A MARGINAL DISTRIBUTION OF LEAD TIME DEMAND BASED ON A DISCRETE LEAD TIME DISTRIBUTION

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1 31 NZOR Volume 11 Number 1 January 1983 A MARGINAL DISTRIBUTION OF LEAD TIME DEMAND BASED ON A DISCRETE LEAD TIME DISTRIBUTION HARRY 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 inventory 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 procurements. INTRODUCTION The m a r g i n a l d i s t r i b u t i o n of lead time d e m a n d has been d e f i n e d by Hadley and Whitin [ 1] as the joint probab i li t y distribution P(x) = p (x y) P (y) (1) y=0 D T 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 that d e m a n d will be x units, g i v e n that the lead time has the v a l u e y, and PT (y) is the p r o b a b i l i t y that lead time d u r a t i o n will be y time units on that occasion. A m o d e l e x t e n s i v e l y u s e d by H a d l e y and Within assumes that 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 and that the probability density for lead time y is the gamma d i s t r i b u tion with 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 lead time 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) where A is the demand rate. This m odel assumes that units are d e m a n d e d one at a time. D emand is t r e a t e d as a d i s c r e t e v a r i able, and lead time as a c o n t i n u o u s variable. Since, in practice, i n v e n t o r y c ontrol t r a n s a c t i o n s are n o r m a l l y r e c o r d e d by date only, i n f o r m a t i o n on past lead times w o u l d be a v a i l a b l e in the form of discrete data, w i t h one day as the b a s i c unit of time. For this Manuscript submitted January 1982, revised July 1982.

2 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 of lead time 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 a l i t y than a c o n t i n u o u s d i s t r i bution. The main a i m of this p a p e r is to 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 the marginal distribution of lead time demand, where both d e m and and lead time are subject to discrete probability d i s t r i b u tions. This m o d e l w i l l be a p p l i e d to the p a r t i c u l a r case of a 'lumpy demand' i n v e n t o r y system, w h i c h is c h a r a c t e r i z e d by v e r y low d e m a n d rates (e.g. f e wer than 20 units sold per annum), high unit v a l u e (of the o r d e r of several h u n d r e d or even t h o u s a n d d o l l a r s per unit), and small p r o c u r e m e n t s (e.g. s t ock r e p l e n i s h m e n t o r ders for one or two units), w h i c h are p l a c e d i n f r e quently. C o n d i t i o n s such as t h e s e are o f t e n e n c o u n t e r e d by suppliers of e x p e n s i v e spare parts for machin e r y and transportation equipment. Demand Function It will be assumed that demand tends to be totally un p r e d i c table, n e i t h e r s e a s o n a l nor c y c l i c a l v a r i a t i o n s b e i n g evident. Demand is not thought to be associated with any preventative m a i n t e n a n c e p r o g r a m s or o t her s chemes w h i c h e n t a i l the r e p l a c e m e n t of p a r t s and c o m p o n e n t s at r e g u l a r intervals. It m a y be reasonable to e x p e c t a secular l o n g - t e r m t r e n d in the d e m a n d rate; however, it m a y not be r e a d i l y p o s s i b l e to d e t e r m i n e the u n d e r l y i n g rate of growth or decline, as the low order frequency and inherent i rregularity of d e mand w ould render any time series analysis of past d e m a n d experience of doubtful value. U n d e r t h ese c i r c u m s t a n c e s, the a s s u m p t i o n that d e m a n d is b eing g ener a t e d by a Poisson process w ould certainly be appropriate. Let <5 be the expected daily demand. The conditional p r o b a b i l i t y of d e m a n d b e i n g j u n its on an o c c a s i o n w h e n the lead time d u r a t i o n is k days is then -k6 (kfi)^ I T " (3) Lead Time Function Let a d e n o t e the r a tio o^/t, w h e r e t is the m e a n and o^. the v a r i a n c e of lead time. The m o d e l a i m e d for s hould permit a wide range of v a l u e s to be u s e d for this r a t i o - c e r t a i n l y i n c luding values less than unity and values greater than one. When s e lecting a d i s c r e t e d i s t r i b u t i o n that will r e a s o n a b l y c l o s e l y resemble the g a m m a d i s t r i b u t i o n, the v alue of a d e t e r m i n e s the choice. If a e x c e e d s unity, the 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 uld be used. For a e q ualling unity, the Poisson distribution could be con s i d e r ed, a n d for the case of a<l, the o r d i n a r y binomial d i s t r i b u t i o n w o u l d be suitable. 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 the m a r g i n a l d i s t r i b u t i o n of lead time d e m a n d in terms of the two lead times p a r a m e t e r s t and a, and

