臺灣博碩士論文加值系統

本研究探討允許機器閒置時間(Machine Idle Time)之不相關平行機臺排程問題(Unrelated Parallel Machine Problem)，目標為總早到及延遲時間(The Total Earliness And Tardiness, ET)最小化。為達此目的「深入瞭解機器閒置時間對於排程績效所帶來效果」和「提出的改良的數學模型是兼具求解品質及效率」，本研究藉由下列方式逐步進行：1) 符合問題前提下，「允許」和「禁止」機臺閒置時間模型被設計出，並透過「混合整數規劃法(Mixed Integer Linear Programming, MIP)」建立，比較獲得之結果，以觀察機器閒置時間在整個排程上所扮演的角色；2)提出新改良之「混合整數規劃法」提升效率，並依此新方法建構原設計之模型，以比較先前研究確保品質及效率；此外「簡易派工法則(Dispatching Rules)」亦配合此新方法同於大型例題下使用；3) 為使效率能再度提升，高性能「時間分割整數規劃法(Time-Indexed Integer Linear Programming, TI)」進而被提出，此方法主要運用於機器允許閒置時間模型上，同時發展嚴格「時程限制(Scheduling Horizon)」且使用，並與新改良混合整數規劃法一同比較。
為此，本研究依各目的設計出數種實驗和例題，並應用專業數學最佳化軟體測試，收集所需之數據進而分析，以驗證所提出方法之正確性。最後，綜合所有實驗結果顯示: 1) 閒置時間在所需的工件之間能妥善的安排，能明顯地有效降低目標函數。然而，以增加大量的機臺數、各工件交期為鬆散且之間差異大及不可避免的閒置時間的做法，對於改善目標也是有意義的；2) 「新改良混合整數規劃法」所建構之模型是具成效和競爭力。它可以減少限制式數量及能夠修正過去研究中的潛在問題(Zhu and Heady, 2000; Omar and Teo, 2006); 3) 各方法所建構之數學模型皆能在合理時間內，對全部例題求解成功並給予最佳解，尤其，「時間分割整數規劃法」更可在短時間達到相同效果; 4) 「簡易派工法則」可使「新改良混合整數規劃法」模型能快速地得到高品質的近似最佳解; 5) 新發展嚴格「時程限制(Scheduling Horizon)」能明顯地強化「時間分割整數規劃法」模型的求解效率。

This study deals with the scheduling problem of minimizing the total earliness and tardiness (ET) on unrelated parallel machines with machine idle time. Its purpose is to gain insight into the effect of machine idle time for job scheduling performances and to test the improved mathematical formulations are very competitive for the quality and efficiency of solutions. The purposes were completed gradually as follows: 1) Two models, where the use of machine idle time is allowed and prohibited, designed to gain insight into the effect of machine idle time for job scheduling performances. The models constructed by mixed integer linear programming (MIP) and compared with each other. 2) The new developed MIP formulations were proposed to obtain the optimal solution for improving solving efficiency. They ensure the quality and efficiency of solution by comparing with the formulations in previous studies. In addition, job dispatching rules also can be applied to the model with allowance machine idle time in large-sized instances. 3) Enhanced Time-Indexed Integer Linear Programming (TI) formulations were proposed for rapidly solving efficiency. At the same time, developed scheduling horizons rigorously were adopted in TI formulations, which compare with the MIP formulations.
Several experiments, which use their respective test instances, were designed to complete the purposes. Results of computational experiments show that 1) The objective function ET can be significantly reduced by allowable machine idle time between jobs. However, the improvement of ET is especially beneficial if adding the number of machines, the due date is loose and broad with range and machine idle time is inevitable. 2) The improved MIP formulations were effective and competitive, where they can reduce the constraint sizes and were capable of reforming the potential problem in the references of Zhu and Heady (2000) and Omar and Teo (2006). 3) The proposed mathematical formulations gives optimal solutions for all test instances in a reasonable time. TI formulations obtain the same solution within a short time; 4) The dispatching rules yield comparably efficient and quality solutions of MIP formulations; 5) The developed scheduling horizons were able to greatly increase the efficiency of TI formulations.

