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Journal of Information Engineering and Applications                                             www.iiste.org
ISSN 2224-5758 (print) ISSN 2224-896X (online)
Vol 1, No.5, 2011

Application of Branch and Bound Method for Optimal Two
  Stage Flow Shop Scheduling Problem with Group Job-
                       Restrictions
                              Deepak Gupta* , Payal Singla, Harminder Singh

            Department of Mathematics Maharishi Markandeshwar University Mullana, Ambala (India)

                     * E-mail of the corresponding author: guptadeepak2003@yahoo.co.in

Abstract:

This paper studies two stage flow shop scheduling problem in which equivalent job for group job,
processing times on each machine are given. The objective of the study is to get optimal sequence of
jobs in order to minimize the total elapsed time. The given problem is solved with branch and bound
method.The method is illustrated by numerical example.

Keywords: Equivalent job, processing time, elapsed time, Branch and Bound.

1. Introduction:

          A flowshop scheduling problem has been one of classical problems in production scheduling
since Johnson [6] proposed the well known Johnson’s rule in the two-stage flowshop makespan
scheduling problem. Yoshida and Hitomi [11] further considered the problem with setup times. Yang
and Chern [10] extended the problem to a two-machine flowshop group scheduling problem. Maggu
and Dass[9] introduced the concept of equivalent job for a job-block when the situations of giving
priority of one job over another arise. Kim, et al.[7] considered a batch scheduling problem for a two-
stage flowshop with identical parallel machines at each stage. Brah and Loo [1] studied a flow shop
scheduling problem with multiple processors. Futatsuishi, et al. [4] further studied a multi-stage
flowshop scheduling problem with alternative operation assignments.

                        Lomnicki [8] introduced the concept of flow shop scheduling with the help of
branch and bound method. Further the work was developed by Ignall and Scharge [5], Chandrasekharan
[3] , Brown and Lomnicki [2], with the branch and bound technique to the machine scheduling problem
by introducing different parameters In this paper we consider a two-stage flowshop scheduling problem
with job-block with the help of branch and bound method. The given method is very simple and easy to
understand. Thus, the problem discussed here has significant use of theoretical results in process
industries.

2. Assumptions:

   i.       No passing is allowed.

  ii.        Each operation once started must performed till completion.

 iii.        Jobs are independent to each other.

 iv.        A job is entity, i.e. no job may be processed by more than one machine at a time.

3.Notations:

We are given n jobs to be processed on three stage flowshop scheduling problem and we have used the
following notations:

Ai          :     Processing time for job ith on machine A
Bi          :     Processing time for job ith on machine B
Cij         :     Completion time for job ith on machines A and B
S0      :        Optimal sequence

                                                      1
Journal of Information Engineering and Applications                                          www.iiste.org
 ISSN 2224-5758 (print) ISSN 2224-896X (online)
 Vol 1, No.5, 2011
  Jr          :         Partial schedule of r scheduled jobs
  Jr′         :         The set of remaining (n-r) free jobs

 4. Mathematical Development:
 Consider n jobs say i=1, 2, 3 … n are processed on two machines A & B in the order AB. A job i
 (i=1,2,3…n) has processing time Ai & Bi on each machine respectively. . Let an equivalent job β is
 defined as (k, m) where k, m are any jobs among the given n jobs such that k occurs before job m in the
 order of job block (k , m).The mathematical model of the problem in matrix form can be stated as :
                                    Jobs Machine A Machine B
                                              i      Ai         Bi
                                              1      A1         B1
                                              2      A2         B2
                                              3      A3         B3
                                              4      A4         B4
                                              -      ----       ---

                                              -       --        --
                                              n      An         Bn

                                               Tableau – 1
 Our objective is to obtain the optimal schedule of all jobs which minimize the total elapsed time, using
 branch and bound technique.

