Optimal feeding profile in fed-batch bioreactors using a genetic algorithm

An optimal feeding profile for a fed-batch process was designed based on an evolutionary algorithm. Usually the presence of multiple objectives in a problem leads to a set of optimal solutions, commonly known as Pareto-optimal solutions. Evolutionary algorithms are well suited for deriving multi-objective optimisation since they evolve a set of non-dominated solutions distributed along the Pareto front. Several evolutionary multi-objective optimisation algorithms have been developed, among which the Non-dominated Sorting Genetic Algorithm NSGA-II is recognised to be very effective in overcoming a variety of problems. To demonstrate the applicability of this technique, an optimal control problem from the literature was solved using several methods considering the single-objective dynamic optimisation problem.     

A new heuristic for the flowshop scheduling problem to minimize makespan and maximum tardiness

This paper presents a new heuristic for solving the flowshop scheduling problem that aims to minimize makespan and maximize tardiness. The algorithm is able to take into account the aforementioned performance measures, finding a set of non-dominated solutions representing the Pareto front. This method is based on the integration of two different techniques: a multi-criteria decision-making method and a constructive heuristic procedure developed for makespan minimization in flowshop scheduling problems. In particular, the technique for order preference by similarity of ideal solution (TOPSIS) algorithm is integrated with the Nawaz–Enscore–Ham (NEH) heuristic to generate a set of potential scheduling solutions. To assess the proposed heuristic's performance, comparison with the best performing multi-objective genetic local search (MOGLS) algorithm proposed in literature is carried out. The test is executed on a large number of random problems characterized by different numbers of machines and jobs. The results show that the new heuristic frequently exceeds the MOGLS results in terms of both non-dominated solutions, set quality and computational time. In particular, the improvement becomes more and more significant as the number of jobs in the problem increases.     

A multi-phase covering Pareto-optimal front method to multi-objective parallel machine scheduling

This paper considers the problem of parallel machine scheduling with sequence-dependent setup times to minimise both makespan and total earliness/tardiness in the due window. To tackle the problem considered, a multi-phase algorithm is proposed. The goal of the initial phase is to obtain a good approximation of the Pareto-front. In the second phase, to improve the Pareto-front, non-dominated solutions are unified to constitute a big population. In this phase, based on the local search in the Pareto space concept, three multi-objective hybrid metaheuristics are proposed. Covering the whole set of Pareto-optimal solutions is a desired task of multi-objective optimisation methods. So in the third phase, a new method using an e-constraint hybrid metaheuristic is proposed to cover the gaps between the non-dominated solutions and improve the Pareto-front. Appropriate combinations of multi-objective methods in various phases are considered to improve the total performance. The multi-phase algorithm iterates over a genetic algorithm in the first phase and three hybrid metaheuristics in the second and third phases. Experiments on the test problems with different structures show that the multi-phase method is a better tool to approximate the efficient set than the global archive sub-population genetic algorithm presented previously.     

Bi-objective partial flexible job shop scheduling problem: NSGA-II,NRGA, MOGA and PAES approaches

This paper deals with a problem of partial flexible job shop with the objective of minimising makespan and minimising total operation costs. This problem is a kind of flexible job shop problem that is known to be NP-hard. Hence four multi-objective, Pareto-based, meta-heuristic optimisation methods, namely non-dominated sorting genetic algorithm (NSGA-II), non-dominated ranked genetic algorithm (NRGA), multi-objective genetic algorithm (MOGA) and Pareto archive evolutionary strategy (PAES) are proposed to solve the problem with the aim of finding approximations of optimal Pareto front. A new solution representation is introduced with the aim of solving the addressed problem. For the purpose of performance evaluation of our proposed algorithms, we generate some instances and use some benchmarks which have been applied in the literature. Also a comprehensive computational and statistical analysis is conducted in order to analyse the performance of the applied algorithms in five metrics including non-dominated solution, diversification, mean ideal distance, quality metric and data envelopment analysis are presented. Data envelopment analysis is a well-known method for efficiently evaluating the effectiveness of multi-criteria decision making. In this study we proposed this method of assessment of the non-dominated solutions. The results indicate that in general NRGA and PAES have had a better performance in comparison with the other two algorithms.     

Multi-objective process planning and scheduling using controlled elitist non-dominated sorting genetic algorithm

The integration of process planning and scheduling is considered as a critical component in manufacturing systems. In this paper, a multi-objective approach is used to solve the planning and scheduling problem. Three different objectives considered in this work are minimisation of makespan, machining cost and idle time of machines. To solve this integration problem, we propose an improved controlled elitist non-dominated sorting genetic algorithm (NSGA) to take into account the computational intractability of the problem. An illustrative example and five test cases have been taken to demonstrate the capability of the proposed model. The results confirm that the proposed multi-objective optimisation model gives optimal and robust solutions. A comparative study between proposed algorithm, controlled elitist NSGA and NSGA-II show that proposed algorithm significantly reduces scheduling objectives like makespan, cost and idle time, and is computationally more efficient.     

