: October 2000.

2 34 Shangyao Yan and Chung-Rey Wang Various models were developed in the past, to help carriers decide on aircraft routes or flight frequencies. For example, [De Vany and Garges, and [Swan, respectively, developed a model to determine flight frequency, that maximizes air carrier profit, for a route without competition. [Powell, developed a similar model for a route considering two competing carriers. [Simpson, introduced a model for simultaneously determining the departure times and flight frequencies on a route. For network operations, [Gordon and de Neufville, and [Teodorovic, developed nonlinear models for determining flight frequencies on a network providing the highest quality of transportation services for the existing transportation capacity. Though these models are more complicated than those handling a single route, they focused on direct flights instead of multi-stop flights. [Carter and Morlok, developed a model determining flight frequencies that maximizes a carrier's direct operating costs for a network, with both direct flights and flights with stopovers. [Abara, and [Konig, developed models to help air carriers solve aircraft assignment problems in practice using different types of aircraft. [Teodorovic, introduced a model that assigns different types of aircraft to routes and determines direct flight frequencies for a network without competition. As the formulation of the above network models, were complicated, and the resolution of large-scale problems was difficult, all the authors suggested using linear programming techmques to solve them. However, the optimal linear solutions may not be integers, which could present difficulties when they are applied to actual operations. More recently, [Anataram et al., developed a mixed integer program that considered the routing of long-haul aircraft from a main base to one or more terminal bases. This model, focusing on a single fleet, helped carriers select good aircraft routes which would facilitate the iterative flight scheduling process and might lead to more profitable timetables. A Lagrangian-based solution procedure was developed to solve the problem. [Teodorovic and Krcmar-Nozic, developed a multicriteria model to determine the flight frequencies in a competitive network. The model, which was formulated as a nonlinear integer problem, focused on maximizing profit, maximizing the number of passengers flown and minimizing the total passenger schedule delay. A Monte Carlo technique was used to solve the problem.

3 The Planning ofaircraft Routes From the above review, an analytic model, which would determine aircraft routes and flight frequencies for a network containing multiple aircraft types, was not found. Besides, to develop such a model for carriers in Taiwan, where international flight operations are involved, certain constraints related to international aviation rights, which were not mentioned before, also have to be considered. Thus our current research aims at developing a model that would help carriers determine aircraft routes and flight frequencies for a network that include multiple fleet operations with non-stop and multi-stop flights. This model should maximize system profits given the projected market demand for all OD pairs, the available aircraft types and their related cost-revenue data, and the frequency constraints for both domestic and foreign airports or for certain flight pairs. Note that in practice the model will provide a good initial solution for the downstream scheduling work which determines the detailed fleet routes and the final timetable pan and Young, The remainder of the paper is organized as follows. First, we introduce our model. The model is then formulated as an integer program and its solution is developed, hereafter. Finally, a case study is performed to test the model. The Model As mentioned in [Teodorovic, 19881, when assigning different types of aircraft to a network with competitive conditions, consideration must be taken of the existing fimctional dependence between the carrier s frequency share and its market share. According to one carrier in Taiwan, the market share of an OD pair service in its operations is generally linearly related to the flight frequency it provides. That is, when the fight frequency decreases, its market share also linearly decreases. Therefore, in our model, we assume that the maximum number of passengers to be transshipped is constrained by the given demand, which is linearly related to the frequency. Given the projected demand (usually a week), the available aircraft types and their operating costbevenue data, the frequency constraints for the airports or the flight pairs, and other information, we use the multi-commodity network flow technique to develop a model that determines both aircraft routes and flight frequencies in order to maximize the system profit. There are two types of networks in the model; one is the flight network and the other is the passenger network. Each flight network, as

