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\title{Optimization of the Flexible Job Shop Scheduling Problem for Economic Sustainability}
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             \author[1]{Md. Riyad  Hossain}

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\date{\small \em Received: 12 December 2017 Accepted: 5 January 2018 Published: 15 January 2018}

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\begin{abstract}
        


The flexible job-shop scheduling problem (FJSP) is one of the challenging optimization problems as they occupy very large search space. Solving this kind of problems with conventional methods are obsolete now as the Internet of Things (IoT) has changed scheduling platform by means of cloud computing and advanced data analytics. Genetic Algorithms (GAs) is a popular modern tool for machine scheduling problems and in this work, a scheduling algorithm has been developed to minimize total tardiness and make span time of parallel machines which is promoting overall economic sustainability. The algorithm consists of a machine selection module (MSM) that helps to select the right machine on the right time with the help of global selection (GS) technique by generating high quality initial population. To represent an optimized solution of the FJSP, an improved chromosome representation is used while adopting uniform crossover and mutation operator. The result showed that proposed algorithm is much more effective and efficient for solving flexible job-shop scheduling problem which is helping to reduce the overall downtime significantly. 

\end{abstract}


\keywords{economic sustainability, flexible job-shop scheduling, genetic algorithm, machine selection module, global search technique}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
owadays sustainable manufacturing is an important issue for every industry and industries are aiming to achieve triple bottom line (TBL) sustainability for maintaining their competitive benchmark. The triple bottom line sustainability means achieving sustainability in every sector and most specifically achieving sustainability in economic, social and environmental sectors (Islam \& AlGeddawy, 2017) (D. Chen, Thiede, Schudeleit, \& Herrmann, 2014). Economic sustainability is one of the major pillars of TBL sustainably which is an expression of economic viability of a company (D. Chen et al., 2014; Islam \& AlGeddawy, 2017; \hyperref[b13]{Kannegiesser \& Günther, 2014)}. In this scheduling problem, the importance of economic sustainability has been highlighted and its mutual relationship between the optimized result has been analyzed.\par
Scheduling is involved with the proper distribution and availability of resources for accomplishing a set of tasks over a specified time  {\ref (Roshanaei,}   \hyperref[b23]{(Zhang, Gao, \& Shi, 2011}). An improved chromosome representation method ''Machine Selection and Operation Sequence" has been introduced to assist the initialization method and to assure the algorithm is performing well \hyperref[b2]{(Carlier \& Pinson, 1989;}\hyperref[b16]{Moon, Lee, Seo, \& Lee, 2002;}\hyperref[b18]{Prakash, Tiwari, \& Shankar, 2008)}. This method helps to find an efficient coding scheme of the individuals which respects all constraints of the FJSP. At the same time, different strategies for crossover and mutation operator are employed. After that, the computational results show that the proposed algorithm could get good solutions. 
\section[{II.}]{II.} 
\section[{Problem Definitions}]{Problem Definitions}\par
The FJSP consists in performing a set of n jobs \{J 1 , J 2 , J 3 ?., j i , ?... J n \} on a set of m machines \{M 1 , M 2 , M 3 ?., M k , ?... M m \}. A job J i is formed by a sequence of operations \{O i1 , O i2 , O i3 ?., O ij \} to be performed one after the other according to the given sequence. Each operation requires one machine selected out of a set of available machines. All jobs and machines are available at time 0, and a machine can only execute one operation at a given time. Preemption is not allowed, i.e., each operation must be completed without interruption once it starts. Machine Setup time and movement time between operations are negligible. The processing time of operation O ij on machine M m is P ijk > 0. Flexibility of FJSP can be categorized into partial flexibility and total flexibility. A system will be called partial if every machine cannot perform every other operation because of some constraint or extremely high cost. Data associated with a partial flexible system are given in Table  {\ref 1}, in which rows stand for operations time and columns is for machines. In this Table, symbol 0 means that a machine cannot be executed the corresponding operation. Problem can be formulated by using following nodes and symbols: Z is the set of all machines n is the number of the total jobs m is the number of the total machines i is the index of the i th job j is the index of the j th operation of job J i k, x is the index of alternative machine set O ij is the j th operation of job J i P ijk is the processing time of operation O ij on machine k S ijk is the start time of operation O ij on machine k E ijk is the end time of operation O ij on machine k Assumptions made in FJSP as follows: ? All machines are available at time t = 0; ? All jobs are released at time t = 0; ? For each job, the sequence of operations is predetermined and cannot be modified; ? Each machine can only execute one operation at a time; ? Each operation Oij must be processed without interruption on one of a set of available machines; ? Recirculation occurs when a job could visit a machine more than once; ? The objective of the FJSP is to minimize the total completion time;\par
