The two-period travelling salesman problem applied to milk collection in Ireland

We describe a new extension to the Symmetric Travelling Salesman Problem (STSP) in which some nodes are visited in both of two periods and the remaining nodes are visited in either one of the periods. A number of possible Integer Programming Formulations are given. Valid cutting plane inequalities are defined for this problem which result in an, otherwise prohibitively difficult, model of 42 nodes becoming easily solvable by a combination of cuts and Branch-and-Bound. Some of the cuts are entered in a ‘pool’ and only used when it is automatically verified that they are violated. Other constraints which are generalisations of the subtour and comb inequalities for the single period STSP, are identified manually when needed. Full computational details of solution process are given.