A variable iterated greedy algorithm with differential evolution for the no-idle permutation flowshop scheduling problem

This paper presents a variable iterated greedy algorithm (IG) with differential evolution (vIG_DE), designed to solve the no-idle permutation flowshop scheduling problem. In an IG algorithm, size d of jobs are removed from a sequence and re-inserted into all possible positions of the remaining sequences of jobs, which affects the performance of the algorithm. The basic concept behind the proposed vIG_DE algorithm is to employ differential evolution (DE) to determine two important parameters for the IG algorithm, which are the destruction size and the probability of applying the IG algorithm to an individual. While DE optimizes the destruction size and the probability on a continuous domain by using DE mutation and crossover operators, these two parameters are used to generate a trial individual by directly applying the IG algorithm to each target individual depending on the probability. Next, the trial individual is replaced with the corresponding target individual if it is better in terms of fitness. A unique multi-vector chromosome representation is presented in such a way that the first vector represents the destruction size and the probability, which is a DE vector, whereas the second vector simply consists of a job permutation assigned to each individual in the target population. Furthermore, the traditional IG and a variable IG from the literature are re-implemented as well. The proposed algorithms are applied to the no-idle permutation flowshop scheduling (NIPFS) problem with the makespan and total flowtime criteria. The performances of the proposed algorithms are tested on the Ruben Ruiz benchmark suite and compared to the best-known solutions available at http://soa.iti.es/rruiz as well as to those from a recent discrete differential evolution algorithm (HDDE) from the literature. The computational results show that all three IG variants represent state-of-art methods for the NIPFS problem.     

A multi-objective electromagnetism algorithm for a bi-objective flowshop scheduling problem

This paper deals with a bi-objective flowshop scheduling problem minimizing the makespan and total weighted tardiness, in which all jobs may not be processed by all machines. Furthermore, we consider transportation times between machines. Obtaining an optimal solution for this type of complex, large-sized problem in reasonable computational time by using traditional approaches and optimization tools is extremely difficult. This paper presents a new multi-objective electromagnetism algorithm (MOEM). The motivation behind this algorithm has risen from the attraction–repulsion mechanism of electromagnetic theories. Along with MOEA, we apply simulated annealing to solve the given problem. A set of experimental instances are carried out to evaluate the algorithm by advanced multi-objective performance measures. The related results show that a variant of our proposed MOEM provides sound performance comparing with other algorithms.     

A three-phase algorithm for flowshop scheduling with blocking to minimize makespan

This paper proposes a three-phase algorithm (TPA) for the flowshop scheduling problem with blocking (BFSP) to minimize makespan. In the first phase, the blocking nature of BFSP is exploited to develop a priority rule that creates a sequence of jobs. Using this as the initial sequence and a variant of the NEH-insert procedure, the second phase generates an approximate solution to the problem. Then, utilizing a modified simulated annealing algorithm incorporated with a local search procedure, the schedule generated in the second phase is improved in the third phase. A pruning procedure that helps evaluate most solutions without calculating their complete makespan values is introduced in the local search to further reduce the computational time needed to solve the problem. Results of the computational experiments with Taillard's benchmark problem instances show that the proposed TPA algorithm is relatively more effective and efficient in minimizing makespan for the BFSP than the state-of-the-art procedures. Utilizing these results, 53 out of 60 new tighter upper bounds have been found for large-sized Taillard's benchmark problem instances.     

The distributed permutation flowshop scheduling problem

This paper studies a new generalization of the regular permutation flowshop scheduling problem (PFSP) referred to as the distributed permutation flowshop scheduling problem or DPFSP. Under this generalization, we assume that there are a total of F identical factories or shops, each one with m machines disposed in series. A set of n available jobs have to be distributed among the F factories and then a processing sequence has to be derived for the jobs assigned to each factory. The optimization criterion is the minimization of the maximum completion time or makespan among the factories. This production setting is necessary in today's decentralized and globalized economy where several production centers might be available for a firm. We characterize the DPFSP and propose six different alternative mixed integer linear programming (MILP) models that are carefully and statistically analyzed for performance. We also propose two simple factory assignment rules together with 14 heuristics based on dispatching rules, effective constructive heuristics and variable neighborhood descent methods. A comprehensive computational and statistical analysis is conducted in order to analyze the performance of the proposed methods.     

