Optimized load planning of trains in intermodal transportation

In this paper the problem of load planning for trains in intermodal container terminals is studied. The objective is to assign load units to wagons of a train such that the utilization of the train is maximized, and setup and transportation costs in the terminal are minimized. Contrary to previous approaches additionally weight restrictions for the wagons are integrated into our model. We present three different integer linear programming formulations and test them on some real-world instances. It is shown that even non-commercial MIP-solvers can solve our models to optimality in reasonable time.

m :

Number of wagons

p :

Number of wagon types

c(j):

Type of wagon j

G :

Total weight limit of the train

n :

Number of load units

q :

Number of load unit types

\({\mathcal{N}_t}\) :

Set of load units belonging to type t

l _i :

Length of load unit i

g _i :

Weight of load unit i

d _ij :

Transportation cost for load unit i to wagon j

\({\mathcal{K}_{\tau}}\) :

Set of physical configurations for wagons of type τ

\({\mathcal{B}_k}\) :

Set of type-weight lines for configuration k (first IP)

\({\mathcal{B}_k'}\) :

Set of weight distributions for configuration k (second IP)

\({\mathcal{S}_j}\) :

Set of slots on wagon j (first and second IP)

\({\kappa_{jk}^0}\) :

Initial configuration k of wagon j

α _tbs :

Feasible load unit length type t for slot s in type-weight line b (first IP)

γ _bs :

Maximum payload for slot s in type-weight line or weight distribution b (first and second IP)

λ _tks :

Feasible load unit fixation-type t for slot s in configuration k (second and third IP)

\({u^+_{i}, u^-_{i}}\) :

overhangs of load unit i (second and third IP)

\({\beta^+_{ks},\beta^-_{ks}}\) :

Feasible overhangs for slot s in configuration k (second and third IP)

\({\mathcal{S}'_j}\) :

Set of slots on wagon j (third IP)

γ _τ :

Maximum bogie payloads for wagons of type τ (third IP)

d _τ :

Distance in between the bogie attachments for wagons of type τ (third IP)

t _τ :

Tare mass for wagons of type τ (third IP)

δ _τs :

Maximum slot payload for wagons of type τ (third IP)

e _τs :

Lever for slot s for wagons of type τ (third IP)

a _j , b _j :

Bogie payloads for wagon j (third IP)

w ₁, . . . , w ₅ :

Weighting factors

Authors and Affiliations

1. Institute of Mathematics, Technical University of Clausthal, 38678, Clausthal-Zellerfeld, Germany

   Florian Bruns & Sigrid Knust

Corresponding author

Correspondence to Sigrid Knust.

Reprints and permissions