T HE ENGINEER¼NG ECONOMIST ¤ 200 1 ¤ V OLUME46 ¤ N UMBER 2 129 -

5f ULTI -I TEM E CONOMIC O RDERQUANTITY M ODEL WITH AN INITIAL

STOCK OF CONVERTIBLE U NITS

E D WA RD A . S E V E R

T hc U ni versi ty of C al gar $ C A N A D A

¦ K Y E 0 N G M O O N

Pusan N at ional U ni ver si ty, K OR E A

A BSTRACT

In th is paper w e co nsi der a group of end items f acing level do nand pang m aMopportuni t ies f or rcgular purchasi ng. I n addit ion, tho -c ar e a num bcr of units that can be

conver ted i nto any one of the em items. but at diHerent unit costs of conversion. A

co rs trai t- d opt im ization modcl is developed f or the problo t1 of how bCS¼ ¼0 al locatc thc

conver tible units am ong the em items. A solution algor ithm i s prcsented along w i th

numer ical i l l us t-aHors -

I NTRODUCTI ON

In t h i s pap er w e COIl si dcr th e si tuat i on of a ser v i ce (or p ur ch asi n g) depar tm cnt

deal i ng w i t h a nu m ber of end i - Ins each sati sf y i n g the assum pt i ons under l y i n g

the cco tto n - o rder q uant i ty d eri v at i on , and par ti cul ar ¯ h av i ng an essenti al l y

k no w n , co nst ant dem and r ate. H ow ever, i n ad d i t i o n t o i n i t i al i nventor i es of thcse

i tem s, t here are al so a num ber o f u ni t s t hat arc not di rect l y u sabl e but w h i ch can

be co m a ted to each cn d i tem at a con stant uni t cost that depends up on the

Spa in c cnd i ta n - E ach of these en d i tem s can be pur chased at uni t costs th at , i n

gener al , arc hi gh cr than the uni t conv er si on cost s. T h i s research w as m ot i v ated

b y a si tuati on , ob served b y onc of thc aut hor s i n a consul t i ng assi gn m ent f or a

tel ecom m u ni cat i ons o rgani zati on, w her e custom er s r etur ned used tel eph one

u ni t s that coul d b e con ver t ed i nt o d hcr usab l e uni ts ( th rough re pai r s, addi ng a

d i Herent col ore d p l ast i c cover, c tc .) . A noth er l ess obv i ous appl i cati on i s i n a

su ppl y ch ai n context w hcr c par t i al l y pr occ sscd i tem s can be con ver t ed to

d i Her o n end i tem s, e.g . Ë B Onal com p uter pr i nters f o r sal e i n d i f f erent cou nt r i es

(sec L ee, et al . [4 ] and Feh u nger and k c [3 ] ) . Pan of th e m anu f actur i ng i s done

centr al l y and uni t s M e shi ppcd t o v ar i ous dest i nat i ons w here l Ä al i zed n ni shi n g

(c .g . ad d i t i on of spcci al po w er sour ce , user m anual , etc.) i s done- Par t i al l y

proccssed u ni t s at a part i cul ar l ocat i on coul d be pr occsscd f ur ther ther e or

t r ansshi p ped to othcr l Ä an ons f or com pl et i on .

l 30 H E ENGWEERING ECONOMIST ¤ 200 l ¤ VOLUME 46 ¤ N UMBER 2 -

T he convertibl e si tuat ion i s clearl y rcl ated to the contexts of servi Ä pa t s

and re¹ rable i tems f or which therc is a substanti al l im ature- Ra ero - es, that

include broad SHR . of model ing efforts i n these contexts, include Brow n [ 1] ,

D iks, ct al . [2], M au ni and Gel dcrs [5] , Nalum as [6] , Sherbrooke [7] , Si l ver, et

d . [9] , and Verup t [ 1Ì .A rum- r closcl y related si tuation i s the so-cal led °special opportuni ty t o

buy± whcre there i s a one-time opportuni ty to acqui - onc or morc cnd i tcms at a

uni t cost lower than wi l l bc the a se i n the nmure- Si l ver, et d [9] present a

hcw ist- solut ion procedurc for the mul ti -i tem case subj ect to a bud getrestrict ion on the total vaIue of th-e uni ts acqui red under the special oppw tuni ty

