Elsevier

Computers & Operations Research

Volume 88, December 2017, Pages 316-323
Computers & Operations Research

Time-dependent green Weber problem

https://doi.org/10.1016/j.cor.2017.04.010Get rights and content

Highlights

  • The green Weber problem (GWP) and the time-dependent GWP (TD-GWP) are introduced.

  • The GWP and TD-GWP are extensions of the classical Weber problem.

  • Both problems involve minimization of the CO2 emitted in the distribution system.

  • Both problems aim to find the optimal vehicle speeds and the facility location.

  • Second order cone programming formulations are proposed for the problems.

Abstract

We consider an extension of the classical Weber problem, named as the green Weber problem (GWP), in which the customers have one-sided time windows. The GWP decides on the location of the single facility in the plane and the speeds of the vehicles serving the customers from the facility within the one-sided time windows so as to minimize the total amount of carbon dioxide emitted in the whole distribution system. We also introduce time-dependent congestion which limits the vehicle speeds in different time periods and call the resulting problem as the time-dependent green Weber problem (TD-GWP). In the TD-GWP, the vehicles are allowed to wait during more congested time periods. We formulate the GWP and TD-GWP as second order cone programming problems both of which can be efficiently solved to optimality. We show that if the traffic congestion is non-increasing, then there exists an optimal solution in which the vehicles do not wait at all. Computational results are provided comparing the locations of the facility and the resulting carbon dioxide emissions of the classical Weber problem with those of the GWP and comparing the GWP with the TD-GWP in terms of carbon dioxide emissions in different traffic congestion patterns.

Section snippets

Introduction and literature review

A single–facility location problem (SFLP) is the problem of finding the location of a single facility which will serve a set of customers so as to minimize an objective function, usually a function of the distances between the facility and the customers. The (discrete) 1-center problem (Agarwal et al., 1998), a discrete facility location problem, and the Weber problem, (Drezner, 1992, Drezner, Hamacher, 2002), a continuous facility location problem, are among the most famous SFLPs.

The Weber

The green Weber problem (GWP)

We assume that the amount of CO2 emitted by a vehicle is proportional to its fuel consumption which is aligned with the related literature, see e.g., Demir et al. (2011). As the fuel consumption model, we use the comprehensive modal emission model (CMEM), suggested in Barth et al. (2005), for heavy-good vehicles. For a review and comparison of different vehicle emission models, the reader is referred to Demir et al. (2011). According to the CMEM, the amount of fuel consumed in liter, f, by a

Illustrative example

In this section, an illustrative example is presented. The solutions of the Weber problem (WP) and the GWP, and the solutions of the GWP and the TD-GWP are compared.

Consider an instance of the Weber problem with four customers in the plane, namely customer 1,2,3, and 4. The coordinates of the customers are (0, 0), (1000, 0), (0, 1000), and (1000, 1000) in kilometers as seen in Fig. 2. We assume identical unit weights for all customers, i.e., w1=w2=w3=w4=1. The optimal location of the facility

Computational experiments

In this section, our aim is to compare the total fuel-emission cost in large scale randomly generated distribution systems with and without traffic congestion. We generate two sets of instances. In the first set, there are 500 customers in the distribution system and in the second one 1000 customers.

For the first set of instances, we consider a square with side length of 234 km. Customers with identical unit weights are distributed in this square in 4 groups. Each group consists of 125

Conclusion

Extensions of the classical Weber problem are introduced in this study. In the first problem considered, namely the GWP, the location of the single facility in the plane and the speeds of the vehicles serving the customers are to be determined that result in the minimum total fuel-emission cost in the distribution system. The customers are assumed to have hard one-sided time windows (time limits) and the vehicles serving the customers should finish their service on or before the time limits.

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