在造紙工業中，許多的研究探討訂單分配和生產損耗(Trim Loss)的最佳化，但這些結果會因某些存貨過多或低於可容忍排程量時，而不被企業所採用。為了讓整個結果能更符合實際的需求，本研究提出排程量的調整與存貨量的限制，取得不同的損耗比例數據與存貨分佈，再利用差異分析篩選損耗比例數據，以供決策者決策的一個新的決策模型，本研究使用一個實際案例說明了該方法的應用，研究結果顯示，我們提出的方法優於人工在排程量與損耗所使用的方法。
本研究也考量一維原料裁切問題（Cutting Stock Problem，簡稱CSP），針對工業用紙所產生非訂單的存貨寬度，且可以在未來使用時，提出了彈性存貨配置與損耗控制，以決定生產的一個新模型。而評估的方法是使用一實務的數據，展示其能夠解決產業排程的問題，同時也考量整體裁切的處理，例如在同一機器上彙整訂單與多種的存貨寬度以及不同的裁切模式。此外，本研究也進行與其他的模型比較，包括損耗最小化問題（Trim Loss Minimization Problem，簡稱TLMP）與原料裁切問題（CSP），研究結果顯示，本模型的整體彈性和損耗率優於這二種模型。

In the paper industry, numerous studies have explored optimizing order allocation and cutting trim loss. But, enterprises may not adopt the resulting solutions because some widths of the inventory are excessive or less than tolerable scheduling. To ensure the results better suit actual requirements, we present a new decision model based on the adjustment of scheduling and limitation of inventory quantity to differentiate data of trim loss and inventory distribution. Differential analysis is used for trim loss data filtering and the information is valuable for decision-making. A numerical example is presented to illustrate the applicability of the proposed method. The results show that our proposed method outperforms the manual method regarding scheduling quantity and trim loss.
In this paper, we also consider a one-dimensional cutting stock problem (CSP) in which the stock widths are not used to fulfill the order but kept for use in the future for the industrial-use paper production. We present a new model based on the flexible stock allocation and trim loss control to determine the production quantity. We evaluate our approach using some illustrative examples and show that we are able to solve industrial-size scheduling problems, while also addressing common cutting considerations such as aggregation of orders, multiple stock widths and cutting different patterns on the same machine. In addition, we compare our model with others, including trim loss minimization problem (TLMP) and cutting stock problem (CSP). The results show that the proposed model outperforms the other two models regarding total flexibility and trim loss ratio.

[1] Erjavec, J., Gradisar, M., Trkman, P., Renovation of the cutting stock process, International Journal of Production Research, vol. 47, no. 14, pp. 3979–3996, 2009.
[2] Gilmore, P. C., Gomory, R. E., A linear programming approach to the cutting stock problem, Operations Research, vol. 9, no. 6, pp. 849–859, 1961.
[3] Gradisar, M., Resinovič, G., Kljajić, M., Evaluation of algorithms for one-dimensional cutting, Computers and Operations Research, vol. 29, no. 9, pp. 1207–1220, 2000.
[4] Gilmore, P. C., Gomory, R. E., A linear programming approach to the cutting stock problem, Part II, Operations Research, vol. 11, no. 6, pp. 863–888, 1963.
[5] Gochet, W., Vandebroek, M., A dynamic programming based heuristic for industrial buying of cardboard, European Journal of Operational Research, vol. 38, no. 1, pp. 104–112, 1989.
[6] Dagli, C. H., Knowledge-based systems for cutting stock problems, European Journal of Operational Research, vol. 44, no. 2, pp. 160-166, 1990.
[7] Foerster, H., Wäscher, G., Simulated annealing for order spread minimization in sequencing cutting patterns, European Journal of Operational Research, vol. 110, no. 2, pp. 272-281, 1998.
[8] Vance, P. H., Branch-and-price algorithms for the one-dimensional cutting stock problem, Computational Optimization and Applications, vol. 9, no. 3, pp. 211–228, 1998.
