在生產環境中常需要面對許多不確定性，其中最主要的便是未來需求的不確定性，預測未來需求對於生產計畫制定者是相當重要地。現今有許多研究提出了不同的隨機模型來解決此問題，其中有多個研究採用情境導向(scenario-based)，也就是將未來需求分成多個配有機率的情境模式，以表示未來需求的不確定性。本研究提出在隨機生產計畫模型中結合情境獨立及情境相依生產量，情境獨立生產量在模型方程式中，以一確定值應付多種情境需求；而情境相依生產量則依照各種情境需求建構出不同情境相對應的生產量。值得注意的是，在規劃生產計畫時，當期的計畫生產量必須為確定值以進行投料生產，也就是必須使用情境獨立生產量來表達。除此之外，在生產計畫的其餘週期中，我們能設定一個界線來分隔情境獨立生產量及情境相依生產量。本研究在隨機模型中變動情境分隔界線，並運用平面滾動法進行模擬，以蒐集不同情境分隔界線下的實驗結果。
實驗結果顯示，在不允許缺貨的假設下，當需求比高時，設定較小的界線值能得到較好的結果。除此之外，多數情況下，將界線設在生產計劃前端會得到較差的結果。
關鍵字：生產計劃、隨機規劃、情境導向預測、情境導向生產量、滾動平面法

A manufacturing company usually deals many uncertain factories. One of the major uncertainties comes from the uncertainty of future demands. Hence, many researchers proposed various stochastic models to deal with these problems. Forecasting future demands becomes an important issue for most production planner before a production plan can be calculated. To consider the uncertainty in demands, many studies adopt scenario-based demand forecast; each of the scenarios specifies a future demand pattern and a probability. This study investigates a demand-scenario-based stochastic production planning models with both scenario-independent and scenario-dependent production variables. A scenario-independent production variable is used in the inventory balance equations for all demand scenarios, while a scenario-dependent production variable is designated to a particular demand-scenario, and is only expressed in the inventory balance equation of the corresponding demand-scenario. To perform the production activity of a particular product in a particular period, a scenario-dependent production variable serves no such a purpose since there are many production variables for a product in a period. Hence, the immediate period can only be modeled as scenario-independent production variable. During the whole planning horizon, a dependence boundary can be set, before which the production variables are scenario-independent and after which the production variables are scenario-dependent. Using simulation experiments incorporating rolling horizon practice, this study tests the performances of the stochastic models with different boundaries between scenario-independent and scenario-dependent production variables.
From the results of simulation experiments, when the demand-to-capacity ratio is high and backorder carryover are not allowable, a small boundary provides better objective value. Other than this case, a small boundary normally renders a worse objective value.

Keywords: Production planning, stochastic programming, scenario-based forecast, scenario-based production variable, rolling horizon.

TABLE OF CONTENTS
摘要 I
Abstract II
LIST OF TABLES V
LIST OF FIGURES VI
1. Introduction 1
2. Literature reviews 6
3. Stochastic programming model 12
3.1. Basic assumptions 12
3.2. Linear programming model – allowable backorder carryover 13
3.3. Linear programming model without backorder carryover 18
4. Experimental design and computational results 19
4.1. Simulation 19
4.2. The generation of actual demand for simulation 20
4.2.1. Notations 20
4.2.2. Generation procedure for actual demands 21
4.3. The generation of forecasted demand for simulation 23
4.3.1. Notations 23
4.3.2. Generate procedure for forecasted demands 24
4.4. Parameters setting 25
4.5. Results and analysis 28
4.5.1. Results of the allowable backorder carryover model 29
4.5.2. Results of the without backorder carryover model 45
5. Conclusion 63
Reference 64

Al-e-hashem, S.M.J. M., Malekly, H. and Aryanezhad, M. B., (2011), “A multi-objective robust optimization model for multi-product multi-site aggregate production planning in a supply chain uncer uncertainty”, International Journal of Production Economics, Vol. 134, No. 1, pp. 28-42.
Awudu, I. and Zhang J., (2013), “Stochastic production planning for a biofuel supply chain under demand and price uncertainties”, Applied Energy, Vol. 103, pp.189-196.
Bakir, M. A. and Bryne, M. D., (2008), “Stochastic linear optimization of an MPMP production planning model”, International Journal Production Economic, Vol. 55, No. 1, pp. 87-96.
Bookbinder, J. H. and H’ng, B. T., (1986), “Rolling horizon production planning for probabilistic time-vary demand”, International Journal of Production Research, Vol. 24, No. 6, pp. 1439-1458.
Bunn, D. W. and Salo, Ahti. A., (1993), “Forecasting with scenarios”, European Journal of Operational Research, Vol. 68, No. 3, pp. 291-303.
