第五章 結論與建議
5.2 未來研究建議
未來可能的研究方向如下：
1.在成本函數可加入在製品的儲存成本及成品的殘值，針對該生產情境重新 建構動態規劃問題，提出求解方法。
2.針對生產週期時間具有不確定性，本論文假設產出情況只有二種：不是一 期產出就是二期產出，未來可擴充為多種產出情況，使模式更具一般化。
3.在多階段生產系統方面，未來也可以考量生產週期時間具不確定性，針對 該生產情境重新建構動態規劃問題，提出求解方法。
參考文獻
英文部份：
[1] Anily, S., “Single machine lot sizing with uniform yields and rigid demands:
robustness of the optimal solution,” IIE Transactions, 27, 625-633, 1995.
[2] Anily, S., A. Beja, and A. Mendel, “Optimal lot sizes with geometric production yield and rigid demand,” Operations Research, 5(3), 424-432, 2002.
[3] Barad, M. and D. Braha., “Control limits for multi-stage manufacturing processes with binomial yield (single an multiple production runs),” Journal of the Operational Research Society, 47, 98-112, 1996.
[4] Beja, A., “Optimal Reject Allowance with Constant Marginal Production Efficiency,” Naval Research Logistics Quarterly, 24, 21-33, 1977.
[5] Ben-Zvi, T. and A. Grosfeld-Nir, “Serial production systems with random yield and rigid demand: a heuristic,” Operations Research Letters, 35, 235-244, 2007.
[6] Bowman, E.H., “Using statistical tools to set a reject allowance,” National Association of Cost Accountants. NACA Bulletin, 36(10), 1334-1342, 1955.
[7] Braha, D., “Manufacturing control of a serial system with binomial yields, multiple production runs, and non-rigid demand: a decomposition approach." IIE Transactions, 31, 1-9, 1999.
[8] Gerchak, Y. and A. Grosfeld-Nir., “Lot-sizing for substitutable, production-to-order parts with random functionality yields,” The International Journal of Flexible Manufacturing Systems, 11(4), 371-377, 1999.
[9] Gerchak, Y. and A. Grosfeld-Nir., “Multiple lotsizing and value of probabilistic information in production to order of an uncertain size,” International Journal of Production Economics, 56–57, 191-197, 1998.
[10] Grosfeld-Nir, A. and B. Ronen., “A single bottleneck system with binomial yield
and rigid demand,” Management Science, 39(5), 650-653, 1993.
[11] Grosfeld-Nir, A. and L.W. Robinson., “Production to order on a two machine line with random yields and rigid demand,” European Journal of Operational Research, 80, 264-276, 1995.
[12] Grosfeld-Nir, A. and Y. Gerchak., “Multiple lotsizing with random common-cause yield and rigid demand,” Operations Research Letters, 9, 383-388, 1990.
[13] Grosfeld-Nir, A. and Y. Gerchak., “Multistage production to order with rework capability,” Management Science, 48(5), 652–664, 2002.
[14] Grosfeld-Nir, A. and Y. Gerchak., “Production to order with random yields:
single-stage multiple lot-sizing,” IIE Transactions, 28, 669-676, 1996.
[15] Grosfeld-Nir, A., “A two-bottleneck system with binomial yields and rigid demand,” European Journal of Operational Research, 165, 231-250, 2005.
[16] Grosfeld-Nir, A., “Multiple lotsizing in production to order with random yields:
review of recent advances,” Annals of Operations Research, 126, 43-69, 2004.
[17] Grosfeld-Nir, A., “Single bottleneck systems with proportional expected yields and rigid demand,” European Journal of Operational Research, 80, 297-307, 1995.
[18] Grosfeld-Nir, A., S. Anily and T. Ben-Zvi., “Lot-sizing two-echelon assembly systems with random yields and rigid demand,” European Journal of Operational Research, 173, 600-616, 2006.