[9] Gradisar, M., Kljajić, M., Resinovič, G., Jesenko, J., A sequential heuristic procedure for one-dimensional cutting, European Journal of Operational Research, vol. 114, no. 3, pp. 557–568, 1999.
[10] Sakawa, M., Interactive fuzzy programming for multi- level 0-1 programming problems through genetic algorithms, European Journal of Operational Research, vol. 114, no. 2, pp. 580-588, 1999.
[11] Suliman, S. M. A., Pattern generating procedure for the cutting stock problem, International Journal of Production Economics, vol. 74, no. 1-3, pp. 293-301, 2001.
[12] Amor, H. B., Desrosiers, J., Valerio de Carvalho, J. M., Dual-optimal inequalities for stabilized column generation, Operations Research, vol. 54, no. 3, pp. 454-463, 2006.
[13] Trkman, P., Gradisar, M., One-dimensional cutting stock optimization in consecutive time periods, European Journal of Operational Research, vol. 179, no. 2, pp. 291-301, 2007.
[14] Weng, W., Sung, T., Optimization of a line-cutting procedure for ship hull construction by an effective tabu search, International Journal of Production Research, vol. 46, no. 21, pp. 5935-5949, 2007.
[15] Alves, C., Valerio de Carvalho, J. M., Accelerating column generation for variable sized bin-packing problems, European Journal of Operational Research, vol. 183, no. 3, pp. 1333–1352, 2008.
[16] Nonås, S. L., Thorstenson, A., Solving a combined cutting-stock and lot-sizing problem with a column generating procedure, Computers and Operations Research, vol. 35, no. 10, pp. 3371–3392, 2008.
[17] Reinertsen, H., Vossen, T. W. M., The one-dimensional cutting stock problem with due dates, European Journal of Operational Research, vol. 201, no. 3, pp. 701-711, 2010.
[18] Matsumoto, K., Umetani, S., Nagamochi, H., On the one-dimensional stock cutting problem in the paper tube industry, Journal of Scheduling, vol. 14, no. 3, pp. 281-290, 2011.
[19] Erjavec, J., Gradisar, M., Trkman, P., Assessment of stock size to minimize cutting stock production costs, International Journal of Production Economics, vol. 135, no. 1, pp. 170–176, 2012.
[20] Jahromi, M. H. M. A., Tavakkoli-Moghaddam, R., Makui, A., Shamsi, A., Solving an one-dimensional cutting stock problem by simulated annealing and tabu search, Journal of Industrial Engineering International, vol. 8 , no. 1, pp. 24-31, 2012.
[21] Mobasher, A., Ekici, A., Solution approaches for the cutting stock problem with setup cost, Computers and Operations Research, vol. 40, no. 1, pp. 225-235, 2013.
[22] Wang, G. C., Li, C. P., Lv, J., Zhao, X. X., Cui, H. Y., An efficient algorithm design for the one-dimensional cutting-stock problem, Advanced Materials Research, vol. 602-604, pp. 1753-1756, 2013.
[23] Menon, S., Schrage, L., Order allocation for stock cutting in the paper industry, Operations Research, vol. 50, pp. 324-332, 2002.
[24] Poltroniere, S. C., Poldi, K. C., Toledo, F. M. B., Arenales, M. N., A coupling cutting stock-lot sizing problem in the paper industry, Annuals of Operations Research, vol. 157, pp. 91-104, 2008.
[25] Tharmmaphornphilas, W., Puemsin, S., Siripongwutikorn, P., A MILP model to select cutting machines and cutting patterns to minimize paper loss, Proceedings of the International MultiConference of Engineers and Computer Scientists, pp.1155-1160, Hong Kong, March 13-15, 2013
[26] Ali, M., Ahn, C. W., Pant, M., Trim loss optimization by an improved differential evolution, Mathematical Problems in Engineering, vol. 2013, Article ID 706350, 8 pages, 2013.
[27] Chauhan, S. S., Martel, A., D’Amour, S., Roll assortment optimization in a paper mill: An integer programming approach, Computers and Operations Research, vol. 35, pp. 614-627, 2008.
