產能限制批量排程問題(CLSP)在製造業中是個很重要的問題，目標是最小化相關生產成本。本研究探討了考量四項重要因素的CLSP，是目前最困難的CLSP。此四項因素包含:(1)順序相依的裝設時間、(2)順序相依的裝設成本、(3)允續裝設延續至下一期，亦即，若下期第一個生產產品與本期最後一個產品相同，不需再重新裝設，以及(4)允許裝設時間跨期，亦即，一次裝設不限定於要在同一期內完成，可以跨越多期。由於此問題的困難度高，目前文獻所提出的解決方法皆附帶許多假設，無法完全符合現實情況。就我們所知，尚無任何相關文獻對於CLSP的研究如同本篇研究，能夠完善的涵蓋此四個重要因素。
本研究針對考量了四項重要因素的單機台CLSP問題，提出一個精確的混合整數規劃(MIP)模型以及一個新穎的啟發式方法。MIP方法能夠得到最佳解，但是在大部分的問題中，需要花費較長的計算時間，因此，本研究同時提出啟發式方法，提供MIP一個好的起始解，提升計算效率。本研究的實驗是根據不同參數來產生亂數問題及模擬，由實驗結果顯示，MIP結合啟發式的求解方法，確實能夠增進求解效率。

Capacitated lot sizing and scheduling problem (CLSP) with the objective of minimizing various relevant costs is an essential decision in the manufacturing industry. This study aims to investigate the CLSP with sequence-dependent setup times, costs, carryovers, and crossovers, which is one of the most complicated CLSP problems being investigated so far. These difficult features include: (1) sequence-dependent setup times, (2) sequence-dependent setup costs, (3) setup carryovers, in which, unlike many existing studies, setup duplication is removed for an identical item in a subsequent period, and (4) setup crossovers, which allows the duration of a setup running over multiple periods. Due to the difficulty of this problem, many existing proposed solution approaches were developed under various simplifying assumptions, which hence made these techniques not applicable in certain practical environments. To the best of our knowledge, no previous studies were able to comprehensively address the four considered features of this study.
In this study, an exact mixed integer programming (MIP) model and a novel heuristic method are proposed for single-machine CLSP which involves the above four features. An exact MIP approach solves for an optimal solution but usually consumes too much computation time for a practical-sized problem. Therefore, a novel heuristic is proposed to generate a good initial solution to the MIP model. The computational experiments on randomly generated problem instances show the effectiveness of the proposed MIP approach.

摘要 I
Abstract II
LIST OF FIGURES IV
LIST OF TABLES V
1. Introduction 1
1.1 Background and Applications 1
1.2. Capacitated Lot Sizing and Scheduling Problem 2
1.3 Extensions of CLSP and Its Complexity 3
1.4 Problem Statement and Research Approach 4
2. Literature Review 6
2.1. Brief History and Classic Papers 6
2.2. Literatures Related to Setup Cost and Time 7
2.3. Literatures Related to Setup Carryover 8
2.4. Literatures Related to Setup Crossover 9
2.5. Literatures Related to Triangular/Non-triangular setups 10
3. Solution Methods 13
3.1. Problem Formulation 13
3.1.1. Assumptions and Notations 13
3.1.2. Mathematical Model 15
3.1.3. Remarks on The Model 19
3.2. Heuristic Method 20
4. Computation Experiments 26
4.1. Experimental Parameters 26
4.2. Problem Generation Procedure 28
4.3. Parameter Setting 28
4.4. Experimental Results 30
4.4.1. Performance Evaluation 30
4.4.2. Factorial Analyses 33
4.4.3. Further Analyses on Heuristic Method 38
4.5. An Example 40
5. Conclusions and Future Research 41
Reference 43

Almada-Lobo, B., James, R. J. (2010). Neighbourhood search meta-heuristics for capacitated lot-sizing with sequence-dependent setups. International Journal of Production Research, Vol. 48, No. 3, pp. 861-878.
Almada-Lobo, B., Klabjan, D., Antónia carravilla, M., & Oliveira, J. F. (2007). Single machine multi-product capacitated lot sizing with sequence-dependent setups. International Journal of Production Research, Vol. 45, No. 20, pp. 4873-4894.
Almada-Lobo, B., Oliveira, J. F., & Carravilla, M. A. (2008). A note on “the capacitated lot-sizing and scheduling problem with sequence-dependent setup costs and setup times. Computers & Operations Research, Vol. 35, No.4, pp. 1374-1376.
