生產計劃問題(production planning problem)充滿了許多不確定因素。大部分的生產計劃模型研究為了簡化問題以快速求得生產計劃，通常以確定性的模式來求解生產計劃。本論文將不確定性因素納入考量，對生產計劃問題進行研究探討。
Thompson and Wayne【1990】所提出的整合模式方法(integrated modeling approach)僅能找出完美資訊期望值(expected value with perfect information, EVwPI)，無法直接找出一組生產計劃的解。因此本論文利用決策樹分析(decision tree analysis)求解隨機生產計劃問題，以具體的方式表達隨機需求的生產計劃問題。當決策樹建構完畢時，再利用由決策樹的末端往前歸納推算的程序(backward induction procedure)，計算出最佳期望目標值，並找出一組生產計劃的解。
每次在做生產計劃時，可以輸入新的產品需求預測，將決策樹重新展開，去找到下一個生產時期的生產計劃。實驗結果顯示，決策樹分析除了能找出一組生產計劃解，且以該方法求得的期望獲利值也非常接近完美資訊期望值。

Production planning problems have many uncertain factors. Most of production planning models use deterministic parameters to simplify production plan problem and in order to find production plan quickly. This study considers uncertainties in production planning problem.
Thompson and Wayne (1990) proposed an integrated modeling approach which can only provide the expected value with perfect information (EVwPI), but it cannot provide a production plan. This study applies decision tree analysis to solve stochastic production planning problem. We express production planning problem with stochastic demand. After a decision tree being constructed, we can compute the expected objective value by using backward induction procedure to find an optimal production plan.
When a production plan must be calculated, we can input new product demand forecast information, and reconstruct the decision tree to find the production quantity of the next period. The result of our experiment shows that the decision tree analysis not only be able to find a set of production plan quickly but also make the expected value of the decision tree analysis close to the expected value with perfect information.

參考文獻
Ballestero E. (2001), “Stochastic goal programming: a mean-variance approach”, European Journal of Operational Research, Vol. 131, pp. 476-481.
Beale E. (1955), “On minimizing a convex function subject to linear inequalities”, Journal of the Royal Statistical Society, B 17, pp. 173-184.
Birge J. R. (1997), “Stochastic programming computation and applications”, INFORMS Journal on Computing, Vol. 9, No. 2, pp. 111-133.
Charnes A. and Cooper W. W. (1959), “Chance-constrained programming”, <a href="http://www.ntsearch.com/search.php?q=Management&v=56">Management</a> <a href="http://www.ntsearch.com/search.php?q=Science&v=56">Science</a>, Vol. 5, pp. 73-79.
Chen X. (2000), “Newton-type methods for stochastic programming”, Mathematical and <a href="http://www.ntsearch.com/search.php?q=Computer&v=56">Computer</a> Modeling, Vol. 31, pp.89-98.
Coles S. and Rowley J. (1995), “Revisiting decision trees”, <a href="http://www.ntsearch.com/search.php?q=Management&v=56">Management</a> Decision, Vol. 33, No. 8, pp. 46-50.
Danzig G. B. (1955), “Linear programming under uncertainty”, <a href="http://www.ntsearch.com/search.php?q=Management&v=56">Management</a> <a href="http://www.ntsearch.com/search.php?q=Science&v=56">Science</a>, Vol. 1, pp. 197-206.
Dupa?ova J. (2002), “Applications of stochastic programming: achievements and questions”, European Journal of Operational Research, Vol. 140, pp. 281-290.
Hillier F. S. and Lieberman G. J. (2001), Introduction to Operations Research, 7th edition, McGraw-Hill, New York, NY, U.S.A..
James M. (1995), “Decision tree analysis - choosing between options by projecting likely outcomes”, Mind <a href="http://www.ntsearch.com/search.php?q=Tools&v=56">Tools</a>, http://www.mindtools.com/dectree.<a href="http://www.ntsearch.com/search.php?q=html&v=56">html</a>.
Liu B. (1997), “Dependent-chance programming: a class of stochastic programming”, <a href="http://www.ntsearch.com/search.php?q=Computers&v=56">Computers</a> Mathematics Application, Vol. 34, No. 12, pp. 89-104.
Liu B. (2001), “Uncertain programming: a unifying optimization theory in various uncertain environments”, Applied Mathematics and Computation, Vol. 120, pp. 227-234.
Masaru T. and Masahiro H. (2003), Genetic algorithm for supply planning optimization under uncertain demand, Genetic and Evolutionary Computation-GECCO 2003: Genetic and Evolutionary Computation Conference, Chicago, IL, U.S.A., pp. 2337-2346.
Papadopoulos C. E. and Yeung H. (2001), “Uncertainty estimation and Monte Carlo simulation method”, Flow Measurement and Instrumentation, Vol. 12, pp. 291-298.
Shapiro J. F. (2001), Modeling the Supply Chain, 1st edition, Duxbury press, CA, U.S.A..
Thompson S. D. and Wayne J. D. (1990), “An integrated approach for modeling uncertainty in aggregate production planning”, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 20, No. 5, pp. 1000-1012.
Wets R. J-B (1996), “Challenges in stochastic programming”, Mathematical Programming, Vol. 75, pp. 115-135.
Zimmermann H. -J. (2000), “An application-oriented view of modeling uncertainty”, European Journal of Operational Research, Vol. 122, pp. 190-198.