31
NZOR Volume 11 Number 1
January 1983
A MARGINAL DISTRIBUTION OF LEAD TIME DEMAND
BASED ON A DISCRETE LEAD TIME DISTRIBUTION
H AR RY
G.
STANTON
GRADUATE SCHOOL OF BUSINESS ADMINISTRATION
UNIVERSITY OF MELBOURNE, PARKVILLE 3052
AUSTRALIA
SUMMARY
In this paper we develop a probability distribution of demand during lead
time in which both demand and lead time are discrete variables.
The choice of
discrete lead times is based on the fact that, in practice, time periods in in­
ventory transactions tend to be recorded by date, rather than as continuous
variables.
The method is illustrated by an example, using a 'lumpy demand'
inventory system, characterized by very low demand rates and infrequent procure­
ments.
INTRODUCTION
The m a r g in a l d i s t r i b u t i o n of lead tim e d e m a n d has bee n d e f i n e d
by H a d l e y a n d W h i t i n [ 1] as t h e j o i n t p r o b a b i l i t y d i s t r i b u t i o n
P(x)
=
£
p (x|y)
y=0
D
P„(y)
T
(1)
w h e r e P^(x|y) r e p r e s e n t s the c o n d i t i o n a l p r o b a b i l i t y t h a t d e m a n d
wi l l be x units, g i v e n t h a t t h e l e a d t i m e h a s t h e v a l u e y, a n d
PT (y) is t h e p r o b a b i l i t y t h a t l e a d t i m e d u r a t i o n w i l l be y t i m e
u n i t s on t h a t o c c a s i o n .
A m o d e l e x t e n s i v e l y u s e d b y H a d l e y a nd
W i t h i n a s s u m e s t h a t d e m a n d is g e n e r a t e d b y a P o i s s o n p r o c e s s a nd
t h a t the p r o b a b i l i t y d e n s i t y for l e a d t i m e y is t h e g a m m a d i s t r i b u ­
t i o n w i t h p a r a m e t e r s (a,b).
The r e s u l t i n g m a r g i n a l d i s t r i b u t i o n
of le a d t i m e is a n e g a t i v e b i n o m i a l d i s t r i b u t i o n
bN (x;a+l,b / ( b + A ) ]
(2)
w h e r e A is the d e m a n d rate.
T h i s m o d e l a s s u m e s t h a t u n i t s a re d e ­
m a n d e d o n e at a time.
D e m a n d is t r e a t e d as a d i s c r e t e v a r i a b l e ,
a n d l e a d t i m e as a c o n t i n u o u s v a r i a b l e .
S i nce, in p r a c t i c e , i n ­
v e n t o r y c o n t r o l t r a n s a c t i o n s a r e n o r m a l l y r e c o r d e d by d a t e only,
i n f o r m a t i o n on p a s t l e a d t i m e s w o u l d be a v a i l a b l e in t he f o r m of
d i s c r e t e data, w i t h o n e d a y as t h e b a s i c u n i t o f time.
F o r this
Manuscript submitted January 1982, revised July 1982.
32
r e a s o n a d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n o f l e a d t i m e w o u l d be a
b e t t e r r e p r e s e n t a t i o n of r e al i t y than a c o n t i n u o u s d i s t r i bution.
T h e m a i n a i m o f t h i s p a p e r is t o d e v e l o p a g e n e r a l e x p r e s s i o n
for t h e m a r g i n a l d i s t r i b u t i o n of l e a d t i m e d e m a n d , w h e r e b o t h d e ­
m a n d and lead time are s ub j e ct to di s c r e t e p r o b a b i l i t y d i s t r i b u ­
ti o n s .
T h i s m o d e l w i l l b e a p p l i e d to t h e p a r t i c u l a r c a s e o f a
' l u m p y d e m a n d ' i n v e n t o r y s y s t e m , w h i c h is c h a r a c t e r i z e d b y v e r y l o w
d e m a n d r a t e s (e.g. f e w e r t h a n 20 u n i t s s o l d p e r a n n u m ) , h i g h un i t
v a l u e (of t h e o r d e r o f s e v e r a l h u n d r e d or e v e n t h o u s a n d d o l l a r s p er
un i t ) , a n d s m a l l p r o c u r e m e n t s (e.g. s t o c k r e p l e n i s h m e n t o r d e r s for
o n e or t w o u n i t s ) , w h i c h a r e p l a c e d i n f r e q u e n t l y .
