Innovative Applications of O.R.
Two heuristic solution concepts for the vehicle selection problem in line haul transports

https://doi.org/10.1016/j.ejor.2011.09.010Get rights and content

Abstract

In this article we will develop a mathematical model for a cost-efficient selection of vehicles with varying capacities for line haul transports with leasing options. For this integer optimization problem, which is a variant of the generalized assignment problem known as NP-hard, we will compare two heuristic solution concepts and try to answer the question in which cases a user should choose an exact or approximate solution concept depending on different data instances of the problem.

Highlights

► A mathematical model for vehicle selection for line haul in hub-and-spoke networks is developed. ► Two different heuristic solution concepts are proposed based on real life problem properties. ► Their efficiency for different generated problems is analyzed. ► Conditions are stated in which cases such heuristic solution concepts lead to good results.

Section snippets

Problem formulation

A transportation network is defined generally as a material flow system characterized by elements such as pickup points, delivery points and consolidation points as well as relationships between these elements. We clarify this definition by using the hub-and-spoke system as an example of a transportation network. A hub-and-spoke system is characterized by a spatial and temporal organization of the material flows within a geographic (trading) area. This transportation network ensures that the

Modeling the vehicle selection problem for line haul transport

Line haul transport comprises transportation of goods from each local depot of a service area to the central hub and the transportation of goods from this hub to each local depot of a service area. Vehicles for the transportation can either be chosen from the own limited fleet or from an unlimited, external fleet from road carriers. This ensures that a feasible solution always exists. The following model assumes that the demand of each service area is symmetric, i.e. incoming and outgoing

Relevant literature

The generalized assignment problem has a lot of applications, among them allocation of memory spaces in computers, assignment of software development tasks to programmers, inventory matching and vehicle routing problems. So the literature is rich in solution concepts and algorithms.

Because of this wide range of real life problems that can be formulated as generalized assignment problems the theoretical attention for developing effective algorithms is high. The generalized assignment problem is

Two solution concepts for the vehicle selection problem

In Section 2 the mathematical formulation for the symmetric case of the vehicle selection problem was given. In this section we present an assignment heuristic which is based on the decision process often found to be performed by practitioners and an adaption of the mathematical model to iteratively solve the problem.

Numerical experiments: description of the test instances

This section shows the design of the test instances which were used. The following list gives an overview of the problem parameters which are changed during the experiments:

  • Number of service areas ∣S = 10, 50, 100, 150, 200, 300, 400.

  • Number of different vehicle types ∣V = 3, 4, 5.

  • Cost difference of the vehicle types: 0.001, 0.1, 0.2, 0.3, 0.4, 0.5.

  • Structure of demand distribution in the interval [10, 100[: uniformly distributed demand in the interval; higher demand for customers located close to the hub;

Comparison of the heuristics with the lower bound of the CPLEX solver

Solutions for the test instances were calculated on Itanium2 processors with 1.6 GHz, 8 GB RAM per processor. Lower bounds are calculated using the IBM ILOG CPLEX solver (version 12.1, single threaded, maximum running time: 3600 s (wall clock time)).

The solution quality of the two heuristic solution concepts is compared using the gap (sh  slb)/slb, where slb is the lower bound calculated by the CPLEX solver and sh is the heuristic solution quality.

We investigate in the experiments the following

Conclusion and further research

This analysis shows that the limited assignment heuristic can lead to good results provided that the problem instance has a suitable structure. The cost difference of the vehicle types is identified as an important factor. In case the relative costs are similar for different vehicle types the solution quality of the limited sequential assignment heuristic deteriorates, whereas the gap of the sequential assignment heuristic is (except in one case) below 0.01, here. For different problem

Acknowledgement

We thank Prof. Gfrerer for his comments on the paper.

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