A mean-risk mixed integer nonlinear program for transportation network protection

Abstract

This paper focuses on transportation network protection to hedge against extreme events such as earthquakes. Traditional two-stage stochastic programming has been widely adopted to obtain solutions under a risk-neutral preference through the use of expectations in the recourse function. In reality, decision makers hold different risk preferences. We develop a mean-risk two-stage stochastic programming model that allows for greater flexibility in handling risk preferences when allocating limited resources. In particular, the first stage minimizes the retrofitting cost by making strategic retrofit decisions whereas the second stage minimizes the travel cost. The conditional value-at-risk (CVaR) is included as the risk measure for the total system cost. The two-stage model is equivalent to a nonconvex mixed integer nonlinear program (MINLP). To solve this model using the Generalized Benders Decomposition (GBD) method, we derive a convex reformulation of the second-stage problem to overcome algorithmic challenges embedded in the non-convexity, nonlinearity, and non-separability of first-and second-stage variables. The model is used for developing retrofit strategies for networked highway bridges, which is one of the research areas that can significantly benefit from mean-risk models. We first justify the model using a hypothetical nine-node network. Then we evaluate our decomposition algorithm by applying the model to the Sioux Falls network, which is a large-scale benchmark network in the transportation research community. The effects of the chosen risk measure and critical parameters on optimal solutions are empirically explored.

Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

“…Finally, the proofs of the error bounds (27) in Theorem 2 and (30) in Theorem 3 for the α-approximationR β α are analogous to the proofs of (26) and (29), respectively. The only difference is that instead of using Theorem 6 from [41], we use Theorem 5 from that reference to obtain an analogue of (40).…”

“…See, e.g., [18,36,43,[46][47][48]54] for applications of mean-CVaR recourse models in this field. Other areas of application include disaster relief planning [5,32,33], (energy) production planning [4,9,21,27], transportation network protection [29], and water allocation [56]. The popularity of CVaR, and of mean-CVaR recourse models in particular, underlines the relevance of the models studied in this paper.…”

“…However, since the bounds forR β andR β α differ only by a factor 2, we present the results for the shifted LP-relaxationR β only. We start by presenting a simplified version of the error bound in (29) from Theorem 3. (4), where v is the SIR value function from (31).…”

Conditional value‐at‐risk beyond finance: a survey

“…The paper by Lu et al. () focuses on protecting, through retrofitting strategies, a transportation network, that includes several bridges, against extreme events, such as an earthquake. Given a budget, the traffic capacity over each bridge, once the extreme event occurred, depends on how well the bridge was retrofitted.…”

Impact of Transportation Problem’s Unbalance on Difference between Initial and Optimal Solution

“…In dealing with both types of TPs, researchers were mainly focused on mathematical improvements of solving methods for gaining initial solutions and comparing them with optimal ones [2][3][4][5][6]. Furthermore, some authors addressed the methods for solving specific TP's with uncertainties in input parameters [7][8][9][10][11][12] and optimization methods for handling TP's with nonlinearities [13][14][15][16][17][18]. However, previous studies have overlooked the questions: does the magnitude of problem's unbalance affect the difference between the initial feasible solution and the optimum one, or does the problem's matrix form influence it as well.…”