A grasp algorithm for m-machine flowshop scheduling problem with bicriteria of makespan and maximum tardiness

In this paper we address the problem of minimizing the weighted sum of makespan and maximum tardiness in an m-machine flow shop environment. This is a NP-hard problem in the strong sense. An attempt has been made to solve this problem using a metaheuristic called Greedy Randomized Adaptive Search Procedure (GRASP). GRASP is a competitive algorithm and is a meta-heuristic for solving combinatorial optimization problems. We have customized the basic concepts of GRASP algorithm to solve a bicriteria flow shop problem and a new algorithm named B-GRASP (Bicriteria GRASP algorithm) is proposed. The new proposed algorithm is evaluated using benchmark problems taken from Taillard and compared with the existing simulated annealing based heuristic developed by Chakravarthy and Rajendran. Computational experiments indicate that the proposed algorithm is much better than the existing one in all cases.     

An efficient memetic algorithm for solving the job shop scheduling problem

The job shop scheduling problem (JSP) is well known as one of the most complicated combinatorial optimization problems, and it is a NP-hard problem. Memetic algorithm (MA) which combines the global search and local search is a hybrid evolutionary algorithm. In this paper, an efficient MA with a novel local search is proposed to solve the JSP. Within the local search, a systematic change of the neighborhood is carried out to avoid trapping into local optimal. And two neighborhood structures are designed by exchanging and inserting based on the critical path. The objective of minimizing makespan is considered while satisfying a number of hard constraints. The computational results obtained in experiments demonstrate that the efficiency of the proposed MA is significantly superior to the other reported approaches in the literature.     

A discrete particle swarm optimization algorithm for the no-wait flowshop scheduling problem

In this paper, a discrete particle swarm optimization (DPSO) algorithm is presented to solve the no-wait flowshop scheduling problem with both makespan and total flowtime criteria. The main contribution of this study is due to the fact that particles are represented as discrete job permutations and a new position update method is developed based on the discrete domain. In addition, the DPSO algorithm is hybridized with the variable neighborhood descent (VND) algorithm to further improve the solution quality. Several speed-up methods are proposed for both the swap and insert neighborhood structures. The DPSO algorithm is applied to both 110 benchmark instances of Taillard [Benchmarks for basic scheduling problems. European Journal of Operational Research 1993;64:278–85] by treating them as the no-wait flowshop problem instances with the total flowtime criterion, and to 31 benchmark instances provided by Carlier [Ordonnancements a contraintes disjonctives. RAIRO Recherche operationelle 1978;12:333–51], Heller [Some numerical experiments for an M×J$\mathrm{M}\times \mathrm{J}$ flow shop and its decision-theoretical aspects. Operations Research 1960;8:178–84], and Revees [A genetic algorithm for flowshop sequencing. Computers and Operations Research 1995;22:5–13] for the makespan criterion. For the makespan criterion, the solution quality is evaluated according to the reference makespans generated by Rajendran [A no-wait flowshop scheduling heuristic to minimize makespan. Journal of the Operational Research Society 1994;45:472–8] whereas for the total flowtime criterion, it is evaluated with the optimal solutions, lower bounds and best known solutions provided by Fink and Voß [Solving the continuous flow-shop scheduling problem by metaheuristics. European Journal of Operational Research 2003;151:400–14]. The computational results show that the DPSO algorithm generated either competitive or better results than those reported in the literature. Ultimately, 74 out of 80 best known solutions provided by Fink and Voß [Solving the continuous flow-shop scheduling problem by metaheuristics. European Journal of Operational Research 2003;151:400–14] were improved by the VND version of the DPSO algorithm.     

A discrete particle swarm optimization algorithm with self-adaptive diversity control for the permutation flowshop problem with blocking

This paper proposes a discrete particle swarm optimization (DPSO) algorithm for the m-machine permutation flowshop scheduling problem with blocking to minimize the makespan, which has a strong industrial background, e.g., many production processes of chemicals and pharmaceuticals in chemical industry can be reduced to this problem. To prevent the DPSO from premature convergence, a self-adaptive diversity control strategy is adopted to diversify the population when necessary by adding a random perturbation to the velocity of each particle according to a probability controlled by the diversity of the current population. In addition, a stochastic variable neighborhood search is used as the local search to improve the search intensification. Computational results using benchmark problems show that the proposed DPSO algorithm outperforms previous algorithms proposed in the literature and that it can obtain 111 new best known upper bounds for the 120 benchmark problems.     