Conditions such
as t h e s e a r e o f t e n e n c o u n t e r e d by s u p p l i e r s of e x p e n s i v e s p a r e
p a r t s for m a c h i n e r y a n d t r a n s p o r t a t i o n e q u i p m e n t .
Demand Function
It w i l l be a s s u m e d t h a t d e m a n d t e n d s t o b e t o t a l l y u n p r e d i c ­
t able, n e i t h e r s e a s o n a l n o r c y c l i c a l v a r i a t i o n s b e i n g e v i d e n t .
D e m a n d is n o t t h o u g h t to be a s s o c i a t e d w i t h a n y p r e v e n t a t i v e m a i n ­
t e n a n c e p r o g r a m s o r o t h e r s c h e m e s w h i c h e n t a i l t he r e p l a c e m e n t o f
p a r t s a n d c o m p o n e n t s at r e g u l a r i n t e r v a l s .
It m a y be r e a s o n a b l e
t o e x p e c t a s e c u l a r l o n g - t e r m t r e n d in t h e d e m a n d rate;
however,
it m a y n o t be r e a d i l y p o s s i b l e t o d e t e r m i n e t h e u n d e r l y i n g r a t e of
g r o w t h o r d e c l i n e , as the l o w o r d e r f r e q u e n c y a n d i n h e r e n t i r r e g u ­
larity of d e m a n d w o u l d render any time series analysis of past d e ­
m a n d e x p e r i e n c e of d o u b t f u l value.
U n d e r t h e s e c i r c u m s t a n c e s , t h e a s s u m p t i o n t h a t d e m a n d is b e i n g
g e n e r a t e d b y a P o i s s o n p r o c e s s w o u l d c e r t a i n l y be a p p r o p r i a t e .
L e t <5 be t h e e x p e c t e d d a i l y d e m a n d .
The c o nd i t i o n a l
prob­
a b i l i t y o f d e m a n d b e i n g j u n i t s on an o c c a s i o n w h e n t h e le a d ti m e
d u r a t i o n is k d a y s is t h e n
-k6
(kfi)^
“ IT"
(3)
Lead Time Function
L e t a d e n o t e the r a t i o o^/t, w h e r e t is the m e a n a n d o^. the
v a r i a n c e o f l e a d time.
T h e m o d e l a i m e d for s h o u l d p e r m i t a w i d e
r a n g e o f v a l u e s t o be u s e d f o r t h i s r a t i o - c e r t a i n l y i n c l u d i n g
v a l u e s l e s s t h a n u n i t y a n d v a l u e s g r e a t e r t h a n one.
When s e lect­
i ng a d i s c r e t e d i s t r i b u t i o n t h a t w i l l r e a s o n a b l y c l o s e l y r e s e m b l e
the g a m m a d i s t r i b u t i o n , t h e v a l u e of a d e t e r m i n e s the c h o i c e .
If
a e x c e e d s u n i t y , t h e n e g a t i v e b i n o m i a l d i s t r i b u t i o n c o u l d be used.
F o r a e q u a l l i n g u n i t y , t h e P o i s s o n d i s t r i b u t i o n c o u l d be c o n s i d e r ­
ed, a n d for t h e c a s e o f a<l, t h e o r d i n a r y b i n o m i a l d i s t r i b u t i o n
w o u l d be s u i t a b l e .
Marginal Distribution of Lead Time Demand
It is c o n v e n i e n t to s p e c i f y t h e m a r g i n a l d i s t r i b u t i o n of lead
t i m e d e m a n d in t e r m s o f t h e t w o l e a d t i m e s p a r a m e t e r s t a n d a, and
33
the d e m a n d r a t e 6.
