Scheduling Algorithm to Optimize Jobs in Shop Floor

Problem statement: The ratio scheduling algorithm to solve the alloca tion of jobs in the shop floor was proposed. The problem was to find an optimal schedule so as to minimize the maximum completion time, the sum of distinct earlin ess and tardiness penalties from a given common due date d. Approach: The objective of the proposed algorithm was to red uc the early penalty and the late penalty and to increase the overall profit of he organization. The proposed method was discussed with different possible instances. Results: The test results showed that the algorithm was rob ust and simple and can be applied for any job size problem. Conclusion: The proposed algorithm gave encouraging result for the bench mark instances whe n t due date is less than half of the total processing time.


INTRODUCTION
In the Job Shop Scheduling Problem (JSP), a finite set of jobs has to be processed for a specified duration on a single machine around a common due date. The machine can process at most one job at a time and no preemption was allowed. Scheduling decision should be made according to a certain measure of performance, or scheduling criterion which is elaborated by Sule (2007). Industries job shop scheduling is a complex phenomenon to be solved with novel computational methods. Early finished jobs can lead to inventory loss and finishing jobs late lead to customer dissatisfaction. In general, a constructive optimization method tries to give the best possible solution. On the other hand, an iterative process improvises an assumed initial solution, which may take a long time, since we do not know which would give the optimal solution in a stipulated time frame. Hence we used the constructive method of optimization, barring the time taken to get the solution. The JSP is NP-hard discussed by Lenstra and Rinnooy Kan (1979) has continuously challenged to computational researchers. As the due date approaches processing speed of the jobs cannot be altered as discussed like in the Bender et al. (2007). Hong et al. (2007) developed LPT and PT algorithms for flexible flow-shop problem. Two-stage scheduling problem which minimizes total completion time was discussed by Yang and Lin (2009). A scheduling algorithm to solve sub models for complex scheduling problem was developed by Ghoul et al. (2007). Ismail and Loh (2009) developed ASO to minimize operational cost of an industry.
A set of n independent jobs has to be scheduled in a single machine, which can handle one job at a time. Assuming that there is no preemption of jobs and the machine is available from time t = 0 onwards. Let Jobs J i (i = 1,2,…n) having processing time p i , earliness penalty α i and tardiness penalty β i are non symmetric. Every job has the Common Due Date d.
If S being the optimal schedule then the objective is to minimize: The Common Due Date d and the ratio between early and late penalties are the decision variables Cheng and Gupta (1989) and Dileepan (1993). The problem is a restricted problem Feldmann and Biskup (2003) studied the restricted Earliness and Tardiness problem in which n i i 1 d p = < ∑ . Hoogeveen and van de Velde (1991) discussed the earliness and tardiness penalties are taken as symmetric α i = β i for all jobs. Bagchi et al. (1987) discussed non symmetric case with all α i are equal and β i are equal. They developed an algorithm which takes O(n log n) time to schedule the non symmetric case. Biskup and Feldmann (2001) created the bench marks for scheduling jobs on a single machine by considering the Common Due Date as decision variable. Figure 1-5 in the Materials and Methods are drawn by using the bench mark test instances. In this study we discussed about restricted problem with common due date and the early and late penalties being non symmetric and distinct. Hemamalini et al. (2010) proposed DMGS algorithm to solve job of scheduling in m machines.
The sequence of jobs J i (i = 1,2.. n) are partitioned to two subsets E(J) and L(J) according to the ratio i i α β .
To minimize F(S) two sets are created E(J) and L(J).
The set E(J) which have jobs with i i 1 α < β , to be completed before the due date in an optimal sequence and the set L(J) which has jobs with i i 1 α > β , to be completed after the due date d.
In section 2 we discussed about job scheduling according to the ratio i i α β and the sum of the processing times of E(J) which is less than the common due date d.
In section 3 we proved the properties of the optimal sequence according to the ratio i i α β and the sum of the processing times of E(J) which is greater than the common due date d. In section 4 ratio scheduling algorithm is developed based on the ratio i i α β which gives an optimal sequence with minimum penalty and the results are illustrated.

MATERIALS AND METHODS
< − ∆ then the starting time of the global optimal sequence is either and τ k is the processing time of a job in L(J) with min γ i .

