Mehdi Mrad

Optimal consumed electric energy while sequencing vehicle trips in a personal rapid transit transportation system

Abstract In this paper, we address the problem of minimizing the consumed electric energy for a personal rapid transit transportation system, in order to fulfil a planned list of trips, performed by a set of powered-batteries vehicles. For that aim, the list of trips is represented by a network, where each trip is associated with a node and the consumed electric energy is assigned to the arcs. Based on this network representation, two mathematical formulations, minimizing the electric energy are established. The first one is a mixed integer programming formulation, which is solved directly with a state-of-the-art LP solver. The second formulation is a 0–1 programming model, which is solved with a constraints generation technique. In addition, if an optimal solution is not obtained within a fixed time limit, the first mathematical formulation provides an upper bound and the second formulation gives a lower bound for the optimal solution. For the unsolved instances, the difference of the upper and lower bound relative to the lower bound gives the so called relative gap. This relative gap measures the maximum deviation from the optimal solution. Finally, extensive computational experiments are presented. The results provide evidence that the proposed procedures are very effective, since, 90% of the instances are solved to optimality and the mean relative gap is 1.41% for the unsolved instances.

Synchronous Routing for Personal Rapid Transit Pods

Personal rapid transit (PRT) is a public and automated transport system in which a fleet of small driverless vehicles operate in order to transport passengers between a set of stations through a network of guided ways. Each customer is carried from one station to another directly with no stop in intermediate stations. This mode of transport can result in a high level of unused capacity due to the empty moves of the vehicles. In this paper, we model the problem of minimizing the energy consumed by the PRT system while assuming predeterministic list of orders; then we solve it using some constructive heuristics. Experiments are run on 1320 randomly generated test problems with various sizes. Our algorithms are shown to give good results over large trip instances.

Optimal solution of the discrete cost multicommodity network design problem

We investigate a multicommodity network design problem where a discrete set of technologies with step-increasing cost and capacity functions should be installed on the edges. This problem is a fundamental network design problem having many important applications in contemporary telecommunication networks. We describe an exact constraint generation approach and we show that the conjunctive use of valid inequalities, bipartition inequalities that are generated using max-flow computations, as well as an exact separation algorithm of metric inequalities makes it feasible to solve to optimality instances with up to 50 nodes and 100 edges.

Enhanced compact models for the connected subgraph problem and for the shortest path problem in digraphs with negative cycles

We investigate the minimum-weight connected subgraph problem. The importance of this problem stems from the fact that it constitutes the backbone of many network design problems having applications in several areas including telecommunication, energy, and distribution planning. We show that this problem is NP-hard, and we propose a new polynomial-size nonlinear mixed-integer programming model. We apply the Reformulation-Linearization Technique (RLT) to linearize the proposed model while keeping a polynomial number of variables and constraints. Furthermore, we show how similar modelling techniques enable an enhanced polynomial size formulation to be derived for the shortest elementary path. This latter problem is known to be intractable and has many applications (in particular, within the context of column generation). We report the results of extensive computational experiments on graphs with up to 1000 nodes. These results attest to the efficacy of the proposed compact formulations. In particular, we show that the proposed formulations consistently outperform compact formulations from the literature.

A Branch-and-Price Algorithm for the Two-Stage Guillotine Cutting Stock Problem

We investigate the two-stage guillotine two-dimensional cutting stock problem. This problem commonly arises in the industry when small rectangular items need to be cut out of large stock sheets. We propose an integer programming formulation that extends the well-known Gilmore and Gomory model by explicitly considering solutions that are obtained by both slitting some stock sheets down their widths and others down their heights. To solve this model, we propose an exact branch-and-price algorithm. To the best of our knowledge, this is the first contribution with regard to obtaining integer optimal solutions to Gilmore and Gomory model. Extensive results, on a set of real-world problems, indicate that the proposed algorithm delivers optimal solutions for instances with up to 809 items and that the hybrid cutting strategy often yields improved solutions. Furthermore, our computational study reveals that the proposed modelling and algorithmic strategy outperforms a recently proposed arc-flow model-based solution strategy.

An Arc Flow-Based Optimization Approach for the Two-Stage Guillotine Strip Cutting Problem

Despite its broad range of industrial applications, the two-stage guillotine restriction has received very scant attention in the strip cutting literature. An integer linear programming model that is based on a special graph structure is devised for this strongly NP-hard problem. In addition to being easy to implement, the empirical study on a large set of instances from the literature and from real industrial world cases shows the efficiency of the proposed method while solving instances with high multiplicity factor.

Scheduling IT Staff at a Bank: A Mathematical Programming Approach

We address a real-world optimization problem: the scheduling of a Bank Information Technologies (IT) staff. This problem can be defined as the process of constructing optimized work schedules for staff. In a general sense, it requires the allocation of suitably qualified staff to specific shifts to meet the demands for services of an organization while observing workplace regulations and attempting to satisfy individual work preferences. A monthly shift schedule is prepared to determine the shift duties of each staff considering shift coverage requirements, seniority-based workload rules, and staff work preferences. Due to the large number of conflicting constraints, a multiobjective programming model has been proposed to automate the schedule generation process. The suggested mathematical model has been implemented using Lingo software. The results indicate that high quality solutions can be obtained within a few seconds compared to the manually prepared schedules.

A new lower bound for minimising the total completion time in a two-machine flow shop under release dates

In this paper, we describe a mixed integer linear programming (MILP) formulation used to model the two-machine flow shop scheduling problem subject to release dates. This MILP formulation is based on the Positional and Assignment Variables(PAVF). The results of the linear relaxation bound derived from the later mathematical formulation shows the performance of this new bound compared with the best known lower bound recently presented for the studied problem.

An optimization-based heuristic for the machine reassignment problem

We address the machine reassignment problem proposed in the context of the ROADEF/EURO challenge 2012 in partnership with Google. The problem consists in reassigning a set of processes to a set of multiple-resource machines so as to minimize a weighted function of the machines load, the resources balance, and the costs of moving processes while satisfying numerous constraints. We propose an optimization-based heuristic that requires decomposing the problem into a sequence of small-sized instances that are iteratively solved using a general MIP solver. To speed-up the solution process several algorithmic expedients are embedded. Extensive computational experiments provide evidence that the proposed approach exhibits a very good performance.

Strong multi-commodity flow formulations for the asymmetric traveling salesman problem

Abstract We provide new compact formulations of polynomial size for the asymmetric traveling salesman problem obtained through the Reformulation-Linearization Technique. The first one is obtained directly by this latter approach while the two others are derived by performing projections of this formulation on the variables of the existing models. We show that the devised formulations are stronger than the state-of-the-art models. Computational experiments conducted on benchmark instances for the classical variant and with precedence constraints confirm the better quality of the relaxations provided by our proposed formulations.