Explicit Dantzig-Fulkerson-Johnson formulation

The library in AIMMS that solves a Capacitated Vehicle Routing Problem (CVRP) contains different formulation options. The formulations have different methods of eliminating subtours. In this article the Explicit Dantzig-Fulkerson-Johnson formulation is discussed. This is an example of a subtour in a route for a CVRP:

../../_images/Subtour.png

Idea behind the formulation

The Dantzig Fulkerson Johnson (DFJ) formulation uses subsets to eliminate subtours. The set of all nodes is V = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. The set of nodes that makes a subtour is S = {3, 4, 5} and is a subset of V.

../../_images/EDFJ1.png

This subset contains three nodes. The number of arcs between the nodes in this subset is also three, which is why it makes a subtour. If the number of arcs should be less than three, one arc should be removed. For example, the arc between node 3 and node 5.

../../_images/EDFJ2.png

Now one arc has to go from node 3 to a node outside of the subset and another arc has to go from node 5 to a node outside of the subset. The following route can then be formed:

../../_images/EDFJ3.png

There are two ways of explaining this way of eliminating subtours:

  1. The number of arcs between nodes in the subset should be less than the number of nodes in that subset.

  2. The number of arcs that connect a node from the subset to a node outside of the subset should be at least 2.

../../_images/EDFJ4.png

The second explanation is used for the CVRP Library. To eliminate all possible subtours, it should apply to every subset that could be a subtour. Subsets containing the depot cannot be subtours, for example S = {1, 9, 10, 11}. This is just a regular tour for one of the vehicles. A subset with 0 or 1 element can also not be a subtour. So it should apply to every possible subset of V that has at least two elements and does not contain the depot.

Subtour elimination constraints

The binary variable \(x_{ijk}\) has a value of 1 if vehicle k drives from node i to node j. The constraint can be formulated as follows:

\[\sum_{i \in S, j \notin S}{x_{ijk}} \geq 2 \qquad S \subset V \setminus \{1\}, \enspace 2 \leq |S| \leq n - 2\]

AIMMS

In the CVRP library this formulation is implemented in the section: Explicit Dantzig Fulkerson Johnson Section. In order to create constraints about subsets, the subsets should be generated first. This happens in the procedure Create_Subsets. The body of this procedure is as follows:

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empty s_CostumerSubset, s_SubsetNumber, p01_Subsets;


repeat
                !add subset (if it contains at least two cities)
                if card(i_selectedCostumer) >= 2 and card(i_selectedCostumer) <=
                card(s_Nodes) - 2 then
                        s_SubsetNumber += card(s_SubsetNumber ) + 1 ;
                        ep_LastSubsetNumber := last(s_SubsetNumber);
                        p01_Subsets( i_SelectedCostumer, ep_LastSubsetNumber ) := 1;
                endif;

                break when s_CostumerSubset = s_Costumers;

        block !generate next subset (using binary counting)
                ep_lastUnselectedCostumer := last( i | not ( i in s_CostumerSubset ) );
                for i | i > ep_lastUnselectedCostumer do
                        s_CostumerSubset -= i ;
                endfor ;
                s_CostumerSubset += ep_lastUnselectedCostumer ;
        endblock ;

endrepeat ;

Every possible subset of s_Nodes is checked using binary counting. All subsets without the depot and with a minimum of two nodes will be created. A number is then added to the set s_SubsetNumber. The binary parameter p01_Subsets indicates which nodes are in that subset.

  • line 15 - line 21: The next subset (s_CostumerSubset) is generated using binary counting.

  • line 6 - line 11: If s_CostumerSubset contains at least two nodes, then that subset is added.

  • line 13: The procedure should stop when s_CostumerSubset contains all costumers. Because with binary counting, all the following subsets would contain the depot.

Generating the constraints

Using the Explicit Dantzig-Fulkerson-Johnson, for every subset, a constraint is generated. It uses subsets to eliminate subtours. The idea behind the formulation is that, for every subset that could form a subtour, at least two arcs should connect nodes from the subsets to nodes outside of the subset. This article (EDFJ) elaborates on this formulation.

\(V\) is the set of all nodes from 1 to n (depot is n = 1). \(S\) is a subset of \(V\). The binary variable \(x_{ijk}\) has a value of 1 if vehicle k drives from node i to node j. The constraint can be formulated as follows:

\[\sum_{i \in S, j \notin S}{x_{ijk}} \geq 2 \qquad S \subset V \setminus \{1\}, \enspace 2 \leq |S| \leq n - 2\]

Note that there are as there are an exponential number of subsets, there are also an exponential number of constraints generated. For instance: All subsets with at least two elements, that do not contain the depot, should be generated. The number of subsets of a set with 10 elements = \(2^{10}\). The number of subsets thereof that contain 0 elements or all elements = 2. The number of subsets thereof that contain 1 element (or all but 1) = 20. So the number of generated subtour elimination constraints is \(2^{10} – 2 – 20 = 1002\).

Last Updated: September, 2020