Aircraft routing for on-demand air transportation with service upgrade and maintenance events: compact model and case study

Transcription

1 Aircraft routing for on-demand air transportation with service upgrade and maintenance events: compact model and case study Pedro Munari, Aldair Alvarez Production Engineering Department, Federal University of São Carlos, Rodovia Washington Luís - Km 235, CEP: , São Carlos-SP, Brazil munari@dep.ufscar.br, aldair@dep.ufscar.br Abstract This paper addresses a problem faced by airline companies that offer on-demand flight services. Given a list of flight requests, the company has to assign its aircraft to these requests while minimizing operational costs. The main issue in this planning process involves the positioning of aircraft when they are not available at the airports of customer departure. The cost of this positioning should be as low as possible, as the customers expenditures are proportional to the requested flight hours only. We propose a compact optimization model to support decision making in this situation. It takes into account the mandatory aircraft maintenance events and the possibility of flight upgrades according to the their impact on the operational costs. One important and novel feature of this model is that it allows the anticipation or postponement of the beginning of flights and maintenance events within a given tolerance, giving more freedom to the decision making process. This research is motivated by a case study carried out with a fractional management airline company that operates in European and Asian countries. Computational experiments using real-life data collected from the company show that the proposed model can quickly be solved using general-purpose optimization software, including open-source alternatives. The results indicate that the obtained solutions lead to significant reductions in the operational costs and hence can be used in practice for effective decision making. Keywords: model. aircraft routing; on-demand air transportation; upgrade; maintenance; optimization; compact 1 Introduction The airline industry is well known for its complexity and costly activities. To operate efficiently, airline companies must plan their operations very carefully. They must rely on powerful information systems based on scientific methods to support decision making, including mathematical models and specialized algorithms (Klabjan, 2005; Van der Zwan et al., 2011). In the past few years, there has been a significant increase in on-demand air transportation services, mainly comprising air taxi and fractional ownership management companies (Yang et al., 2008; Yao et al., 1

2 2008; Van der Zwan et al., 2011). In contrast to the services offered by traditional commercial airlines (Lacasse-Guay et al., 2010), these companies are oriented to individual customers and do not offer flights with prefixed schedules. Customers impose their departure times and airports, and the company has to assign aircraft and crew to each customer request or to a group of them. The flights are point-to-point, are typically outside of hub airports and may be booked up to four hours before departure. This leads to a more dynamic and unpredictable environment than that faced by standard commercial airlines. One big challenge for companies that offer on-demand flights is determining routes that minimize aircraft positioning, which occurs when no suitable aircraft is promptly available at the airport to service a customer request. The cost of positioning flights is covered by the company, as they are not included as customer flying hours. This may incur a high extra cost, as customers pay proportionally to the flying hours corresponding to their requested flights. Typically, positioning may comprise 35% or more of the total flying time (Yao et al., 2008). Therefore, companies seek to optimize their aircraft routes in order to minimize the time spent on positioning flights. In addition, they may offer an upgrade for a customer request by assigning an aircraft of a better type than that required by the customer. This is done as long as the total operational costs are reduced, which is possible when the savings on positioning are larger than the extra costs of using a better (and more costly) aircraft to service a request. Another important concern of these airline companies involves aircraft maintenance. Since the planning process is dynamic, companies use a short horizon that typically ranges from 48 to 72 hours (Van der Zwan et al., 2011). This allows them to determine aircraft maintenance events in advance, and thus, they are pre-scheduled at the route planning process. In practice, it is usually possible to move the starting time of a pre-scheduled maintenance event to vary up to 24 hours, either forward or backward in time. Hence, the planning process can use this freedom to determine more effective routes. In this paper, we address the aircraft routing and scheduling problem that arises in the planning processes of on-demand air transportation services. This research has been motivated by a case study developed with a fractional ownership management company that operates in European and Asian countries. We propose an optimization model that supports decision making regarding aircraft routing, including aircraft maintenance events and service upgrades. The objective is to minimize the total operational cost of flights, which consists of aircraft positioning and extra costs due to upgrade. One important novelty of this model is that the starting times of flights and maintenance events can be anticipated/postponed to a given tolerance if they reduce operational costs. Computational experiments using real-life data show that this model can be solved quickly using general-purpose optimization software, including open-source alternatives. This is a desired feature in practice, as aircraft schedules typically change several times in a day for many different possible reasons, including the arrival of new requests, flight delays and airport congestion. Hence, the aircraft can be quickly rerouted using the model after these events. The model can be used to manage other types of per-aircraft on-demand aviation services, such as air taxis and other, more general types of personalized services. Related aircraft routing and scheduling problems for on-demand air transportation have been addressed in the literature. Keskinocak and Tayur (1998) were the first to study the aircraft 2

