Scheduling of Time-Shared Jet Aircraft 1 PINAR KESKINOCAK AND SRIDHAR TAYUR Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213 Motivated by a real application, we consider the following aircraft scheduling problem. At any time, the aircraft are at different locations or are serving a customer and new customer requests arrive, each consisting of a departure location, departure time, and destination. Our objective is to satify these requests (by subcontracting extra aircraft if necessary) at minimum cost under additional constraints of maintenance requirements and previously scheduled trips. We show that the jet aircraft scheduling problem is NP complete and discuss three special cases. We show that the second and third special cases are also NP complete. We provide a polynomial time network flow based algorithm for the first special case and a pseudo-polynomial time dynamic programming algorithm for the second special case. We formulate the problem as a 0 –1 integer program and observe that most small and medium size problems can be solved by Cplex. For larger and difficult instances, we provide a fast heuristic with good performance. A growing number of executives these days be- lieve that having their own aircraft is really the best flight plan, because using commercial airlines for business trips may sometimes be time consuming and unreliable. On the other hand, a private jet is not always affordable for many companies, espe- cially if they are of small or medium size. Hence, selling shares of aircraft to customers who could otherwise not afford their own is a rapidly growing industry (DEL VALLE (1995) and BRYANT (1995)). Customers become partial owners of aircraft, which would provide them with an allotted number of fly- ing hours per year. A partial owner will call the company that manages these aircraft and book a trip by specifying the departure time, departure lo- cation, destination, and exclusive use information (if a customer request consists of more than one trip, the customer may ask to use the aircraft between these trips). If the customer has enough flying hours left, the company must provide an aircraft to that customer for that trip. Through the sharing of air- craft, customers avoid the high cost of ownership and other associated overheads of establishing a corporate flight department with its own mainte- nance staff and pilots. There are tax advantages, too, as partial owners are allowed to take a depreci- ation expense. For the company which sells shares of aircraft, there are two major types of costs: operating costs (fuel, maintenance, etc.) for flying the aircraft and the penalty costs for not being able to meet some customer requests without subcontracting extra air- craft (usually at a very high cost). Hence, from a cost/profit perspective, we need to solve the follow- ing aircraft scheduling problem: find a feasible schedule for the flights of aircraft, such that the operating costs and the penalty costs are minimized. The model and algorithms we develop here are cur- rently being used by one such company. Briefly, the aircraft scheduling problem has the following features: ● There are n aircraft and each aircraft can serve only one customer at a time. ● Over time, demands for trips come from custom- ers: customer j asks for a trip from a departure location ( j ) to a destination ( j ) with depar- ture time dtime( j ) and total travel time tt ( j ). In fact, a customer may ask for a set of trips, some- times with an exclusive use between them. ● At the beginning of the scheduling horizon, each aircraft is at a known location or is serving a customer. ● Each aircraft has a maintenance history, which provides the number of flight hours and land- ings remaining before the next scheduled main- tenance. In addition, there is a preset time for 1 Accepted by Gilbert Laporte. 277 Transportation Science 0041-1655 / 98 / 3203-0277 $01.25 Vol. 32, No. 3, August 1998 © 1998 Institute for Operations Research and the Management Sciences