3 33 the d e m a n d rate 6. U s i n g e q u a t i o n (1), this d i s t r i b u t i o n takes the following form if the o r d i n a r y b i n o mial, P o i s s o n and 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 are used for the a p p r o p r i a t e ranges of a: (i) for a<l P(j> - q n I e - k6 M p k q" - k k= 1 (4 : w h e r e p = 1 -a; (ii) for a=l -k6 (k6)3 P (j ) = e + I «k=l k! (5) (iii) for a>l P(j) = p n + I n + i-i p n qk (6) k=l 3- K ' w h e r e p = ; n = ^-sc a a-1 When 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 the choice of a, in so far as the last t e r m in the b i n o m i a l e x p a n s i o n p n should have d e c a y e d s u f f i c i e n t l y so as to make any further terms n e g l i g i b l y small. The c o n d i t i o n for this can be w r i t t e n as (1 - a ) T / < 1 - > < E (7) For example, if the v a l u e of E is chosen 10 7 then the m i n i m u m acceptable a value for a g i ven m e a n of the lead time d i s t r i b u t i o n t is set out in the table below: T a min Table J. Minimum acceptable a values. The use of the three types of discrete probability d i s t r i b u tions for r e p r e s e n t i n g lead time is best i l l u s t r a t e d by an example. Suppose that the best e s t i m a t e of the average lead time t is 20 days, and the average d a i l y d e m a n d 6 is 0.04 units. T a b l e 2 shows

4 34 the P (j ) t e rms of the m a r g i n a l d i s t r i b u t i o n of lead time d e m a n d for a w i d e r a nge of p o s s i b l e a values. The n e g a t i v e b i n o m i a l was u s e d for a r a n g i n g from 1.2 to 20, the Poisson d i s t r i b u t i o n for ot= l, and the ordinary binomial for a =0.6 and 0.8. An 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 by the data in Table 2, is that for small 6 values the standard deviation of lead time d e m a n d ox is n o t v e r y s e n s i t i v e to changes in the lead time v a r i a n c e / m e a n r a tio a. F o r example, w h e n a r a nges from 0.6 and 2.0, the v a lue of a x r e m a i n e d w i t h i n and This p r o p e r t y of the l u mpy d e m a n d system, w h i c h is c h a r a c t e r i z e d by v e r y low <5 values, indicates that the Poisson d i s tribution may provide a useful a pproxim ation in such cases. It will n o w be s h o w n that the P (j ) d i s t r i b u t i o n b a s e d on P o i s s o n d i s t r i b u t e d lead time can be c o n v e n i e n t l y solved by manual c a l c u l a t i o n. The m e a n and s t a n d a r d d e v i a t i o n of the m a r g i n a l d i s t r i b u t i o n of lead time demand, as d e fined in equation (5), are: U = fix (8) a x = '/St (1+6) (9) Writing <f) = xe ^, equation (5) can be rewritten P(3) = I ki e-* (10) k=0 and hence P(0) = e^ l. For values of j=l or greater, it is p o s sible to e x p r e s s e q u a t i o n (10) in terms of a jfch o r d e r p o l y n o m i a l in <}>: 1-T ^ P(j) - e * - "?. J a. m (11) m=l where the terms aj (Itl can be found from the recurrence relationship a. = m a., + a.., (12) 3,m ] - 1,m j-1,m-l A d e r i v a t i o n of e q u a t i o n s (11) and (12) is set out in the Appendix. The m a t r i x a^ can be b u i l t up from the known value of the initial t e r m in tne series a ^ (j = 1, and the fact that when the v a lue of m e x c e e d s ( j - 1 ), the c o e f f i c i e n t b e c o m e s zero. A list of a j (m c o e f f i c i e n t s for small v alues of j and m is set out in T a b l e 3.

5 35 r^moo(nrnoovj,cnovdrhrhf-h inhhhhvdtncoconhoo '^)LT)00>rHCTt^J1r H O O LO^r I ^ C N O O O O O O O O incnrhoooooooooo o o o o o o o o o o o o o o o o o o o o o o o ^Onmi^LnOi-HrHa^VDrH I)u) l^- Lf)» 1i IO' O O O COrMmcOOCOO>'X)rHOOOOO oomr^r-ini-ioo^^^^^ fh^cn^fhoooo LDCNJrHOOOOOO o o o o o o o o o o o o o o Ratio of Lead Time Variance/Mean (a,/t) r^h^rcom^ra^co^rocn ^^MCDO^COHOOO CNO^^rromooooo coogrn^rhoooooo ^ r O r H O O O O O O O O ooooooooooo HrOvDvDLOHLDCNlOH hcmno^inh'dho mr^<r»ino>oroooo nfno^ho^fnoooo vd^ro^toooooo ^ r O r H O O O O O O O oooooooooo r^a>r-r^oororocor^fh liingifl^onh^oo a^r^cor-^rcomooo id^^r^roooooo (T»^)00<r»rH0000 ^ m r H O O O O O O O oooooooooo cdhin^h^o^hinh r-or-^v >or^^oo COCN'^fOOr'CNOOO r-cr>rhocr>rhoooo m^t^^roooooo ^ rnr-iooooooo oooooooooo OLnoLncornr^vDLnrH co^vd^rnojinrnoo -S'COCOOCOvOC'JOOO v D O fh O O O rh O O O O LnLn^r^roooooo ^ r r n r H O O O O O O O oooooooooo Table S. Marginal distribution of lead time demand. H idinid ooh r'fm ^ r-^r^rho^rrnmo omoih^mojoo mr\ifno'cohooo - - P 0 O O O O O O ooooooooo r- m o r"~co ro in vdcmvdro ^ co m vdrh<no ^3* CMO O LDLO^rnOOOOO f^m'tn^cohooo ^rronhoooooo ooooooooo OHfMro^, in»x)t^coct'ohrsjrorj, in^)r^