 5. Algorithm:
 Step 1: (i) Aβ       = Ak + Am − min( Ak , Bm )
            (ii)   Aβ = Ak + Am − min( Ak , Bm )
 Step 2: Calculate
 (i)    l1 = t ( J r ,1) + ∑ Ai + min( Bi )
                                       ′
                                   i∈J r
                            ′
                        i∈J r

 (ii)   l1 = t ( J r , 2) + ∑ Bi
                              ′
                          i∈J r

 Step 3: Calculate
        l = max(l1 , l2 )
We evaluate l first for the n classes of permutations, i.e. for these starting with 1, 2, 3………n
 respectively, having labelled the appropriate vertices of the scheduling tree by these values.
 Step 4: Now explore the vertex with lowest label. Evaluate l for the (n-1) subclasses starting with this
   vertex and again concentrate on the lowest label vertex. Continuing this way, until we reach at the end
   of the tree represented by two single permutations, for which we evaluate the total work duration. Thus
   we get the optimal schedule of the jobs.
 Step 5: Prepare in-out table for the optimal sequence obtained in step 3 and get the minimum total
   elapsed time.




                                                            2
Journal of Information Engineering and Applications                                            www.iiste.org
 ISSN 2224-5758 (print) ISSN 2224-896X (online)
 Vol 1, No.5, 2011
 6.Numerical Example:

 Consider 5 jobs 2 machine flow shop problem whose processing time of the jobs on each machine is
 given.
 Job      Machine A      Machine B
  i        Ai             Bi
  1        11             17
  2        15             21
  3        22             25
  4        34             19
  5        27             38
         Tableau – 2
 Our objective is to obtain optimal schedule for above said problem in which jobs 2,4 are to be
 processed as a group job (2,4)

 Solution:
 Step1: Calculate
  (i) Aβ = A2 + A4       − min( A4 , B2 )
           = 28
 (ii)   Bβ = B2 + B4 − min( A4 , B2 )
         = 21
 The problem reduced as in tableau-3.
 Step2: Calculate
 (i)    l1 = t ( J r ,1) + ∑ Ai + min( Bi )
                                       ′
                                   i∈J r
                            ′
                        i∈J r

 (ii)   l1 = t ( J r , 2) + ∑ Bi
                             ′
                         i∈J r

 For J1 = (1).Then J′(1) = {2,3,4}, we get l 1 = 43 , l 2 = 37
 LB(1) l = max(l1 , l2 ) =112                                                                   similarly,
 we have LB(β)= 129, LB(3)= 123, LB(5)= 128
 Step 3 & 4:
 Now branch from J1 = (1). Take J2 =(1β). Then J′2={3,4} and LB(1β) = 123
 Proceeding in this way, we obtain lower bound values as shown in the tableau- 4
Step 5 :
 Therefore the sequence S1 is 1-3-5- β i.e. 1-3-5- 2 - 4 and the corresponding in-out table on sequence
 S1 is as in tableau-5:
 Hence the total elapsed time is 138 units.

 7. Remarks:
     The study may further be extended by considering various parameters such as transportation time,
     mean weightage time etc.

 Refrences:
     [1] Brah, S.A. and Loo, L.L.,(1999), “Heuristics for Scheduling in a Flow Shop with Multiple
         Processors,”European Journal of Operation Research, Vol. 113, No.1, pp.113-122.
     [2] Brown, A.P.G. and Lomnicki, Z.A. (1966), “Some applications of the branch and bound
         algorithm to the machine scheduling problem”, Operational Research Quarterly, Vol. 17,
         pp.173-182.
     [3] Chander Shekharan, K, Rajendra, Deepak Chanderi (1992), “An efficient heuristic approach
         to the scheduling of jobs in a flow shop”, European Journal of Operation Research Vol. 61,pp.
         318-325.




