a mathematical overview of

7/30/2019 A Mathematical Overview Of

1/7

A m athem atical overv iew ofw arehousing sy stem s w ithsing le/du al ord er-pickingcyclesCharles J . Maimborg, Bhaskaran Krishnakumar and G ene R. S imonsDepartment of Decision Sciences and Engineering Systems, Rensselaer PolytechnicInstitute, Tro y, N Y 12181, USA(Received M arch 1987; revised July 1987)

A mu l tid imensional , integrated lot s ize and stock locat ion mo del is ou t l inedto study the em pi r ical p roper t ies o f a wareh ousing s i tuat ion wi th s ingle/dua l - comm and order -p i ck ing cyc les . Normat ive app l ica t ions o f the m ode lare highl ighted, Param eter sensi t iv ity tests are per formed to ide nt i fy de-s i rable parame ter set tings. He ur is tics are then c om pared stat is t ical ly wi threspec t t o t he qu a l i ty o f t he so lu t ion . A con t ro l led exper imenta l f ramew orki s used to s tud y the impac t o f t he l eve l o f de ta i l i n a m ode l representa t ionon the empi r ical per forma nce of the m odel . Exper imen tal resul ts and theresul ts of the stat is t ical a nalys is are presen ted,K e y w o r d s : order p i ck ing , s i ng le- and dua l -command cyc les , mode l rep-resentat ion, cube per order index, inter leaving

IntroductionT h e i m p a c t o f a u t o m a t i o n h a s b r o u g h t a b o u t f a r- re a c h-i n g c h a n g e s i n th e c o n c e p t o f w a r e h o u s i n g . T r a d i t io n a lw a r e h o u s e s a r e b e i n g r e p l a c e d w i t h h i g h l y r e s p o n s i v es y s t e m s w i t h m i n i m u m in v e n t o r y l e v e l s . G r e a t e r e m -p h as i s i s b e in g p l aced o n f l ex ib i l i t y , i n t eg ra t i o n , an dc o n t r o l o f s u c h s y s t e m s . ~ T h e m o s t t i m e - c o n s u m i n ga n d e x p e n s i v e o p e r a t i o n i n a w a r e h o u s e i s u s u a l l y th ep r o c e s s o f f il li n g o r d e r s f o r s h i p m e n t , c o n v e n t i o n a l l yr e f e r r e d t o a s o r d e r p i c k i n g ? A n y o p t i m i z a t i o n e f f o r tsh o u ld t h e re fo re co n s id e r t h i s a r ea o f s i g n i f i can t s av -i n g s p o t e n ti a l . B e s i d e s o r d e r p i c k i n g , t h e o t h e r m a j o rc o s t c o m p o n e n t in w a r e h o u s e o p e r a t i o n s i s th e i n v e n -t o r y c o s t . T h e i n t e g r a l i t y o f t h e s e t w o c o s t c o m p o -n e n t s , i n v e n t o r y a n d o r d e r p i c k i n g , h a s b e e n e s t a b -l i s h e d b y W i l s o n ? A n i m p o r t a n t p r o b l e m , t h e r e f o r e ,i s t o f i n d a m i n i m i z i n g i n t e g r a t e d l o t s i ze a n d s t o c kl o c a t i o n m o d e l . S u c h a m u l t i d i m e n s i o n a l w a r e h o u s em o d e l f o r f in d i n g t h e l e a s t t o t a l o p e r a t i n g c o s t , a s s u m -i n g s i n g l e - a n d d u a l - c o m m a n d o r d e r - p i c k i n g d i s c i -p l i n e s , a n d a s in g l e d o c k o r i n p u t / o u t p u t ( I/ O ) p o i n t ,i s o u t l i n ed h e re .

T h i s p a p e r c o n s i d e r s d e v e l o p m e n t o f t h e e m p i r ic a lp r o p e r t i e s o f t h e w a r e h o u s i n g s i t u a t i o n i l l u s t r a t e d i n