Proof:
In the optimal schedule the m jobs of E(J) has to be completed before the common due date d and n-m jobs of L(J) has to be completed after the common due date d. Since ∆<d and each job of L(J) has lesser processing time than (d-∆), but it is not beneficial if a job J k of L(J) is completed before the due date d (since in the set i i L(J), 1 α > β ) which will increase the penalty.
Therefore the job whose completion time coincides with common due date d belongs to either E(J) or L(J) which depends on  > − ∆ then the starting time of optimal schedule is t = 0 and a job of L(J) with max γ i will be completed before the common due date d.
Proof: Here also ∆<d, but some of the jobs of the set L(J) has the processing time p i >(d-∆). Assuming that there is no job with p i = (d-∆) (which will be discussed in next case). In all the other cases except the case 1, it is advantageous only if the starting time is t = 0. Since there is a time gap d-∆, it is possible to processes a job of L(J) before the due date d, which depends on γ i . i.e., a job of L(J) with max γ i moves to E(J) and then scheduled according to its processing time and early penalty. Therefore some jobs of E(J) will be completed after the due date d (Fig. 2). Therefore in this case, the number of jobs completed before and after the due date d is less than or equal to m. And the remaining jobs of L(J) will be processed only after all the jobs of E(J) along with a job of L(J) with max γ i are completed.

Case 3:
If ∆<d and if at least one job of L(J) with p i = (d-∆) then the starting time of the schedule is t = 0 and the completion time of such a job in L(J) coincides with the common due date d.

Proof:
In the set E(J) if (∆<d), then L(J) be the set of jobs which has to be completed after the due date. Suppose L'(J) be the jobs in which p i = (d-∆) then choose a job in L'(J) with min γ i moves to E(J) and it is scheduled according to its processing time and early penalty. Therefore m jobs will be completed before the due date d and the job of L(J) with min γ i , is completed exactly at the due date d with nil penalty. And combine the remaining jobs L'(J) with L(J) which will be completed after the common due date d (Fig. 3). Now in this case the early penalty is: ∑ then it is obvious that the starting time of the sequence is t = 0 and the completion time of the last job of E(J) in the optimal schedule is after the common due date.
Case 1: If there exists some jobs of E(J) with the processing time p i such that ∆-p i <d, then the job with max γ i will be completed after the due date d in the optimal sequence.
Proof: Let E'(J) be the non empty subset of E(J) such that E'(J) = {J i /∆-p i <d}. Since ∆<d and E'(J) is non empty, exactly one of the job in E'(J) will be completed after the due date d with late penalty L i . The remaining jobs of E'(J) are processed before the common due date d with early penalty E i . Here also γ i is the decision variable that the job of E'(J) with max γ i will move to L(J) and scheduled according to its processing time and late penalty (Fig. 4). Therefore n-m+1 jobs will be completed after the due date d: Corollary 2: If the set E'(J) = {J i /∆p i <d} is empty in E(J), then there exist two or more jobs in E(J), which are completed after the common due date d with late penalty such that γ i of those jobs are greater than γ i of the jobs which are completed before the due date d in an optimal sequence (Fig. 5).
Corollary 3: It is obvious that the above case is true if L(J) is empty (i.e.,) in the given sequence all the jobs with (α i /β i )<1 for the restricted problem.

RESULTS
Results are demonstrated with the set of jobs to be sequenced before the due and after the due date. Jobs displayed before the red mark should be scheduled before due date d. Sequence was highlighted in the Xaxis. Processing time was placed in the Y-axis. Each cell was updated with the penalty. Figure 1-5 illustrated with all possible cases.

DISCUSSION
The existing algorithms like Tabu search and Genetic Algorithm, the time complexity is more i.e., in worst case time complexity is O(n!). But the proposed algorithm time complexity is 2 log (n). So the proposed algorithm outperforms in many instances of the test cases. The proposed algorithm is also suitable for unrestricted problem against single machine, the algorithm can also be extended to m machine scheduling problems.

CONCLUSION
Based on proposed algorithm we present some of the properties of the jobs whose completion time coincides with the common due date and the instances in which early jobs moves after the due date and the late jobs before the due date. The proposed algorithm gives encouraging result for the bench mark instances when the due date is less than half of the total processing time. The algorithm was implemented using Java platform. The authors are gladly willing to distribute the jar file by email.