3 routing and scheduling problem. They developed an integer programming formulation for small and medium-size problem instances and provided a heuristic algorithm for larger instances. Since then, many studies, most of which were motivated by real-life applications, have been presented in the literature. Most of them develop decision support systems based on Operations Research tools. Ronen (2000) presented a set partitioning-based method to solve the aircraft routing and scheduling problem. The proposed method combines feasible schedules for each available aircraft to generate a complete solution. The generation of schedules is performed by a heuristic algorithm controlled by a set of parameters to handle the number of feasible schedules that can be generated. Martin et al. (2003) modeled the problem with a flow-based integer programming formulation that can be solved by general-purpose optimization software. Espinoza et al. (2008a) proposed an multicommodity network flow formulation for the problem. They developed techniques to control the size of the network and to strengthen the linear programming relaxation of the model in order to effectively solve small-size instances with general-purpose optimization software. Espinoza et al. (2008b) developed a local search scheme that was embedded within the core optimization technology of Espinoza et al. (2008a). The developed algorithm uses an integer programming model to explore large neighborhoods of the search space. Yang et al. (2008) proposed a solution method that couples an integer programming formulation and a heuristic algorithm. A generalpurpose optimization software is used to solve the integer programs in the method. Additionally, the authors presented a set partitioning formulation and a branch-and-price algorithm to solve the problem. Yao et al. (2008) modeled the problem as a set partitioning formulation and used a rolling horizon approach to solve it. At each iteration of the approach, a column generation method was used to solve the resulting subproblem. Fagerholt et al. (2009) presented a method that uses a heuristic algorithm and simulation to support strategic decisions for a company planning to establish an air taxervice. The heuristic algorithm uses insertion operators to solve the problem and a local search stage for improvement. To the best of our knowledge, no paper in the literature addresses the possibility of anticipating or postponing the starting times of flights and maintenance events, as proposed in this paper. The remainder of this paper is organized as follows. Section 2 defines the notation and provides the full description of a new optimization model that formulates the aircraft routing and scheduling problem addressed in this paper. The results of computational experiments using real-life data are presented and analyzed in Section 3. Finally, we present the conclusion and topics for future research in Section 4. 2 Optimization model Consider an air company that has a fleet of aircraft of different types distributed over a number of airports. A customer contacts the company and requests a flight by choosing (i) the aircraft type, (ii) the origin and destination airports, and (iii) the departure time of the flight. The origin and destination airports determine a live leg. An aircraft is then assigned to this request to service the requested live leg. A ferry leg (a.k.a. a deadhead or non-revenue flight) is a positioning flight needed when no aircraft of the type requested is promptly available at the departure airport to service the live leg of a given request. 3

4 The data available at the planning process are (1) a list of available airports with their geographical positions; (2) a list of available aircraft and, for each of them, its respective type, travel times between any two airports, waiting time required between flights and the airport where the aircraft is currently available; and (3) a list of a flight requests with their required departure times, aircraft types and origin and destination airports. The goal is to determine the best routes and schedules for the aircraft to service all customer requests and minimize the total operational cost. We propose a compact optimization model for the aircraft routing and scheduling problem. Different formulations have been proposed in the literature to address this problem, and we can classify them into two main categories: compact mixed-integer programming models and set partitioning models (Karaesmen et al., 2005; Yang et al., 2008; Jamili, 2017). While set partitioning models are recognized as the most suitable in practice, they require the implementation of advanced techniques, such as column generation and branch-and-price methods. On the other hand, standard mixed-integer programming models have the benefit of requiring no specific method, as the model can be solved straightforwardly by general-purpose optimization software. The formulation proposed in this paper is a standard mixed-integer programming model that can handle a practical number of flight requests promptly without negatively affecting the running time of the optimization software. As indicated by the results of computational experiments presented later in this paper, problems with more than 100 requests can be solved in a few seconds by general-purpose optimization software. In addition, the model allows the anticipation or postponement of the starting times of flights and maintenance events, a feature not observed in other models in the literature. This feature is relevant and widely used in practice by airline companies. To simplify the definition of the model, we first introduce a basic version (Subsection 2.1), which does not include maintenance or upgrades. We then describe how to incorporate maintenance and upgrades into the formulation (Subsections 2.2 and 2.3, respectively) and finally present the full version of the model (Subsection 2.4). We also propose additional valid inequalities to strengthen the model and speed up the solution times (Subsection 2.5). 2.1 Basic model We first define the notation and parameters required to describe the model. Consider the following sets and input parameters: P = {1,..., P }: set of aircraft types; V = {1,..., N}: set of aircraft. We may partition this set into subsets V 1,..., V P according to the P types of aircraft; R = {0} L: set of requests, where 0 is a dummy request used as the first and last request serviced by any aircraft and L = {1,..., L} is the set of live requests; K = {1,..., K}: set of airports; c p R + : travel cost per time unit of an aircraft of type p P; T p ij R +: travel time between airports i and j for an aircraft of type p P for all i, j K; 4

