A vehicle scheduling problem with fixed trips and time limitations Deniz Tu ¨ rsel Eliiyi a,Ã , Arslan Ornek a , Sadık Serhat Karaku ¨ tu ¨k b a Izmir University of Economics, Izmir 35330, Turkey b Dokuz Eylu ¨l University, Izmir 35100, Turkey article info Article history: Received 22 February 2007 Accepted 8 October 2008 Available online 1 November 2008 Keywords: Vehicle scheduling Fixed job scheduling Time constraints Heuristics abstract We consider minimum-cost scheduling of different vehicle types on a predetermined set of one-way trips. Trips have predetermined ready times, deadlines and associated demands. All trips must be performed. The total time of operations on any vehicle is limited. We develop a mixed integer model to find the optimal number of vehicles at a minimum cost. Based on the hard nature of the problem, we propose six heuristics. Computational results reveal that heuristics return exceptionally good solutions for problem instances with up to 100 jobs in very small computation times, and are likely to perform well for larger instances. & 2008 Elsevier B.V. All rights reserved. 1. Introduction The vehicle scheduling problem (VSP) consists of assigning a set of scheduled trips to a set of vehicles, in such a way that each trip is associated with one vehicle and a cost function is minimized (Baita et al., 2000). VSPs have different objective functions like minimizing the fixed cost or maximizing the utilization of vehicles. VSP is modeled by integer programming. There are efficient algorithms for some versions of the VSP, i.e., when all vehicles are equal and share the same depot. Nevertheless, real-life applications may turn out to be complex due not only to the dimension of the problem but also, and more importantly, to particular requirements which are inherent in practical situations. For example, there may be several vehicle types to choose from. Vehicle types may differ according to capacity or the cost of operation. Another complicating factor may be time windows, where visits to customer locations have to be made at specific times like postal and bank pickups and deliveries. Also, split deliveries may be of concern when one customer has a particular requirement, and it makes sense to have more than one vehicle assigned to that customer. As a result, most real VSPs are difficult for modeling and complex for solving. In this study, we consider the problem of determining the optimal number of different types of vehicles at a minimum cost to meet a given schedule of trips with varying demands. Each vehicle can be run for a given time interval. That is, a spread time constraint imposes an upper bound S on the total time between the start and the finish times of the operations on any vehicle. There may be idle times of the machine between these two time points, yet these times are included in the spread time. Our study is motivated by a real-life VSP, which has many complicating factors. In a games event organization, there are different vehicle types having different capa- cities and costs of operation. The daily starting and ending times of trips are fixed in advance, and each trip has to be started at the exact starting time, otherwise it cannot be performed. There are certain trips which have to be split (because of capacity restrictions of vehicles). On the other hand, the passengers of certain other trips must be carried together; that is, some trips cannot be split. There are predetermined departure and arrival points for each trip. Hence, a trip can be taken by a specific vehicle depending on the previous trips taken by that vehicle. With these Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics ARTICLE IN PRESS 0925-5273/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2008.10.005 Ã Corresponding author. Tel.: +90 2324888383; fax: +90 232 2784700. E-mail address: deniz.eliiyi@ieu.edu.tr (D.T. Eliiyi). Int. J. Production Economics 117 (2009) 150–161