Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

On the time-consistent stochastic dominance risk averse measure for tactical supply chain planning under uncertainty

In this work a modeling framework and a solution approach have been presented for a multi-period stochastic mixed 0–1 problem arising in tactical supply chain planning (TSCP). A multistage scenario tree based scheme is used to represent the parameters’ uncertainty and develop the related Deterministic Equivalent Model. A cost risk reduction is performed by using a new time-consistent risk averse measure. Given the dimensions of this problem in real-life applications, a decomposition approach is proposed. It is based on stochastic dynamic programming (SDP). The computational experience is twofold, a compar- ison is performed between the plain use of a current state-of-the-art mixed integer optimization solver and the proposed SDP decomposition approach considering the risk neutral version of the model as the subject for the benchmarking. The add-value of the new risk averse strategy is confirmed by the compu- tational results that are obtained using SDP for both versions of the TSCP model, namely, risk neutral and risk averse.The authors would like to thank to the two anonymous review- ers for their help on clarifying some concepts presented in the manuscript and strongly improving its presentatio

A Stochastic Benders Decomposition Scheme for Large-Scale Data-Driven Network Design

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a data-driven version of network design where operational costs are uncertain and estimated using historical data. This problem is notoriously computationally challenging, and instances with as few as fifty nodes cannot be solved to optimality by current decomposition techniques. Accordingly, we propose a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless generates deterministically valid cuts (as opposed to the probabilistically valid cuts frequently proposed in the stochastic optimization literature) via a dual averaging technique. We implement both single-cut and multi-cut variants of this Benders decomposition algorithm, as well as a k-cut variant that uses clustering of the historical scenarios. On instances with 100-200 nodes, our algorithm achieves 4-5% optimality gaps, compared with 13-16% for deterministic Benders schemes, and scales to instances with 700 nodes and 50 commodities within hours. Beyond network design, our strategy could be adapted to generic two-stage stochastic mixed-integer optimization problems where second-stage costs are estimated via a sample average

Lagrangian Dual Decision Rules for Multistage Stochastic Mixed Integer Programming

Multistage stochastic programs can be approximated by restricting policies to follow decision rules. Directly applying this idea to problems with integer decisions is difficult because of the need for decision rules that lead to integral decisions. In this work, we introduce Lagrangian dual decision rules (LDDRs) for multistage stochastic mixed integer programming (MSMIP) which overcome this difficulty by applying decision rules in a Lagrangian dual of the MSMIP. We propose two new bounding techniques based on stagewise (SW) and nonanticipative (NA) Lagrangian duals where the Lagrangian multiplier policies are restricted by LDDRs. We demonstrate how the solutions from these duals can be used to drive primal policies. Our proposal requires fewer assumptions than most existing MSMIP methods. We compare the theoretical strength of the restricted duals and show that the restricted NA dual can provide relaxation bounds at least as good as the ones obtained by the restricted SW dual. In our numerical study, we observe that the proposed LDDR approaches yield significant optimality gap reductions compared to existing general-purpose bounding methods for MSMIP problems

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

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