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Vehicle Routing #001

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Vehicle Routing: Models

We are currently preparing this page to outline some of the common (and not so common) models that are used to represent Vehicle Routing Problems.

The surveys page would also be a good place to look if you are looking for a particular model, or just general information about the models that are available.

If you believe that any are missing, please let us know (email:


Capacitated Vehicle Routing Model (Set Partitioning)

This model, for the Capacitated Vehicle Routing Problem, is one of the moset general models. It is based on set-partitioning and the general idea is to minimise the cost, which is represented by cj. The cost could be, for example, the distance. We have a fixed number of customers (V> and also a fixed number of vehicles (K. Note that we are not trying to minimise the number of vehicles used, which is the case in some other formulations.

The model can be formally stated as follows.

CVRP Model


  • q is the the number of feasible routes.
  • cj is the cost associated with route j.
  • xj is a binary variable with xj = 1 if route j is used in the optimal solution; 0 otherwise.
  • aij is a binary variable, where aij = 1 if vertex (customer) i is visited by route j; 0 otherwise.
  • V is the set of vertices (customers).
  • K is the number of vehicles that are available.
  • Constraint (2) ensures that each vertex (customer) is visited exactly once (i.e. it is included in one, and only one, route), with the exception of vertex zero which is usually defined as the depot, where every route starts and ends.
  • Constraint (3) imposes the condition that K circuits (vehicles) are used.
  • Constraint (4) limits the values of xj to zero or one.

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