Scheduling Job Shops with Batch Machines Using the Lagrangian Relaxation Technique
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- 5. CONCLUSIONS
- Appendix A. A List of Symbols
- Acknowledgments.
The feasible schedule shown in Table 2 is obtained in 47 CPU seconds. It has a cost of 44,142 with a relative duality gap 3.0%. According to the schedule, 16 batches of different groups are formed to process the 48 parts. Each batch is fully occupied with three parts of the same group. Since parts in each group are of the same size, optimal batch formation can be obtained by putting the parts into batches in the ascending order of part due dates (Uzsoy, 1995). The feasible schedule in Table 2 is therefore optimal. Table 1.2. Feasible Schedule (A1) for Case 1
Case 2. In this case, there are two standard machine types (M1 and M2) each with two machines, and one batch machine (M3). All machines are available across the time horizon. Ten parts with various due dates are to be scheduled, and are available from time 0. The tardiness weights for all parts are one, and there is no earliness penalty. Each part may consist of one or two standard operations and a batch operation. The batch operations belong to two groups. The sizes of operations in the two groups are three and two, respectively, and the batch volume is six. Data of the ten parts is shown in Table 2.1. The scheduling time horizon is 50, and the total number of multipliers is 250. The feasible schedule (S1) has a cost of 561 with a duality gap 1%, and is obtained within 17 seconds. When keep on running the algorithm to 60 seconds, the feasible cost remains unchanged, and a dual cost of 560.98 is obtained. Since dual cost serves as a lower bound to the optimal feasible cost, and feasible cost should be an integer value here, the schedule obtained is optimal. The resulting operation beginning and completion times are shown in Table 2.2, where M11 and M12 are the two identical machines in machine type M1, and M21 and M22 the two machines in M2. The Gannt chart of the feasible schedule is shown in Figure 1, where four batches are formed and fully occupied by the ten batch operations, and the batch machine is the bottleneck. Parts are processed in batches on batch machine, and are processed individually on standard machines, as shown by the shadowed boxes in the Gannt chart. To illustrate the advantage of explicit modeling batch machines, the problem is re-solved by treating the batch machine as a standard machine in the optimization model, but considering batch formation in heuristics. The resulting schedule (S2) for this simplified model has a cost 585, which is 4% higher than that of schedule S1. Table 2.1. Data for Case 2
Table 2.2. Schedule (S1) for Case 2
Figure 1: Gantt Chart of Feasible Schedule for Case 2 To show the effect of batch processing time on the feasible schedule, Case 2 is modified by doubling all batch operation processing times. By running the algorithm for 34 seconds, a feasible schedule (S3) with a cost of 4,062 and a duality gap of 6.11% is obtained. In the schedule, batches are formed exactly the same as in schedule S1, but have different processing sequence and beginning times. Since the batch machine is the bottleneck, this data change leads to significant increase of the feasible cost. Case 3. This case is created based on the data from a practical job shop. There are four standard machine types with a total of seven machines, and four batch machine types with seven machines. Some machines may not be available during certain periods on the time horizon. Eighty two parts with a total of 752 operations are to be scheduled. Parts have various due dates with two level of tardiness weights (1.0 and 0.5), and there is no earliness penalty. Operations on batch machine types belong to 16 groups with sizes of two or three, and the volume of a batch is six. Operation processing times range from 1 to 24, and part due dates are scattered from 0 to 230. The scheduling time horizon is 2046. The resulting feasible schedule has a cost of 39,944 with a duality gap 33%, and is obtained within 600 CPU seconds. Results at some selected iterations are summarized in Table 3 to show the improvement on both feasible schedule and lower bound as the number of iterations increases. The results show that the method developed keeps on improving schedule quality by reducing the feasible cost and improving the lower bound by maximizing the dual function. A good schedule is obtained within 100 iterations or about 2.5 minutes. Table 3. Testing Results for Case 3
A few practically used performance metrics are provided to measure the schedule quality. The metrics are defined below. Makespan: the duration of time for processing all the parts. Maximum work-in-process inventory: the maximum number of parts in the cell. Average work-in-process inventory: the average number of parts in the shop over the time horizon. Average lead time: the average elapse time between part beginning and completion times. Total processing time/Total lead time: the sum of all operation processing times / the sum of all part lead times
By reducing the number of batch machines to one machine per type, the schedule obtained has a cost of 41,511 with a duality gap of 39% at 600 seconds. It shows that after reducing the number of batch machines, the feasible cost significantly increased. There are more competitions on the batch machines, and the method will take more computation time to find a good schedule. 5. CONCLUSIONS In this paper, the scheduling of job shops with batch machines is considered in an integrated fashion to decide both batch formation and sequencing. A novel "separable" integer programming formulation is developed with manageable numbers of variables and constraints. A Lagrangian relaxation based method is then applied to generate high quality solutions with quantifiable quality. Through the iterative problem resolution, the method keeps on improving the schedule quality as indicated by the feasible costs and performance metrics. Numerical testing shows that the integrated consideration of batch formation and sequencing results in high quality schedules, and the algorithm is computationally efficient to solve practical problems. Appendix A. A List of Symbols ijhk: 0-1 operation-level variable which is one if operation (i, j) is performed on machine type h at time k, and zero otherwise.  : 0-1 operation level variable which is one if operation (i, j) starts on machine type h at time k, and zeor otherwise.
hgnk: 0-1 batch-level variable which is one if batch (h, g, n) is being processed at time k, and zero otherwise.  : 0-1 batch level variable which is one if batch (h, g, n) starts at time k, and zero otherwise.
