I. INTRODUCTION:

In scheduling, there are many objectives or criteria by which the performances of solution methods could be measured. These criteria may be studied singly or combined. Being a major point in scheduling problems, many researchers have carried out extensive study of scheduling criteria [1, 2, 3, 4]. Not less than about 29 different scheduling criteria have been identified [4]. One scheduling criterion that has received the attention of researchers over the years is total elapsed time (also called schedule length or maximum completion time) and is often denoted by C_max. total elapsed time is the completion time of the last scheduled job and is directly proportional to the production costs. Thus, total elapsed time is a very important scheduling criterion [5, 6], hence, its choice in this study.

The recent research addressing group scheduling issues in flow-shop manufacturing environments has been mainly focused on the Flow-Shop Sequence-Dependent Group Scheduling (FSDGS) problem, in which the time required for performing setup operations of each group of jobs depends on the technological features of the previously processed group. The interest towards the FSDGS problem has been basically motivated by the implications of such a scheduling issue in the real industrial practice. Examples of FSDGS problems have been observed in Printed Circuit Boards (PCBs) manufacturing [7], in furniture production [8], in paint and press shops of automobile manufacturers [9].

No-wait flow shop is a popular production system. Because this system relies on continuous production, it is necessary to deploy a continuous production environment. Many industries suffer this constraint, such as the metal melting industry. To prevent deterioration of heated metal, a series of operations must be completed before it cools. Mathematical Problems in Engineering Similarly, plastic, silver, and glass molding also require a series of operations before cooling. However, the related literature is scarce, and so further discussion of this topic is necessary. Besides continuity, setup time is another important consideration in such facilities. These considerations significantly increase the problem complexity.

Allahverdi and Fatih Tatari (1997)[10], in another assignment, performed an empirical investigation on stochastic machine dominance in flow shop problem. Ruiz and Allahverdi (2007)[11] presented some effective heuristics for no-wait flow shop with setup times to minimize total completion time. Ruiz and Stützle (2008)[12] suggested an iterated greedy heuristic for the sequence dependent setup times flow shop problem with total elapsed time (makespan) and weighted tardiness objectives. Sayadi et al. (2010), in another assignment, presented new meta-heuristics called discrete firefly with an adaptation of local search for total elapsed time (makespan) minimization in permutation flow shop scheduling problems. Ancǎu (2010)[13] discussed some weakness and strength of stochastic search in solving flow shop scheduling problem. Naderi- Benia et al. (2012)[14] presented a two-phase fuzzy programming model for a complex bi-objective no wait flow shop scheduling. Wang and Tang (2012)[15] offered a discrete particle swarm optimization algorithm with self-adaptive diversity control for the permutation flow shop problem with blocking. We have taken the problem of FSSP with the objective of total elapsed time (makespan) minimization, we have developed one algorithms and it gives feasible schedule.

II. JV ALGORITHMS:

Step 1: Minimum processing time on machine1 should be greater than or equal to maximum processing time on machine m₂ m₃ … m_n-1.

Step 2: Minimum processing time on machine m should be greater than or equal to maximum processing time on machine m₂ m₃ … m_n-1.

Step 3: If one of these above condition is satisfied then form two hypothetical machines on which we perform the summed up of job’s processing time.

Step 4: Form the group of jobs X that take greater time on the first machine(A) than on the second machine(B) such that X = { i | t_1i t_2i}.

Step 5: Form the group of jobs Y that take greater time on the second machine(B) than on the first machine(A) such that Y = { j | t_2j t_1j}.

Step 6: Sort the jobs in X in descending order of J_i's; if two or more jobs have the same value of J_i, sort them in an arbitrary order.

Step 7: Sort the jobs in Y in ascending order of J_j's; if two or more jobs have the same value of J_i, sort them in an arbitrary order.

Step 8: Schedule the jobs on the machines in the sorted order of Y, then in the sorted order of X. Example: Consider 7 jobs, 4 machine flow shop problem with processing time as shown in table 1.

Table I

JOBS	M1	M2	M3	M4	M5	M6
J1	18	8	7	2	10	25
J2	17	6	9	6	8	19
J3	11	5	8	5	7	15
J4	20	4	3	4	8	12

Makespan calculation using JV’s algorithm:

Step1: Minimum Processing time on machine1 should be greater than or equal to maximum processing time on machine m2, m3,…m5 11 > 10.

Step2: Minimum Processing time on machine m6 should be greater than or equal to the maximum processing time on

machine m2, m3…m5 12 > 10.

Step3: Condition satisfies then we introduce two hypothetical machines A and B respectively.