5 AV v R + : exact time when aircraft v becomes available to fly for the first time in the planning horizon; k v K: initial airport of aircraft v V; t v P: type of aircraft v V; T ATk r R +: turn-around time at airport k K before servicing request r R. This is the waiting time required between flights, which is usually due to the boarding of passengers and aircraft setup; ST r R + : starting time of request r L; L R + : maximum time shift allowed to start servicing a live request; i r, j r V: origin and destination airports, respectively, of request r L; p r P: aircraft type required in request r L. A straightforward strategy to model this problem would be to rely on a network representation of the airports, in which the airports are represented by nodes and decision variables determine the flow of airplanes through these nodes. However, such a formulation leads to a unnecessarily large number of variables, as they depend quadratically on the number of available airports (which can be huge in practice). In addition, the flow between airports of the live leg of a given request is defined a priori, and thus, the flow decision in the network concerns the ferry legs only. One more disadvantage is that the airports can be visited more than once, which increases the size of model in order to account for one additional index in the decision variable to represent the number of the visit. To overcome these drawbacks, we adopt a different type of network, in which the nodes represent the requests in R according to Martin et al. (2003) and Keskinocak and Tayur (1998). The flow of aircraft is defined through requests, and each node has to be visited only once by only one aircraft. Fig. 1 illustrates the network representation based on flight requests. Notice that the nodes are indexed by requests, and hence, the visit to a node r indicates that the aircraft visits the airports i r and j r associated with the request. If the aircraft goes from a node (request) r to another s such that j r, i.e., the destination airport of r is not the same as the origin airport of s, then a ferry leg is implied between these visits. All routes must start and finish at the dummy node 0. The problem is then to find aircraft routes in this network such that all nodes (requests) are visited (serviced) exactly once. Based on the representation using a network of requests presented in Fig. 1, we define the binary decision variable: { 1, if aircraft v services request r and then request s consecutively, y vrs = 0, otherwise, for each v V and r, s R, which determines the sequence of requests serviced by each aircraft. In addition, we define the continuous variable w r that gives the earliest exact time when an aircraft 5

6 1 2 3 i 1 j 1 i 2 j 2 i 3 j 3... R i R j R 0 dummy Fig. 1: Problem representation using a network of requests. starts servicing request r for each r R to be used in the timing constraints of the model. Hence, we obtain a schedule for each aircraft route. The objective is to minimize the total cost with ferry legs, as customers pay proportionally to the live legs only. To simplify the presentation of the model, we define the ferry cost C vrs corresponding to the flying cost of the ferry leg required when using aircraft v to service request r and then request s for v V and r, s R as follows: c tv T tv k v, if r = 0, s > 0 and k v, C vrs = c tv T tv j r, if r > 0, s > 0, r s and j r, 0, otherwise. The first case in (2.1) considers s as the first request serviced by the aircraft v; thus, it computes the flying cost from the initial airport (k v ) of the aircraft to the first airport ( ) of the request. The second case computes the flying cost from the last airport (j r ) of a request r to the first airport ( ) of the next service request s. In both cases, a ferry leg between the requests exists, as k v and j r. In all other cases, there are no flying costs corresponding to ferry legs. (2.1) Using the notation, parameters and decision variables just defined, we obtain the following formulation: min s.t. C vrs y vrs, (2.2) v V r R v V pr y vrs = 1, r L, (2.3) s r y vsr = y vrs, v V, r L, (2.4) s r s r y v0s = 1, v V, (2.5) y vr0 = 1, v V, (2.6) r R ST r w r ST r + L, r L, (2.7) 6

