The effort of searching an optimal solution for **scheduling** problems is important for real-world industrial applications especially for mission-time critical systems. In this paper, a **parallel** GA is employed to solve **flow** **shop** **scheduling** problems to minimize the makespan.According to our experimental results, the proposed **parallel** **genetic** **algorithm** (PPGA) considerably decreases the CPU time without adversely affecting the makespan.

investigating the **problem** of **scheduling** jobs on a single CNC machine to reduce the energy consumption and total completion time. They pointed out that there was a significant amount of energy savings when non-bottleneck machines were turned off until needed; such the relevant savings share on the total energy consumption would sum up to 80%. In addition, they reported that the inter-arrivals were forecasted and more energy-efficient dispatching rules could be adopted for **scheduling**. In further research, Mouzon and Yildirim[15] proposed a greedy randomized adaptive search **algorithm** for **solving** a multi-objective optimization schedule that minimized the total energy consumption and the total tardiness on a machine. Fang et al.[16] provided a new mixed integer linear programming model for **scheduling** a classical **flow** **shop** that combined the peak total power consumption and associated carbon footprint with the makespan. Bruzzone et al. [17] presented an energy-aware **scheduling** **algorithm** based on a mixed integer programming formulation to realize energy savings for a given flexible **flow** **shop** which was required to keep fixed original jobs' assignment and sequencing.

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The Multi-Objective **Genetic** **Algorithm** (MOGA) of Murata et al. [8], being part of evolutionary algorithms, was developed to solve multi-objective ﬂow **shop** **problem**. Other than a modiﬁed selection operator, this **algorithm** was a simple **genetic** approach to **scheduling**. Selection is interrelated with a set of weights assigned to the objectives, which allowed to distribute the search towards di ﬀerent criteria directions. Elitist preservation method was also incorporated, so that several solutions from the actual Pareto frontier were copied to the next generation. The MOGA was furthermore enhanced by Murata et al. [9], by changing the way of weight distribution between objectives. **Using** a cellular structure permitted a better weight selection, which in turn led to ﬁnding a ﬁner approximation of Pareto frontier. New **algorithm** was called CMOGA.

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simulated annealing, **genetic** algorithms etc. In recent years, the adoption of meta-heuristics like GA has led to better results than classical dispatching or heuristic algorithms [2]. **Solving** **scheduling** problems with GA methods have been introduced by many researchers.

In this paper a multi-objective **scheduling** **problem** was studied for a two-stage production system including a hybrid **flow** **shop** and an assembly stage.In this production system it is assumed that several products of differentkinds are ordered to be produced.The parts are manufactured in the hybrid **flow** **shop** and then the products are assembled in the assembly stage after preparing the parts. Two objective functions are considered simultaneously that are: (1) to minimizing the completion time of all products (makespan), and (2) minimizing the sum of earliness and tardiness of all products ( ∑ -E d d ∕ T d ) . Since this **problem** is NP-hard, a new multi-objective **algorithm** based on GA was designed for searching locally Pareto-optimal frontier for the **problem**. Various test problems were designed and the reliability of the proposed **algorithm** was presented in comparison two algorithms WBGA, and NSGA-II. The computational results show that the performance of the proposed algorithms is good in both efficiency and effectiveness.

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Tabu (also called Taboo) search (TS), which was proposed by Glover et al. [23], is a meta-heuristic **algorithm** used for combinatorial optimization problems. The motivation for TS comes from the visited solutions and repeated visits of local search approaches. TS creates a short-memory structure that records forbidden moves called a tabu list. The foundation of tabu search is described as follows. First, the initial solution is generated randomly. Second, a set of neighborhood (candidate) solutions is generated **using** the current solution. Third, the solution with the best admissibility is chosen (the solution with the best admissibility is the one in which the move satisfies the aspiration criterion) and the tabu list is updated. Finally, Steps 2 and 3 are repeated until the stopping criterion is reached.

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Enumeration and heuristic methods have been applied for energy saving in previous studies [10]. Integer programming, branch and bound programming, and MIP are the most widely used enumeration methods that can provide ap- propriate solutions. However, high computational times limit the applicability of enumeration methods to small-scale problems [17]. Thus, heuristics such as the **genetic** **algorithm**, simulated annealing **algorithm**, and ant colony optimization **algorithm** are commonly used for **solving** energy-saving problems. Lian [18] ob- tained the average relative error rates of −28.20% and 60.25% for a combined local and global PSO **algorithm** against PSO and **genetic** algorithms, respectively. Zhang et al. [19] presented an I-ATTPSO **algorithm** with an average effective- ness improvement rate of −14% in small-scale problems and 55% in large-scale problems. Liu et al. [20] obtained an average relative error rate of 0.65% for the PSO-EDA_PI **algorithm** against other algorithms. Zhao et al. [21] found the av- erage relative error rate of their proposed logistic dynamic PSO **algorithm** against other algorithms to be approximately 1.19% - 2.39%.