Multi-objective hierarchical genetic algorithms for multilevel redundancy allocation optimization

Multilevel redundancy allocation optimization problems (MRAOPs) occur frequently when attempting to maximize the system reliability of a hierarchical system, and almost all complex engineering systems are hierarchical. Despite their practical significance, limited research has been done concerning the solving of simple MRAOPs. These problems are not only NP hard but also involve hierarchical design variables. Genetic algorithms (GAs) have been applied in solving MRAOPs, since they are computationally efficient in solving such problems, unlike exact methods, but their applications has been confined to single-objective formulation of MRAOPs. This paper proposes a multi-objective formulation of MRAOPs and a methodology for solving such problems. In this methodology, a hierarchical GA framework for multi-objective optimization is proposed by introducing hierarchical genotype encoding for design variables. In addition, we implement the proposed approach by integrating the hierarchical genotype encoding scheme with two popular multi-objective genetic algorithms (MOGAs)—the strength Pareto evolutionary genetic algorithm (SPEA2) and the non-dominated sorting genetic algorithm (NSGA-II). In the provided numerical examples, the proposed multi-objective hierarchical approach is applied to solve two hierarchical MRAOPs, a 4- and a 3-level problems. The proposed method is compared with a single-objective optimization method that uses a hierarchical genetic algorithm (HGA), also applied to solve the 3- and 4-level problems. The results show that a multi-objective hierarchical GA (MOHGA) that includes elitism and mechanism for diversity preserving performed better than a single-objective GA that only uses elitism, when solving large-scale MRAOPs. Additionally, the experimental results show that the proposed method with NSGA-II outperformed the proposed method with SPEA2 in finding useful Pareto optimal solution sets.     

A decomposition-based approach to flexible flow shop scheduling under machine breakdown

Manufacturing systems in real-world production are generally dynamic and often subject to a wide range of uncertainties. Recently, research on production scheduling under uncertainty has attracted substantial attention. Although some methods have been developed to address this problem, scheduling under uncertainty remains inherently difficult to solve by any single approach. This article considers makespan optimisation of a flexible flow shop (FFS) scheduling problem under machine breakdown. It proposes a novel decomposition-based approach to decompose an FFS scheduling problem into several cluster scheduling problems which can be solved more easily by different approaches. A neighbouring K-means clustering algorithm is developed to first group the machines of an FFS into an appropriate number of machine clusters, based on a proposed machine allocation algorithm and weighted cluster validity indices. Two optimal back propagation networks, corresponding to the scenarios of simultaneous and non-simultaneous job arrivals, are then selectively adopted to assign either the shortest processing time (SPT) or the genetic algorithm (GA) to each machine cluster to solve cluster scheduling problems. If two neighbouring machine clusters are allocated with the same approach, they are subsequently merged. After machine grouping and approach assignment, an overall schedule is generated by integrating the solutions to the sub-problems. Computation results reveal that the proposed approach is superior to SPT and GA alone for FFS scheduling under machine breakdown.     

Multi-objective scheduling problems: Determination of pruned Pareto sets

There are often multiple competing objectives for industrial scheduling and production planning problems. Two practical methods are presented to efficiently identify promising solutions from among a Pareto optimal set for multi-objective scheduling problems. Generally, multi-objective optimization problems can be solved by combining the objectives into a single objective using equivalent cost conversions, utility theory, etc., or by determination of a Pareto optimal set. Pareto optimal sets or representative subsets can be found by using a multi-objective genetic algorithm or by other means. Then, in practice, the decision maker ultimately has to select one solution from this set for system implementation. However, the Pareto optimal set is often large and cumbersome, making the post-Pareto analysis phase potentially difficult, especially as the number of objectives increase. Our research involves the post Pareto analysis phase, and two methods are presented to filter the Pareto optimal set to determine a subset of promising or desirable solutions. The first method is pruning using non-numerical objective function ranking preferences. The second approach involves pruning by using data clustering. The k-means algorithm is used to find clusters of similar solutions in the Pareto optimal set. The clustered data allows the decision maker to have just k general solutions from which to choose. These methods are general, and they are demonstrated using two multi-objective problems involving the scheduling of the bottleneck operation of a printed wiring board manufacturing line and a more general scheduling problem.     