Figure 1, u t i l i z in g a m o d e l s im i l a r t o t h a t r ep o r t ed i nR e f . 2 . T h e o b j e c t i v e i s t o g i v e a g l i m p s e o f t h e m a t h -e m a t i c a l s t r u c t u r e o f th e p r o b l e m , d i s c u s s t h e a v a i la b les o l u t i o n s t r a t e g i e s , a n a l y z e t h e p r o b l e m e m p i r i c a l l y ,a n d d e d u c e c o n c l u s i o n s b a se d o n t h e a n a ly s i s.T h e p a p e r i s o rg a n i z e d i n t o f i v e s e c t i o n s a n d f o l lo w sa l og i ca l d e v e l o p m e n t o f t h e p r o b l e m . S u c c e e d i n g t h ei n t r o d u c t i o n i s t h e f o r m u l a t i o n o f t h e m o d e l . T h e a s -s u m p t i o n s o f t h e m o d e l , th e o p e r a t i n g p a r a m e t e r s , a n dt h e n o t a t i o n u s e d a r e p r e s e n t e d i n t h i s s e c t i o n . N o r -m a t i v e a p p l i c a t i o n s o f t h e f o r m u l a t i o n a r e t h e n d i s -c u s s e d . T h r e e h e u r i s t i c l o t s i z i n g a n d s t o c k l o c a t i o ns o l u t i o n s t r a te g i e s a r e d e s c r i b e d . T h e e m p i r i c a l a n al -y s i s s e c t i o n i n c l u d e s t h e p a r a m e t e r s e n s i t i v i t y t e s t s ,t h e c o m p a r i s o n o f h e u r i s ti c s , a n d t h e e x p e r i m e n t a t i o ni n o p t i m i z a t i o n o f t h e d e t a i l o f m o d e l r e p r e s e n t a t i o n .A s u m m a r y a n d c o n c l u s i o n s ar e p r e s e n t e d .

Outline of the m odelA t y p i c a l w a r e h o u s e c o n f i g u r a t i o n f o r t h e m i x e d s i n -g l e- a n d d u a l - a d d r e s s w a r e h o u s e s y s t e m i s d e p i c t e d i nFigure 1. T h e t h r e e - d i m e n s i o n a l p h y s i c a l s p a c e i s o r -g a n i z e d i n t o o n e - s i d e d r a c k s , s e p a r a t e d b y a i s l e s, f o r

2 App l . Ma th. Mo del l ing, 1988 , Vol . 12, Feb ruary 1988ButterworthPublishers


2/7

Mathem at ical overv iew of wareho using systems: C. Malm borg et a l .

RACK

Conveyor ~ ]

( )OUT I /O Po in t

m 2

FRONT V IEW OF A RACK(i .e . RAC K FACE)

RESERVE STORAGE AREA

F i g u r e 1 Warehousing s i tua t ion

-7

STAGING AREA

Racks numbered 1 Through MConveyor

( )( 1 , R }

1 1 , 2 1 ,

(1~1) (2~ 1) " (C~1)

L _M

Farm rack madeup ofa gr id o f C co lumnsand R rows.

h. Xy

~x

the order -pick ing vehic le to t ravel . Eac h rack i s a s tor -age grid mad e up of rows and column s. I t ems are pickedup f r om t hese r acks by t he o r de r - p i ck i ng veh i c l es andb r o u g h t t o a c o n v e y o r (retrieval), w her e o r de r s a r efdled. I tems remaining after order completion are storedback i n t he r acks (s torage) .A s s u m p t i o n s

T he w ar ehous i ng sys t em unde r d i s cus s i on uses acomb i na t i on o f si ng le - and dua l - addr es s sys t em s . I ns i ng l e -addr es s sys t em s , t he o r de r - p i ck i ng veh ic l e t r av -el s between the input /output ( I /O) point and a s ingleaddress ( locat ion) to execute e i ther a s torage or a re-t ri e va l c o m m a n d , k n o w n a s o u t - a n d - b a c k or de r - p i ck -ing discipl ine. Dual-address system s, on the other hand,a l l ow t he o r de r - p i ck i ng veh i c l e s t o acces s t w o ad-dresses ( locat ions) before re turning to I /O. This prac-t ice i s know n as in ter leaving . E ach dua l - command cyc l ei s a s s u m e d t o b e m a d e u p o f a s t o r ag e c o m m a n d a n da r e t r i eva l command . O t he r a s sumpt i ons a r e :1 . T he r e i s no r e s t ri c t ion on t he a l l oca ti on o f t he spacer equ i r ed f o r i t ems among t he ava i l ab l e s t o r age I o -

ca t i ons . T h i s i mpl ie s t ha t an i t em m ay occu py mor et han one l oca t i on , and a l oca t i on may con t a i n mor ethan a single i tem.2 . S t o r age and r e t ri eva l cos t s a r e l oca t i on dependen t .T h i s is due t o t he a s sum pt i on o f uncons t r a i ned r ec -t i l inear t ravel of the order -picking vehic le .3 . T he r e a r e no con s t r a i n ts o n i t em compa t i b i l it y .4 . E xpec t ed cos t comput a t i ons i gnor e t he shor t - r und y n a m i c s o f s y s t e m o p e r a ti o n s .

