Optial Specially Structured N X 2 Flow Shop Scheduling To Miniize Total Waiting Tie of Jobs ncluding Job Block Concept with Processing Tie Separated Fro Set up Tie Dr Deepak Gupta Professor & Head Departent of Matheatics Maharishi Markandeshwar University Mullana Bharat Goyal Research Scholar Departent of Matheatics Maharishi Markandeshwar University Mullana (Eail id: bhartu89@gailco) ABSTRACT: n the present state of affairs the current engineering and anufacturing built- up units are facing ishash of probles in a lot of aspects such as an power achining tie raw aterial electricity and custoer s constraints The flow-shop scheduling is one of the ost significant anufacturing behaviors particularly in anufacturing planning The creation of every tie adirable schedules has verified to be enorously coplicated This paper involves the fortitude of the order of processing of jobs on 2 achines This paper proposes the specially structured Flow Shop Scheduling proble separated fro set up tie assuing that axiu of the equivalent processing tie on first achine is less than or equal to the iniu of equivalent processing tie on second achine with the objective of getting the optial sequence of jobs for total waiting tie of jobs using the heuristic algorith by taking two of the jobs as a group job The proposed technique is followed by nuerical exaple KEY WORDS: Waiting tie of jobs Job Block Set up tie Flow shop Scheduling Processing tie ***** ntroduction Transfer lines have long been recognized as the ost proficient ethod of producing goods in a high volue/ high variety and id volue/ id variety anufacturing though have always been played with difficulties A lot of the production probles are attributable to probles in the scheduling function: not having the sources when they are needed not having apparatus available when it is needed by eans of surplus inventory to hide probles inflexibility and lack of awareness Scheduling conceivably defined as the proble of deciding when to ipleent a given set of activities subject to chronological constraints and resources capacities with the intention of optiize soe function The flow shop contains different achines arranged in series on which a set of n jobs are to be processed The coon scheduling proble for a usual flow shop gives rise to (n!) possible schedules With the ai to reduce the nuber of possible schedules it is logical to take for granted that all the jobs share the sae processing order on every achine Efforts in the past have been ade by researchers to lessen this nuber of possible schedules as uch as achievable without coproising on optiality condition This paper presents a solution ethodology in a flow shop scheduling proble for iniization the waiting tie of jobs specifically defined as the su of the ties of all the jobs which was devoted in waiting for their turn on both of the achines Literature review Johnson [1] has proved that in a 2 achine flow shop proble an optial schedule for iniizing the total elapsed tie can be constructed t was verified later that achine flow shop scheduling proble (FSSP) is robustly NP- hard for 3 Solution ethods for flow shop scheduling range fro heuristics developed by Paler [2] Capbell et al[3] and Dannenbring [4] to ore coplex techniques such as branch and bound [5] tabu search [6 7] Maggu P L et al[8] introduced the concept of equivalent jobs for job block by taking two of the jobs as a group job Yoshida et al [9] explain two stage production scheduling by taking the set up tie separated fro processing tie Nawaz et al [10] proposed that a job with longer total processing tie should have higher priority in the sequence Singh TP et al [13] considered the proble associated with group job restrictions in a flow shop which engross independent set- up tie and transportation tie Further Gupta D [15] siplified the proble of iniization of Rental Cost in Two Stage Flow Shop Scheduling Proble in which Setup Tie was separated fro Processing Tie JRTCC June 2017 Available @ http://wwwijritccorg 727
