Cover inequalities for a vehicle routing problem with time windows and shifts

Said Dabia, Stefan Ropke, Tom Van Woensel

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This paper introduces the vehicle routing problem with time windows and shifts (VRPTWS). At the depot, several shifts with nonoverlapping operating periods are available to load the planned trucks. Each shift has a limited loading capacity.We solve the VRPTWS exactly by a branch-and-cut-and-price algorithm. The master problem is a set partitioning with an additional constraint for every shift. Each constraint requires the total quantity loaded in a shift to be less than its loading capacity. For every shift, a pricing subproblem is solved by a label-setting algorithm. Shift capacity constraints define knapsack inequalities; hence we use valid inequalities inspired from knapsack inequalities to strengthen the linear programming relaxation of the master problem when solved by column generation. In particular, we use a family of tailored robust cover inequalities and a family of new nonrobust cover inequalities. Numerical results show that nonrobust cover inequalities significantly improve the algorithm.

Original languageEnglish
Pages (from-to)1354-1371
Number of pages18
JournalTransportation Science
Issue number5
Early online date27 Aug 2019
Publication statusPublished - Sept 2019


  • Branch-and-cut-and-price algorithm
  • Column generation
  • Shift capacity
  • Vehicle routing problem


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