[28] Keskinocak, P., Wu, F., Goodwin, R., Murthy, S., Akkiraju, R., Kumaran, S., Derebail, A., Scheduling solutions for the paper industry, Operations Research, vol. 50, pp. 249–259, 2002.
[29] Stadtler, H., A one-dimensional cutting stock problem in the aluminium industry and its solution, European Journal of Operational Research, vol. 44, no. 2, pp. 209-223, 1990.
[30] Gradisar, M., Jesenko, J., Resinovič, G., Optimization of roll cutting in clothing industry, Computers and Operations Research, vol. 24, no. 10, pp. 945–953, 1997.
[31] Morgan, L. O., Morton, A. R., Daniels, R. I., Simultaneously determining the mix of space launch vehicles and the assignment of satellites to rockets, European Journal of Operational Research, vol. 172, no. 3, pp. 747-760, 2006.
[32] Dikili, A., Takinaci, A., Pek, N., A new heuristic approach to one-dimensional stock-cutting problems with multiple stock lengths in ship production, Ocean Engineering, vol. 35, no. 7, pp. 637-645, 2008.
[33] Abuabara, A., Morabito, R., Cutting optimization of structural tubes to build agricultural light aircrafts, Annals of Operations Research, vol. 169, no. 1, pp. 149-165, 2009.
[34] Gramani, M. C., Franca, P. M., The combined cutting stock and lot-sizing problem in industrial processes, European Journal of Operational Research, vol. 174, no. 1, pp. 509–521, 2006.
[35] Ritvirool, A., A trim-loss minimization in a produce-handling vehicle production plant, Journal of Science Technology, vol. 29, no. 1, pp. 157-164, 2007.
[36] Wascher, G., Haubner, H., Schumann, H., An improved typology of cutting and packing problems, European Journal of Operational Research, vol. 183, no. 3, pp. 1109-1130, 2007.
[37] Abbasi, J. A., Sahir, M. H., Development of optimal cutting plan using linear programming tools and MATLAB algorithm, International Journal of Innovation, Management and Technology, vol. 1, no. 5, 483-492, 2010.
[38] Yanasse, H. H., A review of three decades of research on some combinatorial optimization problems, Pesquisa Operacional, vol. 33, no. 1, pp. 11-36, 2013.
[39] Kos, L., Duhovnik, J., Cutting optimization with variable-sized stock and inventory status data, International Journal of Production Research, vol. 40, no. 10, pp. 2289-2301, 2002.
[40] Dimitriadis, S., Kehris, E., Cutting stock optimization in custom door and window manufacturing industry, International Journal of Decision Sciences, Risk and Management, vol. 1, no. 1, pp. 66-80, 2009.
[41] Cherri, A. C., Arenales, M. N., Yanasse, H. H., The one dimensional cutting stock problem with usable leftover – a heuristic approach, European Journal of Operational Research, vol. 196, no. 3, pp. 897–908, 2009.
[42] Poldi, K. C., Arenales, M. N., Heuristics for the one-dimensional cutting stock problem with limited multiple stock lengths, Computers and Operations Research, vol. 36, no. 6, pp. 2074-2081, 2009.
[43] Cui, Y., Yang, Y., A heuristic for the one-dimensional cutting stock problem with usable leftover, European Journal of Operational Research, vol. 204, no. 2, pp. 245-250, 2010.
[44] Araujo, S. A., Constantino, A. A., Poldi, K. C., An evolutionary algorithm for the one-dimensional cutting stock problem, International Transactions in Operational Research, vol. 18, no. 1, pp. 115-127, 2011.
[45] Cherri, A. C., Arenales, M. N., Yanasse, H. H., The usable leftover one-dimensional cutting stock problem – a priority-in-use heuristic, International Transactions in Operational Research, vol. 20, no. 2, pp. 189-199, 2013.
[46] Lingo Software, Version 11, Lindo Systems, Inc., Chicago, IL, USA, 2009.
[47] Wang, F. K., Liu, F. T., A new decision model for reducing trim loss and inventory in the paper industry, Journal of Applied Mathematics, vol. 2014, Article ID 987054, 10 pages, 2014.