A Container Loading Problem MILP-Based Heuristics Solved by CPLEX: An Experimental Analysis

  • Stefano GliozziEmail author
  • Alessandro Castellazzo
  • Giorgio Fasano
Part of the Springer Optimization and Its Applications book series (SOIA, volume 105)


The issue of placing small boxes orthogonally, generally with the possibility of rotations, into a big box, maximizing the loaded volume, is usually referred to as the container loading problem. Despite its being notoriously of an NP-hard typology, a number of algorithms work out this problem very efficiently. The task becomes, nonetheless, even more challenging when additional conditions have to be taken account of. In such cases, a modeling-based approach is supposedly the most suitable and this definitely holds, in particular, when balancing requirements are posed. These, indeed, entail constraints of strong global impact that can hardly be coped with by sequential procedures, based on a step by step incremental loading of items.

MIP (Mixed Integer Programming) models relevant to the container loading problem or possible extensions of it are available in specialized literature. A dedicated MILP (Mixed Integer Linear Programming) formulation, supporting an overall heuristic approach, addressed to non-standard packing issues, is discussed in another chapter of this book. Hereinafter, some relevant computational aspects are looked into, restricting the consideration to the container loading problem, as per its classical statement. An ad hoc heuristics, derived from the above-mentioned overall approach, is outlined. The use of IBM ILOG CPLEX as an MILP optimizer is considered. Case studies concerning the solution of the MILP model tout court, when the instances involved are not of a large-scale nature, are reported first. Outcomes relevant to the ad hoc heuristics are further shown through a number of difficult instances. Examples of container loading issues, involving also balancing conditions, are additionally provided.


Container loading Orthogonal packing with rotations Mixed integer linear programming MILP model Heuristics CPLEX Computational results 



We wish to thank Janos D. Pintér for his accurate review of the manuscript. We are also grateful to Jane Evans for her support. Alessandro Castellazzo acknowledges the grant provided by Lagrange Project – Crt Foundation, Turin, Italy.


  1. 1.
    Bortfeldt, A., Wäscher, G.: Container Loading Problems – A State-of-the-Art Review. FEMM Working Papers 120007, Otto-von-Guericke University Magdeburg, Faculty of Economics and Management (2012)Google Scholar
  2. 2.
    Fasano, G.: A modeling-based approach for non-standard packing problems. In: Fasano, G., Pintér, J.D. (eds.) Optimized Packings and Their Applications. Springer Optimization and Its Applications. Springer Science + Business Media, New York (2015)Google Scholar
  3. 3.
  4. 4.
    Pisinger, D.: Heuristics for the container loading problem. Eur. J. Oper. Res. 141(2), 382–392 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Rothberg, E.: An evolutionary algorithm for polishing mixed integer programming solutions. INFORMS J. Comput. 19(4), 534–541 (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fasano, G.: Solving Non-standard Packing Problems by Global Optimization and Heuristics. SpringerBriefs in Optimization, Springer Science + Business Media, New York (2014)Google Scholar
  7. 7.
    Rietz, J.: Special Difficulties in the Three-Dimensional Container Loading Problem. Universidade do Minho, Braga (2008)Google Scholar
  8. 8.
    Bischoff, E.E., Ratcliff, M.S.W.: Issues in the development of approaches to container loading. OMEGA 23(4), 377–390 (1995)CrossRefGoogle Scholar
  9. 9.
    Fasano, G., Lavopa, C., Negri, D., Vola, M.C.: CAST: a successful project in support of the international space station logistics. In: Fasano, G., Pintér, J.D. (eds.) Optimized Packings and Their Applications. Springer Optimization and Its Applications. Springer Science + Business Media, New York (2015)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Stefano Gliozzi
    • 1
    Email author
  • Alessandro Castellazzo
    • 2
  • Giorgio Fasano
    • 3
  1. 1.IBM Italia S.p.A.RomeItaly
  2. 2.Altran Italia S.p.A. Consultant c/o Thales Alenia Space Italia S.p.A.TurinItaly
  3. 3.Thales Alenia Space Italia S.p.A.TurinItaly

Personalised recommendations