4 36 Shangyao Yun and Chung-Rey Wang shown in Figure 1, represents the routes of a certain aircraft type. If there are k types of aircraft in operation, we have to create k flight networks. As shown in Figure 1, each node denotes an airport and each arc represents an available flight segment for such an aircraft type. If the distance for a segment is too long to be served by this type of aircraft, or the carrier does not have the right to fly in this segment, then we do not add the associated arcs. Generally, three kinds of civil aviation rights are involved in Taiwan carrier operations, known as the third, fourth and fifth freedom rights. For example, if an airline's home base is located at node 1, and it has the third and fourth freedoms at airports 5 and 6, then we add arcs between nodes 1 and 5 and between nodes 1 and 6 (assuming that the distances between them are all within the limits that this type aircraft can fly). If it has the third, fourth and fifth freedoms at airports 2, 3, and 4, then we add arcs between nodes 1 and 2, nodes 1 and 3, nodes 1 and 4, nodes 2 and 3, and nodes 3 and 4. Note that arcs (2,4) and (4,2) are not added, meaning that the fifth freedom between node 2 and node 4, is not available for this carrier. Typically a flight quota, 4v for segment ij or 4l for airpirt i, (or flight indices, where the k* type of aircraft is associated with a weighting index, w,) is defined in a contract. Thus, we have to add side constraints to the flight networks to avoid planning flight quota Fig. 1. A flight network

5 The Planning ofaircraft Routes too many flights (or flight indices) for certain flight pairs or airports. The arc cost in the k* flight network, pki,, represents the fixed operating cost (including fixed fuel expenses, crew expenses, etc.), for using the k* type of aircraft to serve a flight on segment ij. The network flows (yki, for all ij and all k) should satis6 the flow conservation constraints at every node and should be nonnegative integers. A passenger network is constructed for an OD demand to determine the transshipment of these passengers. For example, Figure 2 shows a passenger network for which the demand flows from node 5 to node 2. If there are n OD pairs, then n similar passenger networks are created. Two types of arcs are constructed in each passenger network; one is the passenger flow arc and the other is the demand arc (shown as the dashed arc in Figure 2). A passenger flow arc is created if the associated segment is available for a carrier to provide flight services. The arc cost (e.g., cn.. 'J is for the nth OD pair on segment ij) is set to be the marginal cost (including meals, baggage handling fees, additional fuel costs, etc.) for serving a passenger on this segment, which is normally small compared to the associated passenger fare, and not significantly different for different U" 0 25 / ' \ I' \\ v Fig. 2. A passenger network for OD (5->2)

6 38 Shangyao Yan and Chung-Rey Wang types of aircraft. The camer may use an average incremental cost for simplification. The arc flow upper bound (e.g., uci is for the nth OD pair on segment ij) is equal to infinity. A demand arc, (ij), is created for demand node j to node i. For example, in Figure 2, a demand arc (2,5) is created for the demand from node 5 to node 2. The demand arc cost is set as the negative revenue for serving a passenger for this OD. The demand arc flow, whch is bounded by the given demand, denotes the amount of chosen demand to serve for the purpose of profit maximization. All arc flow lower bounds are set as zero. Any passenger network flows (xni, for all ij and all n) should satisfy the flow conservation constraints at every node and should be nonnegative integers (or real, if the passengers are so numerous that the error for a noninteger solution is not significant). Two types of side constraints should be incorporated into the model in order to reflect the actual operations. The first type of constraint ensures that the number of passengers transshipped on every segment should not be greater than the number of available seats provided on the associated flight segment. Specifically, the sum of all arc flows with respect to segment ij in all passenger networks, CX,;, is less than or equal to the sum of all arc flows with respect to the same segment, in all flight networks multiplying the associated flight capacity (U ) with an available load factor ( $), k k k cy,,u q,,. The second type of constraint ensures that the number of flights satisfies all the carrier's aviation rights. In particular, for flight segment ij with the third or fourth freedoms, a side constraint should be added so that the sum of all arc flows (multiplying the associated weighting indices), in all flight networks associated with segment ij, XW"~:, should be less than or equal to the given quota4,,. k k Similarly, for airport j with the fifth freedom, a side constraint should be added to ensure that the sum of the arc flows (multiplying the associated weighting indices) into (or out of) airport j, ~, ~ w, kis yless 8 than ~ or equal to the given quota 4 I. Note that the fleet size is not considered in this model. If the fleet size is not large enough for the proposed flight frequencies and aircraft routes, then aircraft rental may be considered, or some flights may be canceled, or adjusted later in the scheduling process n i k