Chen, C. L. and Lee, W. C., (2004), “Multi-objective optimization of multi-echelon supply chain networks with uncertain product demands and prices”, Computers and Chemical Engineering, Vol. 28, No. 6, pp.1131-1144
Chen, T.L. and Lu, H.C., (2012), “Stochastic multi-site capacity planning of TFT-LCD manufacturing using expected shadow-price based decomposition”, Applied Mathematical Modelling, Vol. 36, No. 12, pp. 5901-5919.
Domenica, N.D., Mitra, G., Valente, P. and Birbilis, G., (2007), “Stochastic programming and scenario generation within a simulation framework: An information systems perspective”, Science Direct, Vol. 42, No. 4, pp. 2197-2218.
Dupaˇcov´a, J., Gr¨owe-Kuska, N.and R¨omisch. W., (2003), “Scenario reduction in stochastic programming”, Mathematical programming, Vol. 95, No. 3, pp. 493-511.
Eppen, G.D., Martin, R.K. and Schrage, L., (1989), “A scenario approach to capacity planning”, Operations Research, Vol. 37, No. 4, pp. 517-527.
Escudero, L.F., (1993), “Production planning via scenario modelling”, Annals of Operations Research, Vol. 43, No.6, pp. 311-335.
Escudero, L.F. and Kamesam, P.V., (1995), “On solving stochastic production planning problems via scenario modeling”, TOP, Vol. 3, No.1, pp. 69-95.
Frauendorfer, K., (1996), “Barycentric scenario trees in convex multistage stochastic programming”, Mathematical Programming, Vol. 75, No. 2, pp. 277-293.
Barbarosoglu, G. and Arda, Y., (2004), “A two-stage stochastic programming framework for transportation planning in disaster response”, Journal of the Operational Research Society, Vol. 55, No. 1, pp.43-53.
Geng, N. and Jiang, Z., (2009), “A review on strategic capacity planning for the semiconductor manufacturing industry”, International Journal of Production Research, Vol. 47, No. 13, pp. 3639-3655.
Geng, N., Jiang, Z. and Feng, C., (2009), “Stochastic programming based capacity planning for semiconductor Wafer Fab with uncertain demand and capacity”, European Journal of Operational Research, Vol. 198, No. 3, pp. 899-908.
Gelders, L. F. and Van, Luk. N., (1981), “Production planning : a review”, European Journal of Operational Research, Vol. 7, No. 2, pp. 101-110.
Graves, S. C., (1981), “A review of production scheduling”, Operations Research, Vol. 29, No. 4, pp.646-675.
Guide Jr. V. D. R., (2000), “Production planning and control for remanufacturing: industry practice and research needs”, Journal of Operations Management, Vol.18, No. 4, pp467-483.
Günther, S., Wolfgang, S., Michael, S., Annette, R., Stefan, R. and Heiko, L., (2014), “Scenario-based determination of product feature uncertainties for robust product architectures”, German Academic Society for Production Engineering, Vol. 8, No. 3, pp. 383-395.
Hendry L. C. and Kingsman, B. G., (1989), “Production planning systems and their applicability to make-to-order companies”, European Journal of Operational Research, Vol. 40, No.1, pp.1-15.
Ho, C., (1989), “Evaluating the impact of operating environments on MRP system nervousness”, International Journal of Production Research, Vol. 27, No.7, pp. 1115-1135.
Hood, S. J., Bermon S. and Barahona, F., (2003), “Capacity planning under demand uncertainty for semiconductor manufacturing”, IEEE Transactions on Semiconductor Manufacturing, Vol. 16, No. 2, pp. 273-280.
Husbands, P., Mill, F. and Warrington, S., (1991), “Genetic algorithms, production plan optimization and scheduling”, Parallel Problem Solving from Nature, Vol. 496, pp.80.
Jolayemi, J. K. and Olorunniwo, O. F., (2004), “A deterministic model for planning production quantities in a multi-plant, multi-warehouse environment with extensible capacities”, International Journal of Production Economics, Vol. 87, No. 2, pp. 99-113.
Kazaz, B., (2004), “Production planning under yield and demand uncertainty with yield-dependent cost and price”, Manufacturing and Service Operation Management, Vol. 6, No. 3, pp. 209-224.
Katok, E., Tarantino, W., and Harrison, T.P., (2003), “Investment in production resource flexibility: An empirical investigation of methods for planning under uncertainty. Naval Research”, Naval Research Logistics, Vol.50, No. 2, pp.105-129.
Lai, K. K. and Ng, W. L., (2005), “A Stochastic Approach to Hotel Revenue Optimization”, Computer and Operations Research, Vol. 32, No. 5, pp. 1059-1072.