N o t a t i o n sL e tL = a i sle width plus dep th o f a rack ( ft ) ( shown inF i g u r e 1 )h = d i s t ance be t w een an y t w o ad j acen t g r i d po i n t sin a rack face ( f t )f y = ave r age f r equen cy o f t ransac t i ons f o r i tem j( t ransact ions /per iod)t = cos t pe r un i t d i st ance o f tr ave l o f the o r de r -picking vehicle ($/ft)vy = volu me p er uni t o f it em j ( ft )Ma = capac i t y o f loca t i on a ( ft )

App l . Ma th . Mo de l l i ng , 1988 , Vo l . 12 , Feb rua ry 3


3/7

Mathem atical ove rview o f warehousing systems." C. Malmborg et al .N = t o ta l n u m b e r o f i t e m t y p e ss j = a v e r a g e s i z e o f a t r a n s a c t i o n f o r i t e m j( u n i t s / t r a n s a c t i o n )I = i n v e n t o r y c a r r y i n g c h a r g e ( $/ $ i n v e s t e d / p e r i o d )C = v a l u e o f a u n i t o f i te m j ( $ ) a n dA j = r e p l e n i s h m e n t c o s t p e r o r d e r f o r i t emj (S/order)

F o r m u l a t i o nL e t m = 1 . . . . M , c = 1 . . . . C , a n d r = 1 . . . .R b e t h e r a c k , c o l u m n , a n d r o w i n d i c e s f o r t h e c o n -f i g u r a t i o n s h o w n i n F i g u r e I . T o s i m p l i f y t h e n o t a t i o n sl e t a a n d b c o r r e s p o n d t o a n y t w o l o c a t i o n s ( m , c , r )a n d ( x , y , z) . T h e n t h e r e c t i l i n e a r d i s t a n c e f r o m t h e 1 /Opoin t (0 , 0 , 0 ) to loca t ion a i s

Io , = Ll m - 01 + h([c - 01 + Ir - 0l)a n d t h e c o r r e s p o n d i n g t r a v el c o s t o f t h e o r d e r - p i c k in gv e h i c l e i s

Wo~ = t lo ,S i m i l a r l y t h e r e c t i l i n e a r d i s t a n c e a n d c o s t f o r t r a v e lb e t w e e n t h e l o c a t i o n s a a n d b a r eI ,b = LIm - x [ + h ( [c - Y[ + ] r - z l)W . b = t l . b

T h e e x p e c t e d o r d e r - p i c k i n g c o s t f o r o u t - a n d - b a c ko r s i n g l e - c o m m a n d c y c l e s , S CO ST , i s t h e s u m o f t h ec o s t s o f a l l t r i p s o f t h e o r d e r - p i c k i n g v e h i c l e t o e a c hl o c a t i o n f o r e a c h i t e m p e r p e r i o d :N

S c o s T = 2 ~ ~ ( W o j j ( X J Q j ) ) (1 )aJ= l

w h e r e X ,,~, t h e d e c i s i o n v a r i a b l e , i s th e a m o u n t o f i t e mj s to red in loca t ion a , and Qj = E,,X,, i s the i t em lo ts i z e . T h e c o n s t a n t m u l t i p l i e r 2 i n t h e e x p r e s s i o n i s t oa c c o u n t f o r t w o - w a y t r a v e l o f t h e o r d e r - p i c k in g v e h i -c le .I n t h e c a s e o f d u a l - c o m m a n d c y c l e s , t h e e x p e c t e do r d e r - p i c k i n g c o s t , D c o s x , i s t h e s u m o f th e c o s t o f t h es t o r a g e t r a n s a c t i o n t o a n y l o c a t i o n a , t h e i n t e r l e a v i n gt r a v e l c o s t f r o m a t o a n y o t h e r l o c a t i o n b p i c k e d o n arando m se lec t ion o f re t r i eva l t r ansac t ions , and the t r ave lc o s t f r o m b t o t h e I / 0 p o i n t :

( W o o + E ( w o ,+ W ,o ) e ,,) ,. (2 )w h e r e P b , t h e n o r m a l i z e d f r e q u e n c y o f a c c e s s o f l o-c a t i o n b , i s d e f in e d b y t h e r a t i o o f t h e n u m b e r o f t r a n s -a c t i o n s p e r p e r i o d f r o m i t e m s l o c a t e d a t b t o t h e t o t a ln u m b e r o f t r a n s a c t i o n s p e r p e r i o d f o r a l l i t e m s . I t i sg i v e n b y