(and provi de a set of refercnces on thc si ngle i tem si tuation) . h the current paper

the constraint i s in the form of thc total numb er of conver tible uni ts avai lable

and an optima solution algori thm i s pro-dded

Si l va and M oon [8] have addressed a convert ible context that diHcrs in t w o

respects n om thc current paper. Fi rst, a single p Hd horizon i s considered.

Sc ot¤ l $ each end i tem f aces probabi l isti c demand wi th a know n distr ibut i on.

In othcr words, they havc dcah whh thc mul t i-i tem ncwsva t¤ r context .

In thc f i rst secti on wc prcscnt the notati on and assumptions, tha t devel op a

mama nana l model of the decision si tuati on. The soluti on algori thm is thc

suInca of the sccond secti on. A lso included i s a numeri cal exam pl e. Thc f inal

sect ion provides sa ne brief ca d d ite remarks.

PREL- -INARIES

NOTATI0 N

T he notation to bc used i s a fd lows:

n = number of di fferent end i tems

j = i ndex for end i tems O = 1, 2, ¦ , n)

å = unit C0IWa g on cost from thc COIl vcruble i tem to end i tem j , in

dol l arsAnti t

% = unit purchase cost of cnd i tcm j , i n dol l ars/ uni t

ï = im o no¼y ca M ng cost ratc of cnd i tem j , in dol lars/uni t/uni t time

à = f ixed orderi ng cost assÄ iated wi th a reH Olism - nt of i tem j , in dol l ars

Dj = known demand rate for end i tem » in uni ts/unit ti me

a = discount rate per uni t ti lt-

N = avai l - lc number of uni ts of thc cotwa t- lc i tem

L = i nventory of i tem j O * 1) at f = O where f = O is den ned a the moment

where the fi rst i tcm runs out of invcntor y (the i tems Me i ndexed so tha

itcm 1 runs out of i m a nory f i rst)

í = numbcr of conver tiblc units that arc cotw a tcd into end i tem j (deci si on

vaHables)

« = number of units of end i tcm j that are purchased (decision van - ls )

T m ENGINEERING ECONOMIST ¤ 200 l ¤ V OLUME 46 ¤ N UMBER 2 l 3 l

D EVELOPM ENT 0 F THE M AT HEMATICAL M ODEL OF THE D ECISION PROBLEM

A s i ndi cated ear l i er, w e can th i nk o f a ser v i ce depar tm ent (or pu rch asi n g

depar tm ent ) , w hi ch i s i n chargc o f pr ov i d i ng i ta n s to other depan m ens or

f acto r i es, hav i ng N un i ts of m i tcm that M e not di r ect l y usabb - E ach uni t can bc

com a - d t o an end i t cm j O = 1, 2 , . . , n ) at a cost of q . H ½e c nd i tcm can al so

b c purchased at a uni t cost of % Mb nccd to deci de how m any u ni ts shoul d be

convcr tcd for cach i tem , 8 w el l as thc t i m i ng of t he com ers10Its-

F i s t , w c dc r i v e the pr esa u cost of a Sin gh rcgul ar pur chasi ng c y clc f or

i t em j d i scounted to i t s star t i ng poi nt Ä f d l ow s:

.a f h ® * .D l -¤# lî +ç +­ q - í ULem a - a | ­ ® - Ò ®l - e ± | (1)* t a J O l l

Si ru c the- arc i n f i ni te n um bc r of u d cs, w c com pute t he p resc nt val ¤ o f tot al

cost f or i tem j , w i thout con versi on and at the star t o f a cy cl c ( i .e . w i th Ä r o

i nventor y on hand) a f ol low s:

¸ f h ¢ * .D®f -¤# ³ ¼ -¤« -¤* ½pv(Ë ) = | A +Ì L +£ |- £ Å | 1- ¤ ± | | l +e ± +e ¤7+¤¤|® ± l ¯ Ç ¯ q ¹ q ¤ l l R j

f * XA + Q | v + - L |

l ¯ a ½ h D¬ / - × £ L (2 )

- ¤® q 2l - ¤ ± -

W c can prow that P V (Ë ) i s convex i n Ë . If w c Ä m pu- -

d P V Þ ) £

d Qj

w c $ tai n the f ol l ow i ng cquan on .