Lin, J.T., Wu, C.H., Chen, T.L. and Shih, S.H., (2010), “A stochastic programming model for strategic capacity planning in thin film transistor-liquid crystal display (TFT-LCD) industry”, Computers and Operations Research, Vol. 38, No.7, pp. 992-1007.
Meng, L. and Zhou, X., (2011), “Robust single-track train dispatching model under a dynamic and stochastic environment: A scenario-based rolling horizon solution approach”, Transportation Research Part B: Methodological, Vol. 45, No. 7, pp. 1080-1102.
Metters, R., (1997), “Production planning with stochastic seasonal demand and capacitated production”, IIE Transactions, Vol. 29, No. 11, pp. 1017-1029.
Millar, H. H., (1998), “The impact of rolling horizon planning on the cost of industrial fishing activity”, Computers Operation Research, Vol. 25, No. 10, pp.825-837.
Modigliani, F. and Hohn, F.E., (1955), “Production planning over time and the nature of the expectation and planning horizon”, Econometric, Vol. 23, No. 1, pp. 46-66.
Molammad S., Mohammad M. P. and Amin. A., (2013), “Production planning and worker training in dynamic manufacturing systems”, Journal of Manufacturing Systems, Vol. 32, No. 2, pp. 308-314.
Mula, J., Ploer, R., Garcia-Sabater, J.P. and Lario, F.C., (2006), “Models for production planning under uncertainty: A review”, International Journal of Production Economics, Vol. 103, No.1, pp. 271-285.
Mula, J., Diaz-Madronero, M. and Peidro, D., (2014), “A review of discrete-time optimization models for tactical production planning”, International Journal of Production Research, (ahead-of-printf), pp.1-35.
Pastor, R., Altimiras, J. and Mateo, M., (2009), “Planning production using mathematical programming: The case of a woodturning company”, Computers and Operations Research, Vol. 36, No. 7, pp.2173-2178.
Pendharkae, P. C., (1997), “A fuzzy linear programming model for production planning in coal mines”, Computers and Operations Research, Vol. 24, No. 12, pp. 1141-1149.
Phelps, R., Chan, C. and Kapsalis, S.C., (2001), “Does scenario planning affect performance? Two exploratory studies”, Journal of Business Research, Vol. 51, No.3, pp. 223-232.
Rahmani, D., Ramezanian, R. Fattahi, P. and Heydari, M., (2013), “A robust optimization model for multi-product two-stage capacitated production planning under uncertainty”, Applied Mathematical Modelling, Vol. 37, No.20, pp. 8757-8971.
Rockafellar, R.T. and Wets, R.J.-B., (1991), “Scenario and policy aggregation in optimization under uncertainty”, Mathematics of Operations Research, Vol. 16, No. 1, pp. 199-147.
Sethi, S. and Sorger, G., (1991), “A theory of rolling horizon decision making”, Annals of Operations Research, Vol. 29, No. 1, pp. 387-416.
Silver, E. and Meal, H., (1978), “Inventory control under a probabilistic time-varying, demand pattern”, Aiie Transactions, Vol. 10, No. 4, pp. 371-379.
Swoveland, C., (1975), “A deterministic multi-period production planning model with piecewise concave production and holding-backorder costs”, Management Science, Vol. 21, No. 9, pp. 1007-1013.
Tang, L., Liu, J., Rong, A. and Yang, Z., (2000), “A review of planning and scheduling systems and methods for integrated steel production”, European Journal of Operational Research, Vol. 133, No.1, pp.1-20.
Tarim, S. A., and Smith, B. M., (2008), “Constraint programming for computing non-stationary (R,S) inventory policies”, European Journal of Operational Research, Vol. 189, No. 3, pp. 1004-1021.
Vargas, V. and Metters, R., (2011), “A master production scheduling procedure for stochastic demand and rolling planning horizons”, International Journal of Production Economics, Vol. 132, No. 2, pp.296–302.
Volling, T., Matzke, A., Grunewald, M. and Spengler, T. S., (2013), “Planning of capacities and orders in build-to-order automobile production: A review”, European Journal of Operational Research, Vol. 224, No.2, pp.240-260.
Zanjani1, M. K., Nourelfath, M. and Ait-Kadi, D., (2013), “A scenario Decomposition approach for stochastic production planning in sawmills”, Journal of the Operational Research Society, Vol. 64, No. 1, pp. 48–59.
Zäpfel G., (1996), “Production planning in the case of uncertain individual demand Entension for an MRP II concept”, International Journal of Production Economics, Vol. 46, pp. 153-164.
Zhang, X., Prajapati, M. and Peden, E., (2011), “A stochastic production planning model under uncertain seasonal demand and market growth”, International Journal of Production Research, Vol. 49, No. 7, pp. 1957-1975.