Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table

Exact algorithms for the order picking problem

Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and effective algorithms for this problem. Firstly, a sparse formulation in mixed-integer programming is strengthened by preprocessing and valid inequalities. Secondly, a dynamic programming approach generalizing known algorithms for two or three cross-aisles is proposed and evaluated experimentally. Performances of these algorithms are reported and compared with the Traveling Salesman Problem (TSP) solver Concorde

Bidirected minimum Manhattan network problem

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) are axis-parallel and oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this paper, we present a polynomial factor 2 approximation algorithm for the bidirected minimum Manhattan network problem.Comment: 14 pages, 16 figure

Strategic and Tactical Urban Farm Design

International audienceWe use mix integer linear programing to establish a model for fresh fruit and vegetable production. We aim at maximizing the annual farm result, responding to a fixed daily demand. Our main variables are the area to grow using specific management practices, the beginning date of each plot cultivation and the workers to hire to produce. We introduce variables to consider the perishability of the products in different ways. As Ahumada and Villalobos (2009), we use a storage cold room loss function and cold room limited capacity. But we also allow a standing storage on the plot, associated to another loss function: the crops can be harvested some days after the ideal harvest time. The selling price is lowered according to storage time, both on the plot and in the cold room. We force the unsold products left on the plots (losses) to be harvested before a management practice dependent deadline, to limit diseases and insects proliferation. Considering short time perishability (a few days) make us work on a tow-day step. It enables a good combination of different crop management practices on the farm to have the best labor affectation. Perishability, considered at a tactical level, has major consequences on the strategic farming system sizing

Roadef Challenge 2014: A Modeling Approach, Rolling stock unit management on railway sites

We report here our analysis of the challenge Roadef/EURO 2014 problem and propose a methodology strongly based on modeling with MIP and CP technologies. Due to the complexity of the problem formulation, we believe that a robust engineering is easier to achieve by relying on models than dedicated code. Two core modeling ideas are presented for relating the daily maintenances limit and the linked departures to an assignment based MIP model. Additionnaly, a variant of the maximum matching problem lying at the heart of the problem is shown to be NP-Complete. Intermediate experimental results are given along the way to support the ideas reported

An Integer Programming Formulation Using Convex Polygons for the Convex Partition Problem

A convex partition of a point set P in the plane is a planar partition of the convex hull of P into empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition with the minimum number of internal faces. The problem has been shown to be NP-hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points

Good Production Cycles for Circular Robotic Cells

In this paper, we study cyclic production for throughput optimization in robotic flow-shops. We are focusing on simple production cycles. Robotic cells can have a linear or a circular layout: most classical results on linear cells cannot be extended to circular cells, making it difficult to quantify the potential gain brought by the latter configuration. Moreover, though the problem of finding the best one part production cycle is polynomial for linear cells, it is NP-hard for circular cells. We consider the special case of circular balanced cells. We first consider three basic production cycles, and focus on one which is specific to circular cells, for which we establish the expression of the cycle time. Then, we provide a counterexample to a classical conjecture still open in this configuration. Finally, based on computational experiments, we make a conjecture on the dominance of a family of cycle, which could lead to a polynomial algorithm for finding the best 1-cycle for circular balanced cells

Cyclic production in regular robotic cells: A counterexample to the 1-cycle conjecture

Robotic cells consist in a flow-shop where transportation of the parts between machines is handled by a robot. We consider cyclic production of identical parts and optimization of the cell's throughput. Production cycle of 1 part are easier to describe implement and there is a conjecture about their dominance. This conjecture has been studied for linear layout cells, for which the 1-cycles are well known, but not for cells with circular layout, where the input and output buffers are at the same position. We provide a counterexample to the conjecture for this case

Embedding into the rectilinear plane in optimal O*(n^2)

We present an optimal O*(n^2) time algorithm for deciding if a metric space (X,d) on n points can be isometrically embedded into the plane endowed with the l_1-metric. It improves the O*(n^2 log^2 n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by J. Edmonds, our algorithm uses the concept of tight span and the injectivity of the l_1-plane. A different O*(n^2) time algorithm was recently proposed by D. Eppstein (2009).Comment: 12 pages, 13 figure