Dimitri J. Papageorgiou

MIRPLib – A library of maritime inventory routing problem instances: Survey, core model, and benchmark results

This paper presents a detailed description of a particular class of deterministic single product Maritime Inventory Routing Problems (MIRPs), which we call deep-sea MIRPs with inventory tracking at every port. This class involves vessel travel times between ports that are significantly longer than the time spent in port and require inventory levels at all ports to be monitored throughout the planning horizon. After providing a comprehensive literature survey of this class, we introduce a core model for it cast as a mixed-integer linear program. This formulation is quite general and incorporates assumptions and families of constraints that are most prevalent in practice. We also discuss other modeling features commonly found in the literature and how they can be incorporated into the core model. We then offer a unified discussion of some of the most common advanced techniques used for improving the bounds of these problems. Finally, we present a library, called MIRPLib, of publicly available test problem instances for MIRPs with inventory tracking at every port. Despite a growing interest in combined routing and inventory management problems in a maritime setting, no data sets are publicly available, which represents a significant “barrier to entry” for those interested in related research. Our main goal for MIRPLib is to help maritime inventory routing gain maturity as an important and interesting class of planning problems. As a means to this end, we (1) make available benchmark instances for this particular class of MIRPs; (2) provide the mixed-integer linear programming community with a set of optimization problem instances from the maritime transportation domain in LP and MPS format; and (3) provide a template for other researchers when specifying characteristics of MIRPs arising in other settings. Best known computational results are reported for each instance.

Probabilistic Set Covering with Correlations

We consider two variants of a probabilistic set covering (PSC) problem. The first variant assumes that there is uncertainty regarding whether a selected set can cover an item, and the objective is to determine a minimum-cost combination of sets so that each item is covered with a prespecified probability. The second variant seeks to maximize the minimum probability that a selected set can cover all items. To date, literature on this problem has focused on the special case in which uncertainties are independent. In this paper, we formulate deterministic mixed-integer programming models for distributionally robust PSC problems with correlated uncertainties. By exploiting the supermodularity of certain substructures and analyzing their polyhedral properties, we develop strong valid inequalities to strengthen the formulations. Computational results illustrate that our modeling approach can outperform formulations in which correlations are ignored and that our algorithms can significantly reduce overall computation time.

Approximate Dynamic Programming for a Class of Long-Horizon Maritime Inventory Routing Problems

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.

Fixed-Charge Transportation with Product Blending

Numerous planning models within the chemical, petroleum, and process industries involve coordinating the movement of raw materials in distribution networks so they can be blended into final products. The uncapacitated fixed-charge transportation problem with blending (FCTPwB) studied in this paper captures a core structure encountered in many of these environments. We model the FCTPwB as a mixed-integer linear program, and we derive two classes of facets, both exponential in size, for the convex hull of solutions for the problem with a single consumer and show that they can be separated in polynomial time. Furthermore, we prove that, in certain situations, these classes of facets along with the continuous relaxation of the original constraints yield a description of the convex hull. Finally, we present a computational study that demonstrates that these classes of facets are effective in reducing the integrality gap and solution time for more general instances of the FCTPwB with arc capacities and multiple consumers.

An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem

Abstract The multiperiod blending problem involves binary variables and bilinear terms, yielding a nonconvex MINLP. In this work we present two major contributions for the global solution of the problem. The first one is an alternative formulation of the problem. This formulation makes use of redundant constraints that improve the MILP relaxation of the MINLP. The second contribution is an algorithm that decomposes the MINLP model into two levels. The first level, or master problem, is an MILP relaxation of the original MINLP. The second level, or subproblem, is a smaller MINLP in which some of the binary variables of the original problem are fixed. The results show that the new formulation can be solved faster than alternative models, and that the decomposition method can solve the problems faster than state of the art general purpose solvers.

Two-Stage Decomposition Algorithms for Single Product Maritime Inventory Routing

We present two decomposition algorithms for single product deep-sea maritime inventory routing problems (MIRPs) that possess a core substructure common in many real-world applications. The problem involves routing vessels, each belonging to a particular vessel class, between loading and discharging ports, each belonging to a particular region. Our algorithms iteratively solve a MIRP by zooming out and then zooming in on the problem. Specifically, in the “zoomed out” phase, we solve a first-stage master problem in which aggregate information about regions and vessel classes is used to route vessels between regions, while only implicitly considering inventory and capacity requirements, berth limits, and other side constraints. In the “zoomed in” phase, we solve a series of second-stage subproblems, one for each region, in which individual vessels are routed through each region and load and discharge quantities are determined. Computational experience shows that an integrated approach that combines these two...

Recent Progress Using Matheuristics for Strategic Maritime Inventory Routing

This chapter presents an extensive computational study of simple, but prominent matheuristics (i.e., heuristics that rely on mathematical programming models) to find high quality ship schedules and inventory policies for a class of maritime inventory routing problems. Our computational experiments are performed on a test bed of the publicly available MIRPLib instances. This class of inventory routing problems has few constraints relative to some operational problems, but is complicated by long planning horizons. We compare several variants of rolling horizon heuristics, K-opt heuristics, local branching, solution polishing, and hybrids thereof. Many of these matheuristics substantially outperform the commercial mixed-integer programming solvers CPLEX 12.6.2 and Gurobi 6.5 in their ability to quickly find high quality solutions. New best known incumbents are found for 26 out of 70 yet-to-be-proved-optimal instances and new best known bounds on 56 instances.

Deterministic electric power infrastructure planning: Mixed-integer programming model and nested decomposition algorithm

Abstract This paper addresses the long-term planning of electric power infrastructures considering high renewable penetration. To capture the intermittency of these sources, we propose a deterministic multi-scale Mixed-Integer Linear Programming (MILP) formulation that simultaneously considers annual generation investment decisions and hourly operational decisions. We adopt judicious approximations and aggregations to improve its tractability. Moreover, to overcome the computational challenges of treating hourly operational decisions within a monolithic multi-year planning horizon, we propose a decomposition algorithm based on Nested Benders Decomposition for multi-period MILP problems to allow the solution of larger instances. Our decomposition adapts previous nested Benders methods by handling integer and continuous state variables, although at the expense of losing its finite convergence property due to potential duality gap. We apply the proposed modeling framework to a case study in the Electric Reliability Council of Texas (ERCOT) region, and demonstrate massive computational savings from our decomposition.

Pseudo basic steps: bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming

An elementary, but fundamental, operation in disjunctive programming is a basic step, which is the intersection of two disjunctions to form a new disjunction. Basic steps bring a disjunctive set in regular form closer to its disjunctive normal form and, in turn, produce relaxations that are at least as tight. An open question is: What are guaranteed bounds on the improvement from a basic step? In this paper, using properties of a convex disjunctive program’s hull reformulation and multipliers from Lagrangian decomposition, we introduce an operation called a pseudo basic step and use it to provide provable bounds on this improvement along with techniques to exploit this information when solving a disjunctive program as a convex MINLP. Numerical examples illustrate the practical benefits of these bounds. In particular, on a set of K-means clustering instances, we make significant bound improvements relative to state-of-the-art commercial mixed-integer programming solvers.