and each associated with probabilities including Job Block all jobs Equivalent processing ties of k th job on achine Criteria Another approach to study the specially structured R & S are defined as three stage flow shop scheduling to iniize the rental cost = t Sk = t Rk satisfying processing is presented by Gupta D et al[17] ties structural relationship Recently Gupta D et al[19] studied optiality for waiting Max Min S k p and q are any jobs aongst the given tie of jobs in which processing ties are associated with jobs such that job p occurs before job q in the order of job probabilities This study was further extended by including block (p q) the equivalent job β is defined as (p q) job block concept and taking set up tie separated fro set up tie [20 21] The proble discussed here has TABLE 1: MATRX FORM OF THE MATHEMATCAL noteworthy use of conjectural results in process industries or MODEL OF THE PROBLEM in the circustances when the objective is to iniize the total waiting tie of jobs The present paper is an extension t Rk t Sk ade by Gupta D et al [19 20 21] in the sense that we 1 R have taken into consideration the set up tie and the job 1 t R1 S 1 t S1 2 R block criterion 2 t R2 S 2 t S2 3 R 3 t R3 S 3 t S3 PRACTCAL STUATON ndustrial units play a significant role in the financial growth of a nation Flow shop scheduling arises in various organizations service stations banks airports etc n our routine working in factories and anufacturing units different jobs are processed on various achines n textile industry different types of fabric is shaped using different types of yarn Here the axiu equivalent tie taken in dying of yarn on first achine is always less than or equal to the iniu equivalent tie taken in weaving of yarn on the second achine The idea of iniizing the waiting tie ay be an reasonable aspect fro Factory /ndustry anager s view point when he has iniu tie agreeent with a coercial party to coplete the jobs NOTATONS J k : Sequence obtained by applying the algorith proposed : Processing tie of k th job on achine R : Processing tie of k th job on achine S : Equivalent processing tie of k th job on achine R : Equivalent processing tie of k th job on achine S t Rk : Tie for set up of k th job on achine R t Sk : Tie for set up of k th job on achine S T as : The copletion tie of job a on achine S W γ : Waiting tie of job γ W: Total waiting tie of all the jobs R t R S t S ASSUMPTONS n the given flow shop scheduling the following assuptions are ade 1) There are nuber of jobs () and two achines (R & S) 2) The order of sequence of operations in both of the achines is the sae 3) Jobs are independent to each other 4) t is given to sequence r jobs j 1 j 2 j r as a block or group job in the order (j 1 j 2 j r ) showing priority of job j 1 over j 2 etc 5) Machines break down interval transportation tie is not considered for calculating waiting tie 6) Pre- eption is not allowed ie jobs are not being split clearly once a job is started on a achine the process on that achine can t be stopped unless the job is copleted Lea1 Assuing two achines R S are processing jobs in order R S with no passing peritted Let and are the processing ties of job k ( k = 123 ) on each achine correspondingly presuptuous their respective set up ties t Rk and t Sk Equivalent processing ties of k th job on achine R & S are defined as = t Sk = t Rk satisfying processing ties structural relationship Max Min then for the job PROBLEM FORMULATON Assue that R and S are two achines processing jobs in the order R S and are the respective processing ties and t Rk and t Sk are the respective set up ties of the k th job on achines R & S Our intention is to find an optial sequence {J k } of jobs iniizing the total waiting tie of sequence S: γ 1 γ 2 γ 3 γ T γ S = R γ1 + S γ1 + S γ2 + S γ Where T as is the copletion tie of job a on achine S Proof Applying atheatical nduction hypothesis on : Assuing the stateent P : T γ S = R γ1 + S γ1 + S γ2 + S γ JRTCC June 2017 Available @ http://wwwijritccorg 728
1 T γ1 R = R γ1 T γ1 S = R γ1 + S W γ2 = R γ1 + y γr R γ2 γ1 Hence for = 1 the stateent P 1 is true For k = 3 Let for = k the stateent P k be true ie T γk S = R γ1 + S γ1 + S γ2 + S γk Now T γk +1 S = Max T γk +1 R T βk S + S γk +1 As Max Min Hence T γk +1 S = R γ1 + S γ1 + S γ2 + S γk + S γk +1 Hence for n = k + 1 the stateent P k + 1 holds true Since P is true for = 1 = k = k + 1 and k being arbitrary Hence P : T γ S = R γ1 + S γ1 + S γ2 + S γ is true Lea2 With the sae notations as that of Lea1 for - job sequence S: γ 1 γ 2 γ 3 γ k γ W γ1 = 0 W γk k 1 = R γ1 + y γr R γk Where W γk is the waiting tie of job γ k for the sequence (γ 1 γ 2 γ 3 γ ) y γr = S γr R γr γ r є (1 2 3 ) Proof W γ1 = 0 W γk = Max T γk R T γk 1 S T γk R = R γ1 + S γ1 + S γ2 + S γk 1 R γ1 R γ2 R γk k 1 = R γ1 + (S γr k 1 R γr ) R γk = R γ1 + (y γr ) R γk W γ3 = R γ1 + y γr 2 Continuing in this way For k = W γ 1 = R γ1 + y