Problem formulation

This section details a CP formulation of the VRP with quantity of goods and time constraints maintained along each route. As described in section 3.2, CP paradigm is based in three entities: (i) variables, (ii) their corresponding domains, and (iii) the constraints relating these variables. Therefore, the pro-posed CP formulation is defined according to these elements. This model can

be used as a basis for different VRP variants, and is specially suitable for the CVRP and the VRPTW. Nevertheless, one of the main virtues of CP is its versatility, so new constraints may be added without adjustment to this basic model.

Assuming the notation introduced in Chapter 2, we consider a set of n customers and a eet of m vehicles. Then, the variables used in this formulation are:

• C = c₁...c_n are the customers to serve;

• M = m₁...m_m are the available vehicles;

• Q_m = q_m1...q_mm are the vehicles capacities;

• V = v₁...v_n+2m are the visits, with domain V :: [1..m].

It should be noticed that there is one visit per customer and two special visits per vehicle: the starting and ending nodes for the vehicle, usually the depot. Thus, two subsets of V , F and L, are defined as the vehicles departure and arrival nodes:

• F = {n + 1...n + m} is the set of first visits;

• L = {n + m + 1...n + 2m} is the set of last visits.

In addition, two indexes are defined to denote the first and last visit for each vehicle:

• f_k = n + k is the first visit of vehicle k;

• l_k = n + m + k is the last visit of vehicle k.

To deal with routes, the predecessors set P is introduced to model the direct predecessor p_i of each visit i (∀i ∈ V − F ):

• P = p₁...p_n+m is the predecessors set, with domain P :: [1..n + m] ::

(V − L);

Similarly, the set S is introduced to model the direct successor si of each visit i (∀i ∈ V − L):

• S = s₁...s_n+m is the successors set, with domain S :: [1..n, n + m + 1..n + 2m] :: (V − F ).

By convention, each first visit of a vehicle has as predecessor the vehicle's last visit (∀k ∈ M p_fk = l_k). Likewise, each last visit of a vehicle has as successor the vehicle's first visit (∀k ∈ M s_lk = f_k).

The use of the predecessor and successor sets creates a symmetric model.

Although the solution space could be specified using only one of these sets without changing the set of solutions of the problem, defining both variables helps on propagating values more e ciently during the search. Thus, this re-dundant modeling permits making additional inferences which can significantly prune the search tree.

The predecessor and successor variables form a permutation of V and are therefore subject to the di erence constraints (4.17) and (4.18).

p_i 6= p_j ∀i, j ∈ V ∧ i < j (4.17) s_i 6= s_j ∀i, j ∈ V ∧ i < j (4.18) These equations force predecessor and successor sets to contain no repe-titions. Thus, one customer can have one and only one predecessor and suc-cessor. Notice that, when minimizing the number of vehicles is an objective, these constraints domains are restricted to ∀i, j ∈ V₁...V_n∧ i < j to permit the spare vehicles to remain at the depot.

The successor variables are kept consistent with the predecessor variables via the following coherence constraints:

s_pi = i ∀i ∈ V − F (4.19)

p_si = i ∀i ∈ V − L (4.20)

Equations (4.19) and (4.20) connect the concepts successor and predecessor as follows: the former says that i is the successor of its predecessor, and the latter says that i is the predecessor of its successor.

To model multiple vehicles, we let the variables v_i from the set V to have a domain [1..m] for each visit i, representing the vehicle which performs the visit. Along a route, all visits are performed by the same vehicle. This is maintained by the following path constraints:

v_i = v_pi ∀i ∈ V − F (4.21)

vi = vsi ∀i ∈ V − L (4.22)

Equations (4.21) and (4.22) are used to ensure that the vehicle assigned to i is the same as that assigned to its predecessor and successor. Naturally, for the first and last visits, the following constraints (4.23) are imposed:

v_fk = v_lk = k ∀k ∈ M (4.23)

Another three sets of variables are defined to represent visits' demands, cumulated capacities and the time for each visit:

• R = r1...rn is the demands list, determining the amount of goods to be picked up (r_i > 0) or delivered (r_i < 0) at each visit i;

• Q = q₁...q_n is the cumulated capacity list. After every visit i, q_i ≥ 0 is a constrained variable representing the quantity of goods in the vehicle serving the visit;

• T = t₁...t_n is the times list, indicating the time when the visit i is per-formed.