L H ¤a) ® -¤* Ù h ºv +÷ l-e £ ­ ¯ - |v +µ ¤ ¯a =0 G)£ 1 ! D D k ¯ a

E quat i on ( 3) can k rew r i t ten Ä

( * V ¤® | É $ h )v + - L I ¤ ¯ - 1 | - | v + £ |¥ = O (4 )

¯ a 1 ! D , D k j q j

T m ENGINEER¼NG ECONOMIST ¤ 20Ol ¤ V OLUME 46 ¤ N UMBER2 133

B y subst i tut i ng E q. (9) i nto £ . (8) , w e obt ai n

n l 2 A D , | v + *, ½± ± I ± ® J / U J

ç = P V ( q ) ² A , + × £ + U ¬ / ( 1O)

a v a

W c no w com m a the usc o f t he co m m ¤ l c uni t s. W It hou t loss of gc ncr al i t $ WC

reor der i tems i n the order of i nvent or y shm age t i m e. K t f , (j * l ) k thei nv c ntor y l evel of i tcm j w hen i t em l s i nvcnt w y l evel beco rncs O, i .e. f , = O .

T hen i tem ¹ H ) w i l l r un out at t im e U ï , and COIN c ru ble uni ts cm k

replen i shcd at t hat t i m e. A f ter co nv er t i b le u ni t s h ave bec n co nsum ed , the reg ul ar

econo m i c ord er q uant i n g can be rep l eni shed m erc an - - K t P VTR C (¡ , . . . , RU

be thc present v al ue o f t ot al costs i ncl ud ing co nver t i bl e dcciSU Its- T hen the costs

consi st of three c l a nents:

( i ) Co rn e r fh fe Co.sf

. .

- a Í

³ R,¤ ± ( 1 1)

( i i ) H OM in g Cod ± r C onver- - fe M I l o

pR m- -aÒ h .R * n f -¤æ ½ |Aj-- M ±± (B A rk-mdf =¤ ±| é - Á | l - ¤ ± | | (12)£l | Q Q l l |

( i i i ) Ref¤po u f Co s, a/ ?er C a u tum ng Con va - - fe Um n

-¤z n¢ e % ( l 3 )

N ote that at t im e ( L + í ) / q . rcgu l ar eco n om i c ordcr q uant i t y

rcplen i shm ens w i l l k rCSUITE d - T hen, our deci si on pr ob l em can be ex pressed Ä

foH ow s:

M i ni m i ze-p V7 RC ( R , , .. . , Û )

¸ ± | f h R A .D . A .D. -¤# ½ -¤Ì -¤¹ £ |= S ®± | | c R . + £ £ - é é + £ Á e ± |e ± + W e ± | ( l 4 )é éN | l ± q a ® O ® ! ¯ |

sub jec t - Ö R, s N,=l

í Z 0 V j

THE ENGmEERmG ECONOMIST ¤ 2001 ¤ VOLUME 46 ¤ N UMBER2 l 35

f * a w Yl - ¤- + - - £ l

D. | q D. |í = ù l n| h ¯ | (15)

l C, + - £- ll ¯ O l

STEP 2: Start wi th an arbitrary ú >O.

STEP 3: Compute í S Ä fol lows:

| Ã a w || - 4- + - - é- |

D. | q D. |R = - ¹- ln| - ! | fa | / h ¢ -. ¾ | ¤j

| ú +| c, + £ |e W || t ¯ O / l

STEP 4: I f Y í < N , then decrease A and go to STEP 3.

