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Flow-based integer linear programs to solve the weekly log-truck scheduling problem

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Abstract

In this paper we present the solution to a weekly log-truck scheduling problem (LTSP) integrating the routing and scheduling of trucks where all goods are transported in full truckloads. We must take into account pick-up and delivery requirements, multiple products, inventory levels, and lunch breaks. The objective is to minimize the overall transportation cost including wait times and the empty and loaded distance traveled. Our solution is based on a two-phase approach. The first phase involves an integer linear program that determines the destinations of full truckloads. The second phase uses an implicit integer linear program based on an arc formulation to ensure that the trucks are routed and scheduled at a minimum cost. Experiments have been conducted using Cplex 12.4.0, and almost all instances were solved within six hours with a reasonable gap.

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Notes

  1. http://cfs.nrcan.gc.ca/pages/269?lang=en_CA.

  2. http://www.fao.org/forestry/country/57478/en/fin/.

  3. FPInnovation is a private, not-for-profit R&D organization whose mission is to improve Canadian forestry operations.

Abbreviations

B :

Set of all bases

c l :

Cost of waiting time of a log-loader per unit of time

c a :

Cost associated with arc a (empty driven arcs and truck waiting times)

t l :

Loading time of one shipment

t u :

Unloading time of one shipment

H :

Optimization horizon

\(T_{a}^{s}\) :

Start time associated with arc a

\(T_{a}^{e}\) :

End time associated with arc a

K fm :

Number of full truckloads to be delivered from forest area f to wood mill m

L bfm :

Set of loaded-trip arcs linking forest area f and wood mill m and associated with base b

L b :

⋃ f∈F ⋃ m∈M L bfm , set of loaded-trip arcs associated with base b

A +(s b ):

Set of exiting arcs from the source node s b associated with base b

A −(t b ):

Set of entering arcs into the sink node t b associated with base b

A +(i):

Set of exiting arcs from node i

A −(i):

Set of entering arcs into node i

\(A_{bf}^{T}\) :

Set of loading arcs (a b ) loading at forest area f and associated with base b, such that their start times \(T_{a_{b}}^{s} \in[T,T+t^{l} [\), T≤H

\(A_{bm}^{T}\) :

Set of unloading arcs (a b ) unloading at wood mill m and associated with base b, such that their start times \(T_{a_{b}}^{s} \in [T,T+t^{u} [\), T≤H

C bf :

Set of loading arcs associated with forest area f and base b

C b :

⋃ f∈F C bf , set of loading arcs associated with base b

D b :

Set of unloading arcs associated with base b

E b :

Set of empty driven arcs associated with base b

W b :

Set of truck waiting arcs associated with base b

N b :

Set of nodes associated with base b

V b :

Number of trucks for base b

A :

Set of all arcs.

\(x_{a_{b}}\) :

Number of trucks that follow arc a b ∈A

\(t_{f}^{min}\) :

Start time of log-loader associated with forest area f

\(t_{f}^{max}\) :

End time of log-loader associated with forest area f

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Authors and Affiliations

  1. Ecole Mohammadia d’Ingénieurs (Rabat), Université Mohammed V Agdal, Avenue Ibnsina, B.P. 765, Agdal Rabat, Maroc

    Nizar El Hachemi

  2. Département de Mathématiques et Génie Industriel, Ecole Polytechnique de Montréal, C.P. 6079, Succursale Centre-Ville, Montreal, Québec, H3C 3A7, Canada

    Issmail El Hallaoui, Michel Gendreau & Louis-Martin Rousseau

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  1. Nizar El Hachemi
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  3. Michel Gendreau
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  4. Louis-Martin Rousseau
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Correspondence to Nizar El Hachemi.

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El Hachemi, N., El Hallaoui, I., Gendreau, M. et al. Flow-based integer linear programs to solve the weekly log-truck scheduling problem. Ann Oper Res 232, 87–97 (2015). https://doi.org/10.1007/s10479-014-1527-4

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