Title Page i
Abstract in Traditional Chinese ii
Abstract in English iv
Acknowledgement vi
Table of Contents vii
List of Tables ix
List of Figures xi
Chapter 1 Introduction 1
1.1 Background and Motivation 1
1.2 Research Objectives 3
1.3 Organization of this Thesis 4
Chapter 2 Literature Review 5
2.1 Scheduling Problem with Parallel Machines 5
2.1.1 Jobs Scheduling on Unrelated Parallel Machines 5
2.1.2 Objective of Total Earliness and Tardiness 7
2.1.3 Machine Idle Times 8
2.2 Approaches for Scheduling Problem 9
2.2.1 Mixed Integer Linear Programming 10
2.2.2 Time-Indexed Integer Linear Programming 11
2.2.3 Dispatching Rules 12
2.3 Related Literature 13
Chapter 3 Definition and Modeling of the Problem 15
3.1 Description for Problem and the Aim of Modeling 15
3.2 Mixed Integer Linear Programming Formulation 17
3.2.1 MIP Formulation with Triple-Index 17
3.2.2 MIP Formulation with Double-Index 19
3.3 Time-Indexed Formulation 23
3.3.1 TI Formulation with Triple-Index 23
3.3.2 TI Formulation with Double-Index 25
3.4 Collections 27
Chapter 4 Experimental Results and Discussion 29
4.1 Datasets 29
4.2 Experiment: Developed MIP Formulations with Triple-Index and Dispatching Rules 31
4.2.1 Observing the Effect of Machine Idle Time 31
4.2.2 Testing Model A and Dispatching Rules for the Quality of Solution 33
4.2.3 Examining Dispatching Rules for the Solving Efficiency 34
4.3 Experiment: Improved MIP Formulations with Double-Index and Dispatching Rules 35
4.3.1 Verifying the Improved Formulation with the Formulations of Past Studies 36
4.3.2 Investigating Performance Is Affected By Machine Idle Time 37
4.3.3 Examining the Effectiveness of Model A and Dispatching Rules 39
4.4 Experiment: Enhanced TI Formulations with Triple-Index and Double-Index 41
4.4.1 Inspecting the Solution Quality of the TI Formulations 41
4.4.2 Examining the Solution Efficiency of the TI Models with Developed Scheduling Horizons 44
Chapter 5 Conclusion and Future Studies 46
5.1 Conclusion 46
5.2 Future Studies 47
Appendix A: The Results of the Paired T-Tests 49
A.1 Paired T-Test for the Average Lengths of H1 and H3 in Set V 49
A.2 Paired T-Test for the CPUs of T3I with either H1 or H3 in Set V 49
A.3 Paired T-Test for the CPUs of T2I with either H1 or H3 in Set V 50
A.4 Paired T-Test for the CPUs of Two Formulations with H1 in Set V 50
A.5 Paired T-Test for the CPUs of Two Formulations with H3 in Set V 51
A.6 Paired T-Test for the CPUs of Two Formulations with H3 in Set VI 52
Bibliography 53

Anderson, B. E., Blocher, J. D., Bretthauer, K. M., &; Venkataramanan, M. A. (2013). An efficient network-based formulation for sequence dependent setup scheduling on parallel identical machines. Mathematical and Computer Modelling, 57(3), 483-493.
Arnaout, J. P., Musa, R., &; Rabadi, G. (2014). A two-stage Ant Colony optimization algorithm to minimize the makespan on unrelated parallel machines—part II: enhancements and experimentations. Journal of Intelligent Manufacturing, 25(1), 43-53.
Arnaout, J. P., Rabadi, G., &; Musa, R. (2010). A two-stage ant colony optimization algorithm to minimize the makespan on unrelated parallel machines with sequence-dependent setup times. Journal of Intelligent Manufacturing, 21(6), 693-701.
