In this paper, we examine a mixed integer linear programming (MILP) reformulation for mixed integer bilinear problems where each bilinear term involves the product of a nonnegative integer variable and a nonnegative continuous variable. This reformulation is obtained by first replacing a general integer variable with its binary expansion and then using McCormick envelopes to linearize the resulting product of continuous and binary variables. We present the convex hull of the underlying mixed integer linear set. The effectiveness of this reformulation and associated facet-defining inequalities are computationally evaluated on five classes of instances.

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We consider probabilistic constrained linear programs with general distributions for the uncertain parameters. These problems generally involve non-convex feasible sets. We develop a branch and bound algorithm that searches for a global solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partitions. This basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partitions and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires solving linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.

We study a deterministic maritime inventory routing problem with a long planning horizon. For instances with many ports and many vessels, mixed-integer linear programming (MIP) solvers often require hours to produce good solutions even when the planning horizon is 90 or 120 periods. Building on the recent successes of approximate dynamic programming (ADP) for road-based applications within the transportation community, we develop an ADP procedure to generate good solutions to these problems within minutes. Our algorithm operates by solving many small subproblems (one for each time period) and by collecting information about how to produce better solutions. Our main contribution to the ADP community is an algorithm that solves MIP subproblems and uses separable piecewise linear continuous, but not necessarily concave or convex, value function approximations and requires no off-line training. Our algorithm is one of the first of its kind for maritime transportation problems and represents a significant departure from the traditional methods used. In particular, whereas virtually all existing methods are "MIP-centric," i.e., they rely heavily on a solver to tackle a nontrivial MIP to generate a good or improving solution in a couple of minutes, our framework puts the effort on finding suitable value function approximations and places much less responsibility on the solver. Computational results illustrate that with a relatively simple framework, our ADP approach is able to generate good solutions to instances with many ports and vessels much faster than a commercial solver emphasizing feasibility and a popular local search procedure.

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