Heuristics For N Job M Machine Flowshop Batch Processing With Breakdown Times

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1 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) Heuristics For N Job achine Flowshop Batch rocessing With Breadown Times V Senthiil, V S irudhunea rofessor &Head Department of echanical Engineering, Saint eters University, Avadi, Chennai SA Technical Consultant, IB India, DLF tower, Chennai, INDIA Abstract Flow shop scheduling is a typical combinatorial optimization problem, where each job has to go through the processing in each and every machine on the shop floor. Here considered the basic form of flow shop scheduling i.e. Two machine Flow Shop batch processing with type two transportation i.e transportation of jobs from machine shop to dispatch unit. For this problem we investigate the optimal property and propose an algorithm which includes Johnson s algorithm. After the sequences of jobs are formed we implement breadown time at two intervals and form another solution. This problem is extended to N job machine problem to find optimal solution. The performance measure taen here is maes pan and mean weighted flow time of jobs. This type of problem comes under N hard category. Keywords Flow shop scheduling, Breadown times, N hard. I. INTRODUCTION roduction scheduling is generally considered to be the one of the most significant issue in the planning and operation of a manufacturing system. roficient scheduling leads to increase in capacity utilization efficiency and hence thereby reducing the time required to complete jobs. The main focus of this paper is scheduling of N jobs machine flow shop batch processing problem with breadown of machines. Every batch considered here has common due date. Initially an efficient algorithm was proposed for two machine case and the same algorithm was extended to machine case. After the sequences of jobs are formed we implement brea down times of machines which is often called preemption according to algorithm predicted by A.B.Chandramouli. II. ROBLE STATEENT AND NOTATIONS We consider N job machine flow shop scheduling batch processing problems with type transportations. For this we initially consider two machine n job flow shop scheduling problems and a heuristic solution is formed and the same heuristics is extended to m machine N job scheduling problems. The flow shop environment consists of N independent jobs N={J,J,J..J n } to be processed on two machines and.every job comprises two operations associated with respective processing times on both machines. Second operation on cannot be started until the first operation is completed. All the jobs are continuously available at time zero. Every job to be processed is associated with job size S j and also contains identical due date. In this case breadown of machines are occurred in two intervals. After processing in the manufacturing cell, finished jobs are delivered to dispatch section by vehicles. Assume there is only one vehicle available with a limited capacity i.e.) limited batch size. The capacity of vehicle (c) is measured by total physical space that vehicle provides for one delivery. If the sizes of all jobs are equal, the vehicle capacity can be represented by number of jobs. The transporting time from machine shop to dispatch section is denoted by T, which is the sum of t and t, where t is the time taen to travel form machine shop to dispatch section and t is the time taen to travel from dispatch section to machine shop. In most of the cases t and t are equal. Both t and t are independent of the jobs processed. The breadown of machines affects only the mae span and T is independent from breadown times. j rocessing time of j th job in achine j rocessing time of j th job in achine B Batch S j Jobsize. CCapacity of vehicle. eriod of time for processing all jobs in B. Sum of processing times on for jobs in B. Sum of processing times on for jobs in B. Sum of idle times on. Sum of idle times on. III. LITERATURE SURVEY Initially S..Johnson [] in introduced the basic two and three stage flow shop scheduling problems. In this he considered only permutation schedule problems and proposed O (nlogn) algorithm for two stage problem. The complexity of stage problem belongs to N hard (Garey et al []).