U s i n g e q u a t i o n (1), t h i s d i s t r i b u t i o n t a kes
the f o l l o w i n g f o r m if t h e o r d i n a r y b i n o m i a l , P o i s s o n a n d n e g a t i v e
b i n o m i a l d i s t r i b u t i o n s a r e u s e d for t h e a p p r o p r i a t e r a n g e s o f a:
(i)
f o r a<l
P(j>
- q n ♦ I e - k6
k= 1
where
(ii)
(4 :
p = 1 -a;
for a = l
-k6
P (j ) = e
(iii)
M p k q" - k
(k6)3
+ I «
k=l
(5)
k!
for a > l
P(j)
= pn + I
k=l
where
n—
—
3p = —
c
a
;
+— i-I p n q k
(6)
K'
n = — ^-sa-1
W h e n u s i n g the o r d i n a r y b i n o m i a l d i s t r i b u t i o n (4), t h e r e is a
c o n s t r a i n t r e g a r d i n g t h e c h o i c e of a, in so f a r as t h e la s t t e r m
in t he b i n o m i a l e x p a n s i o n p n s h o u l d h a v e d e c a y e d s u f f i c i e n t l y so
as to m a k e a n y f u r t h e r t e r m s n e g l i g i b l y sma l l .
T h e c o n d i t i o n for
this c a n be w r i t t e n as
(1 - a ) T / < 1 - “ >
< E
(7)
F o r e x a m p l e , if the v a l u e o f E is c h o s e n 10 7 t h e n t h e m i n i m u m
a c c e p t a b l e a v a l u e for a g i v e n m e a n of t h e l e a d t i m e d i s t r i b u t i o n
t is set o u t in the t a b l e bel o w :
Table J.
T
a min
1
2
5
10
20
50
0. 8723
0.8002
0.6628
0.5307
0.3889
0.2218
Minimum acceptable a values.
Th e use o f the t h r e e t y p e s o f d i s c r e t e p r o b a b i l i t y d i s t r i b u ­
t i o n s for r e p r e s e n t i n g l e a d t i m e is b e s t i l l u s t r a t e d by an e x a m p l e .
S u p p o s e t h a t the b e s t e s t i m a t e o f t h e a v e r a g e l e a d ti m e t is 20
days, a n d the a v e r a g e d a i l y d e m a n d 6 is 0. 0 4 u n i t s .
Table 2 shows
34
t he P (j ) t e r m s o f t h e m a r g i n a l
f or a w i d e r a n g e o f p o s s i b l e a
u s e d for a r a n g i n g f r o m 1.2 t o
ot= l, and the o r d i n a r y b i n o m i a l
d i s t r i b u t i o n of l e a d t i m e d e m a n d
values.
The n e g a t i v e b i n o m i a l was
20, t h e P o i s s o n d i s t r i b u t i o n for
f o r a = 0 . 6 a n d 0.8.
A n i n t e r e s t i n g c o n c l u s i o n , s u g g e s t e d b y the d a t a in T a b l e 2,
is t h a t for s m a l l 6 v a l u e s t h e s t a n d a r d d e v i a t i o n o f l e a d t i m e d e ­
m a n d o x is n o t v e r y s e n s i t i v e t o c h a n g e s in t h e l e a d ti m e v a r i a n c e /
m e a n r a t i o a.
F o r e x a m p l e , w h e n a r a n g e s f r o m 0.6 and 2.0, the
v a l u e o f a x r e m a i n e d w i t h i n 0 . 9 0 5 a n d 0.930.
T h i s p r o p e r t y o f the
l u m p y d e m a n d s y s t e m , w h i c h is c h a r a c t e r i z e d b y v e r y l o w <5 v a l u e s ,
i n d i c a t e s t h a t the P o i s s o n d i s t r i b u t i o n m a y p r o v i d e a u s e f u l a p p r o x i ­
m a t i o n in s u c h cases.
It w i l l n o w be
Poisson distributed
calculation.
s h o w n t h a t t h e P (j ) d i s t r i b u t i o n b a s e d on
lead time can be c o n v e n i e n t l y so l v e d by ma n u a l
T he m e a n a n d s t a n d a r d d e v i a t i o n o f t h e m a r g i n a l d i s t r i b u t i o n
o f l e a d t i m e d e m a n d , as d e f i n e d in e q u a t i o n (5), are:
U =
Writing
<f) = xe
^ , equation
P(3)
(5)
fix
(8)
a x = '/St (1+6)
(9)
can
=
be
£
rewritten
I ki e - *
k=0
£
(10)
a n d h e n c e P(0) = e^ l.