7 w s w r + T pr j r + T ps j r + M 1 rs( v V y vrs 1), r, s L, r s, j r, (2.8) w s w r + T pr + M 2 rs( v V y vrs 1), r, s L, r s, j r =, (2.9) w s (AV v + T tv k v )y v0s, s L, v V, k v, (2.10) w s (AV v )y v0s, s L, v V, k v =, (2.11) w r 0, r R, (2.12) y vrs {0, 1}, v V, r, s R, (2.13) where M 1 rs = ST r + L + T pr j r + T ps j r and M 2 rs = ST r + L + T pr are sufficiently large values (Big-M constants). The objective function (2.2) consists of minimizing the total travel cost involving ferry legs, as defined by the coefficients in (2.1). Constraints (2.3) ensure that each request r L is serviced by exactly one aircraft and that this aircraft must be of type p r, as required by the customer. Constraints (2.4) guarantee the flow of aircraft through the network of requests. To service request r L, an aircraft must service one request before it and another after it (which can be the dummy request 0 in both cases). Constraints (2.5) and (2.6) impose that the dummy request 0 must be the first and the last request of each aircraft, respectively. This has to be satisfied even when an aircraft v does not service any live request, which results in y v00 = 1. Constraints (2.7) (2.11) guarantee the time consistency in the sequence of requests serviced by each aircraft. Constraints (2.7) ensure that the departure time of the aircraft that services request r satisfies the time imposed by the customer within a tolerance L. Constraints (2.8) determine the exact time when the aircraft can start servicing request s after servicing request r if a ferry leg is needed (i.e., j r ). They arbitrate that if there is an aircraft that services r and s consecutively, then the value of w s is given by the value of w r plus the travel times and turn-around times of the live and ferry legs. Notice that these constraints rely on Big-M values; hence, w r has no direct influence on w s when there is no aircraft visiting r and then s. Constraints (2.9) are similar to (2.8), but they are for the cases in which no ferry leg is needed between r and s (as j r = ). Constraints (2.10) and (2.11) impose the exact time from which the services in a first request of an aircraft can start, as we have y v0s = 1 when s is the first request serviced by aircraft v. Finally, (2.12) and (2.13) state the domain of the decision variables. Model (2.2) (2.13) can be considered a variant of the two-index vehicle flow formulation of the vehicle routing problem with time windows (Desaulniers et al., 2014). In the next subsections, we present how this model is extended to incorporate the characteristics faced in practice. Note that model (2.2) (2.13) can be easily adapted to consider the outsourcing of requests. In this case, a binary variable (e.g., z r r L) indicating whether or not a request is outsource, must be added to the left-hand side of constraints (2.3) and to the objective function with its respective cost. However, since none of the real-life instances in the computational experiments require outsourcing, we decided not to include this feature explicitly in the optimization model. 7

8 2.2 Maintenance events Aircraft have to go through maintenance occasionally. Motivated by the case addressed in this paper, we assume that the planning horizon is short (up to 72 hours), which is a common practice in the planning process of on-demand air transportation services. This allows the modeling of maintenance events as requests defined in advance by the company. Hence, the ideal starting time and duration of each maintenance event are defined a priori and considered input data. Typically, the actual starting time of maintenance can be anticipated or postponed up to a given number of hours, which gives more freedom to the aircraft schedule and hence may lead to a reduction in the operational costs. To include the maintenance events in the model, we consider them as a special type of requests. We define a set M of maintenance requests such that the set of requests becomes R = {0} L M. Maintenance requests differ from live requests because (i) there is only one airport associated to them; (ii) an specific aircraft must service the request (rather than an specific aircraft type), namely, the aircraft in maintenance; (iii) there is a time length for the request in addition to the scheduled starting time. Despite these differences, all maintenance requests must be serviced and scheduled together with other requests to satisfy flow and time constraints. To ensure a uniform notation, we assume that for r M, the airports i r and j r are the same and denote the airport of maintenance, while ST r denotes the scheduled starting time of the maintenance. Additionally, the turnaround time before a maintenance request must be zero, i.e., T AT r k and k K. = 0, for all r M To extend the model (2.2) (2.13) to take into account the maintenance events, we need to define the following additional parameters: v r V: index of the aircraft that must undergo the maintenance r M; T L r R + : duration of the maintenance r M; M R + : maximum tolerance of the anticipation/postponement of a maintenance event. Regarding the constraints, most of them are kept the same, except for a few adjustments. The first modification concerns constraints (2.3), which are still valid for any r L, but for r M, they must ensure that only the aircraft v r services request r. Hence, we add the following constraint to the model: y vrrs = 1, r M. s r For constraints (2.4), we extend the domain from r L to r R, r > 0. Similar changes are required in constraints (2.8) and (2.9), with s L replaced by s R, s > 0. Additionally, we have to include in the model the counterpart of constraints (2.7) (2.11) for r M, which are given by: ST r M w r ST r + M, r M, w s w r + T L r + T pr j r + M 3 rs(y vrrs 1), r M, s R, s > 0, s r, j r, w s w r + T L r + M 4 rs(y vrrs 1), r M, s R, s > 0, s r, j r =, w s (AV vs + T ps k vs )y vs0s, s M, k vs, 8

