866:
denote the minimal cost program for periods 1 to t. If at period t* the minimum in F(t) occurs for j = t** ≤ t*, then in periods t > t* it is sufficient to consider only t** ≤ j ≤ t. In particular, if t* = t**, then it is sufficient to consider programs such that
480:
260:
1081:
EA Silver, HC Meal, A heuristic for selecting lot size quantities for the case of a deterministic time-varying demand rate and discrete opportunities for replenishment, Production and inventory management,
862:
578:
269:
1159:
1112:
Federgruen, Awi, and Michal Tzur. "A simple forward algorithm to solve general dynamic lot sizing models with n periods in 0 (n log n) or 0 (n) time."
146:
643:
1185:
1005:
966:
520:
1121:
Jans, Raf, and Zeger
Degraeve. "Meta-heuristics for dynamic lot sizing: a review and comparison of solution approaches."
1145:
1133:
1114:
1105:
993:
1164:
955:
to the costs of acting optimally for periods 1 to t**-1 determined in the previous iteration of the algorithm
987:
485:
39:
1096:
974:
42:
model that takes into account that demand for the product varies over time. The model was introduced by
1048:, "Dynamic version of the economic lot size model," Management Science, Vol. 5, pp. 89–96, 1958
1169:
891:
73:
59:
1070:
Economic lot sizing: an O (n log n) algorithm that runs in linear time in the Wagner-Whitin case
1045:
629:
Given that I = 0 for period t, it is optimal to consider periods 1 through t - 1 by themselves
47:
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35:
898:
Consider the policies of ordering at period t**, t** = 1, 2, ... , t*, and filling demands
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475:{\displaystyle f_{t}(I)={\underset {x_{t}\geq 0 \atop I+x_{t}\geq d_{t}}{\min }}\left}
17:
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1065:
958:
From these t* alternatives, select the minimum cost policy for periods 1 through t*
91:
638:
The precedent theorems are used in the proof of the
Planning Horizon Theorem. Let
1100:
137:
to order now to minimize the sum of setup cost and inventory cost. Let us denote
1140:
1128:
970:
77:
72:
over a relevant time horizon t=1,2,...,N (for example we might know how many
887:
138:
255:{\displaystyle I=I_{0}+\sum _{j=1}^{t-1}x_{j}-\sum _{j=1}^{t-1}d_{j}\geq 0}
126:
can also vary with time if desired). The problem is how many units
1131:
and T. Whitin, "Dynamic version of the economic lot size model,"
1160:
Solving the Lot Sizing
Problem using the Wagner-Whitin Algorithm
1143:: "Comments on Dynamic version of the economic lot size model",
1072:." Operations Research 40.1-Supplement - 1 (1992): S145-S156.
264:
The functional equation representing minimal cost policy is:
76:
will be needed each week for the next 52 weeks). There is a
857:{\displaystyle F(t)=\min \left \atop s_{t}+F(t-1)}\right]}
488:. Wagner and Whitin proved the following four theorems:
537:
646:
573:{\displaystyle x_{t}=\textstyle \sum _{j=t}^{k}d_{j}}
523:
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There exists an optimal program such that ∀t: either
272:
149:
1101:
A dynamic lot-sizing model with demand time windows
856:
572:
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254:
90:incurred for each order and there is an inventory
969:, a number of authors also developed approximate
992:Constant fill rate for the part being produced:
986:Infinite fill rate for the part being produced:
662:
1004:Several products produced on the same machine:
965:Because this method was perceived by some as
583:There exists an optimal program such that if
8:
909:, t = t**, t** + 1, ... , t*, by this order
492:There exists an optimal program such that I
820:
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616:, t=t**+1,...,t*-1, is also satisfied by
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1123:European Journal of Operational Research
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961:Proceed to period t*+1 (or stop if t*=N)
1027:
1149:, Vol. 50 No. 12 Suppl., December 2004
7:
1095:Lee, Chung-Yee, Sila Çetinkaya, and
890:for finding the optimal solution by
670:
300:
25:
1172:of the Wagner-Whitin algorithm.
1006:Economic lot scheduling problem
27:Mathematical model in economics
1137:, Vol. 5, pp. 89–96, 1958
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38:, is a generalization of the
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994:Economic production quantity
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886:Wagner and Whitin gave an
60:forecast of product demand
1125:177.3 (2007): 1855–1875.
1109:47.10 (2001): 1384–1395.
634:Planning Horizon Theorem
988:Economic order quantity
486:Heaviside step function
40:economic order quantity
1186:Inventory optimization
1165:Dynamic lot size model
858:
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32:dynamic lot-size model
18:Dynamic lot size model
1170:Python implementation
1118:37.8 (1991): 909–925.
975:Silver-Meal heuristic
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594:is satisfied by some
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104:per item per period (
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58:We have available a
1097:Albert PM Wagelmans
977:) for the problem.
894:. Start with t*=1:
892:dynamic programming
1146:Management Science
1134:Management Science
1115:Management Science
1106:Management Science
854:
694:
605:, t**<t*, then
580:for some k (t≤k≤N)
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1058:Wagelmans, Albert
1046:Thomson M. Whitin
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673:
484:Where H() is the
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48:Thomson M. Whitin
16:(Redirected from
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1042:Harvey M. Wagner
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1016:Base stock model
1000:Newsvendor model
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1011:Reorder point
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882:The algorithm
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54:Problem setup
53:
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33:
19:
1144:
1132:
1122:
1113:
1104:
1077:
1066:Antoon Kolen
1053:
964:
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865:
640:
637:
483:
266:
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143:
92:holding cost
57:
31:
29:
1141:H.M. Wagner
1129:H.M. Wagner
973:(e.g., the
967:too complex
1022:References
971:heuristics
78:setup cost
888:algorithm
839:−
801:−
743:∑
734:−
716:∑
682:≤
540:∑
450:−
367:−
337:≥
313:≥
247:≥
229:−
211:∑
207:−
189:−
171:∑
139:inventory
50:in 1958.
1180:Category
981:See also
878:> 0.
74:widgets
1064:, and
912:Add H(
517:=0 or
503:=0; ∀t
1082:1973
1044:and
688:<
115:and
46:and
30:The
1103:."
1099:. "
1068:. "
951:t**
941:t**
930:t**
919:t**
675:min
663:min
623:t**
601:t**
298:min
34:in
1182::
1060:,
1030:^
874:t*
590:t*
141::
947:I
937:i
934:+
926:s
923:)
915:x
905:t
901:d
870:x
851:]
845:)
842:1
836:t
833:(
830:F
827:+
822:t
818:s
811:]
807:)
804:1
798:j
795:(
792:F
789:+
784:k
780:d
774:h
770:i
764:t
759:1
756:+
753:h
750:=
747:k
737:1
731:t
726:j
723:=
720:h
712:+
707:j
703:s
698:[
691:t
685:j
679:1
667:[
660:=
657:)
654:t
651:(
648:F
619:x
612:t
608:d
597:x
586:d
565:j
561:d
555:k
550:t
547:=
544:j
535:=
530:t
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499:t
495:x
469:]
464:)
458:t
454:d
445:t
441:x
437:+
434:I
430:(
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384:(
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355:[
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293:=
290:)
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284:(
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218:=
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202:j
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158:I
154:=
151:I
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100:t
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20:)
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