Lars Magnus Hvattum

Solving a Dynamic and Stochastic Vehicle Routing Problem with a Sample Scenario Hedging Heuristic

The statement of the standard vehicle routing problem cannot always capture all aspects of real-world applications. As a result, extensions or modifications to the model are warranted. Here we consider the case when customers can call in orders during the daily operations; i.e., both customer locations and demands may be unknown in advance. This is modeled as a combined dynamic and stochastic programming problem, and a heuristic solution method is developed where sample scenarios are generated, solved heuristically, and combined iteratively to form a solution to the overall problem.

The robust vehicle routing problem with time windows

This paper addresses the robust vehicle routing problem with time windows. We are motivated by a problem that arises in maritime transportation where delays are frequent and should be taken into account. Our model only allows routes that are feasible for all values of the travel times in a predetermined uncertainty polytope, which yields a robust optimization problem. We propose two new formulations for the robust problem, each based on a different robust approach. The first formulation extends the well-known resource inequalities formulation by employing adjustable robust optimization. We propose two techniques, which, using the structure of the problem, allow to reduce significantly the number of extreme points of the uncertainty polytope. The second formulation generalizes a path inequalities formulation to the uncertain context. The uncertainty appears implicitly in this formulation, so that we develop a new cutting plane technique for robust combinatorial optimization problems with complicated constraints. In particular, efficient separation procedures are discussed. We compare the two formulations on a test bed composed of maritime transportation instances. These results show that the solution times are similar for both formulations while being significantly faster than the solutions times of a layered formulation recently proposed for the problem.

Using scenario trees and progressive hedging for stochastic inventory routing problems

The Stochastic Inventory Routing Problem is a challenging problem, combining inventory management and vehicle routing, as well as including stochastic customer demands. The problem can be described by a discounted, infinite horizon Markov Decision Problem, but it has been showed that this can be effectively approximated by solving a finite scenario tree based problem at each epoch. In this paper the use of the Progressive Hedging Algorithm for solving these scenario tree based problems is examined. The Progressive Hedging Algorithm can be suitable for large-scale problems, by giving an effective decomposition, but is not trivially implemented for non-convex problems. Attempting to improve the solution process, the standard algorithm is extended with locking mechanisms, dynamic multiple penalty parameters, and heuristic intermediate solutions. Extensive computational results are reported, giving further insights into the use of scenario trees as approximations of Markov Decision Problem formulations of the Stochastic Inventory Routing Problem.

Analysis of an exact algorithm for the vessel speed optimization problem

Increased fuel costs together with environmental concerns have led shipping companies to consider the optimization of vessel speeds. Given a fixed sequence of port calls, each with a time window, and fuel cost as a convex function of vessel speed, we show that optimal speeds can be found in quadratic time.

A survey on maritime fleet size and mix problems

This paper presents a literature survey on the fleet size and mix problem in maritime transportation. Fluctuations in the shipping market and frequent mismatches between fleet capacities and demands highlight the relevance of the problem and call for more accurate decision support. After analyzing the available scientific literature on the problem and its variants and extensions, we summarize the state of the art and highlight the main contributions of past research. Furthermore, by identifying important real life aspects of the problem which past research has failed to capture, we uncover the main areas where more research will be needed.

Scenario Tree-Based Heuristics for Stochastic Inventory-Routing Problems

In vendor-managed inventory replenishment, the vendor decides when to make deliveries to customers, how much to deliver, and how to combine shipments using the available vehicles. This gives rise to the inventory-routing problem in which the goal is to coordinate inventory replenishment and transportation to minimize costs. The problem tackled in this paper is the stochastic inventory-routing problem, where stochastic demands are specified through general discrete distributions. The problem is formulated as a discounted infinite-horizon Markov decision problem. Heuristics based on finite scenario trees are developed. Computational results confirm the efficiency of these heuristics.

Finding local optima of high-dimensional functions using direct search methods

This paper focuses on a subclass of box-constrained, non-linear optimization problems. We are particularly concerned with settings where gradient information is unreliable, or too costly to calculate, and the function evaluations themselves are very costly. This encourages the use of derivative free optimization methods, and especially a subclass of these referred to as direct search methods. The thrust of our investigation is twofold. First, we implement and evaluate a number of traditional direct search methods according to the premise that they should be suitable as local optimizers when used in a metaheuristic framework. Second, we introduce a new direct search method, based on Scatter Search, designed to remedy the lack of a good derivative free method for solving problems of high dimensions. Our new direct search method has convergence properties comparable to those of existing methods in addition to being able to solve larger problems more effectively.

Tank allocation problems in maritime bulk shipping

In real world maritime routing problems, many restrictions and regulations influence the daily operations. The effects of several of these restrictions have not yet been studied in depth from an operations research perspective. This paper introduces the problem of allocating bulk cargoes to tanks in maritime shipping. A model and several variations are presented, and it is shown that the main problem consists of a number of complicating constraints. The problem studied is crucial when determining whether a given route is feasible for a given ship, and computational experiments are performed to assess the difficulty of solving realistically sized instances. The proposed formulation is hoped to provide a suitable starting point for research on stowage problems in maritime bulk shipping.

Adaptive memory search for multidemand multidimensional knapsack problems

We describe a simple adaptive memory search method for the 0/1 Multidemand Multidimensional Knapsack Problem (0/1 MDMKP). The search balances the level of infeasibility against the quality of the solution, and uses a simple dynamic tabu search mechanism. A weighting scheme to balance out the differences in the tightness of the constraints is also implemented. Computational results on a portfolio of test problems taken from the literature are reported, showing very favorable results, both in terms of solution quality and the ability of the search to find feasible solutions.

A maritime inventory routing problem with stochastic sailing and port times

This paper describes a stochastic short sea shipping problem where a company is responsible for both the distribution of oil products between islands and the inventory management of those products at consumption storage tanks located at ports. In general, ship routing and scheduling is associated with uncertainty in weather conditions and unpredictable waiting times at ports. In this work, both sailing times and port times are considered to be stochastic parameters. A two-stage stochastic programming model with recourse is presented where the first stage consists of routing, loading and unloading decisions, and the second stage consists of scheduling and inventory decisions. The model is solved using a decomposition approach similar to an L-shaped algorithm where optimality cuts are added dynamically, and this solution process is embedded within the sample average approximation method. A computational study based on real-world instances is presented.