To maintain the load on the vehicles at each point in their route, the following capacity constraints are enforced:

q_i = q_pi+ r_i ∀i ∈ V − F (4.24) q_i = q_si− r_si ∀i ∈ V − L (4.25) Equations (4.24) and (4.25) count the goods picked up in a route, and the goods delivered in that route. Thus, the first constraint says the goods accumulated after visiting customer i is the addition of those accumulated in its predecessor plus those picked up in i (ri 6= 0). The second constraint is similar but using the successor. Furthermore, the maximum capacity Q_k defined for a vehicle k must limit the accumulated capacities for every customer visited by that vehicle. This can be done either with an extra constraint or with domains bounding in case all vehicles have the same maximum capacity:

Q :: [0..Q_k] ←→ 0 ≤ q_i ≤ Q_k ∀i ∈ V, k ∈ M (4.26)

A constraint (4.27) can be added to model that all vehicles begin their routes empty, as in the CVRP or the VRPTW. Leaving these quantities un-constrained would permit modeling different problems.

q_i = 0 ∀i ∈ F (4.27)

Time is maintained in the same manner as vehicle load except that waiting is normally allowed, and so the time constraints (4.28) and (4.29) maintain an inequality rather than an equality.

t_i ≥ 0 ∀i ∈ V

t_i ≥ t_pi+ τ_pii [+τ_i] ∀i ∈ V − F (4.28) ti ≤ tsi − τisi [−τi] ∀i ∈ V − L (4.29) Equations (4.28) and (4.29) bound the accumulated time spent by a vehicle visiting the customer i. This time is, at least, the accumulated time in the predecessor of i, plus the travel time from the predecessor to i (τ_pii). Equally, this time must be, at most, the accumulated time in the successor, minus the travel time from i to its successor (τisi). This constraints may include the time spent at customer i (τ_i) if it is required by the problem instance.

Time windows on customers are specified by adding time windows con-straints on the time variables:

a_i ≤ t_i ≤ b_i ∀i ∈ V (4.30)

Equation (4.30) states that customer i must be visited between times a and b. Usually, depot's time window is known as the scheduling horizon. De-pending on the problem at hand, a customer i may be serviced out of its time window with a related penalty in the objective function (soft time windows).

Minimizing these penalties becomes a secondary goal of the problem. In case this is not permitted, a vehicle is allowed to wait, but it cannot perform the visit i after its latest service time bi (hard time windows). To model both situations, time windows constraints (4.30) are turned into (4.31) and (4.32).

∆a_i ≥ a_i− t_i ∀i ∈ V (4.31)

∆b_i ≥ t_i− b_i ∀i ∈ V (4.32)

The soft time windows are allowed by defining the ∆a and ∆b values as

∆a_i ≥ 0 ∀i ∈ V

∆b_i ≥ 0 ∀i ∈ V (4.33)

while the hard time windows case is obtained by enforcing the ∆b to be equal to zero:

∆a_i ≥ 0 ∀i ∈ V

∆b_i = 0 ∀i ∈ V (4.34)

Finally, the cost function of the problem is based on the total traveled distance. Considering δ_ij as the travel distance from visit i to visit j, the cost function is defined as follows:

It should be noticed the use of both the predecessor and successor variables to constrain the cost, which is usually more effective during search than using one single set.

In general, for CVRP problems the distance is considered to be equal to the traveling time, i.e. time variables may replace distances in the cost function.

Therefore, equations (4.37) and (4.38) may be used to bind the cost function value.

For the VRPTW, in case soft time windows are allowed, the cost function may include penalty terms representing the time windows violations:

d = X

where ρ and ω are penalty weights that can be adjusted according to different modeling purposes.

Eventually, the cost function may also be defined according to the last visit performed by each vehicle, following the equation (4.41).

d = X

k2M

t_lk (4.41)

In order to improve the propagation during the search, equation (4.41) may be used simultaneously to the other defined cost functions, so tighter bounds are obtained on the total cost.