OPTIMISATION OF JOB SHOP
SCHEDULING USING SHIFTING
BOTTLENECK TECHNIQUE
Katuru Phani Raja Kumar
Project Manager, IBG Manufacturing, Mahindra Satyam USA/India phani.katuru@yahoo.com
Dr. V.Kamala
Professor, Mechanical Engineering TKR College Engg & Technology Meerpet Hyderabad drvkamala@yahoo.com
Dr. ACS Kumar
Professor, Mechanical Engineering JNTUH (Rtd) Hyderabad acskumar@yahoo.com
Abstract
In manufacturing system the problem of scheduling machines is a difficult task to reach the due date of the productivity. The Job Shop Scheduling have been solved by different algorithms and methods based on the sequence operation constraints and processing times for small size problems. The JSSP with m machines and n jobs is represented to determine an optimal solution by using the shortest processing time technique and Gantt chart is drawn to visually represent the total makespan. The shifting bottleneck method has been used to reduce the total flow time of the JSSP and arrive at an optimal solution.
Keywords: Job shop scheduling, processing time, shifting bottleneck.
1. Introduction
Job-Shop Scheduling Problem is the hardest optimization problem. JSSP is referred amongst the class of NP-hard problems”[A Jain S. Meeram, 1998], there is a lot of research on the existing techniques. Large solution space is considered to solve JSSP. For n jobs and m machines the number of possible solutions may be given by to (n!)m.
After considering the small size problems, a certain measure of performance is found for the optimal solution. Consider a small problem having 8 machines and 5 jobs gives 4.3x1016alternate number of solutions. Even with a high performance computer, that can evaluate one alternative per second, to find out the optimal solution it would take more than 1000 years of continuous processing.
The Job Shop Scheduling Problem consists of N jobs, set M of machines and a setjof operations, where each one of the jobs is a subset of jin sequential form. Each operation has durationdj. In an assignment of jobs, a start time is defined for each operation such that:
1. A Machine cannot process more than one operation at a time. 2. The execution order of the operations in each job is respected.
The time at which the execution of all the operations of the JSSP is finished is known as the makespan. The main objective of JSSP is to minimize the makespan.
The improved turnaround that can be achieved by using the SPT rule implies lower work-in-process inventories and can provide a competitive sales advantage as well. Intuitively, the SPT rule performs well because the progress of jobs with shorter operation times is accelerated by giving them priority. Thus, shop congestion is reduced as jobs are around fast and machines are made available for delays as they get stuck in the system, When customer service is a dominant concern, then tardiness-based criteria, such as the proportion of jobs tardy or the mean tardiness, may be relevant. Surprisingly, the SPT rule does very well even for tardiness-based criteria. However, the selection of best rule critically depends on such factors as the level of the shop load, the procedure for setting due dates, and the tightness of the due dates.
2. Shifting bottleneck method
The shifting bottleneck was proposed by [Adams, 1998]. This is a powerful heuristic for solving the JSSP. In this method, a one-machine scheduling problem is solved for each machine not yet sequenced, and the outcome
is used to find a bottleneck machine, for. a machine having the longest makespan. Whenever a new machine is sequenced, the sequence of previously sequenced each machine is subjected to re-optimization. The SB consists of two subroutines: the first one repeatedly solves one-machine scheduling problems; the second one builds a path from start to end[D. Applegate, 1992].
2.1 Algorithm The shifting bottleneck procedure Step 1: Set the initial conditions
Set M0=Φ, Graph G is the graph including with the conjunctive arcs and no disjunctive arcs. Set Cmax(M0) represents the longest path in graph G.
Step 2: To analyze the machines for scheduling
For each machine in (M-M0): formulate a single machine problem with all operations subject to release dates and due dates. Release date is the longest path in G from the source to the node. Due date is the path in G from the node to the sink and subtracting Pij.
Minimize Lmaxin each machine. Step 3: Bottleneck selection
The machine having highest cost is designated as bottleneck. Indicate all the corresponding disjunctive arcs in graph. Indicate the machine which is bottleneck in M0. Step 4: Re-sequencing all the machines scheduled earlier
Determine the sequence which minimized the cost and insert the corresponding disjunctive arcs in graph G.
Step 5: Stopping condition
If all machines are scheduled (M0= M) then STOP, else go to step 2. Initial conditions
M0= Φ (set of machines with determined sequence) Set graph G with conjunctive arcs:
3. Results and Discussion
The Job Shop scheduling for m jobs, n machines considered. The sequence of operations and processing time of each job is shown in table 1. For the processing of 15 jobs 17 machines are utilized. Gantt chart is prepared for job shop scheduling problem using shortest processing time technique. The minimum Makespan for processing is obtained by decreasing the idle time of the machine.