9 w s (AV vs )y vs0s, s M, k vs =, where Mrs 3 = ST r + M + T L r + T pr j r + T AT s and Mrs 4 = ST r + M + T L r + T AT s are Big-M constants used to make the corresponding constraints inactive if requests r and s are not served consecutively. 2.3 Upgrade of flights Recall that the aircraft are classified into P different types and that each customer request specifies the required aircraft type. We have assumed in model (2.2) (2.13) that the request cannot be serviced by an aircraft of a different type. As mentioned before, it may be interesting to allow upgrades of aircraft types in practice, i.e., the assignment of an aircraft of a better type than that required by the customer, as long as the total costs are reduced due to these changes. To achieve this, the savings on ferry legs must be larger than the extra costs of using a better (and more costly) aircraft to service a request. For example, consider the situation in which there is a customer request to fly from airport A to B on an aircraft of type 2; however, there are no aircraft of this type available at A. Thus, we need a ferry leg to take an aircraft to airport A and then service the request. Now, suppose that there is an aircraft of type 3 available at A, which is better than the type 2 aircraft but more costly. If the increase in the total cost due to servicing this request using the aircraft of type 3 instead of one of type 2 is smaller than the cost of a ferry leg to take an aircraft of type 2 to airport A, then it may be worthwhile to perform an upgrade and use the aircraft of type 3 promptly available at A. The first step to incorporate upgrades into the model (2.2) (2.13) is to indicate that different aircraft types may service a request. Since we allow upgrades but not downgrades, we assume that the set of aircraft types P follows a non-descending order of quality. Hence, an aircraft of type p is worse than or similar to an aircraft of type p + 1. Therefore, if an aircraft of type 3 is required for a given request, for example, then we may assign any aircraft of types from 3 to P. To add this to the model, we replace constraints (2.3) with: p P p p r y vrs = 1, r L. v V p s r The difference from constraints (2.3) arises from the additional summation over all aircraft types p P that are better than or equal to p r, the aircraft type specified by the request. The time constraints (2.8) and (2.9) must also be adapted, as the travel times must depend on the type of aircraft assigned to the request (t v ) instead of the type specified by the request (p r ). However, because the assignment of aircraft to requests is a decision in the model, we have to use variables to specify the type of the aircraft. Hence, the constraints become: w s w r + v V(T tv j r + T tv j r )y vrs + M 1 rs( v V y vrs 1), r L, s R, r s, s > 0, j r, 9

10 w s w r + v V(T tv )y vrs + M 2 rs( v V y vrs 1), r L, s R, r s, s > 0, j r =. with Big-M constants M 1 rs = M 2 rs = ST r + L. In these two sets of constraints, the travel times are now multiplied by the corresponding decision variables y vrs, which determine the aircraft that is assigned to the request in order for the correct travel times to be used in the constraints when an upgrade happens. Notice that we have also incorporated the changes regarding the inclusion of maintenance events, as described in Section 2.2. Finally, we modify the objective function to incorporate the costs related to upgrades. addition to the summation regarding the ferry costs, as in (2.2), we include the differences in the cost due to the assignment of a better aircraft type to a live leg. The objective function becomes: 2.4 Full model min v V r R C vrs y vrs + r L s r p P p>p r v V p (c tv c pr )T tv y vrs. After incorporating all changes presented thus far into the model, we obtain: min s.t. v V r R C vrs y vrs + r L s r p P p>p r v V p (c tv c pr )T tv y vrs, (2.14) y vrs = 1, r L, (2.15) p P v V p p p r s r y vrrs = 1, r M, (2.16) s r y vsr = y vrs, v V, r R, r > 0,(2.17) s r s r y v0s = 1, v V, (2.18) y vr0 = 1, v V, (2.19) r R ST r w r ST r + L, r L, (2.20) ST r M w r ST r + M, r M, (2.21) w s w r + v V(T tv j r + T tv j r )y vrs + M 1 rs( v V y vrs 1), w s w r + v V(T tv )y vrs + M 2 rs( v V y vrs 1), r L, s R, r s, s > 0, j r, (2.22) r L, s R, r s, s > 0, j r =, (2.23) w s (AV v + T tv k v )y v0s, s L, v V, k v,(2.24) w s (AV v )y v0s, s L, v V, k v =,(2.25) In 10