I f µ ¼ , > N , then i ncreme A and go to STEP 3

If ° í = N , wc ham fa uM att opumal solution

Ex AMPLE:

We asa lty- that the- are l 0OO uni ts of the convertible i tems avai lable. T he

detai led data for thi s a m ple are × ven i n TABLE l . M set a = O.25/uni t time.

TABLEd - D ata fa the Examplc (N = 1000 and ú =O.25)

|Item | q | ± | Ã | AI | c | DI || Item 1 | $20 | $30 | $9 | $75 | m | 2m |

| lÔ 2 | $30 | $50 | $15 | $m | 50 | m |

| Itm 3 | $15 | $20 | $6 | $110 | m | 8m |

| Item 4 | $1m | $m | $36 | $2m | 2m | 5m |

First we compute the unconstrained cotwa t - lc quanti ties using Eq. ( 15),

and obtain (ç , R2 ¡ R4) = ( 1449,467,4260,1750). T he total excceds M 0O (i .e.

M . If we apply the l ine search al gori thm, we f ind the fol l ow ing optimal

convcruble quMIm ics (ç , R2 R3 R4) = (O,213,0,787). wi th the optimalL agrangian multipl i er, ú ¤, being l l .910 for this case. The ¦ value

approx imatel y represents the savings i n costs associated w i th i ncrcasi ng N from

M 0Oto 1Û 1.I f we increase the hol ding cost rate of Item 4 to h = $50, the unconstrained

convert- lc quanti ti es become (ç , R2 R, R4) = ( l 449,467,4260, l 449) . T he

fol l ow i ng optimal convertible quanti t ies are then obtained (¡ , R2 R3 R4) =

(272 36A n n . Hcrc the opti mal l a v a td att multi pl ier) ¦ , becomes l O.762.

1 3 6 H E E N G W E E R I N G E C O N O M I S T ¤ 2 0 O l ¤ V O L U M E 4 6 ¤ N U M B E R 2

W c n o t i c e t h a t R 4 i s d e c ¤ a s c d s o m e w h a t a c m a t e d -

I f w c i n c r c a s e t h e c o n v c r S i o n c o s t o f I t e m 4 t o C 4 = $ 1 1 O , t h e u n c o n s t r a i n e d

c o n v e r t i b l e q u a n t i t i e s b c c o r m - ( ¡ , R 2 R 3 R 4 ) = ( 1 4 4 9 , 4 6 7 , 4 2 6 0 , 9 4 6 ) . TTnIn ®½Ih½Ç

c

f o l l o w i n g o p t i m a l c o n v e r t i b l e q u a n t i t i e s a r e t h e n o b t a i n e d ( R lh , R 2 R 3 R 4 ) =

( 3 7 9 , 2 8 5 , O , 3 2 6 ) . H e r e t h e o p t i m a l L a g r a n g i a n m u l t i p l i e r , ú ¤ , b e c o m e s 7 9 1 7 .

W e n o t i c e t h a t ç i s d e c r e a s e d s o m e w h a t a s a p - - d a n d t h e m a g n a l v a l u e o f

a n e x t r a u n i t o f t h C C O I l V c r t i b l c i t e m i s r e d u c e d s o m e w h a t d u e t o t h e h i g h e r u n i t

c o s t o f c o m a s i o n t o c n d I t c m 4 .

C O N C L U S I O N S

W e h a v e c o n s i d e r e d t h e s i t u a t i o n o f a g r o u p o f e n d i t e m s f a c i n g l e v e l d o n a n d

p a t t e r n s a n d t h e p o s s i b i l i t y o f r e g u l a r p u r c h a s i n g - l n a d d i t i o n , t h e - a r e a n u m b e r

o f u n i t s t h a t c a n b e c o n v e n e d i n t o a n y o n e o f t h e e n d i t a n s , b u t a t d i f f e r c n t u n i t

c o s t s o f c o n v e r s i o n . A n a l g o r i t h m h a s b e e n d e v e l o p e d f o r o p t i m a H y a l l o c a t i n g

t h e c o n v e r t - l c u n i t s t o t h e v a r i o u s c n d i t a n s -

A C K N 0 L V L E D G E n t E N T S

T h e r e s e a r c h o f E d w a r d S i l v e r h a s b e e n s u p p o r t e d b y t h e N a t u r a l S c i e n c e s a n d