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Recently, Artificial Immune System (AIS) is used to solve problems from different fields such as Robotic [3], Anomaly Detection [4], Combinatory Optimization [5], Learning [6], etc. One type of optimization problems is **scheduling**, and one of the very most common models in field of **scheduling** is that of the Job-**shop** **scheduling**. This **problem** belongs to NP-hard problems, whose optimal solution is difficult to achieve [7]. Some evolutionary methods such as **Genetic** Algorithms (GAs), Ant Colony Optimization (ACO), Partial Swarm Optimization (PSO), Tabu Search (TS) etc. must be used to solve this **problem**. But these algorithms have a few lacks. For example GAs have two main drawbacks. One of them is lack of local search ability and the other is the premature

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have been developed for TSP but here we are **using** the concept of **genetic** **algorithm**. Other approximation techniques for finding near optimum solutions for TSP based on heuristics are proposed in the literature such as [1] simulated annealing [2], ant colonies [3], **genetic** algorithms (GA) [4] and [5]. John Holland’s pioneering book “Adaptation in natural Artificial System (1975, 1992) showed how the evolutionary process can be applied to solve a wide variety of problems **using** a highly **parallel** technique that is now called **genetic** **algorithm**. **Genetic** Algorithms have been applied to a large number of real world pro blems. One of the first such applications was a gas pipeline control system created by [6], [7] mentions several GA applications including message routing, **scheduling**, and automated design. Entire conferences have been devoted to applications of **Genetic** Algorithms and evolutionary techniques to specific disciplines, such as Image Analysis, Signal Processing and Telecommunications [8].

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After many reported experiments in the literature, **genetic** algorithms have been found to be efficient, effective and robust **algorithm** for complicated problems. Nevertheless, **genetic** algorithms also have their shortcomings. In fact, if the worst members are discarded after each generation, the population will tend to become homogeneous quickly. And then the crossovers and mutation may not produce offspring of large variation. For this reasons, some authors have suggested inserting another operator, namely, a Boltzmann- type operator, after the crossover and mutation operations. Therefore, it is called this new metaheuristic method as mixed simulated **genetic** **algorithm** (MSGA). In MSGA, new chromosomes are chosen to produce the next generation from parents and offspring according to Boltzeman function. The selection criterion is based on the fitness values of parents and offspring. Chromosomes with higher fitness values have a greater probability of surviving into the next generation. Those with less fitness values are not necessarily discarded. Based on above discussion, the MSGA is now described as follows:

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Since the occurrence of the industrial revolution, manufacturing and industrial construction are growing in size and complexity. Furthermore, timely delivering of products and services to customers is more significant during past decades. Therefore, in order to be more productive and profitable, the need for developing new principles, techniques, and research disciplines is remarkable to survive in this competitive market-place. Operation research (OR), as a systematic and analytical approach to decision-making and **problem**-**solving**, is one of the discipline that has been developed to respond to this challenge (Hillier and Lieberman, 2001). Over the years, many researchers used OR for improving factory layout design, facility location, utilization of resources, **scheduling**, inventory control etc. Among these topics, **scheduling** receives much attention from researchers due to the significant impact on productivity and successful of corporations in terms of reducing cycle time, reducing of cost (or increasing of profit), minimizing work in process (WIP), and so on (Rokni, 2010).

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Journals: Gromicho, J. A. S., van Hoorn, J. J., Saldanha-da-Gama, F., & Timmer, G. T. (2012). “**Solving** the job-**shop** **scheduling** **problem** optimally by dynamic programming”. Computers & Operations Research, 39(12), 2968-2977. doi:http://dx.doi.org/10.1016/j.cor.2012.02.024 Journals: Qing, R., & Wang, Y. (2012). “A new hybrid **genetic** **algorithm** for job **shop** **scheduling**

The job **shop** **scheduling** **problem** is a well known practical planning **problem** in the manufacturing sector. We have considered the JSSP with an objective of minimizing makespan. In this paper, a multi- population based hybrid **genetic** **algorithm** is developed for **solving** the JSSP. The population is divided in several groups at first and the hybrid **algorithm** is applied to the disjoint groups. Then the migration operator is used. The proposed approach, MP-HGA, have been compared with other algorithms for job- **shop** **scheduling** and evaluated with satisfactory results on a set of JSSPs derived from classical job-**shop** **scheduling** benchmarks. We have solved 15 benchmark problems and compared results obtained with a number of algorithms established in the literature. The experimental results show that MP-HGA could gain the best known makespan in 13 out of 15 problems.