6 36 *'~a

7 37 Thus the m a r g i n a l d i s t r i b u t i o n of lead time d e m a n d takes the following form: P(0) = e -T P(l) = e^ T 6 <p P (2) = e ^ ~ T 2 <f>(l+<(») e t c., w h e r e -6 REFERENCE Hadley, G. and whitin, T. M. (1963). Hall, New Jersey. Analysis of Inventory Systems, Prentice- AP P E N D I X D e t e r m i n a t i o n of a. C o e f f i c i e n t s 3,m The m a r g i n a l d i s t r i b u t i o n of lead time d e m a n d was d e f i n e d in E q u a t i o n (10) P (j ) = e ( 1. 1 ) where Wj (<*») k = 0 k k! ( 1. 2 ) The term k 3 in Equation (1.2) can be r e w ritten thus: s=0 (1.3) Since the series e ks e ^ can be regarded as a power series in (es ), it can be d i f f e r e n t i a t e d s u c c e s s i v e l y t e r m - b y - term to give

8 38 W.(<(>) -<b d' = e ds k = 0 s, k (<j) e ) k! s = 0 W. ( e"* ^ l e (* e! 3 d s 3 s = 0 (1.4) Let x = <p e s and the v a l u e of x w h e n s equals zero be d e n o t e d by x (0). x = <j> e x(0) = <f> e s=0 (1.5) (1. 6) W.(<{>) = e ^ - r (ex ) d s 3 s=0 (1.7) The 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 (1.6) can be e x p a n d e d in the f o l l o w i n g way: d, x (e ) 3~1 ru s=0 j-1 ds ds / x vi (e ) ] s=0 J " 1 X dxj j-1 ds d s 1 s=0 ds j-1 (x e ) s=c ds j -2 ds (x e ) ] s=0 ds j-2 [x (1+x) e s=0 and so on. The right h a n d side of the a b ove e q u a t i o n forms a jth order p o l y n o m i a l of the form ex(aj,lx + aj,2x' + aj.3x3 + + aj(jx3) s=0

9 39 U s i n g E q u a t i o n s (1.5) and (1.6), this p o l y n o m i a l becomes -I- CṈ e + fd m -e e^a.. x<j 3 r2 3»J And hence w. (<}>) j = Z m= 1 a 3 -i,m (1.8) + + fd *o - v jin = e ^ T 4 Z a. d> (1.9) 3 I m= 1 n D»n A r e l a t i o n s h i p b e t w e e n the a j (in c o e f f i c i e n t s can be found if we consider the (j+l)th d e r i v a t i v e and c o m p a r e it with the jth derivative. S L L (ex ) ds-^ s=0 e ( a. ix + a. 0x >2 = e Q.(x) 3 s=0 + a..xj ) 3 3 s=0 If Qj (x) is u s e d to d e n o t e the s eries (a-j^x + ajf2x + The next higher derivative of e x is given by i3 +1 (e ) = ds j +1 s=0 d. x. d (e > Q j (x) + e " as Qj (X) L J s=0 2 3, 4. = e ( a. _x + a. x + a. _x...) 3,1 3,2 ],3 + e x (a. x + 2a. x 2 + 3a. x 3 + 4a..x4... ) I n 3/! 3/2 3,e 3,4 S=0 But, from Equation (1.9), the (j+l)th d e rivative w ill be ex (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 " S s=( = 0 and hence, a r e c u r r e n c e r e l a t i o n s h i p can be e s t a b l i s h e d, w h e r e a., = m a. + a.,, w h i c h can a l s o be e x p r e s s e d 3 + 1,m j,m j,m - 1 r as a. = m a. + a., It will be n oted that j,m j-1,m j-lf m - 1 (a j m m = ^ = ^a j m m>^ = ^a j m m = j^= 1/ and h e nce the initial t e r m a 1(1 from w h i c h all o t h e r terms can be derived, e quals unity.

10 40