# A Polyhedral Study of the Elementary Shortest Path Problem with Resource Constraints

## Abstract

The elementary shortest path problems with resource constraints (ESPPRC) in graphs with negative cycles appear as subproblems in column-generation solution approaches for the well-known vehicle routing problem with time windows (VRPTW). ESPPRC is \(\mathcal {NP}\)-hard in the strong sense [8]. Most previous approaches alternatively address a relaxed version of the problem where the path does not have to be elementary, and pseudo-polynomial time algorithms based on dynamic programming are successfully applied. However, this method has a significant disadvantage which is a weakening of the lower bound and may induce a malfunction of the algorithm in some applications [9]. Additionally, previous computational studies on variants of VRPs show that labeling algorithms do not outperform polyhedral approaches when the time windows are wide [13] and may not even be applied in some situations [7]. Furthermore, an integer programming approach is more flexible that allows one to easily incorporate general branching decisions or valid inequalities that would change the structure of the pricing subproblem.

In this paper we introduce an ILP formulation of the ESPPRC problem where the capacity and time window constraints are modeled using path inequalities. Path inequalities have been used by Ascheuer et al. [1] and Kallehauge et al. [13], respectively, in solving the asymmetric traveling salesman problem with time windows and the VRPTW. We study the ESPPRC polytope and determine the polytope dimension. We present a new class of strengthened inequalities lifted from the general cutset inequalities and show that they are facet-defining. Computational experiments are performed on the same ESPPRC instances derived from the Solomon’s data sets [9]. Results compared with previous formulations prove the effectiveness of our approach.

## Keywords

Elementary shortest path with resource constraints Vehicle routing problem with time windows Integer programming Polyhedral study## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Ascheuer, N., Fischetti, M., Grötschel, M.: A polyhedral study of the asymmetric traveling salesman problem with time windows. Networks
**36**(2), 69–79 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Balas, E., Fischetti, M., Pulleyblank, W.R.: The precedence-constrained asymmetric traveling salesman polytope. Mathematical Programming
**68**(1–3), 241–265 (1995)MathSciNetzbMATHGoogle Scholar - 3.Baldacci, R., Mingozzi, A., Roberti, R.: New route relaxation and pricing strategies for the vehicle routing problem. Operations Research
**59**(5), 1269–1283 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Boland, N., Dethridge, J., Dumitrescu, I.: Accelerated label setting algorithms for the elementary resource constrained shortest path problem. Operations Research Letters
**34**(1), 58–68 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Dantzig, G., Fulkerson, R., Johnson, D.S.: Solution of a large-scale traveling salesman problem. Operations Research
**2**(4), 393–410 (1954)MathSciNetGoogle Scholar - 6.Desrochers, M., Desrosiers, J., Solomon, M.: A new optimization algorithm for the vehicle routing problem with time windows. Operations Research
**40**(2), 342–354 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Drexl, M., Irnich, S.: Solving elementary shortest-path problems as mixed-integer programs. OR Spectrum
**36**(2), 281–296 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Dror, M.: Note on the complexity of the shortest path models for column generation in vrptw. Operations Research
**42**(5), 977–978 (1994)CrossRefzbMATHGoogle Scholar - 9.Feillet, D., Dejax, P., Gendreau, M., Gueguen, C.: An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems. Networks
**44**(3), 216–229 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Graves, G.W., Mcbride, R.D., Gershkoff, I., Anderson, D., Mahidhara, D.: Flight crew scheduling. Management Science
**39**(6), 657–682 (1993)CrossRefzbMATHGoogle Scholar - 11.Irnich, S., Villeneuve, D.: The shortest path problem with resource constraints and k-cycle elimination for \(k\ge 3\). INFORMS J. Computing
**18**(3), 391–406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Jepsen, M.K., Petersen, B., Spoorendonk, S.: A branch-and-cut algorithm for the elementary shortest path problem with a capacity constraint. Technical report, Department of Computer Science, University of Copenhagen (2008)Google Scholar
- 13.Kallehauge, B., Boland, N., Madsen, O.: Path inequalities for the vehicle routing problem with time windows. Networks
**49**(4), 273–293 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Lozano, L., Duque, D., Medaglia, A.L.: An exact algorithm for the elementary shortest path problem with resource constraints. Transportation Science
**50**(1), 348–357 (2016)CrossRefGoogle Scholar - 15.Di Puglia Pugliese, L., Guerriero, F.: A survey of resource constrained shortest path problems: exact solution approaches. Networks
**62**(3), 183–200 (2013)MathSciNetzbMATHGoogle Scholar - 16.Righini, M., Salani, M.: New dynamic programming algorithms for the resource constrained elementary shortest path problem. Networks
**51**(3), 155–170 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Rousseau, L.-M., Gendreau, M., Pesant, G., Focacci, F.: Solving vrptws with constraint programming based column generation. Annals of Operations Research
**130**(1), 199–216 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Solomon, M.M.: Vehicle routing and scheduling with time window constraints: Models and Algorithms. PhD thesis, Department of Decision Sciences, University of Pennsylvania (1983)Google Scholar
- 19.Taccari, L.: Integer programming formulations for the elementary shortest path problem. European Journal of Operational Research
**252**(1), 122–130 (2016)MathSciNetCrossRefzbMATHGoogle Scholar