2 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) ore recently the amount of research devoted to minimization of sum of job completion times has increased. Flow time minimization leads to stable or even use of resources, a rapid turn around of jobs and minimization of in process inventory[]. SimchiLevi et al [] proposes a new worst case results for the bin pacing problem. The paper wor provides information about the worst case testing for bin pacing problems. C.Y.Lee et al [] investigated machine scheduling models that impose constraints on both transportation capacity and transportation times. From the journals of Lee et al [], Chang et al [] the complexity status for two machine flow shop is given as F(D),= v=,c C j is strongly N hard. C.Rajendran [] proposes an efficient heuristic for scheduling in a flow shop to minimize total weighted flow time of jobs. This paper wor provides a heuristic approach to find the sequence for optimization of total weighted flow time of jobs. T.C.E.Cheng et al [] explains the parallel machine batching problems. He considers a scheduling problem in which n independent and simultaneously available jobs are to be processed on m identical parallel machines. A.B.Chandramouli [] at considers a flow shop scheduling problems. It provides a new simple heuristic algorithm for a achine, n job flowshop scheduling problem in which jobs are attached with weights to indicate their relative importance and the transportation time and brea down intervals of machine are given. A heuristic approach method to find optimal or near optimal sequence minimizing the total weighted mean production flow time for the problem has been discussed. Guzin ozdagoglu [] includes a sample case of a sequencing problem of flow shop system for which a simulated annealing algorithm is presented. In addition, the results obtained from the simulated annealing algorithm are compared with the results of scheduling software LEKIN for the same problem. Finally, a simulated annealing algorithm is obtained which is very close to the results of LEKIN which is broadly used within the scheduling applications according to the objective under consideration. IV. ROOF OF N HARD Lemma: The scheduling problem (D),= v=,c= C j is N hard in strong sense. artition problem Given a set A={a,a,a,a.a m of m elements with positive integer si e s(a) for each a C such that for integer, ( /) <s(a)< ( /) and s(a)=m decide if can be partitioned in to m disjoint element subsets A,A,A.. m such that s(a)= for each j=,,,.m. Lemma: The scheduling problem F(D) = v=,c= C j is N hard in strong sense roof. With the complexity hierarchy, by setting all of the processing times on or are,the scheduling problem (D) = v=,c= C j is a special case of the more general problem F(D) = v=,c= C j. Since (D) = v=,c= C j is N hard in the strong sense, which is proved in Theorem, the scheduling problem F(D) = v=,c= C j is also Nhard in strong sense. Lemma: The scheduling problem F (D) = v=,c=z C max is N hard in strong sense. Consider a special case of F (D) = v=,c=z C max in which for every job, the processing times of both operations are zero. Hence all of the jobs are ready for delivering to the customer at beginning. In such a case, because of the constant transportation time, minimization of the no. of delivery batches will achieve the optimality and thus the problem can be regarded as a bin pacing problem, which is a wellnown strongly N hard problem. Therefore, the problem that we considered is also N hard in the strong sense. V. HEURISTICS FOR ACHINE FLOW SHO BATCH ROCESSING ROBLES WITH BREAKDOWN TIES Step: Assign jobs into batches by First Fit Decreasing algorithm. Set the total number of resulting batches be b,b n. Step: The jobs within the batches are sequenced using Johnson s algorithm i.e., ST(I)LT(II) schedule. Step: Calculate the following three values for each batch say B in which jobs are scheduled in ST(I)LT(II) order. Δ = Φ = Σ j Δ = ρ = Σ j Δ = max ( T,) Step: a Let S b ={B,B. be the set of batches in which the processing sequence is undecided and another two ordered set S and S which are both initially empty, be the set of scheduled batches. ic the batch from S B with smallest value over all values obtained in step.if the value belongs to Δ, and then set the batch into S as the first element. On the other hand, if the value belongs to Δ or Δ, then set the batch in to S as the last one. Eliminate the chosen batch from S B. Step: b Iterate step until all the batches have its position, i.e., S B is an empty set. Therefore, the permutation of S U S is indeed the processing sequence of the batches.

3 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) Step: Starting with B, assign jobs in B to the machines, for K=,,.b Step: Dispatch each completed but undelivered batch whenever the vehicle becomes available. If multiple batches have been completed when the vehicle becomes available, dispatch the batch with the smallest index. Step:A : The sequence of batches provides the sequence of overall jobs. Consider the sequence of all jobs and C max value. Step: : Denote the sequence of all jobs as flow time in respective machines. Chec for the structural conditions []. Step: : Implement the effect of breadown times in the flow time pattern. Step: : Let us consider (a,b) as the breadown time interval (since deterministic scheduling) then implement the value a to b in the flow processing of the machines. Step:: Following Step leads to addition of breadown times on processing time that the job processing in particular machine. This means that the jobs are processed up to the start of the breadown times and resume the operation after the end of breadown time. Step: : Update the changes in the changes in the upcoming processing times because the breadown interval influences the remaining process if the machines. Step: Calculate the mean flow time by using the following formula Where f i be the flow time of the i th job, W i be the weightage of the i th job VI. EXTENSION OF ALGORITH TO N JOBS ACHINE FLOW SHO BATCH ROCESSING WITH BREAKDOWN TIES. Step: Assign jobs into batches by First Fit Decreasing algorithm. Set the total number of resulting batches be b,b n. Step: The jobs within the batches are sequenced using Software based Local search Heuristics algorithm. Δ = Σ j Δ = Σ j Δ m = Σ mj Δ m+ = max ( T,) Where m denotes the no. of machines. Step: Form Δ and Δ by adding the values of Δ (m/) and Δ (m/). Step: Let S B ={B,B. be the set of batches in which the processing sequence is undecided and another two ordered set S and S which are both initially empty, be the set of scheduled batches. ic the batch from S B with smallest value over all values obtained in step.if the value belongs to Δ, and then set the batch into S as the first element. On the other hand, if the value belongs to Δ or Δ m+, then set the batch in to S as the last one. Eliminate the chosen batch from S B. Step: Iterate step until all the batches have its position, i.e., S B is an empty set. Therefore, the permutation of S U S is indeed the processing sequence of the batches. Step: Starting with B, assign jobs in B to the machines, for K=,,.b Step: Dispatch each completed but undelivered batch whenever the vehicle becomes available. If multiple batches have been completed when the vehicle becomes available, dispatch the batch with the smallest index. Step: : Implement the effect of breadown times in the flow time pattern. Step: A: Let us consider (a,b) as the breadown time interval (since deterministic scheduling) then implement the value a to b in the flow processing of the machines. Step:: Following Step leads to addition of breadown times on processing time that the job processing in particular machine. This means that the jobs are processed up to the start of the breadown times and resume the operation after the end of breadown time. Step:A: Update the changes in the changes in the upcoming processing times because the breadown interval influences the remaining process if the machines. Step:: Calculate the mean flow time by using the following formula Step: Calculate the following values for each batch say B in which jobs are scheduled by using Local search Heuristics.