P b = f j (X h j Q ) w h e r e F = Y . 5j= , F j=~

B e s i d e s t h e o r d e r - p i c k i n g c o s t , t h e m o d e l a l s o c o n -s i d e r s t h e i n v e n t o r y c o s t . A s i m p l e e c o n o m i c o r d e rq u a n t i t y m o d e l w i t h n o s a f e t y s t o c k i s a s s u m e d f o rc o n v e n i e n c e . S t o c k s e t t i n g c o s t s o r r e s t o c k i n g c o s t si n t h e s t a g i n g a r e a ( F i g u r e 1 ) a r e i n c l u d e d i n t h e r e -p l e n i s h m e n t c o s t c o m p o n e n t o f t h e i n v e n t o r y c o s t. T h et o t a l i n v e n t o r y c o s t , / C O S T i s

j = , \ Q iT h e t o t a l c o s t Z f o r t h e m o d e l i s t h e n t h e s u m o f 1 - 3 :

Z = 7Sco sT + (1 - 7)D cosT + /COST (4)w h e r e 3 ' i s t h e p r o p o r t i o n o f s i n g l e - c o m m a n d c y c l e s i nt h e o r d e r - p ic k i n g s p e c t r u m .T h e m o d e l o b j e c t i v e i s t o f i n d a ( t h r e e - d i m e n s i o n a l )l a y o u t w h i c h w o u l d m i n i m i z e t h e t o t a l c o s t Z , s u b j e c tto

Nc a p a c i t y c o n s t r a i n t s ~ X , ,i -< M , , V aj - I

n o n n e g a t i v i t y c o n s t r a i n t s X , j - > 0 V a jD i s c u s s i o n

T h e d e c i s i o n v a r i a b l e s o f th e m o d e l a r e t h e X ' s . F o ra l a y o u t w i t h 1 0 r a c k s , e a c h w i t h I 0 r o w s a n d c o l u m n s ,t h e p r o b l e m o f l o c a t i n g 1 00 i t e m s i s o n e o f c o m p u t i n g1 0 0, 00 0 X ' s , w h i c h d e m o n s t r a t e s t h e d i m e n s i o n a l it y o ft h e p r o b l e m . M o r e o v e r , t h e o b j e c t iv e f u n c t i o n i s n o n -l i n ea r an d n o n c o n v e x . S u c h p r o b l e m s h a v e a l w a y s d e -f i e d c l o s e d - f o r m s o l u t i o n s . P r o b l e m - s p e c i f i c a p -p r o a c h e s s u c h a s h e u r i s t i c s a r e i m p l e m e n t e d t o g e t a" g o o d " s o l u t i o n .N o r m a t i v e a p p l i c a t io n sC o m p u t e r - o r i e n t e d t e c h n i q u e s , s u c h a s s e a rc h h e u r i s-t i c s , a r e i m p l e m e n t e d t o r e a l i z e a l o c a l l y m i n i m i z i n gt h r e e - d i m e n s i o n a l l a y o u t . T h e h e u r i s t i c s u n d e r d i s c u s -s i o n o b t a i n a l o c a l o p t i m u m f o r a n y s p e c i f i c o b j e c t i v ef u n c t i o n i n a d i r e c t a n d i t e r a t i v e f a s h i o n .M o s t n o r m a t i v e a p p l i c a t i o n s o f th e w a r e h o u s i n gp r o b l e m u s e t h e c o n c e p t o f th e c u b e p e r o r d e r i n d e x( C O D . C O I i s t h e r a t i o o f an i t e m ' s s t o r a g e s p a c e ( c u b e )r e q u i r e m e n t t o i t s s t o r a g e / r e t r i e v a l t r a n s a c t i o n s d e -m a n d C o r d er s) . A w i d e l y u s e d p o l i c y i s t o a ll o c a t e t h ei t e m s t o a d d r e s s e s c l o s e s t t o t h e I / 0 p o i n t i n a s e q u e n c eb a s e d o n t h e a s c e n d i n g r a n k o r d e r i n g o f t h e C O I r a t i o s.I t i s a n e s t a b l i s h e d f a c t t h a t C O l p r o d u c e s a n o r d e r -p i c k in g c o s t - m i n i m i z i n g l a y o u t f o r a n L P f o r m u l a t io no f t h e s t o c k l o c a t i o n m o d e l ? O p t i m a l i ty o f th e C O lp o l i cy h a s a l s o b e e n p r o v e n f o r t h e d u a l - a d d re s s s y s -t e m s , a t f i x e d i n v e n t o r y l e v e l s ? T h r e e n o r m a t i v e a p -p l i c a t i o n s a r e d i s c u s s e d .G r a d i e n t s e a r c h h e u r i s t i c