Avella, P., Boccia, M., &; D’Auria, B. (2005). Near-optimal solutions of large-scale single-machine scheduling problems. INFORMS Journal on Computing, 17(2), 183-191.
Azizoglu, M. E. R. A. L., &; Kirca, O. M. E. R. (1999). Scheduling jobs on unrelated parallel machines to minimize regular total cost functions. IIE Transactions, 31(2), 153-159.
Baker, K. R. (1974). Introduction to sequencing and scheduling. John Wiley &; Sons.
Baker, K. R. (2014). Minimizing earliness and tardiness costs in stochastic scheduling. European Journal of Operational Research, 236(2), 445-452.
Baker, K. R., &; Scudder, G. D. (1990). Sequencing with earliness and tardiness penalties: a review. Operations Research, 38(1), 22-36.
Balakrishnan, N., Kanet, J. J., &; Sridharan, V. (1999). Early/tardy scheduling with sequence dependent setups on uniform parallel machines. Computers &; Operations Research, 26(2), 127-141.
Berghman, L., &; Spieksma, F. (2012). Computational experiments with a cutting-plane algorithm for a time-indexed formulation. FEB Research Report KBI 1211, 1-18.
Berghman, L., Spieksma, F., &; T'Kindt, V. (2014). Solving a time-indexed formulation by preprocessing and cutting planes. FEB Research Report KBI_1407, 1-26.
Bigras, L. P., Gamache, M., &; Savard, G. (2008). Time-indexed formulations and the total weighted tardiness problem. INFORMS Journal on Computing, 20(1), 133-142.
Bilyk, A., &; M#westeur055#nch, L. (2012). A variable neighborhood search approach for planning and scheduling of jobs on unrelated parallel machines. Journal of Intelligent Manufacturing, 23(5), 1621-1635.
Blackstone, J. H., Phillips, D. T., &; Hogg, G. L. (1982). A state-of-the-art survey of dispatching rules for manufacturing job shop operations. The International Journal of Production Research, 20(1), 27-45.
Caniyilmaz, E., Benli, B., &; Ilkay, M. S. (2015). An artificial bee colony algorithm approach for unrelated parallel machine scheduling with processing set restrictions, job sequence-dependent setup times, and due date. The International Journal of Advanced Manufacturing Technology, 77(9-12), 2105-2115.
Cappadonna, F. A., Costa, A., &; Fichera, S. (2013). Makespan Minimization of Unrelated Parallel Machines with Limited Human Resources. Procedia CIRP, 12, 450-455.
Chang, P. C., Hsieh, J.C. &; Hsiao, C. H. (2002). Application of genetic algorithm to the unrelated parallel machine scheduling problem. Journal of the Chinese Institute of Industrial Engineers, 19(2), pp. 79-95.
Chen, J. Y., &; Lin, S. F. (2002). Minimizing weighted earliness and tardiness penalties in single‐machine scheduling with idle time permitted. Naval Research Logistics, 49(8), 760-780.
Costa, A., Cappadonna, F. A., &; Fichera, S. (2013). A hybrid genetic algorithm for job sequencing and worker allocation in parallel unrelated machines with sequence-dependent setup times. The International Journal of Advanced Manufacturing Technology, 69(9-12), 2799-2817.
Crama, Y., &; Spieksma, F. C. (1996). Scheduling jobs of equal length: Complexity, facets and computational results. Mathematical Programming, 72(3), 207-227.
Croce, F. D., &; Trubian, M. (2002). Optimal idle time insertion in early-tardy parallel machines scheduling with precedence constraints. Production Planning and Control, 13(2), 133-142.
Cruz-Ch#westeur034#vez, M. A., Ju#westeur034#rez-P#westeur042#rez, F., #westeur002#vila-Melgar, E. Y., &; Mart#westeur046#nez-Oropeza, A. (2009, September). Simulated annealing algorithm for the weighted unrelated parallel machines problem. In Electronics, Robotics and Automotive Mechanics Conference, 2009. CERMA'09. (pp. 94-99). IEEE.