Optimal generator maintenance scheduling using a modified discrete PSO

A modified discrete particle swarm optimisation (MDPSO) algorithm to generate optimal preventive maintenance schedule of generating units for economical and reliable operation of a power system, while satisfying system load demand and crew constraints, is presented. Discrete particle swarm optimisation (DPSO) is known to effectively solve large-scale multi-objective optimisation problems and has been widely applied in power system. The MDPSO proposed for the generator maintenance scheduling optimisation problem generates optimal and feasible solutions and overcomes the limitations of the conventional methods, such as extensive computational effort, which increases exponentially as the size of the problem increases. The efficacy of the proposed algorithm is illustrated and compared with the genetic algorithm (GA) and DPSO in two case studies ? a 21-unit test system and a 49-unit system feeding the Nigerian national grid. The MDPSO algorithm is found to generate schedules with comparatively higher system reliability indices than those obtained with GA and DPSO.     

Two-stage hybrid flow shop scheduling with preventive maintenance using multi-objective tabu search method

This paper investigates an integrated bi-objective optimisation problem with non-resumable jobs for production scheduling and preventive maintenance in a two-stage hybrid flow shop with one machine on the first stage and m identical parallel machines on the second stage. Sequence-dependent set-up times and preventive maintenance (PM) on the first stage machine are considered. The scheduling objectives are to minimise the unavailability of the first stage machine and to minimise the makespan simultaneously. To solve this integrated problem, three decisions have to be made: determine the processing sequence of jobs on the first stage machine, determine whether or not to perform PM activity just after each job, and specify the processing machine of each job on the second stage. Due to the complexity of the problem, a multi-objective tabu search (MOTS) method is adapted with the implementation details. The method generates non-dominated solutions with several parallel tabu lists and Pareto dominance concept. The performance of the method is compared with that of a well-known multi-objective genetic algorithm, in terms of standard multi-objective metrics. Computational results show that the proposed MOTS yields a better approximation.     

A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem

Solving optimization problems with multiple objectives under uncertainty is generally a very difficult task. Evolutionary algorithms, particularly genetic algorithms, have shown to be effective in solving this type of complex problems. In this paper, we develop a simulation-based multi-objective genetic algorithm (SMOGA) procedure to solve the build-operate-transfer (BOT) network design problem with multiple objectives under demand uncertainty. The SMOGA procedure integrates stochastic simulation, a traffic assignment algorithm, a distance-based method, and a genetic algorithm (GA) to solve a multi-objective BOT network design problem formulated as a stochastic bi-level mathematical program. To demonstrate the feasibility of SMOGA procedure, we solve two mean-variance models for determining the optimal toll and capacity in a BOT roadway project subject to demand uncertainty. Using the inter-city expressway in the Pearl River Delta Region of South China as a case study, numerical results show that the SMOGA procedure is robust in generating ‘good’ non-dominated solutions with respect to a number of parameters used in the GA, and performs better than the weighted-sum method in terms of the quality of non-dominated solutions.     

Pareto optimization using the struggle genetic crowding algorithm

Many real-world engineering design problems involve the simultaneous optimization of several conflicting objectives. In this paper, a method combining the struggle genetic crowding algorithm with Pareto-based population ranking is proposed to elicit trade-off frontiers. The new method has been tested on a variety of published problems, reliably locating both discontinuous Pareto frontiers as well as multiple Pareto frontiers in multi-modal search spaces. Other published multi-objective genetic algorithms are less robust in locating both global and local Pareto frontiers in a single optimization. For example, in a multi-modal test problem a previously published non-dominated sorting GA (NSGA) located the global Pareto frontier in 41% of the optimizations, while the proposed method located both global and local frontiers in all test runs. Additionally, the algorithm requires little problem specific tuning of parameters.     

Multi-objective unrelated parallel machine scheduling: a Tabu-enhanced iterated Pareto greedy algorithm

This work proposes a high-performance algorithm for solving the multi-objective unrelated parallel machine scheduling problem. The proposed approach is based on the iterated Pareto greedy (IPG) algorithm but exploits the accessible Tabu list (TL) to enhance its performance. To demonstrate the superior performance of the proposed Tabu-enhanced iterated Pareto greedy (TIPG) algorithm, its computational results are compared with IPG and existing algorithms on the same benchmark problem set. Experimental results reveal that incorporating the accessible TL can eliminate ineffective job moves, causing the TIPG algorithm to outperform state-of-the-art approaches in the light of five multi-objective performance metrics. This work contributes a useful theoretical and practical optimisation method for solving this problem.     

Prudent constraint-handling technique for multiobjective propeller optimisation

The paper presents an alternative constraint-handling technique that converts a nonlinear constrained programming problem into an unconstrained multi-objective optimisation problem. The technique is derived from the behavioural memory constraint-handling method, which was originally implemented for single-objective optimisation with genetic algorithms. We compare our presented technique with two other popular constraint-handling concepts and demonstrate its superiority over them when applied to a propeller optimisation problem. We conclude that the multi-objective behavioural memory constraint-handling technique conjugated with the non-dominated sorting genetic algorithm (NSGA-II) is a prudent method to apply to problems with an infeasible initial design and where constraints have a natural order of satisfaction, which, if not conformed to, would lead to unrealistic designs that impair the search by GA.     