11 w s w r + T L r + T pr j r + M 3 rs(y vrrs 1), w s w r + T L r + M 4 rs(y vrrs 1), r M, s R, r s, s > 0, j r,(2.26) r M, s R, r s, s > 0, j r =,(2.27) w s (AV vs + T ps k vs )y vs0s, s M, k vs, (2.28) w s (AV vs )y vs0s, s M, k vs =, (2.29) w r 0, r R, (2.30) y vrs {0, 1}, v V, r, s R, (2.31) where M 1 rs,..., M 4 rs are sufficiently large values (Big-M constants) defined in the previous sections. Model (2.14) (2.31) expands upon other aircraft routing models available in the literature (Keskinocak and Tayur, 1998; Martin et al., 2003; Karaesmen et al., 2005), as it gives more flexibility to the starting times of live requests and maintenance events. In addition, the objective function (2.14) includes a trade-off regarding flight upgrade costs, which is not included in other models. 2.5 Valid inequalities To strengthen the model and accelerate the solution times for solving it with general-purpose optimization software, we include the following valid inequalities in the model (2.14) (2.31): y vrs = 0, r L, v / V pr V pr+1... V P, (2.32) s r y vrs = 0, r M, (2.33) v V v v r s r v V y vrs = 0, r L, (2.34) s L s r ST r+tirjr P STs+ L y vrs = 0, r L, (2.35) v V s M ST r+tirjr P STs+ M y vrs = 0, r M, (2.36) v V s L ST r M +T L r ST s+ L y vrs = 0, r M. (2.37) v V s M s r ST r M +T L r>st s+ M These constraints are redundant regarding integer solutions but are helpful for cutting off fractional solutions and hence improving the performance of branch-and-cut methods, as indicated by the results of preliminary experiments. 11

12 3 Computational experiments In this section, we present and discuss the results of computational experiments with the optimization model proposed in Section 2.4 using real-life data. A computational implementation was coded in C++ to read the input data, state and solve the model and then provide the optimal solution as output. To solve the model, we used the GLPK library (GLPK, 2012), an open-source general-purpose optimization software that includes procedures for modeling and solving mixed integer programming problems. The model was coded using the algebraic language GMPL and read by a procedure available in the GLPK library. All experiments were run on a computer with an Intel Core i7-2600m 3.40GHz, 16GB of RAM and Linux Operating System. 3.1 Data description We collected real-life data from the operation of the fractional management company previously mentioned in this paper. The company provided the journey logs of its aircraft for four different months. In total, we have 112 requests in the 10 day log provided for the first month, 129 requests in 10 days of the second month, 107 requests in the 8 days of the third month and 578 requests in the 16 days of the last month. We chose a time horizon of 3 days (72 hours), as recommended by the company. Also, this is the usual choice in the planning process of flights for fractional ownership programs (Van der Zwan et al., 2011). Accordingly, we created a set of instances by selecting the requests of every three consecutive days of each month. Hence, in total, we have 36 problem instances. The created instances and their description are presented in Table 1, in which the first column provides the name of the instance and columns 2 to 6 provide the number of live requests (L), the number of maintenance requests (M), the total number of requests (R), the number of airports (K) and the number of aircraft (N), respectively, in the instance. Instance names are in the format MxDy-z, where x is an identifier for the month and y and z represent the first and last day provided in the instance, respectively. For example, M2D3-5 is the instance created with data from the 3rd to the 5th day provided in month 2. The three last columns in Table 1 show the flight information regarding the company s operation. Column Live presents the total flying time in live requests, and column Ferry the total flying time in ferry requests, both of which are in hours. The last column, Cost, gives the corresponding total positioning cost based on ferry legs only. The costs were not provided by the company but instead were estimated based on the aircraft type and cost per hour given in Table 2. After the row corresponding to the last instance of each month, the table gives the average of the values in each column over all the instances of that month. Notice that the characteristics of the instances are rather varied for different months. In the instances of the first month, the number of live and maintenance requests are relatively balanced. This balance is not observed for the instances from the other three months. In month 2, we have significantly more maintenance than live requests, while in months 3 and 4, we observe the opposite. In addition, the number of requests in month 4 is significantly larger than those in the other months. This is an interesting feature of the data because it allows us to observe the behavior of the model in different scenarios. In all instances, we consider the six aircraft types presented in Table 2, as provided by the 12