                                                     3
Journal of Information Engineering and Applications                                            www.iiste.org
ISSN 2224-5758 (print) ISSN 2224-896X (online)
Vol 1, No.5, 2011
      [4]    Futatsuishi, Y., Watanabe, I., and Nakanishi, T. ( 2002), “A Study of the Multi-Stage
           Flowshop Scheduling Problem with Alternative Operation Assignments,”Mathematics and
           Computers in Simulation,Vol. 59, No. 3, pp.73-79.
      [5] Ignall, E. and Schrage, L. (1965), “Application of the branch-and-bound technique to some
           flowshop scheduling problem”, Operations Research, Vol. 13, pp.400-412.
      [6] Johnson, S.M. (1954), “Optimal Two and Three Stage Prouction Schedules with Setup Times
           Include,”Naval Research Logistics Quarterly,Vol. 1 No. 1, pp. 61-68.
      [7] Kim, J.S., Kang, S.H.,and Lee, S.M.(1997), “ Transfer Batch Scheduling for a Two-Stage
           Flowshop with Identical Parallel Machines at Each Stage,” Omega,Vol. 25, No. 1, pp. 547-
           555
      [8] Lomnicki, Z.A. (1965), “A branch-and-bound algorithm for the exact solution of the three-
           machine scheduling problem”, Operational Research Quarterly, Vol. 16, pp.89-100.
      [9] Maggu & Das (1981), “On n x 2 sequencing problem with transportation time of jobs”, Pure
           and Applied Mathematika Sciences, 12-16.
      [10] Yang, D.L. and Chern, M.S. (2000), “Two-Machine Flowshop Group Scheduling
           Problem,”Computers and Operations Research, Vol.27, No.10, pp.975-985.
      [11] Yoshida, T. and Hitomi, K., (1979), “Optimal Two-Stage Production Scheduling with Setup
           Times Separated,”AIIE Transactions, Vol. 11, No. 1, pp. 261-273.

Tables and Figures:
Table 3: The reduced as in tableau-3:


Job    Machine A         Machine B
I         Ai                Bi
1         11                17
Β         28                21
3         22                25
5         27                38

            Tableau –3

Table 4: lower bounds for respective jobs are as in tableau-4

                                                                           Tableau-4

Table 5: In-out table for S1 and the minimum total elapsed time as in tableau-5.



Job     Machine A          Machine B                                                   Node Jr    LB
I       In-out             In-out                                                                 (Jr)
                                                                                       (1)        112
1       0 - 11             11 – 28
                                                                                       (β)        129
3       11 – 33            33 – 58                                                     (3)        123
5       33 – 60            60 - 98                                                     (5)        128
3       60 -75             98 - 119            Tableau- 5                              (1β)       123
4       75 - 109           119 - 138                                                   (13)       117
                                                                                       (15)       122
                                                                                       (13β)      126
                                                                                       (135)      119




                                                   4
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11.application of branch and bound method for optimal two stage flow shop scheduling problem with group job restrictions