bij: beginning time of operation (i, j) bhgn: beginning time of batch (h, g, n) cij: completion time of operation (i, j) chgn: completion time of batch (h, g, n) Di: due date of part i g: group index variable h: machine type variable, hH H: set of all machine types HB: set of all batch machine types  : set of all standard machine types
Hij: set of machine types capable of performing operation (i, j) J: objective function to be minimized Ji: number of operations for part i k: discretized time index K: time horizon Mh: number of identical machines in machine type h Mhk: capacity of machine type h at time k Nhg: number of batches in group g of machine type h Pijh: processing time of operation (i, j) on machine type hH Phg: processing time of a batch within group g of machine type h Ti: tardiness of part i, defined as Ti = max[0, ci,Ji-1-Di] vij: size of part i when performing operation j Vhg: capacity of a batch in group g on machine type h Wi: part tardiness weight Acknowledgments. This work was supported in part by the National Science Foundation under Grant DMI-9500037, and the Advanced Technology Center for Precision Manufacturing, the University of Connecticut. The authors would like to thank Dr. Debra J. Hoitomt of United Airlines for her earlier work on this problem. REFERENCES Ahmadi, J. H., Ahmadi, R. H., Dasu, S., Tang, C. S., 1992, "Batching and Scheduling Jobs on Batch and Discrete Processors," Operations Research, Vol. 40, pp. 750-763. Blackstone, J.H., D.T. Phillips and G.L. Hogg, 1982, "A State-of-the-art Survey of Dispatching Rules for Manufacturing Job Shop Operations," International Journal of Productions Research, Vol. 20, pp. 27-45. Chen, H., Chu, C., Proth, J. M., 1995, "A More Efficient Lagrangian Relaxation Approach to Job Shop Scheduling Problems," Proceeding of IEEE Conference on Robotics and Automation, pp. 496-501. Dessouky, Y. M., Roberts C. A., Dessouky M. M. and Wilson G., 1996, "Scheduling Multipurpose Batch Plants with Junction Constraints," International Journal of Productions Research, Vol. 2, pp. 525-541. Hoitomt, D. J., Luh, P. B., Pattipati, K. R., 1993, "A Practical Approach to Job-Shop Scheduling Problems," IEEE Transactions on Robotics and Automation, Vol. 9, pp. 1-13. Hiriart-Urruty, J. B. and Lemarechal, C., 1993, Convex Analysis and Minimization Algorithms, I and II (Springer-Verlag, Berlin). Kaskavelis, C.A. and Caramanis M.C., 1995, "Efficient Lagrangian Relaxation Algorithms for Real-life-size Job-shop Scheduling Problems," Working Paper, Boston University, Department of Manufacturing Engineering. Lee, C.Y., S.D. Liman and A. Wirakusumah, 1993, "Product Batching and Batch Sequencing for NC Punch Presses," International Journal of Productions Research, Vol. 31, pp. 1143-1156. Lemarechal, C., 1978, "Bundle Methods in Nonsmooth Optimization," Proceedings of a IIASA Workshop, ed. C. Lemarechal and R. Mifflin Pergamon Press, Great Britain, 1978), pp. 77-82. Liao, D. Y., Chang, S. C., Yen, S. R., Chien, C. C., 1993, "Daily Scheduling for R&D Semiconductor Fabrication," Proceeding of IEEE Conference on Robotics and Automation, pp. 77-82. Luh, P.B., D. Chen and L.S. Thakur, 1997, "Modeling Uncertainty in Job Shop Scheduling," Proceedings of the First International Conference on Operations and Quantitative Management, Jaipur, India, pp. 490-497. Luh, P.B., L. Gou, T. Odahara, M. Tsuji, K. Yoneda, T. Hasegawa and Y. Kyoya, 1995, "Job Shop Scheduling with Group-dependent Setups, Finite Buffers, and Long Time Horizon," Proceedings of the 34th Conference on Decision and Control, New Orleans, LA, pp. 4184-4189. Tomastik, R. N. and Luh, P. B., 1993, "The Facet Ascending Algorithm for Integer Programming Problems," Proceedings of 32nd IEEE Conference on Decision and Control (San Antonio, Texas, USA), pp. 2880-2285. Tomastik, R. N. and Luh, P. B., 1996, "A Reduced-Complexity Bundle Method for Maximizing Concave Nonsmooth Functions," Proceedings of the 35th IEEE Conference on Decision and Control (Kobe, Japan). Uzsoy, R., 1995, "Scheduling Batch Processing Machines with Incompatible Job Families," International Journal of Productions Research, Vol. 33, pp. 2685-2708. Wang, J., Luh, P.B., 1994, "Optimization Based Scheduling of a Batch Processing Facility," Proceedings of the Conference on Computer Integrated Manufacturing in the Process Industries, (New Brunswick, NJ, USA), pp. 3-20. Wang, J.H., Luh, P.B., X. Zhao and J.L. Wang, 1997, "An Optimization-based Algorithm for Job Shop Scheduling," Special Issue of SADHANA on Competitive Manufacturing Systems, Indian Academy of Sciences, Bangalore, India. Webster S., and Baker K. R., 1995, "Scheduling Groups of Jobs on a Single Machine," Operations Research, Vol. 43, pp. 692-703. Zijm, W. H. M., 1995, "The Integration of Process Planning and Shop Floor Scheduling in Small Batch Part Manufacturing," Annals of the CIRP, Vol. 44, pp. 429-432. 20. Zhao, X., P.B. Luh and J. Wang, 1997, "The Surrogate Gradient Algorithm for Lagrangian Relaxation Method," Submitted to The 36th IEEE Conf. on Deci & Cont., San Diego, CA, USA. 1 The component of subgradient corresponding to a machine capacity constraint of standard machine type h at time k is  .
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