13 Number of elements Company operation Instance L M R K N Live (h) Ferry (h) Cost M1D M1D M1D M1D M1D M1D M1D M1D Average M2D M2D M2D M2D M2D M2D M2D M2D Average M3D M3D M3D M3D M3D M3D Average M4D M4D M4D M4D M4D M4D M4D M4D M4D M4D M4D M4D M4D M4D Average Tab. 1: Description of the instances created from the real-life data provided by the company and the flying times and costs of the company s operation. 13

14 company. The first column in the table gives the rank of the type (the largest, the best), and the second and third columns give the type name and the total flying cost per hour, respectively. Type (p) Type name Cost/hour (c p) 1 LearJet Challenger Challenger 604/ Challenger Global Global Tab. 2: Description of aircraft types and the corresponding approximate flying costs per hour. 3.2 Results Tables 3 and 4 show the optimal results obtained from the model (2.14) (2.31) using the standard values for the input parameters recommended by the company. They are as follows: L = 15 minutes, M = 1 day, and T AT s i = 20 minutes for all s L and i K. In Table 3, we show the results of the first experiment, in which flight upgrades are not allowed. The results with the full model (allowing flight upgrades) are presented in Table 4. We have divided the results into two tables to enable an easier comparison between them. In both tables, the first column gives the name of the instance; columns 2 and 3 show the total flying time on live and ferry legs, respectively; column 4 shows the flying time on flight upgrades (already included in the flying time of live legs); column 5 gives the total cost of the aircraft routing, which corresponds to the optimal value of the objective function; columns 6 and 7 specify how much of the total cost is due to ferry costs and due to upgrade costs, respectively; and columns 8 to 11 show a comparison of the optimal results (given in columns 3 and 5) relative to the company s data given in Table 1 (columns 8 and 9). The comparisons involve the ferry times and total costs. For both cases, the tables show the absolute difference (Difference), which was computed as v cmp v opt, and the percentage of this difference with respect to the optimal value (Percentage), which was computed as 100 (v cmp v opt )/v opt, where v cmp and v opt are the values corresponding to the company data and the optimal results, respectively. The last column in the tables shows the computational time in seconds required by the optimization software. A time limit of one hour was set for the optimization software. Instances marked with * indicate that the solver could not prove the optimality of the solution found within this time limit. The results in Table 3 show significant reductions in the ferry times of the solutions provided by the model (most of them optimal) compared to the company s operation, even without the possibility of flight upgrades. For months 1, 2, 3 and 4, the aircraft routes provided by the model have on average 2.06, 4.43, 4.06 and hours less than the company s routes in positioning, respectively. A difference of more than 10 hours is observed for two instances in months 2 and 3 (M2D1-3 and M3D2-4), and differences of more than 20 hours are observed for three instances in month 4. The time on ferry legs was the same as in the company s routes in only three instances (M1D6-8, M3D1-3 and M3D6-8). The time reductions in ferry legs contribute to a considerable reduction in the total cost. On average, the company s routes are 12.49%, 17.36%, 13.21% and 47.60% more expensive than the routes provided by the model due to ferry costs only. For 14

15 instance M2D1-3, the total cost of the routes operated in practice was 48.97% larger than that of the optimal routes obtained using the model. For instances M4D7-9 and M4D8-10, the total costs of the routes in practice were more than 100% larger than those of the optimal routes. It can be observed that all instances in months 1, 2 and 3 were solved with short running times, whereas the instances in month 4, as expected, required longer running times, as they contain a significantly larger number of requests. Only two instances, M4D10-12 and M4D11-13, were not solved to optimality by the optimization software within the time limit. Nevertheless, the routes obtained using the model were better than those operated by company for these instances as well. Time (hours) Cost Ferry Time Total Cost Solver Instance Live Ferry Upgrade Total Ferry Upgrade Difference Percentage Difference Percentage Time (sec) M1D M1D M1D M1D M1D M1D M1D M1D Average M2D M2D M2D M2D M2D M2D M2D M2D Average M3D M3D M3D M3D M3D M3D Average M4D M4D M4D M4D M4D M4D M4D M4D M4D M4D10-12* M4D11-13* M4D M4D M4D Average Tab. 3: Results from the model without upgrades (maintenance only). Table 4 shows the results from the full model, including the possibility of flight upgrades. As the results indicate, flight upgrades lead to substantial reductions in the times of ferry legs and in the total costs. By allowing upgrades, the average time reductions in the routes of the model were 4.59 hours in the instances of the first month, 7.26 hours in the second month, 8.56 hours in the third month and hours in the fourth month. For the third and fourth month, the 15

