Luigi Di Puglia Pugliese

A survey of resource constrained shortest path problems: Exact solution approaches

This article surveys the main contributions that have appeared in the scientific literature addressing resource constrained shortest path problems. The aim of this work is twofold: to give a structured survey of the literature on this topic; to provide a starting point for researchers who want to address the problems at hand. The study is focused on the relevant contributions dealing with exact solution approaches.

Shortest path problem with forbidden paths: The elementary version

This paper addresses the elementary shortest path problem with forbidden paths. The main aim is to find the shortest paths from a single origin node to every other node of a directed graph, such that the solution does not contain any path belonging to a given set (i.e., the forbidden set). It is imposed that no cycle can be included in the solution. The problem at hand is formulated as a particular instance of the shortest path problem with resource constraints and two different solution approaches are defined and implemented. One is a Branch & Bound based algorithm, the other is a dynamic programming approach. Different versions of the proposed solution strategies are developed and tested on a large set of test problems.

Optimal drone placement and cost-efficient target coverage

Observing mobile or static targets in the ground using flying drones is a common task for civilian and military applications. We introduce the minimum cost drone location problem and its solutions for this task in a two-dimensional terrain. The number of drones and the total energy consumption are the two cost metrics considered. We assume that each drone has a minimum and a maximum observation altitude. Moreover, the drones energy consumption is related to this altitude. Indeed, the higher the altitude, the larger the observed area but the higher the energy consumption. The aim is to find drone locations that minimize the cost while ensuring the surveillance of all the targets. The problem is mathematically solved by defining an integer linear and a mixed integer non-linear optimization models. We also provide some centralized and localized heuristics to approximate the solution for static and mobile targets. A computational study and extensive simulations are carried out to assess the behavior of the proposed solutions.

Dynamic programming approaches to solve the shortest path problem with forbidden paths

In this paper, the shortest path problem with forbidden paths is addressed. The problem under consideration is formulated as a particular instance of the resource-constrained shortest path problem. Different versions of a dynamic programming-based solution approach are defined and implemented. The proposed algorithms can be viewed as an extension of the node selection approach proposed by Desrochers in 1988. An extensive computational test is carried out on a meaningful number of random instances with the purpose of assessing the behaviour of the developed solution approaches. A comparison with the state-of-the-art method proposed to address the problem under study is also made. The computational results are very encouraging and highlight that the proposed algorithms are very efficient.

Robust constrained shortest path problems under budgeted uncertainty

We study the robust constrained shortest path problem under resource uncertainty. After proving that the problem is NP-hard in the strong sense for arbitrary uncertainty sets, we focus on budgeted uncertainty sets introduced by Bertsimas and Sim 2003 and their extension to variable uncertainty by Poss 2013. We apply classical techniques to show that the problem with capacity constraints can be solved in pseudopolynomial time. However, we prove that the problem with time windows is NP-hard in the strong sense when NP is not fixed, using a reduction from the independent set problem. We introduce then new robust labels that yield dynamic programming algorithms for the problems with time windows and capacity constraints. The running times of these algorithms are pseudopolynomial when NP is fixed, exponential otherwise. We present numerical results for the problem with time windows which show the effectiveness of the label-setting algorithm based on the new robust labels. Our numerical results also highlight the reduction in price of robustness obtained when using variable budgeted uncertainty instead of classical budgeted uncertainty.

Analysis of the Maierato landslide (Calabria, Southern Italy)

On 15 February 2010, a landslide of great dimensions occurred at Maierato (Calabria, Southern Italy) after a long rainy period. Although the zone was continuously affected by ground movements especially during the wet seasons, no monitoring system was installed before the occurrence of the landslide. However, many photos and two videos were taken during the failure process of the slope. In the present study, the available images are used to reconstruct the kinematics of the landslide. In addition, a finite element analysis is performed to define the main factors of triggering and to interpret the failure mechanism of the slope. This analysis is also based on the data from a site investigation carried out after the landslide to characterise the involved soils from a geotechnical viewpoint. The analysis also accounts for the strain-softening behaviour of some soils. The results have shown that the Maierato landslide was the reactivation of a pre-existing landslide body, which was caused by a significant increase in groundwater level.

A Two-stage Stochastic Programming Model for the Resource Constrained Project Scheduling Problem under Uncertainty

Due to the increasing competitiveness of businesses, project planning and scheduling have become a challenging theme in the last years. In this paper, we propose a two-stage stochastic programming model for the resource constrained project scheduling problem, taking into account the stochasticity of activity durations. In this formulation, assuming that some activity duration scenarios are known, resource allocations are taken in the first stage, while scheduling decisions are postponed in the second stage. The resulting problem is a mixed integer problem with recourse, where binary variables appear in the first stage. In order to efficiently solve the problem, a decomposition algorithm is developed, based on the well-known integer L-shaped method. Detailed computational results are presented for a set of benchmark instances taken from the literature.

Last-Mile Deliveries by Using Drones and Classical Vehicles

We address the problem of managing a drone-based delivery process. We consider the specific situation of a delivery company, that uses a set of trucks equipped with a given number of drones. In particular, items of a limited weight and size could be delivered by using drones. A vehicle, during its trip, can launch a drone when serving a customer, the drone performs a delivery for exactly one customer and returns to the vehicle, possibly at a different customer location. Each drone can be launched several times during the vehicle’s route. It is imposed a limit on the maximum distance that each drone can travel and synchronization requirements between vehicle and drone should be ensured. In particular, it is assumed that a vehicle waits for a drone for a maximum period of time. The aim is to serve all customers within their time window. The problem is modeled as a variant of the vehicle routing problem with time windows. The aim of this work is to analyze the delivery process with drones, by taking into account the total transportation cost and highlighting strategical issues, related to the use of drones. The numerical results, collected on instances generated to be very close to reality, show that the use of drones is not economically convenient in the classical terms. However, when considering negative externalities related to the use of classical vehicles and quality of service requirements, the benefit of using drones becomes relevant.

An Algorithm to Find the Link Constrained Steiner Tree in Undirected Graphs

We address a variant of the classical Steiner tree problem defined over undirected graphs. The objective is to determine the Steiner tree rooted at a source node with minimum cost and such that the number of edges is less than or equal to a given threshold. The link constrained Steiner tree problem (\(\mathcal {LCSTP}\)) belongs to the NP-hard class. We formulate a Lagrangian relaxation for the \(\mathcal {LCSTP}\) in order to determine valid bounds on the optimal solution. To solve the Lagrangian dual, we develop a dual ascent heuristic based on updating one multiplier at time. The proposed heuristic relies on the execution of some sub-gradient iterations whenever the multiplier update procedure is unable to generate a significant increase of the Lagrangian dual objective. We calculate an upper bound on the \(\mathcal {LCSTP}\) by adjusting the infeasibility of the solution obtained at each iteration. The solution strategy is tested on instances inspired by the scientific literature.

Heuristics for the local grid scheduling problem with processing time constraints

In this paper, the local job scheduling problem, with processing time constraints (i.e., deadline, earliest start time, reservation) in a computational grid, is addressed. The problem under investigation is formulated as a rectangular packing problem and several heuristic approaches are developed for its solution. The performance of the proposed algorithms are evaluated under different scenarios. Extensive experimental tests demonstrate that the defined solution strategies outperform the state-of-art methods.