First-stage model

We use the following notation for sets, parameters and variables in the first-stage model.

Sets:

R : The set of all types of available vehicles.

Ri : The set of vehicles required for serving customer i.

Ir : The set of customers requiring a type r vehicle. Customers in this set may also need

to be visited by vehicles of other types. Ar : The set of allowed arcs for type r vehicles.

T : The set of available time slots within the scheduling horizon (We suppose that the available time is split into smaller time slots with equal lengths. In the remainder of this paper, wherever we state that an event happens at time slot t, we mean that it occurs at the beginning of the time slot).

T_ridep : The set of time slots at which a type r vehicle may depart the depot to serve customer i with the condition that it can return to the depot before the end of the scheduling horizon. The vehicle may visit some other customers before finishing its tour. This set is inclusive but not necessarily exclusive, i.e., all time slots satisfying the explained condition are members of this set, but there might be some time slots in this set not respecting the condition. This set is not a part of given data. We will discuss at the end of this section how to compute it using the available data.

Parameters: ct

rij : travel cost for a type r vehicle for travelling from customer i to customer j. Super-

script t stands for “travel”.

cv

r : The fixed cost of hiring a type r vehicle. Superscript v stands for “vehicle”.

dri : The required amount of the resource (if there is any) that is consumed by a type r vehicle while serving customer i.

Cr : The capacity of type r vehicles for the resource (if there is any) that they consume

while serving customers. We have Cr = ∞ if type r vehicles do not consume any

resource for serving customers.

qrs : An lower bound on the minimum number of type r vehicles required for serving

customers in set S ⊆ Ir. We compute it by qrs= max

 l P i∈S dir/Cr m , 1  . Variables:

mr : The number of type r vehicles to hire.

xrij : 1 if a type r vehicle visits customer j immediately after customer i ; 0 otherwise. yrjt : 1 if a type r vehicle departs the depot at time slot t to visit customer j ; 0 otherwise.

Based on the given notation, we formulate the first-stage model as follows. In Model (S1), index 0 stands for the depot.

(S1) min x,y,m      X r∈R cv_rmr+ X r∈R X i,j∈Ir∪{0}: (i,j)∈Ar ct_rijxrij+ Q(x, y)      (5.1) Subject to : P i∈Ir:(0,i)∈Ar xr0i= mr r ∈ R (5.2) P i∈Ir∪{0}:(i,j)∈Ar xrij = 1 r ∈ R, j ∈ Ir (5.3) P i∈Ir∪{0}:(i,j)∈Ar xrij = P i∈Ir∪{0}:(j,i)∈Ar xrji r ∈ R, j ∈ Ir∪ {0} (5.4) P i,j∈S:(i,j)∈Ar xrij ≤ |S| − qrs r ∈ R, S ⊆ Ir : |S| ≥ 2 (5.5) P t∈T_ridep yrjt = xr0j r ∈ R, j ∈ Ir (5.6)

xrij ∈ {0, 1} r ∈ R, (i, j) ∈ (Ir∪ {0}) : (i, j) ∈ Ar(5.7) yrjt ∈ {0, 1} r ∈ R, j ∈ Ir, t ∈ T

dep

rj (5.8)

mr ≥ 0, integer r ∈ R (5.9)

Objective function (5.1) consists of the fixed costs of vehicles, the travel costs, and the second-stage cost Q(x, y) that is a function of first-stage decision variables xrij and yrjt.

We define the second-stage cost in Section 5.3.2. Constraints (5.2) and (5.3) are degree constraints for the depot and customers respectively. Constraint (5.4) is the flow conservation constraint. Constraint (5.5) guaranties that for each type of vehicles, the capacity constraints are respected and no subtour is allowed. Constraint (5.6) determines the departure time of vehicles from the depot. Constraints (5.7)-(5.9) represent the integrality constraints for first-stage decision variables. In Model (S1), we point out that constraint (5.5) eliminates subtours composed of arcs traversed by vehicles of the same type for different types of vehicles separately, but not together. We illustrate it by an example depicted in Figure 5.1. In this figure, there is no subtour for type 1 and type 2 vehicles separately, but path A-B-C-D formed by both types of vehicles is a subtour. In our problem, this type of subtour is also illegal and results in infeasibility in the scheduling of customers visits. The infeasibility happens because the start times of services to any pair of customers (i1, i2) in the subtour must satisfy starti1 < starti2 and starti1 > starti2 that is a contradiction. Therefore, although infeasibility cuts from the subproblems of L-shaped algorithm will remove such subtours, we should avoid them beforehand in the first-stage model. As discussed later, we address this issue by adding valid inequalities and also a lower bounding functional developed in Sections 5.4.1 and 5.6 respectively.

Figure 5.1 A subtour formed by two different types of vehicles.