Davis, E., &; Jaffe, J. M. (1981). Algorithms for scheduling tasks on unrelated processors. Journal of the Association for Computing Machinery, 28(4), 721-736.
de CM Nogueira, J. P., Arroyo, J. E. C., Villadiego, H. M. M., &; Gon#westeur040#alves, L. B. (2014). Hybrid GRASP heuristics to solve an unrelated parallel machine scheduling problem with earliness and tardiness penalties. Electronic Notes in Theoretical Computer Science, 302, 53-72.
de Paula, M. R., Mateus, G. R., &; Ravetti, M. G. (2010). A non-delayed relax-and-cut algorithm for scheduling problems with parallel machines, due dates and sequence-dependent setup times. Computers &; Operations Research, 37(5), 938-949.
Dessouky, M. I., Dessouky, Y. M., &; Dessouky, M. M. (1987). A case study in parallel unrelated machine scheduling: A heuristic approach. Journal of Manufacturing Systems, 6(1), 23-36.
Dyer, M. E., &; Wolsey, L. A. (1990). Formulating the single machine sequencing problem with release dates as a mixed integer program. Discrete Applied Mathematics, 26(2), 255-270.
Erschler, J. G. C. F., Fontan, G., Merce, C., &; Roubellat, F. (1983). A new dominance concept in scheduling n jobs on a single machine with ready times and due dates. Operations Research, 31(1), 114-127.
Fleszar, K., Charalambous, C., &; Hindi, K. S. (2012). A variable neighborhood descent heuristic for the problem of makespan minimisation on unrelated parallel machines with setup times. Journal of Intelligent Manufacturing, 23(5), 1949-1958.
Ghirardi, M., &; Potts, C. N. (2005). Makespan minimization for scheduling unrelated parallel machines: A recovering beam search approach. European Journal of Operational Research, 165(2), 457-467.
Giglio, D., Minciardi, R., Sacone, S., &; Siri, S. (2009). Optimal control of production processes with variable execution times. Discrete Event Dynamic Systems, 19(3), 423-448.
Glass, C. A., Potts, C. N., &; Shade, P. (1994). Unrelated parallel machine scheduling using local search. Mathematical and Computer Modelling, 20(2), 41-52.
Go, H., Kim, J. S., &; Lee, D. H. (2013). Operation and preventive maintenance scheduling for containerships: Mathematical model and solution algorithm. European Journal of Operational Research, 229(3), 626-636.
Graham, R. L., Lawler, E. L., Lenstra, J. K., &; Kan, A. R. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287-326.
Hariri, A. M. A., &; Potts, C. N. (1991). Heuristics for scheduling unrelated parallel machines. Computers &; Operations Research, 18(3), 323-331.
Kanet, J. J., &; Sridharan, V. (2000). Scheduling with inserted idle time: problem taxonomy and literature review. Operations Research, 48(1), 99-110.
Kayvanfar, V., &; Teymourian, E. (2014). Hybrid intelligent water drops algorithm to unrelated parallel machines scheduling problem: a just-in-time approach. International Journal of Production Research, 52(19), 5857-5879.
Kayvanfar, V., Komaki, G. M., Aalaei, A., &; Zandieh, M. (2014). Minimizing total tardiness and earliness on unrelated parallel machines with controllable processing times. Computers &; Operations Research, 41, 31-43.
Kedad-Sidhoum, S., Solis, Y. R., &; Sourd, F. (2008). Lower bounds for the earliness–tardiness scheduling problem on parallel machines with distinct due dates. European Journal of Operational Research, 189(3), 1305-1316.
Kerkhove, L. P., &; Vanhoucke, M. (2014). Scheduling of unrelated parallel machines with limited server availability on multiple production locations: A case study in knitted fabrics. International Journal of Production Research, 52(9), 2630-2653.
Kim, D. W., Kim, K. H., Jang, W., &; Chen, F. F. (2002). Unrelated parallel machine scheduling with setup times using simulated annealing. Robotics and Computer-Integrated Manufacturing, 18(3), 223-231.