A basic representation of flexible job shop scheduling is as follows in Table  {\ref 1}.\par
There are 2 jobs and 5 machines, where rows correspond to operations and columns correspond to machines. Each cell denotes the processing time of that operation on the corresponding machine. In the table, the ''0" means that the machine cannot execute the corresponding operation, i.e., it does not belong to the alternative machine set of the operation. So this instance is a Partial FJSP. 
\section[{III. Algorithm For Flexible Job Scheduling}]{III. Algorithm For Flexible Job Scheduling}\par
The proposed algorithm is developed according to the concept of the genetic algorithm. The detailed procedure of the proposed algorithm is as follows: 
\section[{a) Determining operation sequence}]{a) Determining operation sequence}\par
The sequence of operation should be arranged on that way so that every other operation of an individual job must be accomplished before processed to next immediate operation. For example, to perform the job j of table 1 the sequence of operations O 0 should be maintained i.e. O > O 
\section[{b) Global selection}]{b) Global selection}\par
We define that a stage is the process of selecting a suitable machine for an operation. Thus this method records the sum of the processing time of each machine in the whole processing stage. Then the machine which has the minimum processing time in every stage is selected. In particular, the first job and next job are randomly selected. Detailed steps are as follows:\par
1. Create a new array to record all machines' processing time, initialize each element to 0; 2. Select a job randomly and insure one job to be selected only once, then select the first operation of the job; The implementation of GS is given in Fig. \hyperref[fig_0]{1}.We assume that the sequence of job is J >j >j from table 1, as mentioned earlier processing time '0' means the machine is unable to perform the operation. So, we easily see that the processing time on M is the shortest in the alternative machine set of operation O . Hence, machine M is selected to process the operation O of job J , and set corresponding allele in MS to the index of M .Then the processing time is added to the corresponding position in time array. The selection process continued till finding a new chromosome as described above 8 steps.  
\section[{c) Generate the Initial Population}]{c) Generate the Initial Population}\par
As shown in Fig \hyperref[fig_0]{1}, the structure of the chromosome used in this paper consists of two components. The first component of the chromosome represents the list of machines while the second component contains the sequence of operations to be processed used in executing the operations in the first component.\par
For example, consider the 3-job, 5-machine problem given in Table  {\ref 1}. Initial chromosomes are randomly created as shown in The goal of the crossover is to obtain better chromosomes to improve the result by exchanging information contained in the current good ones. In our work we carried out uniform crossover operator to generate new chromosomes. 
\section[{i. Machine Selection part}]{i. Machine Selection part}\par
The crossover operation of MS is performed on two Machine Selection parts and generates two new Machine Selection parts each of which corresponds to a Machine selection Operation sequence1 4 1 3 2 1 1 2 2 2\par
new allocation of operations to machines. Each machine of the new Machine Selection parts must be effective, i.e., the machine must be included in the alternative machine set of the corresponding operation. We adopt uniform crossover (Gao, Sun \& Gen, 2008). MS crossover operator only changes some alleles, while their location in each individual i.e., their preceding constraints are not changed. Therefore, the individuals after crossover are also feasible. The procedure could be illustrated in Fig.  {\ref 3}. 
\section[{ii. Operation Sequence part}]{ii. Operation Sequence part}\par
The crossover operation of OS is different from that of MS. During the past decades, several crossover operators have been proposed for permutation representation. Here we apply a Precedence preserving order-based crossover (POX) for the Operation Sequence. Detailed implementing steps of POX are as follows:\par
1. Generate two sub-job set Js1/Js2 from all jobs and select two parent individuals as p1 and p2; 2. Copy any allele in p1/ p2 that belong to Js1/Js2 into two child individuals c1/c2, and retain in the same position in c1/c2; 3. Delete the alleles that are already in the sub-job Js1/Js2 from p2/ p1; 4. Orderly fill the empty position in c1/c2 with the alleles of p2/p1 that belongs to in their previous sequence. In Table  {\ref 1}, there are only four jobs. So it is difficult to present the process of POX clearly. 
\section[{Fig. 3: MS uniform crossover operator e) Mutation operator}]{Fig. 3: MS uniform crossover operator e) Mutation operator}\par
Mutation introduces some extra variability into the population to enhance the diversity of population. Usually, mutation is applied with small probability. Large probability may destroy the good chromosome. 
\section[{i. Machine Selection part}]{i. Machine Selection part}\par
The machine selection mutation operation is performed by following steps:\par
1. MS mutation operator only changes the assignment property of the chromosomes. We select the shortest processing time from alternative machine set to balance the workload of the machines. Taking the chromosome from Fig.  {\ref 3} for example, MS mutation is described as follows: Select one individual from the population;\par