F o r v a l u e s o f j=l or g r e a t e r , it is p o s ­
s i b l e to e x p r e s s e q u a t i o n (10) in t e r m s of a j fch o r d e r p o l y n o m i a l
in <}> :
P(j)
1-T
- e*-
^
"
J
?
.
a.
„m
(11)
m=l
where
the terms
aj (Itl c a n
be
found
from the
recurrence
a.
= m a . ,
+ a . .
,
3,m
] - 1 ,m
j-1,m-l
A derivation
of equations
(11)
and
(12)
is
set
relationship
(12)
out
in the A p p e n d i x .
T h e m a t r i x a^
c a n be b u i l t u p f r o m the k n o w n v a l u e of the
i n i t i a l t e r m in t n e s e r i e s a ^ (j = 1, a n d the fact that w h e n the
v a l u e of m e x c e e d s ( j - 1 ) , t h e c o e f f i c i e n t b e c o m e s zero.
A list
o f a j (m c o e f f i c i e n t s for s m a l l v a l u e s o f j a n d m is set o u t in
T a b l e 3.
35
r^moO(NrnoOvJ,CNOvDrHrHf-H
inHHHhvDtNCOCOnHOO
'^)LT)00>rHCTt^J1r H O O O O O
L O^ r —
I^ C NOOOOOOOO
oo oo
i n c N r H o o o o o o o o o o oo oo
oo
oo
oo
oo
oo
oo
o o o o o o o o o o o o o o o o o o
o o o o o o o o o o
r^a>r-r^oororocor^fH
lai^irn^Gc iofrl-^^orncH^
OO
omooo
i( T»^)
D ^ ^ 00<r
r ^ r o»ro H0000
oooo
•^mrHOOOOOOO
oooooooooo
c DHi n ^ h ^ o ^ Hi n H
rC -OoC rN-'^^vf £O O>ro'rC^N^OoOoO
r-cr>rHOcr>rHoooo
m
roooooo
^ • ^r nt r^- ^i OOOOOOO
o o o o o o o o o o
OLnoLncornr^vDLnrH
co^vD^rnojinrnoo
-S'COCOOCOvOC'JOOO
vD O fH O OO rH O OO O
LnLn^r^roooooo
^ r r n r H O O O O O O O
o o o o o o o o o o
H iDiniDooHr'fM ^
ro-m
^ ro^irhHo
^ ror nj omo
^m
o
OOOOOO
o o o o o o o o o
mr\ifNO'COHOOO
- - — P 0 O O O O O
r- m o r"~ co ro
in
co m vDrH<No
vD^cmvDro
^3* CM O O
fLDLO^r
^M' t N^COHOOO
nOOOOO
^ r r OnHOOOOOO
o o o o o o o o o
O H f M r o ^ , in»x)t^cocT'OHrsjrorj, in^)r^
time
lead
of
HrOvDvDLOHLDCNlOH
hm rc ^M< nr »oi^nion>ho'rDo H
O
ooo
nfNO^HO^fNOOOO
v• ^D^
r
o
^
T
OOOOOO
rOrHOOOOOOO
distribution
o o o o o o o o o o o
Marginal
CNO^^rromooooo
coogrn^rHOOOOOo
•^rOrHOOOOOOOO
S.
Ratio
of
Lead
Time
Variance/Mean
(a,/T)
^ ^ MC D O ^ C O H O O O
Table
r^H^rcom^ra^co^rocN
demand.
^Onmi^LnOi-HrHa^VDrH
I) u) l^- Lf) »—1 i—I O'
O O O
COrMmcOOCOO>'
X)rHOOOOO
oomr^r-ini-ioo^^^^^
fL H^
CN^
f
HOOOO
DCNJ r HOOOOOO
o o o o o o o o o o o o o o
36
*'~a
£
37
Thus the ma r g i n a l
f o l l o w i n g form:
distribution
of
lead time
demand
takes
the
-T
P(0)
= e
P(l)
= e^
P (2)
= e^~T
T 6 <p
2
<f>(l+<(»)
-6
e t c . , where
REFERENCE
Hadley, G. and whitin, T. M.
Hall, New Jersey.
Analysis of Inventory Systems, Prentice-
(1963).
APPENDIX
Determination
o f a.
