Journal of Computational Science 34 (2019) 1–10 Contents lists available at ScienceDirect Journal of Computational Science journal homepage: www.elsevier.com/locate/jocs Vehicle routing problem with time windows having stochastic customers demands and stochastic service times: Modelling and solution Rajeev Goel ∗ , Raman Maini, Sandhya Bansal Government College, Naraingarh, 134101, India a r t i c l e i n f o Article history: Received 8 August 2018 Received in revised form 6 January 2019 Accepted 8 April 2019 Available online 27 April 2019 Keywords: Vehicle routing problem Stochastic demands Service times Ant colony system Firefly optimization a b s t r a c t On-time and foolproof demand satisfaction has gained attention in today’s transportation and logis- tics industries due to their direct impact on business success. However, the uncertainty in customer’s demand, travel times or service times make it difficult to achieve 100% accuracy of these objectives. In the present study, one such issue has been considered as vehicle routing problem with time windows having stochastic demands and stochastic service times. To address this problem, a mathematical model has been developed which tries to maximize customer’s satisfaction and at the same time minimize total transportation cost. A modified ant colony system is proposed to solve the developed mathematical model. Numerical results on suitably modified datasets of Solomon benchmarks show that the pro- posed approach provides low cost solutions while maximizing the number of customers served and minimizing the penalties imposed due to late deliveries with little increase in total travelled distance. Cer- tain penalties formulations provide several managerial insights to decision makers in the transportation industry. © 2019 Elsevier B.V. All rights reserved. 1. Introduction Classical vehicle routing problem (VRP) is concerned with ser- vicing of a number of geographically distributed locations with a limited fleet of available vehicles. An optimal number of minimum vehicles has to be determined such that each of the vehicle starts from the central warehouse, serves the customers allocated to it and then returns back to the warehouse within a specified time horizon (Laporte [16], Goel & Maini [11]). Vehicle routing problem with time windows (VRPTW) is a variant of VRP, which has addi- tional constraints of servicing of each customer within its specified time window. Primarily, the aim of logistic companies is to min- imize the total transportation cost, involving the cost of vehicles used and the fuel cost for total traveled distance ensuring at same time, the timely delivery of goods to the customers in their spec- ified windows. Further, the dispatcher companies primarily focus on not only reducing the distribution costs but at the same time improving the customer satisfaction by timely delivery of goods (Baldacci [2], Toth & Vigo [31]). In the classical deterministic version of VRP, all of the variables like travel time, the demand of each cus- tomer etc. are assumed to be known in advance while the route is ∗ Corresponding author. E-mail address: rcse123@gmail.com (R. Goel). being planned. However, such models do not simulate the real-life behaviors, where some or all of the component variables may not be precisely known in advance. Due to random nature of travel times because of traffic etc. and unpredictable demands of customers etc., the planned routing decisions may become inappropriate or result in poor quality service. This paper deals with the stochastic version of VRPTW known as SVRPTW, where the demands of customers and their servicing times are not known precisely. However, from the past experience, it has been observed that in general demand follow some known statistical distribution (Gendreau et al. [9]). The main issue with VRPSD is that it introduces another uncertainty in the situation i.e. whether the given vehicle will be able to fulfill the total demand of its assigned customers or not. Moreover, it has been assumed that service times are also correlated to demand, which addition- ally imparts uncertainty in arrival time at customer’s location with its associated time windows. Ignorance of such uncertainties while planning the route may result in poorly planned solutions. Arrival of a vehicle at a customer with insufficient supply so that it can- not satisfy the total actual demand of that customer is considered a route failure. In general, a commonly adopted action in such cases has been that the vehicle returns back to the depot to reload and then resumes the service from the point of failure. However, due to the additional time consumed in such an action, the vehicle may miss the time windows of the next customer/s along the sched- https://doi.org/10.1016/j.jocs.2019.04.003 1877-7503/© 2019 Elsevier B.V. All rights reserved.