E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a u n d e r G r a n t A 1 4 8 5 , b y t h e e a r n 1a

C h a i r a t t h e U n h c r s i t y o f C a l g a r y . T h c r e s e a r c h o f I l k - o n g M o o n h a s b c e n

s u p p o r t c d b y t h e I z a d i n g R e s e a r c h e r F u n d ( G r a n t N o . 2 0 0 0 - O 4 1 - E 0 0 1 3 O )

h o s t c d b y K o r e a R o c a - h F o u n d a t i o n i n K o r c a .

R E F E R E N C E S

[ l ] B R O W N , RR .£Gä .±¯ °%w®%/A4 d vm a ,n1 C° ed d S& e ,rn±qvtvud®®®C° e P am¿ ,r±m,**f°±3s . fh ± v² em ±¹ fmOm ,rn±À3T ,y , C oa ,n¹ t° f* rm Od fLC,c ,±, M a t e r i a l s

M a n a g e m e n t S y S t a n s 1 n c . , 2 n d E d . , N w w i c k V W I t t o l H , 1 9 8 2 .

[ 2 ] D I K s , E . B . , A . G . D E K 0 K , M d A . G . L N O D I M 0 S , ° M u l t i - e c h e l o n s y S t a n s : a

s e r v i c c I n c a s u r e p e r s p c t i v c , ± E u r o p e a n J o u r n a l q f O p e r a f I o n a f R e s e a r c h ,

V o l - 9 5 , 1 9 9 6 , p p . 2 4 1 2 6 3 .

[ 3 ] F E H a m a - E . , a n d H . L . L E E , ° M a s s c u s t o m i z a n o n a t H e W i e n - P a c k a r d : t h e

p o w e r o f p o s t p o n e m e n t , ± H a n a r d B u s i n e s s R e d e - - , , V o l - 7 5 , 1 9 9 7 . p p . l l 6 -

1 2 1 .

[ 4 ] L E E , H . L . , C . B 1L L I N G T 0 N , a n d B . C A R T E R , ° H c w l e u - P a c k a r d g a n s c o n t r o l o f

i l M I n o r y a n d s e r v i c e t h r o u g h d e s i g n f o r l o c a l i z a t i o n , ± f n r e d u c e s , V o l u m e

2 3 , 1 9 9 3 , p p . 1 - 1 1 .

[ 5 ] M A B R I , M . C . , M d L . E G E L D E R S , ° R e p a i r a b l e i t e m i n v e n t o r y s y s t e m : A

l i m a t u r e r c v i e w , ± B C I g a r t J o t u n a f q f O p e r a f ² f L y R e s e a r c h , SS&×SsmYvfmad¾¿ f°ÈmÁ ®LbÈ®¹sgmy±mÁf°È ®®CG s a n d

C o m p a r - r S c i e n c e , V o l - 3 O , 1 9 9 0 , p p . 5 7 6 9 .

[ 6 ] N A H M I A S , S . , ° M a n a g i n g r e p a i r a b l c i t c m i m a n o r y s y s t e m s : A r c v i e w - ± i n :

S c h w a r z - LL .B BB . (ï cd d¥ .J )L , AM f z°d ®d f*M f°M ®À -Ì Lk eaw tvw ¯zd ed f P rm oa ®d4

¾ fú* zu¿ ®°d Cdì f° ®mm oam ,±° t°d /â¹ f*h ,±mw tn± tv² ¯zø eam fnm 1° fm o¿ rnn ½y , C - a r r - f S$* $* ½y ¯q¶U :sg y±Ò f¿Î eam ,±m ,±mm 1nU°

»sÌ .ö :

T H E E N G I m E R I N G E C O N O M I S T ¤ 2 0 0 l ¤ V O L U M E 4 6 ¤ N U M B E R 2 l 3 7 -

T h e o r y m d P r a t i c c , N o r t h - H o l l a n d . A m s t e r d a m , 1 9 8 1 , p p . 2 5 3 2 7 7 .