7 The Planning ofaircraft Routes van and Young, The model effectively consolidates all OD demands in network operations, in order to improve the efficiency of using conveyances so that the system profit can be optimized. The model can be formulated into the following integer program. Cwkyyl; IT,J, Vij E FS k (1-5) u; 2x; 20, Vn,Vij E OD x; 20, Vn,Vij eod Y; 20, Vk,Vij x; 1, Vn, Vij (1-8) (1-9) (1-10) (1-11) Yfj 1, Vk, Vij (1-12) where; n: the nth OD pair k: the kth aircraft type xn.. 1Jy cn... 1J. arc (ij) flow and cost in the n* passenger network ykij, pkij: arc (ij) flow and cost in the k* flight network 4, : the quota of flights (or flight indices) on segment ij 4, (or 4, ): the quota of flights (or flight indices) out of airport i (or into airport j)

8 40 Shangyao Yan and Chung-Rey Wang w : the weighting index for a kth type aircraft (unit: indicedflight) FS: the set of all segments for which the carrier has contracts of third and fourth freedoms OD: the set of all arcs associated with all origindestination pairs in all passenger networks Uk : the capacity of seats for a kth type aircraft (seatdflight) 7,; : the planned load factor for the kth type aircraft on segment ij u,; : arc (ij) flow upper bound for the nth passenger network The model is a multi-commodity network flow problem. Objective ( 1-1) is to find a flow pattern for each passenger network and each flight network, to minimize the total system cost. Since passenger revenues have been formulated as negative costs, the objective is equivalent to a maximization of system profit. Constraints (1-2) and (1-3) are the flow conservation constraints at every node, in every passenger network and flight network, respectively. Constraint (1-4) ensures that passengers in each segment are served by flights. Constraints (1-5), (1-6) and (1-7) are usefid for aviation rights. In particular, constraint (1-5) is associated with flight segments (third and fourth freedoms); constraints (1-6) and (1-7) are associated with airports (fifth freedom). Constraint (1-8) ensures that all demand arc flows in the passenger networks are nonnegative and within the given demand. Constraints (1-9) and (1-10) ensure that all arc flows in the passenger networks (except for the demand arc flows) and flight networks are nonnegative. Constraints (1-1 1) and (1-12) ensure that all arc flows are integers in all networks. Note that constraints (1-9, (1-6) and (1-7) are subject to a carrier's needs. For example, if the model is applied to a domestic environment, where no aviation constraints exist and no flow control constraints are required by airports, then these three constraints will not appear in the formulation. For another example, if certain flow controls for the carrier are applied to some airports, then suitable constraints, (1-6) and (1-7), should be added into the program. Sometimes, considering the level of service, carriers may set a minimum frequency on certain flight segments. Thus, additional constraints, to ensure a minimum frequency on a certain segment, can be extended from this model. After all, the model is flexible enough to be modified for different operating environments.

9 The Planning of AircraJ Routes Solution Methods Since the problem is characterized as an NP-hard problem [ Garey and Johnson, 19791, referring to [Fisher, 19811, van and Yang, 19961, wan and Young, 19961, pan and Lin, 19971, and van and Tu, 19971, we suggest using a Lagrangian relaxation with subgradient methods (LRS) for an approximation of the near-optimal solutions in order to avoid long computation times in practice. The LRS application in this research is summarized below in three parts. (1) A lower bound for each iteration: The side constraints (i.e. constraints (1-3), (1-4), (1-5) and (1-6)) are relaxed using a nonnegative Lagrangian multiplier and are added to the objective function, resulting in a Lagrangian program which composes two types of independent pure network problems. One being the minimum cost network flow problem associated with flight networks, and the other the shortest path problem associated with passenger networks. The former can be solved using the network simplex method, while the latter can be solved using the label correcting algorithm [Ahuja et al., The sum of these network objectives can be proven to be the lower bound of the original problem [Fisher, (2) An upper bound for each iteration: A Lagrangian heuristic is developed to find a feasible solution (an upper bound of the optimal solution), from the lower bound solution (typically an infeasible solution), for each iteration. In particular, the least cost flow augmenting path algorithm (LCFAP) van and Lin, is applied through the following five steps, to find a feasible flow. step 1: Check constraints (1-4), (13, and (1-6) sequentially; if any of them are violated, then we use LCFAP to augment flows in the flight networks to make it feasible. Note that after the augmentation, constraints (1-4), (1-5), and (1-6) will be satisfied, because at worst, the flight frequencies for all segments are reduced to be zero, which obviously satisfies constraints (1-4), (1-5), and (1-6). step 2: Check the flight segments sequentially; if there are excessive flights carrying no passengers, then use LCFAP in the flight