T h i s i s a n i t e r a t i v e i m p r o v e m e n t a p p r o a c h w h i c hu s e s a r e a r r a n g e m e n t s c h e m e t o i m p r o v e t h e c o s t fu n c -t i o n b y m o v i n g in t h e d i r e c t i o n s p e c i f i e d b y t h e g r a d i e n to f t h e c o s t f u n c t i o n a s d e r i v e d i n t h e a p p e n d i x . T h e

4 Appl . Math. Mo del l ing, 1988 , Vol . 12, Feb ruary


4/7

Mathemat icals o l u t io n a p p r o a c h s t a r t s w i t h t h e e c o n o m i c o r d e r q u a n -t i t i e s ( EOQs ) a n d mo v e s s e q u e n t i a l l y f r o m o n e s e t o fi n v e n t o r y l e v e l s t o t h e n e x t s o a s t o mi n i m i z e th e c o s tf u n c t i o n Z . A t e a c h s t a g e ( i .e . i n v e n t o r y l e v e l ) t h e C O Ip o l i c y s p e c i f i e s t h e " b e s t " a s s i g n me n t , a n d a s c a l e ds tep s i ze based on use r inpu t in i t i a l i za t ion va lues g ivest h e a mo u n t t o mo v e i n t h e d i r e c t i o n s p e c i f i e d b y t h eg rad ien t .Th e g r a d i e n t o f t h e c o s t f u n c t io n , G j , is o b t a in e d b yc o mp u t i n g t h e r a te o f c h a n g e i n t h e u n c o n s t r a i n e d c o s tf u n c t i o n w i t h r e s p e c t t o X~ j , OZ/OX,j, a n d a d d i n g t o i tt h e i n c r e me n t a l p r o d u c t r e l o c a t i o n c o s t , AR ' .

aZ* OZG ~ - - - ~ - ~ L R 'oQ~ ox, ,Th e i n c r e me n t a l p r o d u c t r e l o c a t i o n c o s t i s i n t r o d u c e di n t h e a b o v e e x p r e s s i o n t o t a k e c a r e o f t h e c a p a c i t ycons t ra in t s . I t i s de f ined a s

aR' -- v:Zaj

wh e r e a i s t h e f a r t h e s t l o c a ti o n a f o r wh i c h X,,~ > O.Th e s t e p s i z e ma y b e a r b i t r a r i l y c h o s e n a s f o l l o ws :S T E P = AQm axlmaxlGjI j E ( 1 , N)

wh e r e ~ Qm , x i s t h e m a x i m u m c h a n g e i n Q j i n a n y o n ed i r e c ti o n , i . e . , t h e p r e v i o u s l y me n t i o n e d u s e r s p e c i f i edi n i t i a l i z a t i o n v a l u e . Th e s o l u t i o n p r o c e d u r e ma y b es u mma r i z e d a s f o l l o ws :1. Star t by set ting Qj equal to EO Q and i teration cou nter ,K = I .2 . As s i g n i t e ms u s i n g t h e C OI p o l i c y . C o mp u t e Z* ,t h e f u n c t i o n a l v a l u e f o r t h e a s s i g n me n t .3 . C o m p u t e t h e g r a d i e n t G j a n d t h e s t e p s i z e S TE P .4 . Th e nex t se t o f inven to ry l eve l s i s QJ~+ ~ = Q~ -S T E P * G j V j . S e t K = K + 1.5 . R e p e a t s t e p s 2 - - 4 u n ti l Z* s h o w s a n i n c r e a s e .6 . H a l v e ~Qma x .7 . R e p e a t s t e p s 2 - 6 u n t i la . ~Q , , ,~ _< E (a p re spec i f i ed cons tan t )

orb . K i s g rea te r than the i t e ra t ion l imi t .Pattern search heuristic

Th i s h e u r i s t i c i s a " d i r e c t s e a r c h " s e q u e n t i a l t e c h -n i q u e i n v o l v i n g a s e q u e n t i a l e x a m i n a t i o n o f e a c h t r i a ls o l u t i o n a n d c o mp a r i n g i t t o t h e b e s t o b t a i n e d s o f a ri n o r d e r t o f i x t h e n e x t s o l u t i o n p o i n t . Th e me t h o d o fH o o k e a n d J e e v e s 6 i s a we l l - k n o wn , p o p u l a r p a t t e r ns e a r c h me t h o d . I t is c o mp r i s e d o f t wo mo v e s : t h e ex -p l o r a t o r y a n d t h e p a t t e r n . Th e e x p l o r a t o r y mo v e e x -plores the local behavior of the object ive funct ion; whilet h e p a t t e r n mo v e g u i d e s t h e s o l u t i o n f r o m o n e b a s ep o i n t t o a n o t h e r .

