Minimizing fleet operating costs for a container transportation company☆
Section snippets
Introduction and problem statement
A core problem faced by container trucking companies deals with a set of transportation orders at minimum cost. The essential decisions to be taken are: how to partition the set of transportation orders so that each subset can be executed by a single driver; to whom to assign such subsets of orders; how to reduce the misplacement of containers produced by the two previous operations; see, e.g., Crainic and Laporte (1997) and Powell et al. (1995).
This paper addresses the problem that a container
Mathematical formulation and solution approach
In this section an approximate solution approach for Problem 1 is proposed.
From a mathematical point of view, Problem 1 could be formulated as:where J0 represents the overall cost, function of the initial state, the decisions taken and the orders to execute; expectation of J0 is evaluated, since future transportation orders can only be estimated.
We are interested in dynamically solving (1), i.e., at each day t, we aim at determining the optimal decisions ut
Solution of the single day problem
Problem (4a), (4b), (4c), (4d), (4e), (4f), (4g), (4h) is too difficult to be solved exactly for typical real-world instances (with hundreds of transportation orders and, consequently, tens of thousands of pairings). Hence, we decided to decompose it by separating the different decisions that the fleet manager has to take (i.e., which pairings to execute, which resources to use, and which containers to reposition). In order to perform this decomposition, constraints (4d), (4e), (4g) have been
Lower and upper bounds
In this section, a lower and an upper bound of Problem (4a), (4b), (4c), (4d), (4e), (4f), (4g), (4h) are presented.
Implementation details
The test algorithm (in the following also referred to as Ctr_opt module) was coded in ANSI C programming language, with calls to Cplex Callable Library (Cplex Linear Optimizer 4.0.7). Ctr_opt module was implemented on a COMPAQ AlphaServer DS20e running HP Tru64 Unix V4.0F operating system.
Fig. 4 shows a high level flow chart of the test algorithm. During the subgradient iterations, all the Lagrangian multipliers involved (vectors λ, μ and α) have been updated using a scalar stepsize that has
Summary and conclusions
This paper describes and tackles a fleet management problem that arises in container transportation industry. We have focused on the minimization of the present and future operating costs incurred by the container carrier. By means of Lagrangian relaxation, the integer programming model has been decomposed into three sub-problems.
We have set up numerical experiments using randomized data sets as well as a real-world data set of an Italian container transportation company, in order to assess the
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The research described in this paper has been partially supported by C.N.R. (National Research Council of Italy) contracts 00.00333.ST74 and 00.00339.ST74, and by “Fondo Trieste-Progetti di ricerca scientifica e tecnologica-Anno 1998”.