4 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) Where f i be the flow time of the i th job W i be the weightage of the i th job. Since we are dealing with mono weightage jobs sum of W i corresponds to the total number of jobs. Job s Table rocessing time and job size for m machines S j VII. ILLUSTRATION OF HEURISTIC ROCEDURE In the previous chapters we proposed the algorithms for m machine flow shop batch processing with breadown times. To illustrate the algorithm we select an environment of machine and jobs with common due date and distinct job size. The problem considered here is deterministic nature and all the values are nown. The typical data input for the problem is shown here. No. of the machine:. achine availability: at zero Breadown time: (), () No. of the jobs:. Job type: Independent jobs. Job availability: all are available at zero. Environment: Flow shop. Sequence: Sequenced as Earliness cost: /unit Tardiness cost: /unit. Weightage of the jobs: ono weightage jobs. rocessing: Batch processing. Batch size:. Transportation time to dispatch center:. Transportation time between the machines: not considered. Complexity of the problem: N hard [chapter ] erformance ratio of the Algorithm: [chapter ] Sub algorithms used:. First fit Decreasing Algorithm. Software based Local Search Heuristic. The processing time of each job on machine and machine and the job size of the respective jobs are tabulated as follows.

5 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) The above data are used as input in the algorithm to find Cmax and mean weighted flow time. The major steps in the algorithm are Formation of batches FFD algorithm. Sequence of Jobs Johnson s algorithm. arameter calculation Sequence of batches Breadown implementation Flow time calculation. Step: : Formation of batches FFD algorithm: Arrange the jobs in nonincreasing order of sizes. J J J J J J J J J J J J J J J J J J J J. Formation of batches by First Fit Decreasing algorithm B ={J,J,J,J,J } Batch size= B ={J,J,J,J,J,J } Batch size= B ={J,J,J,J,J,J,J,J,J } Batch size= In the jobs from to J and J are the jobs having highest job sizes hence are placed in the top places of the batch.the total batch size of the processing is.hence jobs forms three batches exactly. The last batch B is the only batch which has the batch size as.but during the transportation B is also considered equivalent to other batches. The batches B, consists of only jobs while the batches B consists of jobs and B consists of jobs. In this the batch B consists the jobs with very low job size in the processing cell. Step:: Sequence of jobs within the batches. Hence the problem is Fm C max and from many researches it is proved that Local search Heuristic provides the optimal solution to these ind of problems. So we use Local search Heuristic rule for the sequence of jobs within each batch.sequence of Batch B : B ={J,J,J,J,J }, Batch size= By applying Local search Heuristic we obtain C max = Job sequence = J J J J J. Sequence of Batch B : B ={J,J,J,J,J,J } Batch size= Fig Gantt chart for sequence of jobs within the batch By applying Local search Heuristic we obtain C max =. Job sequence = J J J J J J. Sequence of Batch B: B = {J, J, J, J, J, J, J, J, J } Batchsize= Fig Gantt chart for sequence of jobs within the batch By applying Local search Heuristic we obtain C max =. Job sequence = J J J J J J J J J. Step:: arameter Calculation: Calculate the following values for each batch say B in which jobs are scheduled by using Local search Heuristics.. Δ = Σ j Δ = Σ j Fig Gantt chart for sequence of jobs within the batch Δ m = Σ mj Δ m+ = max ( T,) Where m denotes the no. of machines. Form Δ and Δ by adding the values of Δ (m/) and Δ (m/). Where, eriod of time for processing all jobs in B.