Th e e x p l o r a t o r y mo v e l o g i c i s a s f o l l o ws :I . S t a rt w i t h a s o lu t io n N E W Q = ( E O Q , E O Q z . . . .EOQN) .2 . As s u m e s t e p l e n g t h s AQI i n d i r e c t io n s i = I , 2 . . . .N .

overv iew o f w arehousing systems: C. Malm borg et a l .3 . C o m p u t e c u r r e n t v a l u e o f t h e c o s t f u n c t i o nZCUR R = f ( N EW Q) a n d l et i = 1 .4 . I f i -< N , t h e n t h e n e w t r ia l p o i n t is Q = ( Qb Q2 ,

. . . . Q ,. + A Q i . . . . Q N ) ; Z N EW = f ( Q ) . I f / > N ,t h e n S T O P .5 . ] f ZN Ew < Zc u R a , t h e n ZCURR = Zr~ EW ; NE W Q =Q; i = i + 1 . A ccep t th is a s the s t a r t ing po in t and

r e p e a t f r o m s t e p 4 .6 . I f ZN E W - - Z C UR R , t h e n Q - - ( Q , Q z . . . . Qi -2 A Oi . . . . Q N ); ZNEW = f ( a ) .7 . I f ZNEW < ZC UR a t h e n Zc o R R = ZNEW ; NE W Q =Q ; i = i + 1. G o t o s t e p 4 .8 . I fZN Ew > ZCUR a s t il l, t hen Q , r em a ins unchang ed .Se t i = i + 1 , and go to s t ep 4 .

Th e e x p l o r a t o r y mo v e y i e ld s a n e w b a s e p o in t NE W Q .Th e o v e r a l l l o g i c i s s h o wn i n Figure 2. Th e s t a r t i n g

Star t AtBase Po i n t

MakeExploratoryMoves

No

y

( S e t )New BasePo i n t

iMakePatternMove

Step

Ye s

MakeExploratoryMoves

Final

Y e s ~ V a l u e B e lo wT h a t A t BaseP o i n t ,

Figure 2 F l ow c har t f o r H ook e and J eev es method

A p p l . M a t h . M o d e l l in g , 1 9 8 8 , V o l . 1 2 , F e b r u a r y 5


5/7

Mathem at ical overv iew o f wa rehousing systems: C. Malmb org e t e l .po in t i s t he s t a r t i ng base by de f in i t i on . T he pa t t e rn Table 1 Parameter se ns i t i v i t y testa bm o ve m ake s a s im ple s t ep f rom the cu r r en t basepo in t GAM MAC U R R Q t o a n o t h e r p o i n t Q ': 0,9 0.5 0.25 0.1 0Q ' : 2 C U R R Q - P R E V Q % redu ct ion in cos t 17 .57 20 .52 18 .70 22 .24 21 .50w h e r e P R E V Q i s t h e p r e v io u s b a s e p o i n t. E x p l o r a t o r y No. of i te ra t ion s to 62 33 16 37 14m o v e s a r e p e r f o r m e d o n c e a g a i n t o y i e l d N E W Q . I f r e a c h "opt imum"t h e f u n c t i o n a l v a l u e a t N E W Q i s b e t t e r t h a n a t Q ' , " Responsesbased on gradient search technique.t hen N E W Q becom es t he cu r r en t basepo in t and P R E V Q b Execut ionon IBM 3081 (


6/7

Table 3 Test problemsProblem No. 1 2 3 4 5

No. of i tems 50 60 80 125 150No. of locations 10 10 10 10 10No. of var iables 500 600 800 1250 1500

Ma thema t ica l overv iew o f warehous ing sys tems: C . Ma lmbo rg e t a l .Table 5 Control exper imenta,b~

heuristic seems to p erform as well as, if not be tter than,the well-defined schemes such as the gradient search.From a n empirical standpoint, this reinforces the un-predictability of classical solution schem es on highlynonc onv ex and nonl inear surfaces.Comparison o f heuristics

