wagner whitin method

8/18/2019 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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"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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"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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"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

wagner whitin method

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