6 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) Sum of processing times on for jobs in B. Sum of processing times on for jobs in B. Δ Sum of idle times on. Δ Sum of idle times on. Δ m Sum of idle times on m. T Transportation time From these values only we can sequence the set of batches formed through FFD algorithm. Here the transportation time is.hence for the m machine flow shop scheduling batch processing problem the values are tabulated as follows. Table List of sequence of batch and processing time of jobs in respective machines B Seq,,,,,,,,,,,,,,,,, Table: calculated values for Batches Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Let us form Δ and Δ by adding the values of Δ (m/) and Δ (m/) Table Summation of Δ (m/) and Δ (m/) Δ Δ Δ Step:: Sequence of batches Establish S B = {B, B, B }.S U S = {} which is an empty set. From the table, the smallest value is which lies in both. Hence place B at the end of S i.e.) S = {B }. Remove that batch from S B. Then the repeat the procedure until S B becomes an empty set. Hence the optimal batch sequence is S U S i.e.) B B B.It means that the overall sequence of the jobs to be processed in the machines is J J J J J J J J J J J J J J J J J J J J J.. By following this a Gantt chart is obtained. The value of C max obtained through the proposed algorithm is. Hence C max =. Fig Overall sequence of jobs in three batches. Completion time for B =. Completion time for B =. Completion time for B =. Common Due date of the batches = Early Batches = B. Tardy Batches =B, B. JIT Batch = B Early cost = Tardy cost = +=. Total cost = +=. Total completion time =. ean weighted flow time without breadown =..

7 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) Step: : Breadown implementation: For the implementation of breadown we have to formulate the optimal sequence as typical flow pattern. In our problem breadown times are implemented in two intervals as () and () [since deterministic scheduling].initial step of brea down calculation is sequence of jobs optimally. That we have done at the end of Step:. as is J J J J J J J J J J J J J J J J J J J J J. Second step is to denote the sequence as flow time in the machines Jo b Table: Flow time of jobs in respective machines

8 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June )..

9 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) The jobs which are affected by implementation of breadown times are highlighted. J on,j on,j on,j on,j on,j on,j on The initial breadown causes some changes in second one. Implementation of Breadown times: Jo b Table: Flow time of jobs with breadown in respective machines..

10 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) w i f i = +()+()+()+()+()+( )+()+( )+()+()+( )+()+()()+()+()+( )+()+(). = =. w i f i = F w = / =.. Calculation of Total flow time for m machine with breadown: Hence C max = Completion time for B =. Completion time for B =. Completion time for B =. Common Due date of the batches = Early Batches = B. Tardy Batches =B, B. Early cost = Tardy cost = +=. Total cost = +=. Calculation of Total flow time for m machine without breadown:. w i f i = +()+()+()+()+( )+()+()+()+()+()+( )+()+()+()+()+( )+()+()+()= = w i f i =. F w = / =.. VIII. CONCLUSION In our paper we considered the major constraint as breadown times of the machines. We are dealing with the jobs which are having the distinct job size. The major objective considered here is mae span and mean weighted flow time of jobs. From our results it is proved that the mean weighted flow time varies slightly while compared between the processing with breadown times and without breadown times. rofessor ichael L. inedo s Lein Scheduling system. contributes more in our paper.

11 Website: (ISSN, ISO : Certified Journal, Volume, Issue, June ) All the Gantt charts are made with the help of Lein scheduling systems. Lein provides the solution for m machine problems through Local search Heuristics method. Another software tool used in our paper is Lisa (Library of Scheduling Systems) to find the optimality. Hence we complete our objective in our paper successfully. REFERENCE [ ] Johnson.S.. Optimal two and three stage production schedules with setup time included., Naval Research Logistics Quartly(). [ ] Garey..R,Johnson.D.S.,The Complexity of Flow shop and Job shop scheduling, athematics of operations research (). [ ] Rajendran.C.,Zeigler.H., An efficient heuristics for scheduling in a flow shop to minimize total weighted flow time of jobs. [ ] Lee.C.Y., Chen.Z.L. achine scheduling with deliveries to multiple customer locations, European Journal of Operational Research (). [ ] Simchi Leve.D., New worst case results for binpacing problem, Naval Research Logistics (). [ ] A.B.Chandramouli., Heuristic approach for N job machine flow shop scheduling problem involving transportation time, Breadown times and Weightage of jobs., athematical and Computer application vol pp.,. [ ] T.C.E. Cheng, Z.L. Chen,.Y. Kovalyov and B..T. Lin, arallelmachine batching and scheduling to minimize total completion time, IIE Transactions (). [ ] Chang.Y.C et al achine scheduling with job deliver coordination, European Journal of Operational Research (). [ ] Guzin ozdagoglu A simulated annealing application on flow shop sequencing problem: A comparative case study.