2. Read the chromosomes of the individual from left to right and generate a probability value randomly; if all the chromosomes have been read, then end the procedure; 3. If the probability value is less than or equal to the mutation probability then go to Step 4; otherwise, go to Step 2; 4. Select the shortest processing time from the alternative machine set and assign it to the mutation position;\par
An illustrative instance is shown in Fig.  {\ref 4}. Suppose the mutative operation is O , before the mutation, O is processed on M , which is the fourth machine in the alternative machine set, so the allele is 2. In the mutation, the rule that selecting the machine of the shortest processing time is obeyed, so M4 is  
\section[{Global}]{Global} 
\section[{Machine selection}]{Machine selection}\par
Operation sequence Fig.  {\ref 4}: MS after mutation operation.\par
IV. 
\section[{Numerical Illustration And Computational Result}]{Numerical Illustration And Computational Result}\par
The genetic algorithm was coded in ANSI C++ programming language using only simple array data structure and implemented on a PC (Intel(R) Pentium(R), CPU B960 Intel processor of 2.20 GHz).   Firstly, we tested the performance of the algorithm using GS method. In Fig.  {\ref 6} we draw the generation vs time completion curve that shows the decreased average time required to complete each and every operation. The optimal solution for a 4×5 problem can be achieved at 17th generation. In order to obtain meaningful results, we ran our algorithm for several times on different instance. The parameters used in this GA are chosen experimentally in order to get a satisfactory solution in an acceptable time span.\par
V. 
\section[{Conclusion And Future Study}]{Conclusion And Future Study}\par
In this paper, a genetic algorithm for solving the flexible job-shop scheduling problem (FJSP) is proposed. A new chromosome representation scheme is proposed and an effective decoding method interpreting each chromosome into a feasible active schedule is designed. In order to enhance the quality of initial solution, a new initial assignment method i.e., global search technique is used to generate high-quality initial population integrating different strategies to improve the convergence speed and the quality of final solutions. Then different strategies for crossover, selection and mutation operator are adopted. This makes it possible to solve the problem of trade-off resources allocation. Some realistic problems are solved by using this new algorithm. The computational results show that the proposed genetic algorithm leads to an effective scheduling considering time and quality compared with other genetic algorithms which is an indicator of gaining economic profit. The proposed algorithm can enhance the convergence speed and the quality of the solution. This algorithm can be used in solving large size flexible job shop scheduling problem with a focus on achieving economic sustainability throughout the system. In future, it will be interesting to investigate a better search technique which can be developed along with global search technique to generate better results. Also, there is more scope of adjusting an appropriate relation between crossover and mutation probability to enhance the chance of better solution. 
\section[{Author Declaration}]{Author Declaration}\par
Authors declare that we don't have any conflict of interest. This work has been funded by the Department of Industrial Engineering and Management of Khulna University of Engineering and Technology. All the data and information were collected through the existing lab facilities from this department.\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
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\noindent\textbf{} \par 
\begin{longtable}{P{0.595\textwidth}P{0.05525\textwidth}P{0.059500000000000004\textwidth}P{0.08075\textwidth}P{0.05525\textwidth}P{0.00425\textwidth}}
Job J 1 J 2 21 6. O/P M 1 O 11 O 12 2 0 O 21 O 22 O 23 3 4 0\tabcellsep M 2 6 8 0 6 7\tabcellsep M 3 5 0 6 5 11\tabcellsep M 4 3 4 0 0 5 21 22\tabcellsep M 5 4 0 5 0 8\tabcellsep 2\\
\multicolumn{6}{l}{5. Set the allele which corresponds to the current}\\
\multicolumn{2}{l}{operation in the MS part to k;}\tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.1117808219178082\textwidth}P{0.3236986301369863\textwidth}P{0.21191780821917808\textwidth}P{0.20260273972602738\textwidth}}
Generation Number\tabcellsep Average Objective Function value (min)\tabcellsep Best Objective Function value (min)\tabcellsep \\
1\tabcellsep 113.433\tabcellsep 107\tabcellsep \\
2\tabcellsep 111.76\tabcellsep 103\tabcellsep \\
3\tabcellsep 111.375\tabcellsep 103\tabcellsep \\
4\tabcellsep 111.1\tabcellsep 103\tabcellsep \\
5\tabcellsep 115.88\tabcellsep 103\tabcellsep \\
6 7 8\tabcellsep 108.21 107.35 105.21\tabcellsep 103 103 103\tabcellsep Year 2018\\
9 10\tabcellsep 104.83 104.83\tabcellsep 103 103\tabcellsep 27\\
11 12 13 14 15 20\tabcellsep 104.83 104.16 103.5 102.83 102.85 101\tabcellsep 103 103 103 101 101 101\tabcellsep XVIII Issue I Version I\\
\tabcellsep \tabcellsep \tabcellsep Volume\\
\tabcellsep \tabcellsep \tabcellsep ( ) A\\
\tabcellsep \tabcellsep \tabcellsep Journal of Researches in Engineering\\
\tabcellsep \tabcellsep \tabcellsep Global\end{longtable} \par
  {\small\itshape [Note: © 2018 Global Journals]} 
\caption{\label{tab_3}}\end{figure}
 			\footnote{Year 2018 © 2018 Global Journals Optimization of the Flexible Job Shop Scheduling Problem for Economic Sustainability} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}
