A robust model for recharging station location problem

Abstract

Due to the harmful effects of greenhouse gas emissions emitted by petroleum-based vehicles, the transition to alternative fuel vehicles, running on cleaner sources of energy such as electricity, is an important trend. At the beginning of the transition period to alternative-fuel vehicles, due to the lack of comprehensive recharging infrastructure, it is essential to optimally locate recharging stations so that the network is covered as much as possible. Most existing studies on this topic generally assume that the driving range of vehicles and the flow volume are known, however, in practice, they are highly stochastic. The first aim of this paper is to incorporate this type of uncertainty into the problem and formulate it as a robust scenario-based model in which the expected number of drivers who can complete their trip without running out of charge is maximized. The development of a solution algorithm, capable to solve large instances of the model, is another aim followed in this paper. Computational results over different instances taken from the literature or generated randomly confirm the efficiency of the proposed model and algorithms.

Appendix: Calibration of parameters

In the “Appendix”, the Taguchi method is described to calibrate some parameters including $\left| T \right|$ and $E$, as well as the parameters $\lambda$ and $\,w$. Table

Table 10 Parameters and their levels

10 shows the levels at which these parameters are varied.

To design experiments, instead of considering full factorial combinations, i.e., 81 $\left( {3^{4} } \right)$ trials, the Taguchi method utilizes the orthogonal array $\,L_{9} \left( {3^{4} } \right)$ presented in Table

Table 11 The orthogonal array $\,L_{9} \left( {3^{4} } \right)$ and the S/N ratios

11 in which the numbers 1, 2, and 3 refer to the levels introduced in Table 10. Experiments are executed five times. For each experiment of Table 11, the signal-to-noise (S/N) ratio is calculated using Eq. (44), where $\gamma$ represents the number of repetitions of each experiment and $f_{\gamma }$ is the objective value of the best solution obtained in the $\gamma^{th}$ run of FA on that experiment.

$$S/N\,ratio = - 10 \times \log \left( {\frac{1}{\Gamma }\sum\limits_{\gamma = 1}^{\Gamma } {\frac{1}{{f_{\gamma }^{2} }}} } \right)$$

(44)

The mean S/N ratio for different levels of each parameter is represented in Fig. 

Fig. 6

The mean S/N ratio plot

6 Since a larger S/N ratio indicates better performance, the appropriate levels of parameters $\left| T \right|$, $E$, $\lambda$, and $\,w$ would be 30 (Level-3), 5 (Level-3), 1 (Level-1), and 50,000 (Level-2), respectively. Additionally, the analysis of variance (ANOVA), reported in Table

Table 12 ANOVA results

12, indicates the relative significance of each parameter in terms of its effect on the solution quality. As can be seen, the parameter w has the most effect, and the parameters $E$, $\lambda$, and $\left| T \right|$ are in the second, third, and fourth positions, respectively.