Coefficients
3 ,m
The m a r g i n a l d i s t r i b u t i o n
Equation
(10)
of
lead
time
demand was
defined
P (j ) = e
in
( 1. 1)
where
k
Wj (<*»)
£
k=0
The
t e r m k 3 in E q u a t i o n
( 1. 2)
k!
(1.2)
can be r e w ritten
thus:
(1.3)
s=0
Since
the
series
s e r i e s in (es ),
t e r m to gi v e
£ e ks e ^
it can
can
be
r e g a r d e d as a p o w e r
be d i f f e r e n t i a t e d
successively term-by-
38
W . (<(>) = e
-<b d '
ds
W.(«
3
» e"*
(<j) e
k!
k =0
s , k
)
s= 0
(1.4)
^
l e (* e!
ds3
L e t x = <p e s a n d t h e v a l u e
b y x (0) .
s=0
of x w h e n
s equals
z e r o be d e n o t e d
(1.5)
x = <j> e
x(0)
= <f> e ‘
( 1 . 6)
s=0
(1.7)
W . (<{>) = e ^ -— r (eX )
ds3
s=0
T h e d i f f e r e n t i a l t e r m in E q u a t i o n
t h e f o l l o w i n g way:
d
|3~1
, x
(e )
s=0
ds
j-1
(1.6)
ru
ds
J"1
ds
j-1
ds j-1
ds
j -2
c a n be e x p a n d e d
‘
in
/ x vi
(e ) ]
s=0
X dXj
ds1
(x e
s=0
)
s=C
ds (x e
)]
s=0
ds
The right h a n d side of the
p o l y n o m i a l o f the f o r m
j-2
[ x (1+x)
above equation
e
s=0 a n d
forms
eX(aj , l X + aj, 2X' + aj . 3x3 + ••• + aj(jx3)
so on.
a jth o r d e r
s=0
39
Using Equations
(1.6),
polynomial
3 »J
3
w. (<}>) = Z
-i ,™
m
j
m= 1 a 3
3-3
becomes
*o
-©•
fd
+
3 r2
this
+
m
-e
fd
And hence
and
+
CN
-e
x<j
-I-
e^a..
(1.5)
(1 .8 )
v
j in
Z
a .
d>
n
D »n
m= 1
= e ^ T 4
3I
(1.9)
A r e l a t i o n s h i p b e t w e e n t h e a j (in c o e f f i c i e n t s can b e f o u n d if
w e c o n s i d e r the (j+l ) t h d e r i v a t i v e a n d c o m p a r e it w i t h the
j th d e r i v a t i v e .
S L L (ex )
e ( a . ix + a . 0x
ds-^
3
s=0
=
e
3
ds j +1
d . x.
d ¥ (e >
=
(e )
3
3
Q j (x)
(
a.
_x
3,1
2
+ a.
(1.9),
the
„x
3,2
3
s=0
J
,
+ a.
_x
4
] ,3
where
(a j m | m = ^
= m
relationship
=
a.
+ a. ,
j-1,m
j- lf m - 1
^a j m | m > ^ =
t h e i n i t i a l t e r m a 1(1
e q u a l s unity.
...)I n
|S=0
( j + l ) t h d e r i v a t i v e w i l l be
from which
)
,
S
s=(
=0
c a n be e s t a b l i s h e d ,
a. ,
= m a .
+a.
,, w h i c h
3 + 1,m
j,m
j ,m - 1
as a .
j,m
.
...)
3a.
x 3 + 4a. .x4
3 ,e
3,4
e x ( a . il ,x + a.,, „ x 2 + a.,, „ x 3 + a. , Ax 4 ...
‘j+l , 1X + a j + l , 2V
j + 1,3
j
3+
+l
1,4
,4
"■
a recurrence
+
Qj (X)
+ e " as
+ e x (a.
x + 2a.
x2 +
3/!
3/2
a n d henc e ,
s=0
t h e s e r i e s (a-j^x + a j f2x
o f e x is g i v e n b y
L
from Equation
’
s =0
s=0
= e
But,
+ a . .xJ )
Q . (x)
If Qj (x) is u s e d to d e n o t e
The next h i g h e r d e r i v a t i v e
i 3 +1
+
3 >2
-1
can also be e x p r essed
r
It w i l l
be n o t e d
^a j m | m = j^ =
that
1 / and hence
all o t her terms
can b e d e r i v e d ,
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