Summed up the processing time first three machines namely M1+M2+M3+M4+M5 = A Summed up the processing time last three machines namely M2+M3+M4+M5+M6= B

JOBS	A	B
J1	45	52
J2	46	48
J3	36	40
J4	39	31

Step 4 :Form group X as X={J4}

Step 5 :Form group Y as Y={J1,J2,J3}

Step6 :Ordered Group in Descending X={J4}.

Step7 :Ordered Group in Ascending Y= {J3,J2,J1}

Step8 :Form a schedule such that { J3,J2,J1,J4 }

Table V: Makespan calculation using JV Algorithms

JOBS	M1	M2	M3	M4	M5	M6
J3	0-11	11-16	16-24	24-29	29-36	36-51
J2	11-28	28-34	34-43	43-49	49-57	57-76
J1	28-46	46-54	54-61	61-63	63-73	76-101
J4	46-66	66-70	70-73	73-77	77-85	101-113

Makespan is 113. Idle time of M1=0, M2=11+12+12+12=47, M3=24+10+11+9=54, M4=24+14+12+10=60, M5=29+13+6+4=52, M6=0.

CONCLUSION:

This paper proposed a new permutation flow shop scheduling problem can be solved using JV’s algorithm. The proposed algorithm gives near optimal solution , rather than a feasible one. If provides minimum makespan as well as better partial makespan, proposed algorithm Performing better than Sandeep kumar et al[16] algorithms(makespan=116). The proposed JV’s algorithm not only minimizing the makespan but also it reduces the resource idle time too.

REFERENCES:

1. Conway, R. W., Johnson, B. M., Maxwell, W. L. (1960). An experimental investigation of priority dispatching, J. Ind. Eng., 11, 221-230.

2. Gere, W. S. (1962). A heuristic approach to job shop scheduling. PhD. Thesis, Carnegie Institute of Technology.

3. Beenhakker, H. L. (1963). Development of alternate criteria for optimality in the machine sequencing problem, PhD. Thesis, Purdue University.

4. Oyetunji, E.O. (2009). Some common performance measures in scheduling problems. Research Journal of Applied Sciences, Engineering and Technology, 1(2), 6-9.

5. Oluleye, A.E., Oyetunji, E.O. (1999). Performance of some static flowshop scheduling heuristics. Directions in Mathematics, 315-327.

6. Shahzad, A. (2010). A Single Machine Scheduling Problem with Individual Job Tardiness based Objectives, available at: http://oro.univ-nantes.fr/sujets-09-10/shahzad.pdf; (accessed on 13^th August, 2010).

7. Pinedo, M. Scheduling: Theory, Algorithms and Systems, 4th ed.; Springer: New York, NY, USA, 2012.

8. Wilson, A.D.; King, R.E.; Hodgson, T.J. Scheduling non-similar groups on a flow line: Multiple group setups. Robot. Comput.-Integr. Manuf. 2004, 20, 505–515.

9. Salmasi, N.; Logendran, R.; Skandari, M.R. Total flow time minimization in a flowshop sequence-dependent group scheduling problem. Comput. Oper. Res. 2010, 37, 199–212.

10. Allahverdi, A., & Fatih Tatari, M. (1997). Stochastic machine dominance in flowshops. Computers and Industrial Engineering, 32(4), 735-741.

11. Ruiz, R., & Allahverdi, A. (2007). Some effective heuristics for no-wait flowshops with setup times to minimize total completion time. Annals of Operations Research, 156(1), 143-171.

12. Ruiz, R., & Stützle, T. (2008). An iterated greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives. European Journal of Operational Research, 187(3), 1143-1159.

13. Ancǎu, M. (2010). Weakness and strength of stochastic search in solving flowshop scheduling problem. Academic Journal of Manufacturing Engineering, 8(4), 6-10.

14. Naderi-Benia, M., Tavakkoli-Moghaddamb, R., Naderic, B., Ghobadiana, E., & Pourroustaa, A. (2012). A two-phase fuzzy programming model for a complex bi-objective no-wait flow shop scheduling. International Journal of Industrial Engineering, 3, 617-626.

15. Wang, X., & Tang, L. (2012). A discrete particle swarm optimization algorithm with self-adaptive diversity control for the permutation flowshop problem with blocking. Applied Soft Computing Journal, 12(2), 652-662.

16. Sandeep Kumar et al, / (IJCSIT) International Journal of Computer Science and Information Technologies, Vol. 5 (4) , 2014, 5057-5061 a novel hybrid algorithm for permutation flowshop scheduling.

Received on 23.03.2019 Accepted on 25.05.2019

Int. J. Tech. 2018; 9(1):05-07.

DOI: 10.5958/2231-3915.2019.00002.6