For comparing the heuristics, test problems with afair dimension of realism n eed to be used. Five testproblems of reasonable m agnitude, and within the rangeof compu tational capabili ty o f the heuristics, are de-veloped using a simple pseudorandom problem gen-erator . Th e configuration of the wareho use is kept con-stant . The level of deta i l of model representat ion,character ized by the num ber of storage locat ions in themodel (i .e., th e cap acity of individual locations), is alsomainta ined the same for the test problems. The num berof i tems vary from problem to problem.Each of the five test problems listed in Table 3 is"solved" using the three heurist ics. The responses,i .e . , the percentage reduct ions in tota l cost , are pre-sented in Table4. A two-w ay factorial design, as shownin Table 5, i s used to com pare the heurist ics. The testproblems form the blocking variable of the experi -mental design, and the heuristics form the treatments.At a 5% level of significance, the analysis of variancegives no reason to reject the null hyp othes is--that thereis no significant differe nce in the th ree heuristics viaquality of the solution.Experimentation in optimization of m odelrepresentation

Since the three heuristics do n ot differ in terms ofthe quality of solution, it is sufficient from the limitedanalysis if any o f the heuristics is conside red for furth eranalysis. The gradient search heuristic is chosen tostudy the impact of the level of detail in model rep-resentat ion, i .e . , the num ber of locations o n the variouscost components. The control experiment i s se t upusing test problem number I , and varying the modelrepresentation, as l isted in Table 5. This corresponds

No. of Capaci ty No. oflocations 15,000/i M columnsi (ft3) (ft} y10 1500 19 .4 520 750 13 .7 1030 500 11 .2 1550 300 8 .7 25

8 50 i tems; i tem cha racteristics fixed.b Volume required (base d on EOQ):15,000L = 10 + 4 = 14 f t.to va rying the cap acity of individual storage locations.In actual problems individual location capacities aresmall. An extre me ly precise solution is obtained w henthe exa ct locations are specified, b ut this can result inprohibitively high problem dimensionality. Ver y largelocation capacities can greatly reduce the number ofdecision variables but also yield less p recise solutionswhere a single locat ion in the m odel could correspondto a po tentially large nu mb er of pallet rack addresses.Each problem in the cont ro l expe r iment i s " so lv ed"using the gradient s earch heuristic, and the level of thecost components a t EOQ, and a t the opt imum, aregraphed v ersus the nu mb er of storage locations (i .e.,location capacity) in the model representation (Figure3). I t can be seen from Figure 3 that inventory cost i sinsensitive to model representation. This is intuitivesince the inventory cost formulation depends only onthe inventory levels Q 's, and not on the X's. The order-picking costs are sensitive to model representation,since they depend on the X's. How ever, out of the twocost compo nents the single-command cost compon entseems to b e the m ost sensitive to model representation.This is reasonable since the travel distance involvedin a simple out-and-back schem e is definitely m ore thanthat in the case of dual commands fo r the same order-picking demand.Obviously the mod el represen tation influences thecomputational severity of the problem. Hence, for suchmodeling efforts i t is nece ssary to carefully choo se themodel representation to match any particular imple-men tation, since m odel effectiven ess (cost reductionachieved) as well as computational requirements maybe affected. A simple experimentation similar to theone described above is sufficient to gain insight intothe inf luence of the model representat ion on the em-pirical perform ance of the model. Such an experimen-tation m ay be viewed a s an effo rt in optimization of

Table 4 ANOVA resul ts for com par ison of heur ist ics*Blocks

Treatments P rob 1 P rob2 P rob 3 Prob 4 Prob 5 Total/MeanGrad ient search 9 .2 0 !1 .20 9 .40 9 .6 8 1 0 .6 8 50.16/10.03Hooke and Jeeves 9 .2 0 11.20 9 .57 9 .5 0 1 0 .6 8 50.15/10.03Sequent ial search 6. 60 11.5 0 9.53 9.85 9. 67 47.15/9.43T o ta l 25 33 . 90 28 . 5 0 29 . 0 3 31 . 03 147 .46M e a n 8 . 33 11 . 30 9 .50 9 . 68 10 . 34 9 .83* E xecution on IBM 3081 and IBM PC-XT.