Kim, D. W., Na, D. G., &; Chen, F. F. (2003). Unrelated parallel machine scheduling with setup times and a total weighted tardiness objective. Robotics and Computer-Integrated Manufacturing, 19(1), 173-181.
Kim, D. W., Na, D. G., Jang, W., &; Chen, F. F. (2006). Simulated annealing and genetic algorithm for unrelated parallel machine scheduling considering set-up times. International Journal of Computer Applications in Technology, 26(1), 28-36.
Kim, J. G., Kim, J. S., &; Lee, D. H. (2012). Fast and meta-heuristics for common due-date assignment and scheduling on parallel machines. International Journal of Production Research, 50(20), 6040-6057.
Lancia, G. (2000). Scheduling jobs with release dates and tails on two unrelated parallel machines to minimize the makespan. European Journal of Operational Research, 120(2), 277-288.
Larson, R. E., Dessouky, M. I., &; Devor, R. E. (1985). A forward-backward procedure for the single machine problem to minimize maximum lateness. IIE Transactions, 17(3), 252-260.
Lee, Y. H., Bhaskaran, K., &; Pinedo, M. (1997). A heuristic to minimize the total weighted tardiness with sequence-dependent setups. IIE Transactions, 29(1), 45-52.
Lenstra, J. K., Shmoys, D. B., &; Tardos, #westeur010#. (1990). Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, 46(1-3), 259-271.
Li, X., Chen, Y., &; Sun, Y. (2010). Minimizing job completion time variance for service stability on identical parallel machines. Computers and Industrial Engineering, 58(4), 729-738.
Liaw, C. F., Lin, Y. K., Cheng, C. Y., &; Chen, M. (2003). Scheduling unrelated parallel machines to minimize total weighted tardiness. Computers &; Operations Research, 30(12), 1777-1789.
Lin, B. M. T., &; Jeng, A. A. K. (2004). Parallel-machine batch scheduling to minimize the maximum lateness and the number of tardy jobs. International Journal of Production Economics, 91(2), 121-134.
Lin, Y. K. and Lin, C. W. (2013). Dispatching rules for unrelated parallel machine scheduling with release dates. The International Journal of Advanced Manufacturing Technology, 67(1-4), 269-279.
Logendran, R., McDonell, B., &; Smucker, B. (2007). Scheduling unrelated parallel machines with sequence-dependent setups. Computers &; Operations Research, 34(11), 3420-3438.
Mathirajan, M., Bhargav, V., &; Ramachandran, V. (2010). Minimizing total weighted tardiness on a batch-processing machine with non-agreeable release times and due dates. The International Journal of Advanced Manufacturing Technology, 48(9-12), 1133-1148.
M'Hallah, R. &; Al-Khamis, T. (2012). Minimising total weighted earliness and tardiness on parallel machines using a hybrid heuristic, International Journal of Production Research, 50(10), 2639-2664.
Mokotoff, E. (2001). Parallel machine scheduling problems: A survey, Asia-Pacific Journal of Operational Research, 18(2), 193-242.
Monch, L. (2008, August). Heuristics to minimize total weighted tardiness of jobs on unrelated parallel machines. In Automation Science and Engineering, 2008. CASE 2008. IEEE International Conference on (pp. 572-577). IEEE.
M#westeur055#nch, L., &; Unbehaun, R. (2007). Decomposition heuristics for minimizing earliness–tardiness on parallel burn-in ovens with a common due date. Computers &; Operations Research, 34(11), 3380-3396.
Morton, T. E., &; Pentico, D. W. (1993). Heuristic Scheduling Systems. John Wiley &; Sons, Inc., New York.
Omar, M. K., &; Teo, S. C. (2006). Minimizing the sum of earliness/tardiness in identical parallel machines schedule with incompatible job families: An improved MIP approach. Applied Mathematics and Computation, 181(2), 1008-1017.