Evolutionary multi-objective concurrent maximisation of process tolerances

Concurrent tolerancing which simultaneously optimises process tolerance based on constraints of both dimensional and geometrical tolerances (DGTs), and process accuracy with multi-objective functions is tedious to solve by a conventional optimisation technique like a linear programming approach. Concurrent tolerancing becomes an optimisation problem to determine optimum allotment of the process tolerances under the design function constraints. Optimum solution for this advanced tolerance design problem is difficult to obtain using traditional optimisation techniques. The proposed algorithms (elitist non-dominated sorting genetic algorithm (NSGA-II) and multi-objective differential evolution (MODE)) significantly outperform the previous algorithms for obtaining the optimum solution. The average fitness factor method and the normalised weighting objective function method are used to select the best optimal solution from Pareto optimal fronts. Two multi-objective performance measures namely solution spread measure and ratio of non-dominated individuals are used to evaluate the strength of the Pareto optimal fronts. Two more multi-objective performance measures namely optimiser overhead and algorithm effort are used to find the computational effort of the NSGA-II and MODE algorithms. Comparison of the results establishes that the proposed algorithms are superior to the algorithms in the literature.     

Non-dominated archiving multi-colony ant algorithm for multi-objective optimization: Application to multi-purpose reservoir operation

Multi-objective optimization using heuristic methods has been established as a subdiscipline that combines the fields of heuristic computation and classical multiple criteria decision making. This article presents the Non-dominated Archiving Ant Colony Optimization (NA-ACO), which benefits from the concept of a multi-colony ant algorithm and incorporates a new information-exchange policy. In the proposed information-exchange policy, after a given number of iterations, different colonies exchange information on the assigned objective, resulting in a set of non-dominated solutions. The non-dominated solutions are moved into an offline archive for further pheromone updating. Performance of the NA-ACO is tested employing two well-known mathematical multi-objective benchmark problems. The results are promising and compare well with those of well-known NSGA-II algorithms used in real-world multi-objective-optimization problems. In addition, the optimization of reservoir operating policy with multiple objectives (i.e. flood control, hydropower generation and irrigation water supply) is considered and the associated Pareto front generated.     

A multi-objective particle swarm optimisation algorithm for unequal sized dynamic facility layout problem with pickup/drop-off locations

This paper deals with a multi-objective unequal sized dynamic facility layout problem (DFLP) with pickup/drop-off locations. First, a mathematical model to obtain optimal solutions for small size instances of the problem is developed. Then, a multi-objective particle swarm optimisation (MOPSO) algorithm is implemented to find near optimal solutions. Two new heuristics to prevent overlapping of the departments and to reduce ‘unused gaps’ between the departments are introduced. The performance of the MOPSO is examined using some sets of available test problems in the literature and various random test problems in small, medium, and large sizes. The percentage of improvements on the initial solutions is calculated for small, medium and large size instances. Also, the generation metric and the space metric for non-dominated solutions are examined. These experiments show the good performance of the developed MOPSO and sensitivity analysis show the robustness of the obtained solutions.     

Using response surface design to determine the optimal parameters of genetic algorithm and a case study

Genetic algorithms (GAs) are efficient stochastic search techniques for approximating optimal solutions within complex search spaces and used widely to solve NP-hard problems. Genetic algorithm includes a number of parameters whose different levels strictly affect the performance of the algorithm. The general approach to determine the appropriate parameter combination of GA depends on too many trials of different combinations, and the best one of them that produces good results is selected for the programme, which would be used for problem solving. A few researchers studied on the parameter optimisation of GA. In this paper, response surface-dependent parameter optimisation is proposed to determine the optimal parameters of GA. Results are tested for benchmark problems that are most common in mixed-model assembly line balancing problems of type-I.     

Minimisation of total tardiness for identical parallel machine scheduling using genetic algorithm

In recent years research on parallel machine scheduling has received an increased attention. This paper considers minimisation of total tardiness for scheduling of n jobs on a set of m parallel machines. A spread-sheet-based genetic algorithm (GA) approach is proposed for the problem. The proposed approach is a domain-independent general purpose approach, which has been effectively used to solve this class of problem. The performance of GA is compared with branch and bound and particle swarm optimisation approaches. Two set of problems having 20 and 25 jobs with number of parallel machines equal to 2, 4, 6, 8 and 10 are solved with the proposed approach. Each combination of number of jobs and machines consists of 125 benchmark problems; thus a total for 2250 problems are solved. The results obtained by the proposed approach are comparable with two earlier approaches. It is also demonstrated that a simple GA can be used to produce results that are comparable with problem-specific approach. The proposed approach can also be used to optimise any objective function without changing the basic GA routine.