Choice Rules with Size Constraints for Multiple Criteria Decision Making

In outranking methods for Multiple Criteria Decision Making (MCDM), pair-wise comparisons of alternatives are often summarized through a fuzzy preference relation. In this paper, the binary preference relation is extended to pairs of subsets of alternatives in order to define on this basis a scoring function over subsets. A choice rule based on maximizing score under size constraint is studied, which turns to formulate as solving a sequence of classical location problems. For comparison with the kernel approach, the interior stability property of the selected subset is discussed and analyzed.Combinatorial optimization; Fuzzy preferences; Integer Programming; Location; Multiple Criteria Decision Aid

An Exact Method for Assortment Optimization under the Nested Logit Model

We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a Nested Logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. For the NP-hard cases, we provide a general exact method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. Contrary to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed, and hence our approach can solve the most general problem setting. We show that the parameterized subproblem of the fractional programming scheme, which is a binary highly non-linear optimization problem, is decomposable by nests, which is a main advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we identify those products that are undoubtedly beneficial to offer, or not, which can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with up to 5,000 products per nest. The most challenging instances for our approach are those in which the dissimilarity parameters of nests can be either less or greater than one

Space minimization in agricultural production planning by column generation

We deal in this paper with an agricultural production planning problem where crops must be scheduled on land plots so as to satisfy crop demands every period of time and to minimize the overall surface of land used for cultivation. This problem can be formulated as a covering integer program with a huge number of variables. A resolution scheme based on column generation is thus proposed, where the resulting pricing problem is efficiently solved by dynamic programming. The numerical experiments show that the method is all the more so efficient and robust as the planning horizon is long and plot sizes are small

A Column Generation Based Heuristic for the Multicommodity-ring Vehicle Routing Problem

AbstractWe study a new routing problem arising in City Logistics. Given a ring connecting a set of urban distribution centers (UDCs) in the outskirts of a city, the problem consists in delivering goods from virtual gates located outside the city to the customers inside of it. Goods are transported from a gate to a UDC, then either go to another UDC before being delivered to customers or are directly shipped from the first UDC. The reverse process occurs for pick-up. Routes are performed by electric vans and may be open. The objective is to find a set of routes that visit each customer and to determine ring and gates-UDC flows so that the total transportation and routing cost is minimized. We solve this problem using a column generation-based heuristic, which is tested over a set of benchmark instances issued from a more strategic location-routing problem

Approximation of the Constrained Path Covering Problem

We study a generic covering problem frequently met in transportation planning. The problem is to cover a set of tasks by constrained paths, representing routings of commodities, so that the total cost of selected paths is minimal. In this paper, paths are constrained to have limited weight and limited number of tasks. We show that the generic problem is NP-hard and that a greedy heuristic using dynamic programming at each step achieves an approximation ratio of ln(d), where d is the maximum number of tasks of a path

Improved approximation of the general Soft-Capacitated Facility Location Problem

The Soft-Capacitated Facility Location Problem, where each facility is composed of a variable number of fixed-capacity production units, has been recently studied in several papers, especially in the metric case. 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 subproblem is approximately solved by a Fully Polynomial-Time Approximation Scheme based on cost scaling and dynamic programming, achieves a logaritmic approximation ratio of (1 + ɛ)H(n) for the problem, where n is the number of customers to be served and H is the harmonic series. This improves the previous bound of 2H(n) for this problem. Key-words: facility location, set covering, dynamic programming, FPTAS.