Bitran, G. R., & Yanasse, H. H. (1982). Computational complexity of the capacitated lot size problem. Management Science, Vol. 28, No.10, pp. 1174-1186.
Buschkühl, L., Sahling, F., Helber, S., & Tempelmeier, H. (2010). Dynamic capacitated lot-sizing problems: a classification and review of solution approaches. OR Spectrum, Vol. 32, No. 2, pp. 231-261.
Chen, W. H., & Thizy, J. M. (1990). Analysis of relaxations for the multi-item capacitated lot-sizing problem. Annals of operations Research, Vol. 26, No. 1, pp. 29-72.
Clark A, Mahdieh M, Rangel S (2014). Production lot sizing and scheduling with non-triangular sequence dependent setup times. International Journal of Production Research, Vol.52, No.8, pp.2490–2503
Clark, A. R., & Clark, S. J. (2000). Rolling-horizon lot-sizing when set-up times are sequence-dependent. International Journal of Production Research, Vol. 38, No. 10, pp. 2287-2307.
Coleman, B. (1992). Technical note: a simple model for optimizing the single machine early/tardy problem with sequence‐dependent setups. Production and Operations Management, Vol. 1, No. 2, pp. 225-228.
Copil, K., Wörbelauer, M., Meyr, H., & Tempelmeier, H. (2016). Simultaneous lotsizing and scheduling problems: a classification and review of models. OR Spectrum, pp. 1-64, DOI 10.1007/s00291-015-0429-4.
Drexl, A., & Kimms, A. (1997). Lot sizing and scheduling—survey and extensions. European Journal of Operational Research, Vol. 99, No.2, pp. 221-235.
Fleischmann, B. (1990). The discrete lot-sizing and scheduling problem. European Journal of Operational Research, Vol. 44, No. 3, pp. 337-348.
Fleischmann, B., & Meyr, H. (1997). The general lotsizing and scheduling problem. Operations-Research-Spektrum, Vol. 19, No. 1, pp. 11-21.
Florian, M., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1980). Deterministic production planning: Algorithms and complexity. Management science, Vol. 26, No. 7, pp. 669-679.
Gopalakrishnan, M. (2000). A modified framework for modelling set-up carryover in the capacitated lotsizing problem. International Journal of Production Research, Vol. 38, No. 14, pp. 3421-3424.
Gopalakrishnan, M., Ding, K., Bourjolly, J. M., & Mohan, S. (2001). A tabu-search heuristic for the capacitated lot-sizing problem with set-up carryover. Management Science, Vol. 47, No. 6, pp. 851-863.
Gopalakrishnan, M., Miller, D. M., & Schmidt, C. P. (1995). A framework for modelling setup carryover in the capacitated lot sizing problem. International Journal of Production Research, Vol. 33, No. 7, pp. 1973-1988.
Guimarães, L., Klabjan, D., & Almada-Lobo, B. (2014). Modeling lotsizing and scheduling problems with sequence dependent setups. European Journal of Operational Research, Vol. 239, No. 3, pp. 644-662.
Gupta, D., & Magnusson, T. (2005). The capacitated lot-sizing and scheduling problem with sequence-dependent setup costs and setup times. Computers & Operations Research, Vol. 32, No. 4, pp. 727-747.
Haase K (1994) Lotsizing and scheduling for production planning. Springer, Berlin, Germany.
Haase, K. (1996). Capacitated lot-sizing with sequence dependent setup costs. Operations-Research-Spektrum, Vol. 18, No. 1, pp. 51-59.
Haase, K. (1998). Capacitated lot-sizing with linked production quantities of adjacent periods. In Beyond Manufacturing Resource Planning (MRP II). Springer, Berlin, Germany. pp. 127-146.
Haase, K., & Kimms, A. (2000). Lot sizing and scheduling with sequence-dependent setup costs and times and efficient rescheduling opportunities. International Journal of Production Economics, Vol. 66, No. 2, pp. 159-169.
Harris, F. W. (1990). How many parts to make at once. Operations Research, Vol. 38, No. 6, pp. 947-950.
Hung, Y. F., Bao, J. S., & Cheng, Y. E. (2016). Minimizing earliness and tardiness costs in scheduling jobs with time windows. Computers & Industrial Engineering.