  • 1. Journal of Information Engineering and Applications www.iiste.org ISSN 2224-5758 (print) ISSN 2224-896X (online) Vol 1, No.5, 2011 Application of Branch and Bound Method for Optimal Two Stage Flow Shop Scheduling Problem with Group Job- Restrictions Deepak Gupta* , Payal Singla, Harminder Singh Department of Mathematics Maharishi Markandeshwar University Mullana, Ambala (India) * E-mail of the corresponding author: guptadeepak2003@yahoo.co.in Abstract: This paper studies two stage flow shop scheduling problem in which equivalent job for group job, processing times on each machine are given. The objective of the study is to get optimal sequence of jobs in order to minimize the total elapsed time. The given problem is solved with branch and bound method.The method is illustrated by numerical example. Keywords: Equivalent job, processing time, elapsed time, Branch and Bound. 1. Introduction: A flowshop scheduling problem has been one of classical problems in production scheduling since Johnson [6] proposed the well known Johnson’s rule in the two-stage flowshop makespan scheduling problem. Yoshida and Hitomi [11] further considered the problem with setup times. Yang and Chern [10] extended the problem to a two-machine flowshop group scheduling problem. Maggu and Dass[9] introduced the concept of equivalent job for a job-block when the situations of giving priority of one job over another arise. Kim, et al.[7] considered a batch scheduling problem for a two- stage flowshop with identical parallel machines at each stage. Brah and Loo [1] studied a flow shop scheduling problem with multiple processors. Futatsuishi, et al. [4] further studied a multi-stage flowshop scheduling problem with alternative operation assignments. Lomnicki [8] introduced the concept of flow shop scheduling with the help of branch and bound method. Further the work was developed by Ignall and Scharge [5], Chandrasekharan [3] , Brown and Lomnicki [2], with the branch and bound technique to the machine scheduling problem by introducing different parameters In this paper we consider a two-stage flowshop scheduling problem with job-block with the help of branch and bound method. The given method is very simple and easy to understand. Thus, the problem discussed here has significant use of theoretical results in process industries. 2. Assumptions: i. No passing is allowed. ii. Each operation once started must performed till completion. iii. Jobs are independent to each other. iv. A job is entity, i.e. no job may be processed by more than one machine at a time. 3.Notations: We are given n jobs to be processed on three stage flowshop scheduling problem and we have used the following notations: Ai : Processing time for job ith on machine A Bi : Processing time for job ith on machine B Cij : Completion time for job ith on machines A and B S0 : Optimal sequence 1
  • 2. Journal of Information Engineering and Applications www.iiste.org ISSN 2224-5758 (print) ISSN 2224-896X (online) Vol 1, No.5, 2011 Jr : Partial schedule of r scheduled jobs Jr′ : The set of remaining (n-r) free jobs 4. Mathematical Development: Consider n jobs say i=1, 2, 3 … n are processed on two machines A & B in the order AB. A job i (i=1,2,3…n) has processing time Ai & Bi on each machine respectively. . Let an equivalent job β is defined as (k, m) where k, m are any jobs among the given n jobs such that k occurs before job m in the order of job block (k , m).The mathematical model of the problem in matrix form can be stated as : Jobs Machine A Machine B i Ai Bi 1 A1 B1 2 A2 B2 3 A3 B3 4 A4 B4 - ---- --- - -- -- n An Bn Tableau – 1 Our objective is to obtain the optimal schedule of all jobs which minimize the total elapsed time, using branch and bound technique. 