[19] Grosfeld-Nir, A., Y. Gerchak, and Q.M. He., “Manufacturing to order with random yield and costly inspection,” Operations Research, 48(5), 761–767, 2000.
[20] Guu S.M. and A.X. Zhang., “The finite multiple lot sizing problem with interrupted geometric yield and holding costs,” European Journal of Operational
Research, 145, 635-644, 2003.
[21] Guu, S.M. and F.R. Liou., “An algorithm for the multiple lot sizing problem with rigid demand and interrupted geometric yield,” Journal of Mathematical Analysis and Applications, 234, 567-579, 1999.
[22] Guu, S.M., “Properties of the multiple lot-sizing problem with rigid demand, general cost structures, and interrupted geometric yield,” Operations Research Letters, 25, 59-65, 1999.
[23] Lee, H.L. and C.A. Yano, “Production control in multistage systems with variable yield losses,” Operations Research, 36(2), 269-278, 1988.
[24] Llewellyn, R.W., “Order sizes of job lot manufacturing,” The Journal of Industrial Engineering, 10, 176-180, 1959.
[25] Pentico, D.W., “An evaluation and proposed modification of the Sepheri-Silver-New heuristic for multiple lot sizing under variable yield,” IIE Transactions, 20(4), 360-363, 1988.
[26] Pentico, D.W., “Multistage production systems with random yield: heuristics and optimality,” International Journal of Production Research, 32(10), 2455-2462, 1994.
[27] Sepheri, M., E.A. Silver, and C. New., “A heuristic for multiple lot sizing for an order under variable yield,” IIE Transactions, 18, 63-69, 1986.
[28] Wang, Y. and Y. Gerchak., “Input control in a batch production system with lead times, due dates and random yields,” European Journal of Operational Research, 126, 371-385, 2000.
[29] Wein, A.S., “Random yield, rework and scrap in multistage batch manufacturing environments,” Operations Research, 40(3), 551-563, 1992.
[30] Yano, C.A. and H.L. Lee., “Lot sizing and random yields: a review,” Operations Research, 43(2), 311-334, 1995.
[31] Yano, C.A., “Setting planned leadtimes in serial production systems with tardiness costs,” Management Science, 33(1), 95-106, 1987.
[32] Zhang, A.X. and S.M. Guu., “Properties of the multiple lot-sizing problem with rigid demands and general yield distributions,” Computers and Mathematics with Applications, 33(5), 55-65, 1997.
