Download - Wagner Whitin Method
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The Wagner-Whitin Model
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EOQ Assumptions
1. Instantaneous production.
2. Immediate delivery.
3. Deterministic demand.
4. Constant demand.
5. Known fied setup costs.
!. "in#le product or separa$le products.
WW model relaxes this one
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Dynamic %ot "i&in# 'otation
t a period (e.#.) day) wee*) mont+,- we will consider t 1) / )T )w+ere T represents t+e planning horizon.
Dt demand in period t (in units,
ct unit production cost (in dollars per unit,) not countin# setup or
inventory costs in period t
At fied or setup cost (in dollars, to place an order in period t
ht +oldin# cost (in dollars, to carry a unit of inventory from
period t to period t 01
Qt t+e unknown si&e of t+e order or lot si&e in period t
decision variables
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a#ner+itin Eample
Data
%otfor%ot "olution
t 1 2 3 4 5 ! 6 7 18
Dt
28 58 18 58 58 18 28 48 28 38
ct
18 18 18 18 18 18 18 18 18 18
At
188 188 188 188 188 188 188 188 188 188
ht 1 1 1 1 1 1 1 1 1 1
t 1 2 3 4 5 ! 6 7 18 9otal
Dt 28 58 18 58 58 18 28 48 28 38 388
Qt
28 58 18 58 58 18 28 48 28 38 388
I t 8 8 8 8 8 8 8 8 8 8 8
"etup cost 188 188 188 188 188 188 188 188 188 188 1888
:oldin# cost 8 8 8 8 8 8 8 8 8 8 8
9otal cost 188 188 188 188 188 188 188 188 188 188 1888
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a#ner+itin Eample (cont.,
;ied Order Quantity "olution
t 1 2 3 4 5 ! 6 7 18 9otal
Dt
28 58 18 58 58 18 28 48 28 38 388
Qt
188 8 8 188 8 8 188 8 8 8 388
I t
68 38 28 8 28 18 78 58 38 8 8
"etup cost 188 8 8 188 8 8 188 8 8 8 388
:oldin# cost 68 38 28 8 28 18 78 58 38 8 488
9otal cost 168 38 28 18 28 18 178 58 38 8 88
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a#ner+itin
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=asic Idea of a#ner+itin Al#orit+m
By WW Property I, either t !" or t !#t +$+#% for some % . If
& % * = last period of prodution in a % period pro!lem
then "e "ill produe e#atly #% +$#' in period & % *.
We an then onsider periods 1( $ ( & % ) -1 as if they are an independent & %
) -
1 period pro!lem.
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a#ner+itin Eample
"tep 1> $!%iously, &ust satisfy #' (note "e are negleting prodution
ost, sine it is fi#ed).
"tep 2> T"o hoies, either & * = ' or &
* = .
1
188
?
1
1
?
1
=
==
j
A Z
1
158
288188188
158,58(1188
min
2in produce )@
1in produce)min
?
2
2
?
1
211?
2
=
=
=+
=+
=
+
+=
j
A
Dh A Z
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a#ner+itin Eample (cont.,
"tep3> Three hoies, & +* = ', , +.
1
18
258 188158218 18,1(188188
1818,11(,58(1188
min
3in produce )@
2in produce )@
1in produce ),(
min
?
3
3?2
322
?
1
321211?
3
=
=
=+=++=+++
=
+
++
+++
=
j
A
Dh A
Dhh Dh A
Z
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a#ner+itin Eample (cont.,
"tep 4> our hoies, & 4* = ', , +, 4.
4
28
28 18818
388 58,1(188158
318 58,11(18,1(188188
32858,111(18,11(,58(1188
min
4in produce )@3in produce )@
2in produce ),(@
1in produce),(,(
min
?
4
4
?
3
433
?
2
432322
?
1
4321321211
?
4
=
=
=+
=++
=++++
=++++++
=
+++
++++
++++++
=
j
A Dh A
Dhh Dh A
Dhhh Dhh Dh A
Z
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a#ner+itin Eample (cont.,
"tep 5> $nly t"o hoies, & * = 4, .
"tep !> Three hoies, & * = 4, , .
And so on.
4
328
38 18828328,58(118818min
5in produce )@
4in produce)min
?
5
5
?
4
544
?
3?
5
=
=
=+=++=
+
++=
j
A
Dh A Z Z
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a#ner+itin Eample "olution
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a#ner+itin Eample "olution (cont., Optimal
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Inventory vs 9ime in (Q)r , Codel
I n
% e n t or y
Time
r
l