5. Algorithm: Step 1: (i) Aβ = Ak + Am − min( Ak , Bm ) (ii) Aβ = Ak + Am − min( Ak , Bm ) Step 2: Calculate (i) l1 = t ( J r ,1) + ∑ Ai + min( Bi ) ′ i∈J r ′ i∈J r (ii) l1 = t ( J r , 2) + ∑ Bi ′ i∈J r Step 3: Calculate l = max(l1 , l2 ) We evaluate l first for the n classes of permutations, i.e. for these starting with 1, 2, 3………n respectively, having labelled the appropriate vertices of the scheduling tree by these values. Step 4: Now explore the vertex with lowest label. Evaluate l for the (n-1) subclasses starting with this vertex and again concentrate on the lowest label vertex. Continuing this way, until we reach at the end of the tree represented by two single permutations, for which we evaluate the total work duration. Thus we get the optimal schedule of the jobs. Step 5: Prepare in-out table for the optimal sequence obtained in step 3 and get the minimum total elapsed time. 2
  • 3. Journal of Information Engineering and Applications www.iiste.org ISSN 2224-5758 (print) ISSN 2224-896X (online) Vol 1, No.5, 2011 6.Numerical Example: Consider 5 jobs 2 machine flow shop problem whose processing time of the jobs on each machine is given. Job Machine A Machine B i Ai Bi 1 11 17 2 15 21 3 22 25 4 34 19 5 27 38 Tableau – 2 Our objective is to obtain optimal schedule for above said problem in which jobs 2,4 are to be processed as a group job (2,4) Solution: Step1: Calculate (i) Aβ = A2 + A4 − min( A4 , B2 ) = 28 (ii) Bβ = B2 + B4 − min( A4 , B2 ) = 21 The problem reduced as in tableau-3. Step2: Calculate (i) l1 = t ( J r ,1) + ∑ Ai + min( Bi ) ′ i∈J r ′ i∈J r (ii) l1 = t ( J r , 2) + ∑ Bi ′ i∈J r For J1 = (1).Then J′(1) = {2,3,4}, we get l 1 = 43 , l 2 = 37 LB(1) l = max(l1 , l2 ) =112 similarly, we have LB(β)= 129, LB(3)= 123, LB(5)= 128 Step 3 & 4: Now branch from J1 = (1). Take J2 =(1β). Then J′2={3,4} and LB(1β) = 123 Proceeding in this way, we obtain lower bound values as shown in the tableau- 4 Step 5 : Therefore the sequence S1 is 1-3-5- β i.e. 1-3-5- 2 - 4 and the corresponding in-out table on sequence S1 is as in tableau-5: Hence the total elapsed time is 138 units. 7. Remarks: The study may further be extended by considering various parameters such as transportation time, mean weightage time etc. Refrences: [1] Brah, S.A. and Loo, L.L.,(1999), “Heuristics for Scheduling in a Flow Shop with Multiple Processors,”European Journal of Operation Research, Vol. 113, No.1, pp.113-122. [2] Brown, A.P.G. and Lomnicki, Z.A. (1966), “Some applications of the branch and bound algorithm to the machine scheduling problem”, Operational Research Quarterly, Vol. 17, pp.173-182. [3] Chander Shekharan, K, Rajendra, Deepak Chanderi (1992), “An efficient heuristic approach to the scheduling of jobs in a flow shop”, European Journal of Operation Research Vol. 61,pp. 318-325. 3
  • 4. Journal of Information Engineering and Applications www.iiste.org ISSN 2224-5758 (print) ISSN 2224-896X (online) Vol 1, No.5, 2011 [4] Futatsuishi, Y., Watanabe, I., and Nakanishi, T. ( 2002), “A Study of the Multi-Stage Flowshop Scheduling Problem with Alternative Operation Assignments,”Mathematics and Computers in Simulation,Vol. 59, No. 3, pp.73-79. [5] Ignall, E. and Schrage, L. (1965), “Application of the branch-and-bound technique to some flowshop scheduling problem”, Operations Research, Vol. 13, pp.400-412. [6] Johnson, S.M. (1954), “Optimal Two and Three Stage Prouction Schedules with Setup Times Include,”Naval Research Logistics Quarterly,Vol. 1 No. 1, pp. 61-68. [7] Kim, J.S., Kang, S.H.,and Lee, S.M.