γr Hence total waiting tie W = i=1 W γi W = R γ1 1 R γ3 R γ + z γr Where z γr = r y γr Equivalent Job Block Theore Theore 2 n processing a schedule = (123 ) of jobs on two achines R and S in the order R S with no passing allowed A job k (k = 123 ) has processing tie and on each achine respectively The job block (p q) is equivalent to the single job β (called equivalent job β) Now the processing ties of job β on the achines R and S are denoted respectively by R β S β are given by R β = R p + R q in R q S p S β = S p + S q in R q S p Theore1 Let two achines R S are processing jobs in order R S with no passing allowed Let and are the processing ties of job k ( i = 123 ) on each achine respectively assuing their respective set up ties t Rk and t Sk Equivalent processing ties are defined as = t Sk = t Rk satisfying processing ties structural relationship Max Min then for any job sequence S: γ 1 γ 2 γ 3 γ the total waiting tie W (say) W = R γ1 z γr 1 + z γr = r y γr ; γ r 1 2 3 Proof Fro Lea 2 we have W γ1 = 0 For k = 2 The proof of the theore is given by Maggu PL et al [8] ALGORTHM Step 1: Equivalent processing ties and on achine R & S respectively be calculated in first step as defined in the lea 1 Step 2: Take equivalent job β = (p q) and define processing ties using equivalent job block theore and replace the pair of jobs p q in this order by the single job Fill up the values in the following table: JRTCC June 2017 Available @ http://wwwijritccorg 729
1 R 1 2 R 2 3 R 3 y k = β R β p R p S 1 S 2 S 3 S β S p y 1 y 2 y 3 y β y p Step 3: Asseble the jobs in increasing order of y i R γ1 in{ } ove on to step 5 Assuing the sequence found be (γ 1 γ 2 γ 3 γ p ) Step 4: Locate in{ For the following two possibilities R γ1 = in{ required optial sequence } } Schedule according to step 3 is the Step 5: Consider the different sequence of jobs J 1 J 2 J 3 J p Where J 1 is the sequence obtained in step 3 Sequence J ( = 23 p) can be obtained by placing th job in the sequence J 1 to the first position and rest of the sequence reaining sae Step 6: Calculate the entries for the following table TABLE 2 z ki = ( i)y k R i S i y k i = 1 i= 2 i = 3 i = n-1 1 R 1 S 1 y 1 z 11 z 12 z 13 z 1 1 2 R 2 S 2 y 2 z 21 z 22 z 23 z 2 1 3 R 3 S 3 y 3 z 31 z 32 z 33 z 3 1 R S y z 1 z 2 z 3 z 1 Step 7: Copute the total waiting tie W for all the sequences J 1 J 2 J 3 J p using the following forula: W = R b R b 1 + z ak = Equivalent processing tie of the first job on achine R in sequence J The sequence with iniu total waiting tie is the required optial sequence TABLE 3: PROCESSNG TME MATRX t Rk t Sk 1 6 2 10 1 2 5 4 11 3 3 8 5 15 4 4 5 1 7 3 5 6 2 12 4 6 7 1 12 3 NUMERCAL LLUSTRATON Assue 6 jobs 1 2 3 4 5 6 has to be processed on two achines R & S with processing ties and and set up ties t Rk and t Sk respectively Our objective is to obtain optial string iniizing the total waiting tie for the jobs Solution As per step 1- Equivalent processing tie achine R & S given in the following table JRTCC June 2017 Available @ http://wwwijritccorg & on 730
`TABLE 4: EQUVALENT PROCESSNG TME MATRX y k = Max Min As per step 2- Take equivalent job β = (2 5) Calculating the processing ties for equivalent job β R β = 2 S β 1 5 8 2 2 7 3 4 10 4 2 6 5 2 10 6 4 11 = 15 1 5 8 3 β 2 15 13 3 4 10 6 4 2 6 4 6 4 11 7 As per step 3- Arrange the jobs in increasing order of y i ie the sequence found be 1 4 36 β As per step 4- Min = 2 R 1 As per step 5- Consider the following different sequences of jobs J 1 : 1 4 36 β ; J 2 : 41 36 β ; J 3 : 31 46 β ; J 4 : 61 4 3 β ; J 5 : β 1 4 36 As per step 6- Fill up the values in the following table TABLE 5 z ki = (6 i)y k y k i = 1 i = 2 i = 3 i = 4 i=5 = 1 5 8 3 15 12 9 6 3 2 2 7 5 25 20 15 10 5 3 4 10 6 30 24 18 12 6 4 2 6 4 20 16 12 8 4 5 2 10 8 40 32 24 16 8 6 4 11 7 35 28 21 14 7 As per step 7- The total waiting tie for the sequences obtained in step 5 can be calculated 5 Here i=1 = 19 For the sequence J 1 : 1 4 36 β or J 1 : 1 4 3625 For the sequence J 5 : β 1 4 36 or J 5 : 251 4 36 Total waiting tie W = 73 Hence schedule J 2 : 41 362 5 is the required schedule with iniu total waiting tie Total waiting tie W = 79 Conclusion For the sequence J 2 : 41 36 β or J 2 : 41 362 5 Total waiting tie W = 62 For the sequence J 3 : 31 46 β or J 3 : 31 462 5 Total waiting tie W = 78 For the sequence J 4 : 61 4 3 β or J 4 : 61 4 32 5 Total waiting tie W = 81 The present study deals with the flow shop scheduling proble with the ain idea to iniize the total waiting tie of jobs However it ay increase the other costs like achine idle cost or penalty cost of the jobs yet the idea of iniizing the waiting tie ay be an econoical aspect fro Factory /ndustry anager s view point when he has iniu tie contract with a coercial party to coplete the jobs The work can be extended by introducing various paraeters like transportation tie break down interval etc JRTCC June 2017 Available @ http://wwwijritccorg 731
