Vehicle routing problem with hard time windows and stochastic service time

In this paper we consider the Vehicle Routing Problem with Hard Time Windows and Stochastic Service Times (VRPHTWSST). We can describe the problem as follows. Consider a directed graph G = (V,A) where V = {0, 1, . . . , n} is the node set, and A = {(i, j) | i, j ∈ N} is the arc set. Node 0 represents a depot where a fleet of homogeneous vehicles is initially located and Vc = {1, . . . , n} is the customer set. A time window [ai, bi] and a stochastic service time are associated with each customer i ∈ Vc. Service time probability distributions are supposed to be known and mutually independent. A non-negative travel cost cij and travel time tij are associated with each arc (i, j) in A. Furthermore, a global “reliability” threshold 0 < α < 1 is given. The VRPHTWSST consists of finding a set of vehicle routes such that: (i) Routes start and end in node 0; (ii) All customers are served; (iii) Service at customers starts within the given time window. Vehicles are however allowed to arrive before the beginning of a time window. In this case vehicles must postpone the beginning of the service until the customer’s time window opens. In no case vehicles are allowed to arrive after the end of the time window. (iv) The global probability that the route plan is feasible with respect to customers’ time windows once the customers’ service time becomes known, is higher than the reliability threshold; (v) The travel distance is minimized. For convenience, we use