Laurent Alfandari

Improved approximation of the general soft-capacitated facility location problem

An NP-hard variant of the single-source Capacitated Facility Location Problem is studied, where each facility is composed of a variable number of fixed-capacity production units. This problem, especially the metric case, has been recently studied in several papers. In this paper, we only consider the general problem where connection costs do not systematically satisfy the triangle inequality property. We show that an adaptation of the set covering greedy heuristic, where the sub-problem is approximately solved by a Fully Polynomial-Time Approximation Scheme based on cost scaling and dynamic programming, achieves a logarithmic approximation ratio of (1+ƒO)H(n) for the problem, where n is the number of clients to be served, and H is the harmonic series. This improves the previous bound of 2H(n) for this problem.

A path relinking algorithm for the generalized assignment problem

The Generalized Assignment Problem (GAP) consists in finding a maximum-profit assignment of tasks to agents with capacity constraints. In this paper, a path relinking heuristic is proposed for the GAP. The main feature of our path relinking is that both feasible and infeasible solutions are inserted in the reference set of elite solutions, trade-off between feasibility and infeasibility being ruled through a penalty coefficient for infeasibility. Since exploration of the solution space is very sensitive to this penalty coefficient, the coefficient is dynamically updated at each round of combinations so that a balance is kept between feasible and infeasible solutions in the reference set. Numerical experiments reported on classical testbed instances of the OR-library show that the algorithm compares favorably to several other methods in the literature. In particular, more than 95% of the instances in the test-file were solved to optimality with short computation time.

Master-Slave Strategy and Polynomial Approximation

A lot of minimization covering problems on graphs consist in covering vertices or edges by subgraphs verifying a certain property. These problems can be seen as particular cases of set-covering. If the number of subgraphs is polynomial in the order n of the input-graph, then these problems can be approximated within logarithmic ratio by the classical greedy set-covering algorithm. We extend the class of problems approximable by this approach to covering problems where the number of candidate subgraphs is exponential in n, by revisiting an old technique called “master-slave” and extending it to weighted master or/and slave problems. Finally, we use the master-slave tool to prove some positive approximation results for two network-design and a VLSI-design graph-problems.

The Multicommodity-Ring Location Routing Problem

The multicommodity-ring location routing problem (MRLRP) studied in this paper is an NP-hard minimization problem arising in city logistics. The aim is to locate a set of urban distribution centers (UDCs) and to connect them via a ring in which massive flows of goods will circulate. Goods are transported from gates located outside the city to a UDC, and either join a second UDC through the ring before being delivered in electric vans to the final customers or are delivered directly to the customers from the first UDC. The reverse trip with pickup and transportation to the gates is also possible. A delivery service path starts at a particular UDC, then visits a subset of customers and ends at the same UDC, another UDC, or a self-service parking lot (SPL). A pickup route can start from an SPL or a UDC and ends at a UDC. The objective is to minimize the sum of the installation costs of the ring, flow transportation costs, and routing costs. The MRLRP belongs to the class of location-routing problems (LRP). We model it with a set-partitioning-like representation of delivery and pickup trips and arc-flow elements to describe goods transportation in the ring and between the ring and the gates. We present three approaches to solving the MRLRP: an exact method for small-size instances, a matheuristic for instances of a larger size, and a hybrid approach that applies the exact method to the columns output by the matheuristic. Numerical results are provided for an exhaustive set of instances, obtained by extending benchmark instances of the capacitated LRP with additional MRLRP features.

A branch-and-price-and-cut approach for sustainable crop rotation planning

In this paper, we study a multi-periodic production planning problem in agriculture. This problem belongs to the class of crop rotation planning problems, which have received increased attention in the literature in recent years. Crop cultivation and fallow periods must be scheduled on land plots over a given time horizon so as to minimize the total surface area of land used, while satisfying crop demands every period. This problem is proven strongly NP-hard. We propose a 0-1 linear programming compact formulation based on crop-sequence graphs. An extended formulation is then provided with a polynomial-time pricing problem, and a Branch-and-Priceand- Cut (BPC) algorithm is presented with adapted branching rules and cutting planes. The numerical experiments on instances varying the number of crops, periods and plots show the effectiveness of the BPC for the extended formulation compared to solving the compact formulation, even though these two formulations have the same linear relaxation bound.