REFERENCES [1] Johnson [1954] Optial two and three stage production schedule with set up ties included Naval Research Logistics Quarterly Vol 1 pp 61-68 [2] Paler DS [1965] Sequencing jobs through a ulti stage process in the iniu total tie- a quick ethod of obtaining a near optiu Operation Research Quarterly Vol 16 pp 101-107 [3] Capbell HG; Dudek RA; Sith ML [1970] A Heuristic Algorith for the n- job -achine Sequencing Proble Manageent Science Vol16 No10 pp 630-637 [4] Dannenbring David G [1977] An Evaluation of Flow Shop Sequencing Heuristics Manageent Science Vol23 No11 pp 1174-1182 [5] Brucker P; Jurisch B; Sievers B [1994] A branch and bound algorith for the job shop scheduling proble Discrete Applied Matheatics Vol 49 No 1 pp 109 127 [6] Gendreau M; Laporte G; Seet F [1998] A tabu search heuristic for the undirected selective travelling salesan proble European Journal of Operational Research Elsevier Vol106 No 2-3 pp 539-545 [7] Nowicki E; Sutnicki C [1996] A fast taboo search algorith for the job shop proble Manageent Science Vol 42 No6 pp 797 813 [8] Maggu PL; Dass G [1977] Equivalent jobs for job block in job sequencing Operations Research Vol 14 No 4 pp 277-281 [9] Yoshida; Hitoi [1979] Optial two stage Production Scheduling with set- up tie separated AE Transactions Vol pp 261-26 [10] Nawaz M; Enscore E; Ha [1983] A heuristic algorith for the achine n job flow shop sequencing proble OMEGA Vol 11 No1 pp 91-95 [11] Singh TP [1985] on n 2 flow shop proble involving job block Transportation ties and Break-down Machine ties PAMS Vol21 No1-2 [12] Rajendran C; Chaudhuri D [1992] An efficient heuristic approach to the scheduling of jobs in a flow-shop European Journal of Operational Research Vol 61 pp 318-325 [13] Singh TP;Gupta D [2004] Optial two stage production schedule with group jobs restrictions having set up ties separated fro processing tie associated with probabilities Presented in nternational Conference on ndustrial & Applied Matheatics at ndia nternational Centre New Delhi 4-6 Dec(2004) and published in the journal reflections des ERA (JMS) VOL 1 pp 53-70 [14] Singh TP; Gupta D; Kuar R [2006] Optial two stage production schedule with Group job-restrictions having set up ties separated fro processing tie associated with probabilities Reflections des ERA (JMS) Vol 1 pp 53-70 [15] Gupta D [2011] Miniizing rental cost under specified rental policy in two stage flowshop the processing tie associated with probabilities including break down interval and job block criteria European Journal of Business and Manageent (USA) Vol 3 No2 pp 85-103 [16] Gupta D; Bala S; Singla P[2012] Optial Two Stage Open Shop Specially Structured Scheduling To Miniize the Rental Cost processing tie Associated with Probabilities including transportation tie OSR Journal of Matheatics Vol 3 No 3 pp 01-06 [17] Gupta D; Shara S; Bala S; [2012] Specially Structured Three Stage Flowshop Scheduling to iniize the rental cost nternational Journal of Applied Physics and Matheatics Vol 2 No 3 pp 156-158 [18] Gupta D;Bala S; Singla P;Shara S [2015] 3- stage Specially Structured Flow Shop Scheduling to Miniize the Rental Cost ncluding Transportation Tie Job Weightage and Job Block Criteria European Journal of Business and Manageent Vol 7 No4 [19] Gupta D; Goyal B[2016] Optial Scheduling For Total Waiting Tie Of Jobs n Specially Structured Two Stage Flow Shop Proble Processing Ties Associated With Probabilities Aryabhatta Journal of Matheatics & nforatics Vol8 No 1 pp 45-52 [20] Gupta D; Goyal B[2016] Job block concept in two stage specially structured Flow shop scheduling to iniize the total waiting tie of jobs nternational Journal of Latest Trends in Engineering and Technology Vol 7 No 3 pp 287 295 [21] Gupta D; Goyal B[2016] Miniizing total waiting tie of jobs in specially structured two stage Flow Shop Scheduling processing tie separated fro set up tie Journal of Matheatics and Syste Sciences Vol 12 No 1-2 pp 73-76 JRTCC June 2017 Available @ http://wwwijritccorg 732