Pan, Y., &; Shi, L. (2007). On the equivalence of the max-min transportation lower bound and the time-indexed lower bound for single-machine scheduling problems. Mathematical Programming, 110(3), 543-559.
Park, Y. J. (2008). Minimization of total weighted earliness and tardiness on a single burn‐in oven using a genetic algorithm. Journal of the Society of Korea Industrial and Systems Engineering. 31(4), 21-28.
Park, Y. J., &; Sun, H. (2011). Application a heuristic approach to minimization of total weighted earliness and tardiness on a single burn-in oven. ICIC Express Letters, 5(5), 1691-1695.
Pearn, W. L., Chung, S. H., Chen, A. Y., &; Yang, M. H. (2004). A case study on the multistage IC final testing scheduling problem with reentry. International Journal of Production Economics, 88(3), 257-267.
Piersma, N., &; van Dijk, W. (1996). A local search heuristic for unrelated parallel machine scheduling with efficient neighborhood search. Mathematical and Computer Modelling, 24(9), 11-19.
Pinedo, M. L. (1995). Scheduling: theory, algorithms, and systems. Englewood Cliffs: Prentice-Hall.
Plateau, M. C., &; Rios-Solis, Y. A. (2010). Optimal solutions for unrelated parallel machines scheduling problems using convex quadratic reformulations. European Journal of Operational Research, 201(3), 729-736.
Polyakovskiy, S., &; M'Hallah, R. (2014). A multi-agent system for the weighted earliness tardiness parallel machine problem. Computers &; Operations Research, 44, 115-136.
Potts, C. N. (1985). Analysis of a linear programming heuristic for scheduling unrelated parallel machines. Discrete Applied Mathematics, 10(2), 155-164.
Rabadi, G., Moraga, R. J., &; Al-Salem, A. (2006). Heuristics for the unrelated parallel machine scheduling problem with setup times. Journal of Intelligent Manufacturing, 17(1), 85-97.
Rios-Solis, Y. A., &; Sourd, F. (2008). Exponential neighborhood search for a parallel machine scheduling problem. Computers &; Operations Research, 35(5), 1697-1712.
Rodriguez, F. J., Lozano, M., Blum, C., &; Garc#westeur046#a-Mart#westeur046#nez, C. (2013). An iterated greedy algorithm for the large-scale unrelated parallel machines scheduling problem. Computers &; Operations Research, 40(7), 1829-1841.
Sels, V., Coelho, J., Dias, A. M., &; Vanhoucke, M. (2015). Hybrid tabu search and a truncated branch-and-bound for the unrelated parallel machine scheduling problem. Computers &; Operations Research, 53, 107-117.
Şen, H., &; B#westeur061#lb#westeur061#l, K. (2015). A strong preemptive relaxation for weighted tardiness and earliness/tardiness problems on unrelated parallel machines. INFORMS Journal on Computing, 27(1), 135-150.
Sousa, J. P., &; Wolsey, L. A. (1992). A time indexed formulation of non-preemptive single machine scheduling problems. Mathematical Programming, 54(1-3), 353-367.
Sridharan, S. V., &; Zhou, Z. (1996). Dynamic non-preemptive single machine scheduling. Computers &; Operations Research, 23(12), 1183-1190.
Srivastava, B. (1998). An effective heuristic for minimising makespan on unrelated parallel machines. Journal of the Operational Research Society, 49(8), 886-894.
Supithak, W., &; Plongon, K. (2011). Memetic algorithm for non-identical parallel machines scheduling problem with earliness and tardiness penalties. International Journal of Manufacturing Technology and Management, 22(1), 26-38.
Suresh, V., &; Chaudhuri, D. (1994). Minimizing maximum tardiness for unrelated parallel machines. International Journal of Production Economics, 34(2), 223-229.
Tanaka, S., &; Araki, M. (2008). A branch-and-bound algorithm with Lagrangian relaxation to minimize total tardiness on identical parallel machines. International Journal of Production Economics, 113(1), 446-458.