[33] Zhang, A.X. and S.M. Guu., “The multiple lot sizing problem with rigid demand and interrupted geometric yield,” IIE Transactions, 30(5), 427-431, 1998.
中文部份：
[34] 劉芳如(1997)，『中斷式幾何分配生產模式之最適下料演算法研究』，碩士 論文，元智大學工業工程與管理研究所。
[35] 謝偉志(2000)，『多批量下料問題在中斷式幾何分配且有設置次數及庫存成 本下之研究』，碩士論文，元智大學工業工程與管理研究所。
附錄 A
求解生產週期時間具不確定性，單階段生產系統有交期限制的多次投料問 題，該問題可建構成動態規劃問題，其演算法如下：
歩驟3.1：根據 3.2.4 節第一個邊界條件。當1 t T≤ ≤ ，狀態為st =(0 ,Rt(kt+1))，設 定最佳成本與最佳投料量。
FOR 1, t= 2, ..., T FOR Rt(kt+1)=0 ,1 ,...,D
SET ))st =(0 ,Rt(kt+1 , 0Ct*(st)= , 0Nt(st)= /*參見公式(3-2)*/
ENDFOR ENDFOR
歩驟3.2：根據 3.2.4 節第二個邊界條件。當t =0，狀態為s0 =(D0 ,R0(k1))，設定 最佳成本與最佳投料量。
FOR d=1, 2, ...,D FOR R0(k1)=0 ,1 ,...,D
SET ))s0 =(d ,R0(k1 , C0*(s0)=md, 0N0(s0)= /*參見公式(3-3)*/
ENDFOR ENDFOR
歩驟3.3：根據定理 3.3。當Dt =1，1≤t ≤T ，狀態為st =(1, (R kt t+1) 0)= ，計算 最佳成本與最佳投料量。
FOR t =1 ,2,...,T
IF Ct*−1(st−1 =(1,0)) [≤ + +α β ph t( 1)−
θ
+ −(1 p C) t*−1(st−1 =(1,1))] [1/ −p(1−θ
)]THEN (N st t =(1,0)) 0= , and C st*( t =(1,0))=Ct*−1(st−1 =(1,0)) ELSE
( (1,0)) 1
t t
N s = = , and
* *
1 1
( (1,0)) ( 1) (1 ) ( (1,0))
t t t t
C s = = + +α β ph t−
θ
+p −θ
C− s− = + −(1 p C) t*−1(st−1=(1,1)) ENDIFENDFOR
歩驟3.4：根據定理 3.4。當Dt =1，1≤t ≤T ，狀態為st =(1 ,Rt(kt+1)>0)，計算 最佳成本與最佳投料量。
FOR t =1 ,2,...,T
IF Ct*−1(st−1 =(1,0)) [≤ + +α β ph t( 1)−
θ
+ −(1 p)(1−θ
)Ct*−1(st−1 =(1,1))] {(1/ −θ
)[1−p(1−θ
)]}THEN N st( t =(1, (R kt t+1))) 0= ,
1
* *
1 ( ) 1 1
( (1, ( ))) ( 1) [ ] (1 ) ( (1,0))
t t
t t t t R k t t
C s = R k+ =h t− E Y + + −θ C− s− = ELSE N st( t =(1, (R kt t+1))) 1= ,
1
*
1 ( )
( (1, ( ))) ( 1) [ ] ( 1)
t t
t t t t R k
C s = R k+ = + +α β h t− E Y + +ph t− θ
2 * *
1 1 1 1
(1 ) t ( t (1,0)) (1 )(1 ) t ( t (1,1)) p
θ
C− s− pθ
C− s−+ − = + − − =
ENDIF ENDFOR
歩 驟 3.5 ： 根 據 公 式 (3-1) 和 定 理 3.2 。 當 1≤t≤T−1 ， 狀 態 為
歩驟3.6：根據公式(3-1)和定理 3.2。在期初( )T ，狀態為sT = D( ≥2 ,0)，計算最
附錄 B
OPS 演算法如下：
歩驟4.1：根據 4.2.4 節第一個邊界條件。當1 t T≤ ≤ ，狀態為st =(0 ,Bt)，設定最 佳成本與最佳投料量。
FOR 1, t= 2, ..., T
FOR 0, Bt = 1, ..., (T t D− ⋅ )
SET N st*( t =(0, )) (0, 0)Bt = , C st*( t =(0, )) 0Bt = /*參見公式(4-2)和(4-3)*/
ENDFOR ENDFOR
歩驟4.2：根據 4.2.4 節第二個邊界條件。當t =0，狀態為s0 =(D0,B0))，設定最 佳成本與最佳投料量。
FOR D0 =1, 2, ...,D FOR ...B0 =0 ,1 , ,T D⋅
SET C0*(s0 =(D0,B0))=mD0, )N0*(s0 =(D0,B0))=(0,0 /*參見公式(4-4)*/
ENDFOR ENDFOR
歩驟4.3：根據定理 4.1。當t=1，狀態為s1=(D1≥1, B1 ≥ ，計算最佳成本與第0)
歩 驟 4.4 ： 根 據 公 式 (4-1) 、 定 理 4.3 及 定 理 4.4 。 當 2≤ ≤ − ，狀態為t T 2
歩 驟 4.5 ： 根 據 公 式 (4-1) 、 定 理 4.3 及 定 理 4.4 。 當 t T= −1， 狀 態 為
歩驟4.6：根據公式(4-1)、定理 4.3 及定理 4.4。當t T= ，狀態為