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As organizations move from creating plans for individual production lines to entire supply chains, it is increasingly important to recognize that planning decisions impact the quality of products and the production output. Suboptimal planning give rise to increasing lead times, increased costs and also unsatisfied customers waiting for their products. In order to avoid all these operation issues, proper production planning and **scheduling** of the process is required. Usually GANTT charts and PERT/CPM network techniques are the basic tools used to solve these problems. Gantt charts were meant to help individual managers make better decisions. In practice, PERT and CPM **scheduling** techniques have proven to be only helpful when the project deadline is not fixed and the resources are not constrained by either availability or time (Eknarin Sriprasert and Nashwan Dawood, 2003) [4]. For **scheduling** a process, such that the make span is minimized, the precedence constraints and the resource requirements of various activities of the process need to be considered. The **scheduling** with above mentioned constraints makes it a combinatorial **problem** and hence difficult to solve **using** traditional tools. Based on the works done by researchers in this area, the use of **genetic** **algorithm** (GA) can result in an optimal solution.

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Zuzana Čičková and Stanislav Števo wrote that **flow** **shop** **scheduling** processing systems with two machines, where the aim is to minimize the makespan, can be solved by a Johnsons **algorithm**, but there is no polynomial **algorithm** for **solving** the **problem** for three and more machines [8][9][10]. A heuristic is used for deciding the number of blocks, Johnson’s and NEH **algorithm** for sequencing the parts and finally **Genetic** **Algorithm** and Simulated Annealing for sizing the blocks. Four algorithms are presented by combination of this method. Three lower bounds presented and improved to evaluate the performance of algorithms [11]. The **algorithm** of Johnson is a classic method which solves to optimum the **problem** of sequencing n jobs on two machines, in a polynomial time. Assume that there are n jobs on three machines, then the problems become NP-complete (which is cannot be solved optimally in polynomial time) and the Johnson’s **algorithm** can be applied only for some kind of cases that obey some primary conditions [12].

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Nagar et al. [7] developed a distinct method for the minimization of mean **flow** time along with the minimization of make span in a **flow** **shop** environment. They used two very different methods to reach to the required objective. These two methods are branch and bound method and the evolutionary **algorithm** i.e. **genetic** **algorithm**. Nowicki and Smutnicki [8] implemented tabu search to solve the **flow**-**shop** **scheduling** **problem**. Neppalli et al. [9] used the basic evolutionary **algorithm** for **solving** the two machine **problem**. It used the **genetic** algorithms approach to carry out the same, which aimed at minimizing the make span.

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impossible to directly solve the model of practical sizes within a reasonable amount of time, suitable decomposition can be applied to achieve good performance. In the first part of the decomposition, only the train type related constraints stay active. In the second part, the remaining constraints are satisfied **using** relaxation technique. This decomposition idea provides a cornerstone for an **algorithm** integrating cutting plane and branch-and-bound to optimise the railway networks in Germany and the Netherlands. Zhou and Zhong [8] dealt with a double-track train **scheduling** **problem** with multiple objectives. Focusing on a high-speed passenger rail line in an existing network, the **problem** is to minimise both, 1) the expected waiting times for high-speed trains (efficiency criterion); and 2) the total travel times of high-speed and medium-speed trains (effectiveness criterion). By applying two practical priority rules to model acceleration and deceleration times, the **problem** is formulated as a multi-mode **flow**-**shop** **scheduling** **problem**. A branch-and-bound **algorithm** with effective dominance rule is developed for the bicriteria **scheduling** **problem**, and a beam search **algorithm** with utility evaluation rules is used to construct non-dominated solutions. The authors illustrated the methodology and evaluated the performances of the proposed **algorithm** by a case study based on Beijing-Shanghai high-speed railway in China.

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If initial population is diverse enough then it is possible to choose best solutions for recombination operations and this may reduce the computational time required. Dispatching Rules (DRs) have been applied consistently to **scheduling** problems. They are procedures designed to provide good solutions to complex problems in real time. Many authors claim that priority dispatching rules can be successfully used in **solving** large JSSPs and even oth- er **scheduling** problems [11]. Mahanim et al. [12] used **Genetic** **Algorithm** (GA) with some modifications to deal with **problem** of job **shop** **scheduling** which generated an initial population randomly including the result ob- tained by some well-known priority rules such as shortest processing time and longest processing time. Kuc- zapski et al. [13] presented an efficient method of enhancing **Genetic** Algorithms (GAs) for **solving** the Job- **Shop** **Scheduling** **Problem** (JSSP), by generating near optimal initial populations.

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We consider the N P-hard **problem** of **scheduling** n jobs in m two-stage **parallel** ﬂow shops so as to minimize the makespan. This **problem** decomposes into two subproblems: assigning the jobs to **parallel** ﬂow shops; and **scheduling** the jobs assigned to the same ﬂow **shop** by use of Johnson’s rule. For m = 2, we present a 3 2 -approximation **algorithm**, and for m = 3, we present a 12 7 -approximation **algorithm**. Both these algorithms run in O(n log n) time. These are the ﬁrst approximation algorithms with ﬁxed worst-case performance guarantees for the **parallel** ﬂow **shop** **problem**.

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