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Recursive Algorithm for finding Optimal Product Categories in Inventory Management
Introduction
In [1] p.21 describes the problem of optimal categories in inventory management. The overall optimal solution is to create a category for each product with its optimal order frequency. Normally an inventory contains several 10.000 products. So the overall optimal solution (with a large number of categories) never is useable is practice. The highest number of categories possible to administrate lies somewhere around 10. The problem is to find the optimal categories for a given maximal number of categories. Nowadays most inventory administrators use 3 categories result from ABCanalysis. A better approach is the Kcurve method computing the optimal categories for a given set of order frequencies the inventory administrator wants to implement. Normally the input set of order frequencies is a geometrical sequence with factor 2. For special distibutions of the products in the inventoy the Kkurve method returns the optimal categories.
([1] p. 61 chap 4.3) The algorithm described here determinesd optimal categories without any preassumptions on the distribution of the products. Input is the size of the first or last category. The other categories are computed recursively.
Optimal Categories
Let P be the set of all products in an inventory and α(p) be the usage value for a product p ∈P. The products can be ordered such that p<q iff α(p) > α(q). So without loss of generality we can assume that P is a closed interval in the set of real numbers R starting with 0 and ending with p_{max }. The Function α(p) is a non continious step function in R.
Lemma 1
^{α}ε(p∈For every e>0 exists a smooth function ) C°°(P), such that
^{ε}αα20(x)(x)dxpmax ∫ < * ε .
Proof First construct a continious function α'(p) (easily to do for a stepwise linear function) with
^{}'(x)(x)dxpmax∫ <
0
ε
α
Then use the Standardtheorem from Differential Topology stating that C°°(P) lies dense in C°°(P) (s.[2] p.44).
α
1
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Using the lemma above we now can assume that α(p) is a smooth function and all operations used here are defined.
A partition of the inventory into m+1 categories is given by a tupel of m values in P. Let P =(p_{1},...,p_{m}) be such a partition. We denote
1
1
0 0  ^{=}==+  i
i
i
max
^{p}:,p:p,d:pm p and
p
α
The costfunction defined in [1] of the inventory to minimize is:
Zc= order costs + storage value = n_{ord} + s. For every category j the cost function is:
)
p
(
=
∫
α
(
dx
)
x
p
(
dx
)
x
α
α
=

p
p
i
i
∫
0
1
α
(
)
=
α
)
p
(
i
i
i

1

F
F
d
K
s
n
Z α
The optimal order frequence Fj* for every category ([1] (3.3) p.21) is:
j
j
j
j
j
ord
c
j
+
+
=
⋅
⋅
j
j
Z
Z
=
j
c
c
j
∑=
j
*
j
d
K
F
⋅
= α
.
If the optimal order frequency is used then
j
j
j
^{n}sKod d
j
=
=
⋅
⋅
α
.
For a specific inventory the constant K is fixed, depending on ordering cost rates and capital rents. So the function to to optimize is
d
(
Z
d
,...,
m
i
∑
i
i
m
^{)})=+ ⋅
+
=
1
1
d
(
α
^{1}1 where
(a)
j
max...