James, R. J., & Almada-Lobo, B. (2011). Single and parallel machine capacitated lotsizing and scheduling: New iterative MIP-based neighborhood search heuristics. Computers & Operations Research, Vol. 38, No. 12, pp. 1816-1825.
Jans, R., & Degraeve, Z. (2008). Modeling industrial lot sizing problems: a review. International Journal of Production Research, Vol. 46, No. 6, pp. 1619-1643.
Karimi, B., Ghomi, S. F., & Wilson, J. M. (2003). The capacitated lot sizing problem: a review of models and algorithms. Omega, Vol. 31, No. 5, pp. 365-378.
Kovács, A., Brown, K. N., & Tarim, S. A. (2009). An efficient MIP model for the capacitated lot-sizing and scheduling problem with sequence-dependent setups. International Journal of Production Economics, Vol. 118, No. 1, pp. 282-291.
Kwak, I. S., & Jeong, I. J. (2011). A hierarchical approach for the capacitated lot-sizing and scheduling problem with a special structure of sequence-dependent setups. International Journal of Production Research, Vol. 49, No. 24, pp. 7425-7439.
Mahdieh, M., Clark, A., & Bijari, M. (2017). A novel flexible model for lot sizing and scheduling with non-triangular, period overlapping and carryover setups in different machine configurations. Flexible Services and Manufacturing Journal, pp.1-40.
Menezes, A. A., Clark, A., & Almada-Lobo, B. (2011). Capacitated lot-sizing and scheduling with sequence-dependent, period-overlapping and non-triangular setups. Journal of Scheduling, Vol. 14, No. 2, pp.209-219.
Mohan, S., Gopalakrishnan, M., Marathe, R., & Rajan, A. (2012). A note on modelling the capacitated lot-sizing problem with set-up carryover and set-up splitting. International Journal of Production Research, Vol. 50, No. 19, pp. 5538-5543.
Monma, C. L., & Potts, C. N. (1989). On the complexity of scheduling with batch setup times. Operations Research, Vol. 37, No. 5, pp. 798-804.
Porkka, P., Vepsäläinen, A. P. J., & Kuula, M. (2003). Multiperiod production planning carrying over set-up time. International Journal of Production Research, Vol. 41, No. 6, pp. 1133-1148.
Quadt, D., & Kuhn, H. (2005). Conceptual framework for lot-sizing and scheduling of flexible flow lines. International Journal of Production Research, Vol. 43, No. 11, pp. 2291-2308.
Quadt, D., & Kuhn, H. (2008). Capacitated lot-sizing with extensions: a review. 4OR, Vol. 6, No. 1, pp. 61-83.
Robinson, E. P., & Sahin, F. (2001). Economic production lot sizing with periodic costs and overtime. Decision Sciences, Vol. 32, No. 3, pp. 423-452.
Sox, C. R., & Gao, Y. (1999). The capacitated lot sizing problem with setup carry-over. IIE Transactions, Vol. 31, No. 2, pp. 173-181.
Suerie, C. (2006). Modeling of period overlapping setup times. European Journal of Operational Research, Vol. 174, No. 2, pp. 874-886.
Sung, C., & Maravelias, C. T. (2008). A mixed-integer programming formulation for the general capacitated lot-sizing problem. Computers & Chemical Engineering, Vol. 32, No. 1, pp. 244-259.
Sürie C, Stadtler H (2003). The capacitated lot-sizing problem with linked lot-sizes. Manage Science. Vol. 49, No. 8, pp.1039–1054
Toy, A. Ö., & Berk, E. (2006). Dynamic lot sizing problem for a warm/cold process. IIE Transactions, Vol. 38, No. 11, pp. 1027-1044.
Trigeiro, W. W., Thomas, L. J., & McClain, J. O. (1989). Capacitated lot sizing with setup times. Management science, Vol. 35, No. 3, pp. 353-366.
Ullah, H., & Parveen, S. (2010). A literature review on inventory lot sizing problems. Global Journal of Research In Engineering, Vol. 10, No. 5, pp. 21-36.
Wagner, H. M., & Whitin, T. M. (1958). Dynamic version of the economic lot size model. Management science, Vol. 5, No. 1, pp. 89-96.
Xiao, J., Zhang, C., Zheng, L., & Gupta, J. N. (2013). MIP-based fix-and-optimise algorithms for the parallel machine capacitated lot-sizing and scheduling problem. International Journal of Production Research, Vol. 51, No. 16, pp. 5011-5028.