Minimize the Rental Cost in Two Stage Flow Shop Scheduling Problem in Which Set Up Time Separated From Processing Time with Branch and Bound Technique

(1)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 3, Issue 8, August 2013)

315

Minimize the Rental Cost in Two Stage Flow Shop Scheduling

Problem in Which Set Up Time Separated From Processing

Time with Branch and Bound Technique

Deepak Gupta

1

, Payal Singla

2

, Shashi Bala

3

1_{Prof. & Head,}2,3_{Research Scholar, Department of Mathematics, Maharishi Markandeshwar University}

Mullana, Ambala (India)

Abstract--This paper studies two stage flow shop scheduling problem in which set up times are separated from processing times. Further set up times and processing times are associated with their respective probabilities. The objective of the study is to get optimal sequence of jobs in order to minimize the rental cost of the machines. The given problem is solved with branch and bound method. The method is illustrated by numerical example.

Keywords-- Set up time, processing time, Rental Cost, Branch and Bound.

I. 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 [17] considered the problem with setup times. Yang and Chern [16] extended the problem to a two-machine flowshop group scheduling problem. Maggu and Das[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 flow shop with identical parallel machines at each stage. Brah and Loo [1] studied a flow shop scheduling problem with multiple processors. Futatsuishi, et al. [4] 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. Sawarc [11] studied special cases of the flow shop problem, Smith [11] discussed Various Optimizers for single stage production, Temiz Izzettin and Serpil Erol [14] studied Fuzzy branch and bound algorithm for flow shop scheduling, V.A.

Strusevich [15] discussed Two-machine open-shop scheduling problem with setup, processing and removal times separated. In scheduling problems, setup can include execution of different preparation jobs such as getting of tools, calibrating of tools, etc., before starting the actual tasks of maintenance activities. In centralized maintenance system where sets of tasks may be scheduled on equipments distributed on different geographic sites, the travelling time between two sites is considered as a setup time. In certain cases, the setup may be neglected or included in the job processing time. However, in several real life situations it should be considered apart from jobs, which allows having a more accurate solution.

Two types of setup are distinguished. We call it sequence independent setup when it depends only on the job to be processed and sequence dependent setup when it depends both on the job to be processed and the preceding job. The problem investigated in this article is scheduling n jobs on 2 machines in order to minimize the total elapsed time. We consider a sequence – independent setup time.

(2)

International Journal of Emerging Technology and Advanced Engineering

316

If there is a machine that is perfect for ones practice, renting will allow to using the machine without having to pay high purchase cost. Renting can enable small business to acquire sophisticated technology, such as a voice over internet protocol phone system that might be otherwise unaffordable. The renting is better able to keep up with longer competition without financial resources.

In this paper we consider a two-stage flowshop scheduling problem with set up time separated from processing times with the help of branch and bound method. Branch and bound is an exact method usually used in scheduling problems to find optimal solutions. This method requires three components: a lower bound (LB), an upper bound and a branching strategy. Branch and bound provides a systematic enumeration procedure that considers bounds on the objective function for different subsets of solutions. Subsets are eliminated when their lower bound is dominated by the bound of other subset. The procedure is repeated until the search tree is exhausted and the optimal solution is found. The given method is very simple and easy to understand. Thus, the problem discussed here has significant use of theoretical results in process industries.

II.ASSUMPTIONS

1. No passing is allowed.

2. Each operation once started must performed till completion.

3. Jobs are independent to each other.

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

5. 0



p

_i



1

,

0 

p

_iA



1

, 0

 

q

_i

1

,

i

0 

q

B



1

.

Rental Policy(P):

The machines will be taken on rent as and when they are required and are returned as and when they are no longer required. i.e. the first machine will be taken on rent in the starting of the processing of jobs, 2nd machine will be taken on rent at time when 1st job is completed on 1st machine.

III. 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

pi : Probability associated to the processing time ai.

qi : Probability associated to the processing time bi.

Ai : Expected Processing time for ith job on machine A

Bi : Expected Processing time for ith job on machine B.

si A

: Set up time of i th

job on machine A

siB : Set up time of ith jobon machine B

piA : Probability associated to the set up time siA

qiB : Probability associated to the set up time siB

SiA : Expected set up time for ith job on machine A

SiB : Expected set up time for ith job on machine B

Cij : Completion time for job ith on machines A and B

S0 : Optimal sequence

Jr : Partial schedule of r scheduled jobs

Jr′ : The set of remaining (n-r) free jobs

IV. MATHEMATICAL DEVELOPMENT

Consider n jobs say i=1, 2, 3 … n are to be processed on two machines A & B in the order AB. Let ai and bi be the processing times of ith job on machine A and machine B with probability pi and qi respectively. Let siA & siB be the set up times of ith job on machine A and machine B with probabilities piA and piB respectively. Ai and Bi be the expected processing times of ith job with SiA & SiB be the expected set up times on each machine respectively. Let C1 & C2 be the rental cost of machines A and B respectively.

(3)

International Journal of Emerging Technology and Advanced Engineering

317

Tableau – 1

Jobs

Machine A

Machine B

i

a

i

p

i

s

iA

p

iA

b

i

q

i

s

iB

q

iB

1 a

1

p

1

s

1A

p

1A

b

1

q

1

s

1B

q

1B

2 a

2

p

2

s

2A

p

2A

b

2

q

2

s

2B

q

2B

3 a

3

p

3

s

3A

p

3A

b

3

q

3

s

3B

q

3B

-

n

a

n

p

n

s

nA

p

nA

b

n

q

n

s

nB

q

nB

Our objective is to obtain optimal or near optimal sequence S0 of the jobs which minimize the rental cost of all the machines, using branch and bound technique.

