A Disjunctive Graph for the Job-Shop with Several Robots Philippe Lacomme , Mohand Larabi Université Blaise Pascal, LIMOS (UMR CNRS 6158), 63177 Aubière Cedex, France, {placomme, larabi }@isima.fr Nikolay Tchernev IUP « Management et gestion des entreprises », Université d’Auvergne, LIMOS (UMR CNRS 6158), 26, av. Léon Blum, Clermont Ferrand, France (tchernev@isima.fr) This paper addresses the scheduling problem in a job-shop where the jobs have to be transported between the machines by several transport robots. The problem can be efficiently modeled by a disjunctive graph and any solution can be fully defined by an orientation of the graph. The objective is to determine a schedule of machine and transport operations as well as an assignment of robots to transport operations with minimal makespan. We present: (i) a problem representation using an appropriate disjunctive graph; (ii) a solution representation based on 3 vectors consisting of machine disjunctions, transport disjunctions and robots assignments; (iii) a new problem-specific properties to define local search algorithms. Computational results are presented for test data arising from Bilge and Uluzoy’s benchmark instances enlarged by transportation, empty moving times for each robot. This paper is a step forward definition of an efficient approach arising a job-shop with several robots and it is an attempt to extend the Hurink and Knust proposal dedicated to one single transport robot. Keywords: scheduling, job-shop, transport. 1. Problem settings and definition A job-shop scheduling problem with transportation times and several robots can be formulated, in general, as a problem of processing a set of n jobs { } n J J J , , 1 K = on a set of machines , transported by a set of m { m M M M , , 1 K = } r transport robots . Each job is an ordered set of operations denoted , , …, for which precedence constraints are defined. Each machine can process (without preemption) only one job at one time and each job can be processed by only one machine at the same time. Additionally, transportation times are taken into account considered each time a job changes from one machine to another one. Transportation operation must be considered for robot when machine operation is processed on machine { r R R K , 1 } R = i n , r R i J i n r i R t μ ik 1 i O 2 i O i O k i k 1 , , , + μ ik O μ and machine operation is processed on machine 1 , k i O + 1 + , k i μ . These transportation times are job-independent and robot-dependent. Each transportation operation is assumed to be processed by only one transport robot which can handle at most one job at one time. For convenience, is used to denote both a transportation operation and a transportation time. r k i k i R t 1 , , , + μ μ i M Also empty moving times have to be considered while one robot moves from machine to without carrying a job. It is possible to assume, for each robot , that and t . As in job-shop problems, we assume that sufficient buffer space exists between machines. This assumption is also stated as an “unlimited input/output buffer capacity”. Jobs processed on one machine are assumed to wait until the robot affected to this transport operation is available to do it. No additional time is required to transfer job from machine to the r R j i v , r R i M j M r R j i v , ≥ r R 0 , = R i i v r R j i , 285 Regular Papers