The Stochastic Multipath Traveling Salesman Problem with Dependent Random Travel Costs

Abstract

The objective of the stochastic multipath traveling salesman problem is to determine the expected minimum-cost Hamiltonian tour in a network characterized by the presence of different paths between each pair of nodes, given that a random travel cost with an unknown probability distribution is associated with each of these paths. Previous works have proved that this problem can be deterministically approximated when the path travel costs are independent and identically distributed. Such an approximation has been demonstrated to be of acceptable quality in terms of the estimation of an optimal solution compared with consolidated approaches such as stochastic programming with recourse, completely overcoming the computational burden of solving enormous programs exacerbated by the number of scenarios considered. Nevertheless, the hypothesis regarding the independence among the path travel costs does not hold when considering real settings. It is well known, in fact, that traffic congestion influences travel costs and creates dependence among them. In this paper, we demonstrate that the independence assumption can be relaxed and a deterministic approximation of the stochastic multipath traveling salesman problem can be derived by assuming just asymptotically independent travel costs. We also demonstrate that this deterministic approximation has strong operational implications because it allows the consideration of realistic traffic models. Computational tests on extensive sets of random and realistic instances indicate the excellent efficiency and accuracy of the deterministic approximation.

Optimization Problems Under Uncertainty in Smart Cities

“…For a detailed and exhaustive presentation of the various theoretical steps that lead to the above results, the reader is referred to Perboli et al (2014), Tadei et al (2009), andFadda et al (2020). For the sake of completeness, it is interesting to remark that, in the case of a maximization problem, the same formula in (9) holds with some minor changes regarding the computation of e q i , i.e.,…”

“…Therefore, this assumption is a very mild assumption on the distribution of the random oscillations, and thus it does not significantly preclude the deployment of the EVTDA for practical applications. It is important to remark that Assumption 3, in the general form presented above, has been derived only recently in Fadda et al (2020). Therefore, in some other previous papers, the reader could find a more restrictive assumption on the distribution of the oscillations.…”

“…The most exciting direction toward the use of the EVTDA in a more generalized context regards the relaxation of the assumption of independence among the stochastic costs. In Fadda et al (2020), the authors study the distribution of uncertain travel times associated with the different alternative paths or transportation modes available for a traveler to move from one place to another. The study reveals that travel times in realistic road networks cannot be considered independent, if it is assumed that the network can involve traffic congestions, while asymptotic independence better characterizes those stochastic processes.…”

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“…It is important to notice that the assumption needed to apply the proposed EVTDA approach are quite mild and, in any case, less restrictive of those required by the SP paradigm. Interesting enough, Fadda et al (2020) have recently proved that the EVTDA can still be deployed when assumption ( 18) is relaxed to the following condition:…”

“…Thus, this is a very mild assumption and, therefore, does not significantly restrict the deployment of the EVTDA approach. It is essential to notice that some empirical results presented in Tadei et al (2017), Fadda et al (2020), and Roohnavazfar et al (2019) have demonstrated the effectiveness of the EVTDA framework even when the distribution of random oscillations does not satisfy the assumption ( 22) like the Uniform distribution.…”