Inés González-Rodríguez

Semantics of Schedules for the Fuzzy Job-Shop Problem

In the sequel, we consider the fuzzy job-shop problem, which is a variation of the job-shop problem where duration of tasks may be uncertain and where due-date constraints are allowed to be flexible. Uncertain durations are modeled using triangular fuzzy numbers, and due-date constraints are fuzzy sets with decreasing membership functions expressing a flexible threshold ldquoless than.rdquo Also, the objective function is built using fuzzy decision-making theory. We propose the use of a genetic algorithm (GA) to find solutions to this problem. Our aim is to provide a semantics for this type of problems and use this semantics in a methodology to analyze, evaluate, and, therefore, compare solutions. Finally, we present the results obtained using the GA and evaluate them using the proposed methodology.

Genetic tabu search for the fuzzy flexible job shop problem

This paper tackles the flexible job-shop scheduling problem with uncertain processing times. The uncertainty in processing times is represented by means of fuzzy numbers, hence the name fuzzy flexible job-shop scheduling. We propose an effective genetic algorithm hybridised with tabu search and heuristic seeding to minimise the total time needed to complete all jobs, known as makespan. To build a high-quality and diverse set of initial solutions we introduce a heuristic method which benefits from the flexible nature of the problem. This initial population will be the starting point for the genetic algorithm, which then applies tabu search to every generated chromosome. The tabu search algorithm relies on a neighbourhood structure that is proposed and analysed in this paper; in particular, some interesting properties are proved, such as feasibility and connectivity. Additionally, we incorporate a filtering mechanism to reduce the neighbourhood size and a method that allows to speed-up the evaluation of new chromosomes. To assess the performance of the resulting method and compare it with the state-of-the-art, we present an extensive computational study on a benchmark with 205 instances, considering both deterministic and fuzzy instances to enhance the significance of the study. The results of these experiments clearly show that not only does the hybrid algorithm benefit from the synergy among its components but it is also quite competitive with the state-of-the-art when solving both crisp and fuzzy instances, providing new best-known solutions for a number of these test instances.

A Memetic Approach to Fuzzy Job Shop Based on Expectation Model

In the sequel we consider a job shop problem with uncertain processing times modelled using triangular fuzzy numbers. A scheduling model based on the expected value of the makespan is introduced. Later, a genetic algorithm based on codification of permutations with repetitions, a decoding algorithm to generate possibly active schedules and a local search schema are defined in order to solve the job shop problem. Experimental results illustrate the potential of the proposed methods.

A bipolar model of assertability and belief

Valuation pairs are introduced as a bipolar model of the assertability of propositions. These correspond to a pair of dual valuation functions, respectively, representing the strong property of definite assertability and the dual weaker property of acceptable assertability. In the case where there is uncertainty about the correct valuation pair for a language then a probability distribution is defined on possible valuation pairs. This results in two measures, @m^+ giving the probability that a sentence is definitely assertable, and @m^- giving the probability that a sentence is acceptable to assert. It is shown that @m^+ and @m^- can be determined directly from a two dimensional mass function m defined on pairs of sets of propositional variables. Certain natural properties of @m^+ and @m^- are easily expressed in terms of m, and in particular we introduce certain consonance or nestedness assumptions. These capture qualitative information in the form of assertability orderings for both the propositional variables and the negated propositional variables. On the basis of these consonance assumptions we show that label semantics, intuitionistic fuzzy logic and max-min fuzzy logic can all be viewed as special cases of this bipolar model. We also show that bipolar belief measures can be interpreted within an interval-set model.

Coevolutionary makespan optimisation through different ranking methods for the fuzzy flexible job shop

In this paper we tackle a variant of the flexible job shop scheduling problem with uncertain task durations modelled as fuzzy numbers, the fuzzy flexible job shop scheduling problem or FfJSP in short. To minimise the schedules fuzzy makespan, we consider different ranking methods for fuzzy numbers. We then propose a cooperative coevolutionary algorithm with two different populations evolving the two components of a solution: machine assignment and task relative order. Additionally, we incorporate a specific local search method for each population. The resulting hybrid algorithm is then evaluated on existing benchmark instances, comparing favourably with the state-of-the-art methods. The experimental results also serve to analyse the influence in the robustness of the resulting schedules of the chosen ranking method.

An efficient hybrid evolutionary algorithm for scheduling with setup times and weighted tardiness minimization

We confront the job shop scheduling problem with sequence-dependent setup times and weighted tardiness minimization. To solve this problem, we propose a hybrid metaheuristic that combines the intensification capability of tabu search with the diversification capability of a genetic algorithm which plays the role of long term memory for tabu search in the combined approach. We define and analyze a new neighborhood structure for this problem which is embedded in the tabu search algorithm. The efficiency of the proposed algorithm relies on some elements such as neighbors filtering and a proper balance between intensification and diversification of the search. We report results from an experimental study across conventional benchmarks, where we analyze our approach and demonstrate that it compares favorably to the state-of-the-art methods.

Heuristic local search for fuzzy open shop scheduling

We consider the fuzzy open shop scheduling problem, where task durations are assumed to be ill-known and modelled as triangular fuzzy numbers. We propose a neighbourhood structure for local search procedures, based on reversing critical arcs in the associated disjunctive graph. We provide a thorough theoretical study of the structure and, in particular, prove that feasibility and asymptotic convergence hold. We further illustrate its good behaviour with experimental results obtained by incorporating the local search procedure to an existing genetic algorithm from the literature and provide a new benchmark of problem instances.

Schedule generation schemes for job shop problems with fuzziness

We consider the job shop scheduling problem with fuzzy durations and expected makespan minimisation. We formally define the space of semi-active and active fuzzy schedules and propose and analyse different schedule-generation schemes (SGSs) in this fuzzy framework. In particular, we study dominance properties of the set of schedules obtained with each SGS. Finally, a computational study illustrates the great difference between the spaces of active and the semi-active fuzzy schedules, an analogous behaviour to that of the deterministic job shop.

Robust swarm optimisation for fuzzy open shop scheduling

Abstract In this paper we consider a variant of the open shop problem where task durations are allowed to be uncertain and where uncertainty is modelled using fuzzy numbers. Solutions to this problem are fuzzy schedules, which we argue should be seen as predictive schedules, thus establishing links with the concept of robustness and a measure thereof. We propose a particle swarm optimization (PSO) approach to minimise the schedule’s expected makespan, using priorities to represent particle position, as well as a decoding algorithm to generate schedules in a subset of possibly active ones. Our proposal is evaluated on a varied set of several benchmark problems. The experimental study includes a parametric analysis, results of the PSO compared with the state-of-the-art, and an empirical study of the robustness of taking into account uncertainty along the scheduling process.

A Genetic Algorithm for the Open Shop Problem with Uncertain Durations

We consider a variation of the open shop problem where task durations are allowed to be uncertain and where uncertainty is modelled using fuzzy numbers. We propose a genetic approach to minimise the expected makespan: we consider different possibilities for the genetic operators and analyse their performance, in order to obtain a competitive configuration. Finally, the performance of the proposed genetic algorithm is tested on several benchmark problems, modified so as to have fuzzy durations, compared with a greedy heuristic from the literature.