A p p l . M a t h . M o d e l l i n g , 1 9 88 , V o l . 1 2 , F e b r u a r y 7


7/7

Mathem at i ca l overv iew o f warehous ing systems ." C. Ma lmbo rg et al .=

=

o

o

8r- O0

+ Single Command + 113 Dual Comm and JInventory2 A t Opt imum1 A t E O Q

+

, $ f -

2

I I I t I10.00 20.00 30.00 40.00 50.00Number of Locat ions

Figure 3 Costcurvest h e m o d e l r e p re s e n ta t io n fo r b e s t im p le m e n ta t io n o f ah e u r i s t i c fo r a g iv e n p ro b le m .Summ ary and conclusionsT h e in t e n t o f th i s e f fo r t is t o a n a ly z e th e w a re h o u s in gp ro b le m e m p i r i c a l ly , b a s e d o n a m u l t id im e n s io n a l s in -g le /d u a l - a d d re s s m o d e l . T h e m a th e m a t i c a l s t ru c tu re o fth e p ro b le m re v e a l s a n o b je c t iv e fu n c t io n w h ic h m a p sin to a n o n l in e a r a n d n o n c o n v e x s u r fa c e . T h i s m a k e si t imposs ib le to a t tempt an y c losed-form so lu t ions . Threen o rm a t iv e a p p l i c a t io n s o f t h e m o d e l a re o u t l in e d . T h ep a ra m e te r s e n s i t i v i ty t e s t s a re u s e d to fix th e p a ra m e te rs e t t i n g s o f t h e m o d e l . T h e h e u r i s t i c s a re th e n c o m p a re dfo r th e c h o s e n p a ra m e te r s e t t i n g s . S t a t i s t i c a l ly , t h eh e u r i s t i c s d o n o t d i f f e r s ig n i f i c a n t ly in t e rm s o f t h eq u a l i ty o f t h e s o lu t io n . F ro m a n im p le m e n ta t io n s t a n d -p o in t t h e a rb i t r a r y s e q u e n t i a l s e a rc h h e u r i s t i c i s p re f -e ra b le o v e r th e o th e r tw o h e u r i s t i c s w h ic h r e q u i re th ec o m p u t a t i o n o f t h e g r a d i e n t o f t h e c o s t f u n c ti o n . T h ec o n t r o l e x p e r i m e n t f r a m e w o r k i s u s e d t o s t u d y t h eim p a c t o f m o d e l r e p re s e n ta t io n o n th e e m p i r i c a l p e r -fo rm a n c e o f t h e m o d e l . T h e e m p i r i c a l a n a ly s i s s u g g e s t sa n o p t im iz a t io n a p p ro a c h fo r c h o o s in g th e m o d e l r e p -

r e s e n t a t i o n f o r t h e b e s t i m p l e m e n t a t i o n o f t h e h e u r i s -t ic .T h e s ig n i f i c a n c e o f t h i s e m p i r i c a l o v e rv ie w o f th ew a re h o u s in g p ro b le m i s t h e n o t io n th a t r e a l i s t i c a l lys i z e d p ro b le m s c a n b e h a n d le d w i th a c o m p u ta t io n a l lye f f i c i e n t a n d o th e rw is e s u i t a b le m o d e l r e p re s e n ta t io n .W i t h w a r e h o u s i n g f u n c t io n s b e c o m i n g i n c r e a s i n g l y d y -n a m ic , t h e r e i s a p re s s in g n e e d fo r m u l t i c r i t e r i a o p t i -m iz a t io n m o d e l s , s u c h a s th e o n e s tu d ie d , t o e n h a n c eu t i l i z a t io n a n d r e d u c e o p e ra t in g c o s t s .

Referencesl W hite , . A. Warehousing n a changing world, Proc. Fifth Int.Conf . on Automation in W arehous ing. Atlanta,Geor gia, 1983,pp. 3-62 Malmborg,C. J., Balachandran, S., and K yle, D. M. A m odelbased estimationof a commonly used rule of thumb for ware-house layout.App l . Math. Model l ing, 1986, 10, 133-1383 Wilson, H. G. Order quantity, product popularity,and the lo-cation of stoc k in a warehouse. AII E Trans. 1977, 9, 230-2374 Harmatuck, D. J. A comparison of two approaches to stock

location. Logistics T ransport. Rev. 1976, 12, 282-2855 Malmborg,C. J. and Krishnakumar,B. On the optimality f thecube per order index or warehouses with dual commandcycles.J. Mater ial Flow 1987, 4, 169-1756 Hooke ,R. and Jeeves,J. A. D irect search solutionof num ericaland statistical problems. J. Assoc. Comput . Maeh. 1961, 8,212-229Appendixo z e = (1 + ~ ) w o o a j - w J ~ - (J - y )

( f ] ] A s f s j + I C

w h e r eK o j = ~ , ( W . ~ + Wo o ) X . j

b#u

N o t e :1 . i i s t h e l a s t i t e m to b e lo c a te d a t l o c a t io n a ; i . e . , i t

h a s th e h ig h e s t C O I o f a l l i t e m s a t l o c a t io n a .2 . K re fe r s t o th e i t e ra t io n c o u n te r .

8 App l . Ma th . M ode l l ing , 1988, Vo l . 12 , Febru ary

a mathematical overview of

Documents