一种快速解决PFSP问题的混合遗传算法

流水车间调度问题属于NP难问题,并且和实际问题联系很紧。但是因为它的解空间太大,一般的算法很容易过早的陷入局部最优或者计算时间太长,提出了一种比较快速的混合遗传算法,能够在很短时间内计算出比较优的结果。详细介绍了这种算法的效果,并与两种常用来解决此类问题的算法进行了比较,总结出了这个算法的特点。     

A two-machine flowshop makespan scheduling problem with deteriorating jobs

In traditional scheduling problems, the job processing times are assumed to be known and fixed from the first job to be processed to the last job to be completed. However, in many realistic situations, a job will consume more time than it would have consumed if it had begun earlier. This phenomenon is known as deteriorating jobs. In the science literature, the deteriorating job scheduling problems are relatively unexplored in the flowshop settings. In this paper, we study a two-machine flowshop makespan scheduling problem in which job processing times vary as time passes, i.e. they are assumed as increasing functions of their starting times. First, an exact algorithm is established to solve most of the problems of up to 32 jobs in a reasonable amount of time. Then, three heuristic algorithms are provided to derive the near-optimal solutions. A simulation study is conducted to evaluate the performances of the proposed algorithms. In addition, the impact of the deterioration rate is also investigated.     

A branch-and-bound algorithm for the three-machine flowshop scheduling problem with bicriteria of makespan andtotal flowtime

This paper addresses the three‐machine flowshop scheduling problem with a bicriteria of minimizing a weighted sum of makespan and total flowtime. Three lower bounds, an upper bound, and several dominance relations are developed. The upper bound is developed using a two‐phase hybrid heuristic method. A branch‐and‐bound algorithm, incorporating the developed bounds and dominance relations, is presented. An extensive computational analysis on randomly generated problems is conducted. The analysis indicates that the proposed bounds, dominance relations, and branch‐and‐bound algorithm are efficient.     

Makespan distribution of permutation flowshop schedules

The makespan distribution of permutation flowshop schedules has been a topic of debate for almost fifty years. Many researchers have confirmed or doubted the famous claim that the makespan distribution of permutation flowshop schedules is asymptotically normal if the number of jobs is sufficiently large. This paper theoretically and empirically investigates the makespan distribution of permutation flowshop schedules and shows that the normality claim is not valid for the job-dominated and machine-dominated flowshops. Errors in the proof of normality of the makespan distribution of permutation flowshop schedules are pointed out. It is shown that the makespan distribution of a permutation flowshop scheduling problem depends on the number of jobs as well as the number of machines.     

A Self-guided Genetic Algorithm for permutation flowshop scheduling problems

In this paper we develop a Self-guided Genetic Algorithm (Self-guided GA), which belongs to the category of Estimation of Distribution Algorithms (EDAs). Most EDAs explicitly use the probabilistic model to sample new solutions without using traditional genetic operators. EDAs make good use of the global statistical information collected from previous searches but they do not efficiently use the location information about individual solutions. It is recently realized that global statistical information and location information should complement each other during the evolution process. In view of this, we design the Self-guided GA based on a novel strategy to combine these two kinds of information. The Self-guided GA does not sample new solutions from the probabilistic model. Instead, it estimates the quality of a candidate offspring based on the probabilistic model used in its crossover and mutation operations. In such a way, the mutation and crossover operations are able to generate fitter solutions, thus improving the performance of the algorithm. We tested the proposed algorithm by applying it to deal with the NP-complete flowshop scheduling problem to minimize the makespan. The experimental results show that the Self-guided GA is very promising. We also demonstrate that the Self-guided GA can be easily extended to treat other intractable combinatorial problems.     

Polynomial time algorithms for the UET permutation flowshop problem with time delays

This paper addresses the problem of scheduling a set of n unit execution time (UET) jobs on an m-permutation flowshop with arbitrary time delays, so as to minimize the makespan criterion. A polynomial time algorithm is exhibited for the three-machine and four-machine cases, respectively.     

A new Lagrangian relaxation algorithm for hybrid flowshop scheduling to minimize total weighted completion time

We investigate the problem of scheduling n jobs in s-stage hybrid flowshops with parallel identical machines at each stage. The objective is to find a schedule that minimizes the sum of weighted completion times of the jobs. This problem has been proven to be NP-hard. In this paper, an integer programming formulation is constructed for the problem. A new Lagrangian relaxation algorithm is presented in which precedence constraints are relaxed to the objective function by introducing Lagrangian multipliers, unlike the commonly used method of relaxing capacity constraints. In this way the relaxed problem can be decomposed into machine type subproblems, each of which corresponds to a specific stage. A dynamic programming algorithm is designed for solving parallel identical machine subproblems where jobs may have negative weights. The multipliers are then iteratively updated along a subgradient direction. The new algorithm is computationally compared with the commonly used Lagrangian relaxation algorithms which, after capacity constraints are relaxed, decompose the relaxed problem into job level subproblems and solve the subproblems by using the regular and speed-up dynamic programming algorithms, respectively. Numerical results show that the new Lagrangian relaxation method produces better schedules in much shorter computation time, especially for large-scale problems.     