Tanaka, S., &; Fujikuma, S. (2012). A dynamic-programming-based exact algorithm for general single-machine scheduling with machine idle time. Journal of Scheduling, 15(3), 347-361.
Toksarı, M. D., &; G#westeur061#ner, E. (2010). Parallel machine scheduling problem to minimize the earliness/tardiness costs with learning effect and deteriorating jobs. Journal of Intelligent Manufacturing, 21(6), 843-851.
Tran, T. T., &; Beck, J. C. (2012) Logic-based Benders decomposition for alternative resource scheduling with sequence dependent setups. Frontiers in Artificial Intelligence and Applications, 242, 774-779.
Tsai, C. Y., &; Wang, Y. C. (2015a). Efficient mixed integer programming formulations and dispatching rules for parallel machine scheduling with allowing machine idle times. International Journal of Industrial and Systems Engineering.
Tsai, C. Y., &; Wang, Y. C. (2015b). An improved MIP-based approach to minimize total earliness and tardiness in unrelated parallel machines scheduling with machine idle time. International Journal of Manufacturing Technology and Management.
Unlu, Y., &; Mason, S. J. (2010). Evaluation of mixed integer programming formulations for non-preemptive parallel machine scheduling problems. Computers &; Industrial Engineering, 58(4), 785-800.
Vallada, E., &; Ruiz, R. (2012). Scheduling unrelated parallel machines with sequence dependent setup times and weighted earliness–tardiness minimization. In: R#westeur046#os-Mercado R.Z.Z., R#westeur046#os-Sol#westeur046#s Y.A.A. (Eds.). Just-in-Time Systems, 60, 67–90. New York: Springer.
van den Akker, J. M., Hoogeveen, J. A., &; van de Velde, S. L. (1999). Parallel machine scheduling by column generation. Operations Research, 47(6), 862-872.
van den Akker, J. M., Hurkens, C. A., &; Savelsbergh, M. W. (2000). Time-indexed formulations for machine scheduling problems: Column generation. INFORMS Journal on Computing, 12(2), 111-124.
Waterer, H., Johnson, E. L., Nobili, P., &; Savelsbergh, M. W. (2002). The relation of time indexed formulations of single machine scheduling problems to the node packing problem. Mathematical Programming, 93(3), 477-494.
Xie, Z., Wang, P., Gui, Z., &; Yang, J. (2012). Integrated scheduling algorithm based on dynamic essential short path. Advances in Intelligent and Soft Computing, 169(2), 709-715.
Yaghubian, A.R. Hodgson, T.J. Joines, J.A. Culbreth, C.T. and Huang, J.C. (1999). Dry kiln scheduling in furniture production. IIE Transactions, 31(8), 733–738.
Yavuz, M. (2013). Iterated beam search for the combined car sequencing and level scheduling problem. International Journal of Production Research, 51(12), 3698-3718.
Yilmaz Eroglu, D., Ozmutlu, H. C., &; Ozmutlu, S. (2014). Genetic algorithm with local search for the unrelated parallel machine scheduling problem with sequence-dependent set-up times. International Journal of Production Research, 52(19), 5841-5856.
Yin, N., Kang, L., Sun, T. C., Yue, C., &; Wang, X. R. (2014). Unrelated parallel machines scheduling with deteriorating jobs and resource dependent processing times. Applied Mathematical Modelling, 38(19), 4747-4755.
Ying, K. C., Lee, Z. J., &; Lin, S. W. (2012). Makespan minimization for scheduling unrelated parallel machines with setup times. Journal of Intelligent Manufacturing, 23(5), 1795-1803.
Zhou, H., Li, Z., &; Wu, X. (2007, August). Scheduling unrelated parallel machine to minimize total weighted tardiness using ant colony optimization. In Automation and Logistics, 2007 IEEE International Conference on (pp. 132-136). IEEE.
Zhu, Z., &; Heady, R. B. (2000). Minimizing the sum of earliness/tardiness in multi-machine scheduling: a mixed integer programming approach. Computers &; Industrial Engineering, 38(2), 297-305.