10 42 Shangyao Yan and Chung-Rey Wang step 3: step 4: step 5: networks to reduce the number of excessive flights while avoiding the violation of constraints ( 1-4), ( 1-5) and ( 1-6). Check the flight segments sequentially; if the flight frequency of a segment is not enough to carry the associated passenger flows, then use LCFAP in the flight networks to increase the frequency in this segment while avoiding the violation of constraints (1-4), (1-5)and(1-6). Check the flight segments sequentially; if the flight frequency of a segment is not enough to carry the associated passenger flows, then use LCFAP in the passenger networks to make it feasible. Note that after this augmentation, constraint (1-3) will be feasible, because at worst, the arc flows in the passenger flow networks are augmented to be zero which will make constraint (1-3) feasible. Check the flight segments sequentially; if there are excessive seats for a segment, then use LCFAP in the passenger networks to increase the associated arc flows while avoiding the violation of constraint (1-3). We note that the Lagrangian heuristic always finds a feasible solution. The reasons are as follows. First, after step 1, the side constraints (1-4), (1-5) and (1-6) are satisfied. After steps 2 and 3, the side constraints ( 1-4), (1-5) and (1-6) also remain satisfied. After step 4, the side constraint (1-3) is satisfied, while constraints (1-4), (1-5) and (1-6) also remain satisfied. After step 5, the side constraint (1-3) remains satisfied while constraints (1-4), (1-5) and (1-6) are not violated. Moreover, all other constraints are not violated by using LCFAP to modify the arc flows van and Lin, As a result, we obtain a feasible solution. Note that the five steps can be suitably altered. However, we found that the above sequence performed best in the case study presented next. (3) Solution process: the solution steps are listed below. Step 0: Set the initial Lagrangian multipliers. Step 1: Solve the Lagrangian problem optimally using the network simplex method and the label correcting algorithm to get a lower bound. Update the lower bound. Step 2: Apply the Lagrangian heuristic to find an upper bound and update the upper bound. Step 3: If the gap between the lower bound and the upper bound is within 1 %, or the number of iterations reaches 1000, stop the algorithm.

11 The Planning ofaircrafr Routes Step 4: Adjust the Lagrangian multipliers using the subgradient method van and Young, Step 5: Set m = m + 1. Go to Step 1. Case Study Our case study is based on data from a major Taiwan airline's international operations [China Airlines, There are 22 cities involved in the test. There are several types of aircraft involved in its operations, including B737's, AB3's, AB6's, MD1 I's, B74L's, B744's and B747's. For simplification, we used three types of aircraft in our case study. (1) type A indicates B737's (3 airplanes) with 120 seats and a maximum flying distance of 5000 km; (2) type B includes AB3's, AB6's, MD1 1's and B74L's (17 airplanes) with an average of 260 seats and a maximum flying distance of km and; (3) type C includes B744's and B747's (6 airplanes) with an average of 400 seats and a maximum distance that is available for all segments in its operations. Seventy-two OD pairs with a total demand of 108,s 10 passengerdweek are tested. For ease in testing, all the cost parameters were set according to the airline's reports and Taiwan government regulations, with reasonable simplifications [China Airlines, and [Civil Aeronautics Administration, C programs were coded for (1) the analysis of raw data; (2) the building of the model; (3) the solution algorithm; and (4) the output of data. The case study was implemented on an HP735 workstation with three flight networks and 72 passenger networks. The problem size was substantial with 5218 variables, 1642 flow balance constraints and 1 13 side constraints. We tried 20 different combinations of parameters using the subgradient method proposed by van and Young, to test the problem. The computation time for each test was around 8 minutes. We found that the convergence gaps were between 8% and 9%. The best one was 8.63% with an optimal lower bound of , and an optimal upper bound of From the best solution, we found that 15 flights per week were served by type A aircraft, 8 flights were served by type C aircraft, and 332 flights were served by type B aircraft, meaning that, considering the objective of maximizing total profits, type B aircraft is more economical than other types of aircraft. Besides, 100,518 out of 108,810 passengers were served. Among the 72 OD pairs, the passengers from 45 pairs were