(1997), “ Transfer Batch Scheduling for a Two-Stage Flowshop with Identical Parallel Machines at Each Stage,” Omega,Vol. 25, No. 1, pp. 547- 555 [8] Lomnicki, Z.A. (1965), “A branch-and-bound algorithm for the exact solution of the three- machine scheduling problem”, Operational Research Quarterly, Vol. 16, pp.89-100. [9] Maggu & Das (1981), “On n x 2 sequencing problem with transportation time of jobs”, Pure and Applied Mathematika Sciences, 12-16. [10] Yang, D.L. and Chern, M.S. (2000), “Two-Machine Flowshop Group Scheduling Problem,”Computers and Operations Research, Vol.27, No.10, pp.975-985. [11] Yoshida, T. and Hitomi, K., (1979), “Optimal Two-Stage Production Scheduling with Setup Times Separated,”AIIE Transactions, Vol. 11, No. 1, pp. 261-273. Tables and Figures: Table 3: The reduced as in tableau-3: Job Machine A Machine B I Ai Bi 1 11 17 Β 28 21 3 22 25 5 27 38 Tableau –3 Table 4: lower bounds for respective jobs are as in tableau-4 Tableau-4 Table 5: In-out table for S1 and the minimum total elapsed time as in tableau-5. Job Machine A Machine B Node Jr LB I In-out In-out (Jr) (1) 112 1 0 - 11 11 – 28 (β) 129 3 11 – 33 33 – 58 (3) 123 5 33 – 60 60 - 98 (5) 128 3 60 -75 98 - 119 Tableau- 5 (1β) 123 4 75 - 109 119 - 138 (13) 117 (15) 122 (13β) 126 (135) 119 4
  • 5. International Journals Call for Paper The IISTE, a U.S. publisher, is currently hosting the academic journals listed below. The peer review process of the following journals usually takes LESS THAN 14 business days and IISTE usually publishes a qualified article within 30 days. Authors should send their full paper to the following email address. More information can be found in the IISTE website : www.iiste.org Business, Economics, Finance and Management PAPER SUBMISSION EMAIL European Journal of Business and Management EJBM@iiste.org Research Journal of Finance and Accounting RJFA@iiste.org Journal of Economics and Sustainable Development JESD@iiste.org Information and Knowledge Management IKM@iiste.org Developing Country Studies DCS@iiste.org Industrial Engineering Letters IEL@iiste.org Physical Sciences, Mathematics and Chemistry PAPER SUBMISSION EMAIL Journal of Natural Sciences Research JNSR@iiste.org Chemistry and Materials Research CMR@iiste.org Mathematical Theory and Modeling MTM@iiste.org Advances in Physics Theories and Applications APTA@iiste.org Chemical and Process Engineering Research CPER@iiste.org Engineering, Technology and Systems PAPER SUBMISSION EMAIL Computer Engineering and Intelligent Systems CEIS@iiste.org Innovative Systems Design and Engineering ISDE@iiste.org Journal of Energy Technologies and Policy JETP@iiste.org Information and Knowledge Management IKM@iiste.org Control Theory and Informatics CTI@iiste.org Journal of Information Engineering and Applications JIEA@iiste.org Industrial Engineering Letters IEL@iiste.org Network and Complex Systems NCS@iiste.org Environment, Civil, Materials Sciences PAPER SUBMISSION EMAIL Journal of Environment and Earth Science JEES@iiste.org Civil and Environmental Research CER@iiste.org Journal of Natural Sciences Research JNSR@iiste.org Civil and Environmental Research CER@iiste.org Life Science, Food and Medical Sciences PAPER SUBMISSION EMAIL Journal of Natural Sciences Research JNSR@iiste.org Journal of Biology, Agriculture and Healthcare JBAH@iiste.org Food Science and Quality Management FSQM@iiste.org Chemistry and Materials Research CMR@iiste.org Education, and other Social Sciences PAPER SUBMISSION EMAIL Journal of Education and Practice JEP@iiste.org Journal of Law, Policy and Globalization JLPG@iiste.org Global knowledge sharing: New Media and Mass Communication NMMC@iiste.org EBSCO, Index Copernicus, Ulrich's Journal of Energy Technologies and Policy JETP@iiste.org Periodicals Directory, JournalTOCS, PKP Historical Research Letter HRL@iiste.org Open Archives Harvester, Bielefeld Academic Search Engine, Elektronische Public Policy and Administration Research PPAR@iiste.org Zeitschriftenbibliothek EZB, Open J-Gate, International Affairs and Global Strategy IAGS@iiste.org OCLC WorldCat, Universe Digtial Library , Research on Humanities and Social Sciences RHSS@iiste.org NewJour, Google Scholar. Developing Country Studies DCS@iiste.org IISTE is member of CrossRef. All journals Arts and Design Studies ADS@iiste.org have high IC Impact Factor Values (ICV).