16 ferry legs in the company s routes accounted for around 50% and 57% more hours than those in the routes obtained from the full model, respectively. For instance M3D6-8, the total cost of the executed routes by the company was 78.10% larger than that of the optimal routes. This is an interesting result, as no improvement is observed for this instance in Table 3. By allowing upgrades, the time spent on positioning is reduced from 5.13 to 2.67 hours at no extra cost, as the upgrade is from an aircraft of type 1 (LearJet 60) to type 2 (Challenger 350). Notice that a slight increase was observed in the running times required to solve the instances of months 1, 2 and 3. For the instances of the fourth month, the average running time increased by approximately, 20%. In this case, two instances (M4D10-12 and M4D12-14) could not be solved to optimality by the optimization software within the enforced time limit. Nonetheless, for all these instances, the software provided a better solution than the company s routes. Time (hours) Cost Ferry Time Total Cost Solver Instance Live Ferry Upgrade Total Ferry Upgrade Difference Percentage Difference Percentage Time (sec) M1D M1D M1D M1D M1D M1D M1D M1D Average M2D M2D M2D M2D M2D M2D M2D M2D Average M3D M3D M3D M3D M3D M3D Average M4D M4D M4D M4D M4D M4D M4D M4D M4D M4D10-12* M4D M4D12-14* M4D M4D Average Tab. 4: Results with the full model (including maintenance and upgrades). In Figure 2, we show the ferry times of the company operation and the solutions of the model without and with upgrades. It can be observed that the solutions provided by the optimization 16

17 M1D1-3 M1D2-4 M1D3-5 M1D4-6 M1D5-7 M1D6-8 M1D7-9 M1D8-10 M2D1-3 M2D2-4 M2D3-5 M2D4-6 M2D5-7 M2D6-8 M2D7-9 M2D8-10 M3D1-3 M3D2-4 M3D3-5 M3D4-6 M3D5-7 M3D6-8 M4D1-3 M4D2-4 M4D3-5 M4D4-6 M4D5-7 M4D6-8 M4D7-9 M4D8-10 M4D9-11 M4D10-12 M4D11-13 M4D12-14 M4D13-15 M4D14-16 model always have smaller ferry times than the company operation. For all but one instance (M4D10-12), the model considering upgrades on flights finds solutions with smaller ferry times than the model in which upgrades are not considered (recall that this solution is not optimal, as indicated in Table 4) Company Without upgrades With upgrades Fig. 2: Ferry times of the solutions. An additional experiment was carried out, in which all instances of month 4 were also solved with the commercial general-purpose optimization solver IBM CPLEX v The purpose was to verify the difficulty of obtaining the optimal solutions of the instances for which GLPK could not. The results are summarized in Table 5. In the table, the first and second columns show the instance name and the total cost of the company operation (ferry cost), respectively; columns 3 and 4 show the total cost of the solutions obtained with both solvers when upgrades are not allowed in the model; and the last two columns show the results with both solvers when flight upgrades are allowed in the model. CPLEX found and proved the optimal solutions to all instances in 2.71 and seconds on average using the model without and with upgrades, respectively. We emphasize in boldface the instances when GLPK could not find the optimal solution to the corresponding instance. For three instances, GLPK found solutions with higher costs than the optimal one after one hour of running, while CPLEX obtained the optimal solutions in a few seconds. This shows that a commercial solver may offer advantages when solving large-scale problems in practice. 4 Conclusion In this paper, we described and modeled an aircraft routing and scheduling problem that occurs in the planning process of companies offering on-demand transportation services. A compact mixedinteger programming formulation was presented for the problem, including aircraft maintenance events and service upgrades. Additionally, the model allows the anticipation or postponement of the starting time of flights and maintenance events, affording more flexible decision making in practice. This study was motivated by the case of a fractional ownership management company that provided real-life data based on the operation of its aircraft. Computational experiments 17