12 44 Shangvao Yan and Chung-Rey Wang completely served, those from 5 pairs were not served, and those from 22 pairs were partially served. Among the 67 pairs where passengers were completely or partially served, 44 pairs were served used direct flights, while 23 using one-stop flights, indicating that stopovers would be usefid for consolidating passengers in transshipment, from the system perspectives. Conclusions and Continuing Work We developed a model to help carriers maximize system profit for determining aircraft routes and flight frequencies in network operations. The model employed network flow techniques to effectively "collect" or "deliver" passenger flows from all origins to all destinations with non-stop and multi-stop flights, using multiple aircraft types. We formulated the model as a special multi-commodity network flow problem. We also developed a Lagrangian-based algorithm to solve the problem, based on Lagrangian relaxation, the network simplex method, the shortest path algorithm, the least cost flow augmentation algorithm and the subgradient method. To test the model in practice, we performed a case study with substantial-sized problems, regarding the international operations of a major Taiwan airline. The preliminary results show that the model is potentially useful for airlines in Taiwan. In particular, the convergence gap is about 8 % and the computation time about 8 minutes on an HP735 workstation. We also found that the solution proposes the consolidation of passengers and flight stopovers to improve the operations from a system perspective. Since the convergence gaps in the case study are not small, the quality of the best solution obtained is not guaranteed. Therefore, the later work should focus on reducing the gap. The reason for such gaps might be as follows: the Lagrangian sub-problem may lose its lower bound due to the inherent duality gap, or the Lagrangian heuristic may lose its upper bound due to a significant adjustment (instead of a perturbation) on the lower bound solution. For the former, it might be useful to develop another approach to find a better lower bound; for example, by relaxing flow conservation constraints or using LP relaxation. For the latter, it might be helpful to reduce the gap by developing another Lagrangian heuristic to find a better upper bound; for example, by using more effective rules for flow augmentation. Or, modem meta-heuristics, such as simulated

13 The Planning ofaircraji Routes annealing, threshold accepting, a tabu search and a genetic algorithm, might be used to find better upper bounds. In particular, after finding an upper bound solution using the proposed Lagrangian heuristic, at each iteration, or at the end of the algorithm, we might use this solution as an initial solution and use meta-heuristics to improve it for better upper bound solutions. Finally, if there is a significant duality (relaxation) gap after the above improvements, then some form of enumeration procedure, such as the branch and bound approach, using the Lagrangian lower bound to help reduce the amount of concentration required could be developed. Nevertheless, the model and the solution developed here are flexible enough to be extended and improved, and therefore are potentially useful in practice. Acknowledgments This research was supported by the National Science Council of Taiwan under grant NSC E We thank China Airlines for providing the test data and their valuable opinions used in this research. We also thank the anonymous referees for their valuable comments and suggestions. References Abara. J Applying integer linear programming to the fleet assignment problem. Interfaces 19: Ahuja, R. K., Magnanti, T. L. and Orlin, J. B Network Flows, Theory, Algorithms, and Applications, Prentice Hall. Balakrishnan, A., Chien, T. W. and Wong, R. T Selecting Aircraft Routes for Long-Haul Operations : A Formulation and Solution Method. Transportation Research 24B: Carter, E.C. and Morlok, E. K Planning An Air Transport network in Appalachia. Transportation Engineering Journal of ASCE 10 1 : Chna Airlines China Airlines Annual Report Taipei, Taiwan. De Vany, A. S. and Garges, E. H A Forecast of Air Travel and Airport and Airway Use in Transportation Research 6: 1-18.

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