[nm 7± ] Sm HmÁ E«Á Rm Bm RÄ OÄ Oê Kòç Eí , Cc .CL CÒ . °± o OÆé fp±§ ,ï첩 f°m¹Ô ®*¹¹ ,±mÁ ,±m§ 1mî ad f f a y - f - f o u AMÎ 4± od d*° eddÎ f°h¹ ®*¹¦ ,±¼ . g q® f S¡à¡ ,y,±M ,®sd² isg²Èy±ÒÈ f¿mÇ eammm ,±mmm ,nîmÏ 1LUÏ ®sÌ y : M u f f ® e d d o n

D M a i m - s , ± J o h n W H e y & S o n s , N e w Y w L 1 9 9 Z

[ 8 ] S I L V E R , E . , a n d I . M O O N , ° T h c m u l t i - - m S i n g h p e r i o d p r o b l e m w i t h a n i n i t i a l

s t o c k o f c o n v e r t i b l e u n i t s , ± W o r k i n g P a p e r , F a c u l t y o f M a n a g e m e n t , T h e

U n i v a s h y o f C a l g a r y , E u m p e a n J o a n - a f q f O p e r a f M n a f R - s e a - e h , V o l - l 3 3 ,

2 0 0 1 , p p . 4 6 6 4 7 7 .

[ 9 ] S E V E R , E . , D . P Y K E , a n d R . P E T E R S 0 N , W z p a t r o n M a n a g - m - H f a n d P m d u m a t

P f a r m M g a n d & A d d i n g , ± 3 r d E d . J d m W H e y & S o n s , N c w Y o r k , 1 9 9 8 .

[ 1 Ì V E R R - T , J . H . C . M . , ° D e s i g n a n d C o n n ¤ f q f s - n i c e P m ¤ f D i m - i b z d o n

S y g e m s , ± U n p u b l i s h e d P h . D . D i s s e r t a t i o n , T e c h n i c a l U n i v e r s i t y o f

E i n d ¤ v c n , 1 9 9 7

B I O G R A P H I C A L S K E T C H E S

E D W A R D A . S I L V E R h o l d s t h e C a n n a C h a i r a t t h c U n i v e r s i t y o f C a l g a r y . H e r c c c i v e d a

B a c h e l o r o f C i v i l E n g i n e e r i n g f r o m M c G i l l U n i v e r s i t y , a m s u b s e q u e n t l y . a S d e m e

D Ä t o r a t e i n O p e r a t i o n s R c s e a r c h a t M . I T . P r i o r t o t h e U n i v e r s i t y o f C a l g a r y , h i s w o r k

e x p e r i c n c e i n c l u d e d p o s i t - m w i t h A r t h u r D . L i t t l e I n c . , a n d t h e U n i v e r s i t y o f W a m b o -

D r . S i l v a h Ä h e l d s e v e r a l v i s i t i n g p o s i t - m i I n t u i t - a t S t a r t - r d U n i v e r s i t y . t h e S w i s s

F e d e r a l I m o t u - o f T e c h n o l o g y , t h e U n i v a s h y o f C a n t a - m r y , N e w Z e a l a n d ( V i s i t i n g

E r s k i m F C H o w ) a n d t h e U n i v e r s i t y o f A u c k l a n d , N e w Z c a l m - d ( A u c k l a n d F o u n d a t i o n

V i s i t o r ) . P r o f c s s o r S i l v e r h a s c o n s u l t e d a m p r o v i d e d e x e c u t i v e t r a i n i n g a m o t h e r

w o r k s h o p s f o r a w i d e r a n g e o f o r g a n z a - m . T h c s c a c t i v i t i e s h a v e c o v e r e d a b r o a d r a n g e

o f t o p i c s i n d u d i n g i m a n o r y m a n a g e m e n t , s u p p l y c h a i n m a n a g a n a u , b u s i n g s p r o c e s s

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