A MIP flow model for crop-rotation planning in a context of forest sustainable development

We propose a Mixed-Integer Linear Programming model for a class of multi-period crop rotation optimization problems with demand constraints and incompatibility constraints between cultivation and fallow state on a land plot. This model is applied to a case study on Madagascan farms in the scope of a sustainable development campain against deforestation, where the objective is to better control agricultural space while covering seasonal needs of farmer. We propose an efficient upper bound computation and study the variation of the minimum number of plots and total space needed in function of the unitary surface area of a plot. Numerical results associated with the Madagascan case are reported.

Solving the electricity production planning problem by a column generation based heuristic

This paper presents a heuristic method based on column generation for the EDF (Electricité De France) long-term electricity production planning problem proposed as subject of the ROADEF/EURO 2010 Challenge. This is to our knowledge the first-ranked method among those methods based on mathematical programming, and was ranked fourth overall. The problem consists in determining a production plan over the whole time horizon for each thermal power plant of the French electricity company, and for nuclear plants, a schedule of plant outages which are necessary for refueling and maintenance operations. The average cost of the overall outage and production planning, computed over a set of demand scenarios, is to be minimized. The method proceeds in two stages. In the first stage, dates for outages are fixed once for all for each nuclear plant. Data are aggregated with a single average scenario and reduced time steps, and a set-partitioning reformulation of this aggregated problem is solved for fixing outage dates with a heuristic based on column generation. The pricing problem associated with each nuclear plant is a shortest path problem in an appropriately constructed graph. In the second stage, the reload level is determined at each date of an outage, considering now all scenarios. Finally, the production quantities between two outages are optimized for each plant and each scenario by solving independent linear programming problems.

Edge cover by connected bipartite subgraphs

We consider the problem of covering the edge set of an unweighted, undirected graph with the minimum number of connected bipartite subgraphs (where the subgraphs are not necessarily bicliques). We show that this is an NP-hard problem, provide lower bounds through an integer programming formulation, propose several constructive heuristics and a local search, and discuss computational results. Finally, we consider a constrained variant of the problem which we show to be NP-hard, and provide an integer programming formulation for the variant.

Optimizing healthcare network design under reference pricing and parameter uncertainty

Healthcare payers are exploring cost-containing policies to steer patients, through qualified information and financial incentives, towards providers offering the best value. With Reference Pricing (RP), a payer determines a maximum amount paid for a procedure, and patients selecting a provider charging more pay the difference. In a Tiered Network (TN), providers are stratified according to criteria such as quality and cost, each tier having a different out-of-pocket price. Motivated by a program recently implemented in California, we design an optimization model for payers combining both RP and TN, filling the gap of quantitative research on these novel payment policies. The main decision is to select which providers to exempt from RP, whose patients will face no out-of-pocket costs. Patients’ choice of a provider is modeled with a Multinomial Logit model. The objective is to minimize the payer’s cost, while constraints provide decision makers with levers for a trade-off between cost reduction and providers quality. We build a robust counterpart of our model to account for parameter uncertainty. Numerical experiments provide insights into how tiers are scattered on a price/quality plane. We argue that this system has strong potential in terms of costs reduction, quality increase for patients and visibility for high-value providers.

Hybrid column generation for large-size Covering Integer Programs: Application to transportation planning

The well-known column generation scheme is often an efficient approach for solving the linear relaxation of large-size Covering Integer Programs (CIP). In this paper, this technique is hybridized with an extension of the best-known CIP approximation heuristic, taking advantage of distinct criteria of columns selection. This extension uses fractional optimization for solving pricing subproblems. Numerical results on a real-case transportation planning problem show that the hybrid scheme accelerates the convergence of column generation both in terms of number of iterations and computational time. The integer solutions generated at the end of the process can also be improved for a significant proportion of instances, highlighting the potential of diversification of the approximation heuristic.