Mathematically, the problem is stated as:

Minimize R(S0) = 1

n

i i

A





(S0) × C1 + U2B (S0) × C2

Subject to constraint : Rental policy P, with the objective is to minimize rental cost of machines.

Algorithm: Step 1: Calculate

(i) Ai = ai × pi (ii) Bi = bi × qi

Step 2: Calculate

(i) SiA= aiA× piA (ii) SiB= biB× qiB

Step 3: Calculate

(i)

A

_i



= Ai − Si

B (ii)

i

B



= Bi − SiA Step 4: Calculate

(i) ₁

(

,1)

min( )

r r

r i i

i J i J

l

t J

A

B

   











(ii) ₂

(

, 2)

r

r i

i J

l

t J

B

 









Step 3: Calculate

l



max( , )

l l

₁ ₂

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. Compute the total completion time CT(S0) by computing in-out table.

Step 6: Calculate utilization time U2B(S0) of 2nd machine.

1

n

i i

A



(4)

International Journal of Emerging Technology and Advanced Engineering

318

Step 7: Find rental cost R(S0) = 1

n

i i

A





(S0) × C1 + U2B (S0) × C2 where C1 & C2 are the rental cost per unit time of A and B machine respectively.

Numerical Example:

Consider 5 jobs 2 machine flow shop scheduling problem whose processing time and set up time of the jobs on each machine is given. The rental cost per unit time for machine A & B is 10units & 20 units respectively.

Tableau – 1

Jobs Machine A Machine B

i ai pi siA piA bi qi siB qiB

1 150 0.2 6 0.1 125 0.2 4 0.2

2 220 0.1 5 0.3 120 0.3 7 0.2 3 90 0.2 4 0.3 120 0.1 6 0.1

4 130 0.3 7 0.2 150 0.3 3 0.2

5 80 0.2 3 0.1 100 0.2 2 0.3

Our objective is to obtain optimal schedule for above said problem.

Solution: As per Step1& step 2 Expected processing time and expected set up time:

Tableau – 2

Jobs Machine A Machine B

i Ai SiA Bi SiB

1 30 0.6 25 0.8

2 22 1.5 36 1.4 3 18 1.2 12 0.6

4 39 1.4 45 0.6

5 16 0.3 20 0.6

As per step 3

Tableau – 3

Jobs MachineA MachineB i

i

A



B

_i



1 29.2 24.4

2 20.6 34.5 3 17.4 10.8

4 38.4 43.6

(5)

International Journal of Emerging Technology and Advanced Engineering

319

Step2: Calculate

(i) ₁

(

,1)

min( )

r r

r i i

i J i J

l

t J

A

B

   











(ii) ₂

(

, 2)

r

r i

i J

l

t J

B

 









For J1 = (1).Then J′(1) = {2,3,4}, we get LB(1):

1 2

max( , )

l



l l

=160.2

Similarly, we have LB(2)= 151.6, LB(3)= 148.4, LB(4) = 169.4,

LB(5)= 147.2

Step 3 & 4:

Now branch from J1 = (5). Take J2 =(51). Then J′2={2,3,4} and LB(51) = 157.9

Tableau – 4

Proceeding in this way, we obtain lower bound values as shown in the tableau- 4

[image:5.612.119.202.376.571.2]

The complete schedule tree is shown as in Figure 1

Figure 1: Lower bounds

Step 5 : Therefore the sequence S1 is 5-3-1-2--4 and the corresponding in-out table on sequence S1 is as follows:

Tableau- 5

Job

i

Machine A

In-out

Machine B

In-out

5 0 -16 16 – 36

3 16.3 – 34.3 36.6 – 48.6

1 35.5 – 65.5 65.5- 90.5

2 66.1 – 88.1 91.3 – 127.3

4 89.6 – 129.7 129.7 – 174.7

total elapsed time is 174.7 units

.

5

1

i i

A





= 129.7 , U2B (S0) = 174.7 – 16 = 158.7 units

Rental Cost =129.7 × 10 + 158.7 ×20 = 4471 units Node Jr LB

(Jr) (1)

(2) (3) (4) (5) (51) (52) (53) (54) (531) (532) (534) (5312) (5314)

(6)

International Journal of Emerging Technology and Advanced Engineering

320

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. [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, pp. 12-16.

[10] Narian L Bagga P C (2005), “Two machine flow shop problem with availability constraint on each machine”, JISSOR, Vol. XXIV 1-4, pp.17-24.

[11] Sawarc, W. (1977), “Special cases of the flow shop problem”, Naval Research Logistic Quarterly, Vol. 24, pp 483-492.

[12] Singh, T P, Gupta Deepak (2005), “Minimizing rental cost in two stage flow shop , the processing time associated with probabilities including job block”, Reflections de ERA, Vol 1, No.2, pp.107-120. [13] Smith, W.E. (1956), “Various Optimizers for single stage production

“, Naval Research Logistic, Vol. 2, pp 59-66.

[14] Temiz Izzettin and Serpil Erol (2004),Fuzzy branch and bound algorithm for flow shop scheduling, Journal of Intelligent Manufacturing,Vol.15,pp.449-454.

[15] V.A. Strusevich (1993), Two-machine open-shop scheduling problem with setup, processing and removal times separated, Comput. Oper. Res. 20 597–611.

[16] 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.