Gantt charts is developed using the SPT shown in figure 2. It is prepared by determining the set of jobs waiting for each machine. When the processing of a job on one machine is completed, that job is added to the waiting list for the machine it needs to be processed by next. The procedure is repeated until all the jobs are scheduled.
According to the shifting bottleneck heuristic the Makespan is minimized in job shop.
• Jm - m machines and each job has its own predetermined route to follow.
• Processing of the job j in machine, i is operation (i, j) with duration pij. • J2| | Cmaxcan be reduced to 2 problems F2| | Cmax.
• Heuristic: SPT (1)–LPT (2) (Provides optimal solution)
Fig. 1. Sequence of operations for processing the jobs.
Fig. 2. Gantt Chart drawn for the JSSP using Shortest Processing Time.
J1 M1 M3 M2 M7 M4 M5 M8 M9 M6 M11 M13
48 38 48 35.1 23.3 96 7 113.3 29 11 2
J2 M2 M5 M8 M16 M10 M3 M7 M11 M12
336 3.15 9.3 4.15 72 12 9.3 12.3 13.5
J3 M1 M3 M2 M4 M5 M8 M7 M16 M10 M9 M11
216 34 120 4 96 28.3 11.34 3 24 5 4.3
J4 M1 M3 M4 M6 M7 M5 M8 M9
96 20.3 26.3 9.3 24.08 144 7.5 7
J5 M2 M8 M14 M5 M15 M16 M3 M10 M11 M17
216 44.3 3 128 96 6.3 1.3 24 4 1.3
J6 M1 M3 M4 M13 M5 M9 M10
48 7 2.3 2 72 15.3 48
J7 M1 M3 M4 M13 M5 M9 M10
96 12.3 2 1 48 5 48
J8 M1 M3 M4 M5 M9
96 8.3 2.3 48 5
J9 M1 M3 M4 M5 M10
48 8.3 3 72 48
J10 M1 M3 M4 M5 M10
48 9 3.3 72 48
J11 M1 M3 M6 M13
72 96 45.39 2
J12 M1 M3 M6 M4 M13
72 2 6.43 3 1
J13 M2 M5 M14 M15 M3 M16
96 96 216 168 1 4
J14 M1 M3 M6 M13
72 96 9.54 1
J15 M1 M4 M3
96 9 5
RM RT
SGS J9(8.3) J6(7) J10(9) J15(5) J5(1.3)
M J9(3) J7(2) J3(4)
VHT
2D B/AHT J4(9.3)
3D B/AHT J2(9.3)
CG EDM J4(7) N P ENG TAP GC
J12(72) J4(96) J3(216) J8(96) J7(96) J14(72) J9(48) J1(48) J6(48) J10(48) J15(96) J11(72)
J13(1) J1(38) J11(96)
J3(120) J13(96) J1(48) J5(216) J2(336)
J3(34) J8(8.3)) J7(12.3) J12(2) J4(20.3) J2(12)
J8(2.3) J12(3) J4(26.3)
J6(2.3) J1(23.3) J10(3.3) J15(9)
J14(9.54) J7(48) J3(96) J8(48) J13(96) J6(72) J1(96) J10(72) J5(128) J4(144)
J3(11.34)
J1(35.10) J4(28)
J1(29) J11(45.39) J12(6.43)
J3(28.3) J1(7) J5(44.3) J2(18) J4(7.5)
J3(5) J8(5)
J6(15.3) J1(113.3) J7(5)
J2(72) J7(48) J3(24) J8(48)
J9(48) J6(48) J10(48) J5(24)
J14(1)
J6(2) J12(1) J7(1)
J3(4.3) J2(13.5)
J1(11) J5(4) J2(12.3)
J13(216)
Fig. 3. Conjunctive graph based on SPT
Fig. 4. This is the caption for the figure. If the caption is less than one line then it needs to be manually centered.
Initially the release date and due date of each operation are determined. all sequence constraints activities in M-M0 are removed. The critical path is determined using CPM to minimize the start time and maximize ending time for each activity. The total flow time for the JSSP obtained was 1008 hrs.
Initially the M0is empty, due to which conjunctive arcs appear in the graph. The machine with highest Lmaxis determined by:(Lj= Cj–dj) and Lmax= Max(Lj).
The machine with the highest Lmax is added to machine M0. The second iteration is repeated with above procedure for the remaining machines. When M0 = M, stop the iterations. The Cmax for all the processing machines is determined. The total flow time for the JSSP problem using Shifting Bottleneck method was 846.52 hrs. The total flow time decreased by 161.08 hrs when compared to SPT method.
4. Conclusion
The Shortest Processing Time rule minimized shop congestion as measured by the mean flow time for the number of jobs in the system. The Gantt chart was a convenient way of visually representing a solution of the JSSP. The total flow time of the JSSP decreased by 16 percent by using shifting bottleneck method. The sophisticated method shifting bottleneck has been very successful in determining the optimal makespan for the JSSP.
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