New simple constructive heuristic algorithms for minimizing total flow-time in the permutation flowshop scheduling problem

This paper develops a set of new simple constructive heuristic algorithms to minimize total flow-time for an $n$-jobs×$m$-machines permutation flowshop scheduling problem. We first propose a new iterative algorithm based on the best existing simple heuristic algorithm, and then integrate new indicator variables for weighting jobs into this algorithm. We also propose new decision criteria to select the best partial sequence in each iteration of our algorithm. A comprehensive numerical experiment reveals that our modifications and extensions improve the effectiveness of the best existing simple heuristic without affecting its computational efficiency.     

An improved particle swarm optimization algorithm for flowshop scheduling problem

The flowshop scheduling problem has been widely studied and many techniques have been applied to it, but few algorithms based on particle swarm optimization (PSO) have been proposed to solve it. In this paper, an improved PSO algorithm (IPSO) based on the “alldifferent” constraint is proposed to solve the flow shop scheduling problem with the objective of minimizing makespan. It combines the particle swarm optimization algorithm with genetic operators together effectively. When a particle is going to stagnate, the mutation operator is used to search its neighborhood. The proposed algorithm is tested on different scale benchmarks and compared with the recently proposed efficient algorithms. The results show that the proposed IPSO algorithm is more effective and better than the other compared algorithms. It can be used to solve large scale flow shop scheduling problem effectively.     

A Constructive Genetic Algorithm for permutation flowshop scheduling

The general flowshop scheduling problem is a production problem where a set of n jobs have to be processed with identical flow pattern on m machines. In permutation flowshops the sequence of jobs is the same on all machines. A significant research effort has been devoted for sequencing jobs in a flowshop minimizing the makespan. This paper describes the application of a Constructive Genetic Algorithm (CGA) to makespan minimization on flowshop scheduling. The CGA was proposed recently as an alternative to traditional GA approaches, particularly, for evaluating schemata directly. The population initially formed only by schemata, evolves controlled by recombination to a population of well-adapted structures (schemata instantiation). The CGA implemented is based on the NEH classic heuristic and a local search heuristic used to define the fitness functions. The parameters of the CGA are calibrated using a Design of Experiments (DOE) approach. The computational results are compared against some other successful algorithms from the literature on Taillard’s well-known standard benchmark. The computational experience shows that this innovative CGA approach provides competitive results for flowshop scheduling problems.     

A new algorithm for online uniform-machine scheduling to minimize the makespan

We consider the online scheduling problem with m−1, m?2, uniform machines each with a processing speed of 1, and one machine with a speed of s, 1?s?2, to minimize the makespan. The well-known list scheduling (LS) algorithm has a worst-case bound of [Y. Cho, S. Sahni, Bounds for list schedules on uniform processors, SIAM J. Comput. 9 (1980) 91-103]. An algorithm with a better competitive ratio was proposed in [R. Li, L. Shi, An on-line algorithm for some uniform processor scheduling, SIAM J. Comput. 27 (1998) 414-422]. It has a worst-case bound of 2.8795 for a big m and s=2. In this note we present a 2.45-competitive algorithm for m?4 and any s, 1?s?2.     

An iterated local search for the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times

Due to its simplicity yet powerful search ability, iterated local search (ILS) has been widely used to tackle a variety of single-objective combinatorial optimization problems. However, applying ILS to solve multi-objective combinatorial optimization problems is scanty. In this paper we design a multi-objective ILS (MOILS) to solve the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times to minimize the makespan and total weighted tardiness of all jobs. In the MOILS, we design a Pareto-based variable depth search in the multi-objective local search phase. The search depth is dynamically adjusted during the search process of the MOILS to strike a balance between exploration and exploitation. We incorporate an external archive into the MOILS to store the non-dominated solutions and provide initial search points for the MOILS to escape from local optima traps. We compare the MOILS with several multi-objective evolutionary algorithms (MOEAs) shown to be effective for treating the multi-objective permutation flowshop scheduling problem in the literature. The computational results show that the proposed MOILS outperforms the MOEAs.