Stochastic programming models assumes that the stochastic data can be estimated by a probability distribution. More generally, these models are formulated and solved analytically in order to provide information to decision maker. The most widely and studied programming model are the sample average approximation (SAA) that solving stochastic problems by using MCS. Shapiro et al. (2009) present a lit- erature review of the most recent studies related with this solving approach.
2.6 Stochastic programming 33
Figure 2.6: The SAA approach
In Figure 2.6 is described the SAA model which relies on two stages: firstly the sampling process and secondly the problem optimization. A MCS is performed to generate the sample of conditions which follow a probability distribution. From the sample a set of feasible solutions are defined assuming unknown the set of scenar- ios. In SAA, for each scenario in the sample an occurrence probability is assigned. Assuming x s the set of feasible solutions and ω the set of scenarios which refer to the most likely conditions for problem. Thus, the value of f(xi, ωi) represents the performance of the solution xi under the condition ωi. Therefore, the complexity relies on the estimation of the expected objective function value. The complexity of this solving approach relies on the estimation of the expected objective function value. For a finite number n of scenarios in ω with an occurrence probability (p(ωi)) the expected value is computed as the f(x, ω)= Pn
i=1f (xi, ωi)*p(ωi). Finally, the value of f(x, ω) indicates whether number of scenarios are suitable for solving the problem. Evidently, the number of scenarios grows exponentially the problem data, however the quality of f(x, ω) depends on the sample, particularly on the variability between scenarios. The candidate solutions are given by a comparison between the obtained solution and the optimal one in the scenario ωi. Thus, the best solution is given by the one with the best expected value for f(x, ω).
34 Methodology
2.6 Stochastic programming 35
3
The sustainable multi-depot vehicle routing
problem
This chapter focuses on the distribution process in urban zones modeled as a multi- depot vehicle routing problem with several cost dimensions: economic, social, and environmental. A metaheuristic-based approach is proposed for tackling an enriched multi-depot vehicle routing problem in which economic, environmental, and social dimensions are considered. A series of computational experiments illustrates how the aforementioned dimensions can be integrated in realistic transport operations.
The work presented in this chapter has been published in the Journal of Heuris- tics:
• Reyes-Rubiano, L., Calvet, L., Juan, A. A., Faulin, J., Bove, L. (2018). A biased-randomized variable neighborhood search for sustainable multi-depot vehicle routing problems. Journal of Heuristics, 1-22.
Part of the contents of this chapter has been presented at the following confer- ences:
• Reyes-Rubiano, L., Calvet, L., Juan, A. A., Faulin, J., (2017). Sustainable Urban Freight Transport Considering Multiple Capacitated Depots. In MIC 2017, Barcelona, Spain.
• Reyes-Rubiano, L., Calvet, L., Juan, A. A., Faulin, J. (2017). Solving the Multi-Depot Vehicle Routing Problem with Sustainability Indicators. In VeRoLog 2017, Amsterdam, Netherlands.
38 The sustainable multi-depot vehicle routing problem
3.1
A BR-VNS simheuristic algorithm for solving
the sustainable multi-depot vehicle routing prob-
lem
This chapter focuses on the distribution process in urban zones modeled as a multi- depot vehicle routing problem (MDVRP) with several cost dimensions: economic, social, and environmental. The MDVRP is a challenging combinatorial optimization problem, which has been widely studied (Pisinger and Ropke 2007; Cordeau and
Maischberger 2012; Vidal et al. 2012; Subramanian et al. 2013; Vidal et al. 2014;
Escobar et al. 2014;Juan et al. 2015b). A metaheuristic-based approach is proposed
to tackle this problem when distribution costs depend on multiple sustainability indicators. As a consequence, for large instances of the problem –as the ones we might find in urban transport–, using a metaheuristic approach becomes a rational alternative (Talbi, 2009; Salhi, 2017). The proposed solving approach relies on the integration of biased-randomized (BR) techniques (Grasas et al., 2017) into a variable neighborhood search (VNS) framework (Hansen et al.,2010). In addition, a mixed-integer mathematical formulation is presented to define the problem and get optimal solutions for some small-size instances. The computational experiments also allow to compare the BR-VNS algorithm with an exact solver. More computational experiments are performed adapting benchmark MDVRP instances to gain insights into the problem and the relationships among the sustainable indicators.
To the best of our knowledge, this is the first work addressing a rich MDVRP including sustainability indicators (Caceres-Cruz et al., 2015). Accordingly, the main contributions of this chapter are: (i) the proposal of a rich MDVRP exten- sion considering different sustainability dimensions; (ii) a BR-VNS algorithm which introduces biased-randomization techniques into a metaheuristic framework to bet- ter guide the solution-construction process; (iii) a mathematical formulation of the problem, which enables a direct comparison between exact methods and a heuristic- based approach; and (iv) a comprehensive analysis (including suitable visualization techniques) of the trade-off among the different sustainability indicators.
The rest of the chapter is structured as follows: Section3.2 provides a literature review. Section 3.3 offers a detailed description of the problem analyzed, including a mathematical formulation. Section 3.4 proposes a BR-VNS algorithm. The com- putational experiments are explained in Section 3.5, while Section3.6 discusses the results. Finally, Section 3.7 gathers the contribution of the chapter.