a model for scheduling postal distribution

8/7/2019 A model for scheduling postal distribution

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374 E u r o p e a n J o u r n a l o f O p e r a t i o n a l R e s e a r c h 5 8 ( 1 9 92 ) 3 7 4 - 3 8 5

N o r t h - H o l l a n d

Theory and M ethodology

A model for scheduling postal distr ibutionemployees

Manoj K. Malhotra

Department of M anagement Science, College of B usiness Administration, University o f South Carolina,

Columbia, SC 29208, USA

L a r r y P . R i t z m a n , W . C . B e n t o n a n d G . K e o n g L e o n g

Academic Fac ulty o f Management Sciences, The Ohio State University, 177 5 College Ro ad, Columbus,

OH 43210, USA

R e c e i v e d J u n e 1 9 9 0 ; r e v i s e d N o v e m b e r 1 9 9 0

Abst rac t : Th is paper desc r ibes the des ign and ana lys is o f a genera l izab le l inea r p rogramm ing mode l fo r

schedu l ing personne l in Uni ted S ta tes pos ta l d is t r ibu t ion s ta t ions . The schedu l ing mode l incorpora tes

param ete rs such as d i f fe ren t work loads a long rou tes , equ i tab le ass ignment o f overt ime be tw een fu l l - time

employees des i r ing over t ime , t ra in ing leve ls o f pa r t - t ime employees , and management p re fe rences fo r

different levels of custom er service. A nea r optim al sche dule specifying daily assignm ent of ful l- t ime and

part- t ime employees to different task categories is developed. The model is i l lustra ted using data from atypical week in a large postal d is tr ibution s ta t ion located in a representat ive American ci ty . Analysis of

the resu l t s shows tha t the p roposed mode l cap tu res the essence o f schedu l ing dec is ions under taken by

pos ta l superv iso rs and mid- leve l managers f rom the same da ta se t . The schedu les deve loped by the

mode l a re cons idered feas ib le from an im plem enta t ion v iewpoin t , and a lso requ i re le ss over time than the

ma nagem ent schedu les . The ' impac t o f chang ing ce r ta in long- te rm po l icy is sues ( such as the mix be tw een

par t - t ime and fu l l -t ime employees , a ss ignment po l ic ies , and t ra in ing leve ls o f pa r t - t ime employees ) can

be eva lua ted by conduc t ing a sys temat ic sens i tiv ity ana lys is o f the mod e l ' s pa ram ete rs . A noth er poss ib le

fu tu re bene f i t l ie s in using the mod e l on a m ic rocomp ute r fo r deve lop ing ac tua l day- to -day schedu les o f

postal d is tr ibution employees.

K e y w o r d s : Service, post office, scheduling, distribution

1 . I n t r o d u c t i o n

Research in service sector scheduling can be

classif ied along three levels of disaggregation as

sugges ted by Kra jewsk i and R i tzm an (1977): the

staff s iz ing problem, shif t scheduling and crew

ass ignment p rob lem, and shor t - te rm ind iv idua l

assignment problem. The third level of disaggre-

gation, the operat ional level , has been scanti ly

addressed in the service sector l i terature even

though several theoret ical and applicat ion ori-

ented s tudies in a varie ty of industr ies have been

reported at the f irs t two levels . Past research at

the f i r s t two leve ls has cen te red on opera to r

scheduling in the te lephone industry (Segal , 1974;

H end erso n a nd Berry, 1976 ; Buffa e t a l ., 1976;

and Keith , 197 9), shif t scheduling in the ba nking

indus t ry (Ma ber t and R aede ls , 1977 ; and Kra jew-

ski e t a l . , 1980 ), shif t scheduling in the postal

depa r tme nt (R i tzm an e t a l ., 1976 ; and Show al te r

0 3 7 7 - 2 2 1 7 / 9 2 / $ 0 5 . 0 0 © 1 9 9 2 - E l s e v i e r S c i e n c e P u b l i s h e r s B . V . A l l r i g h t s r e s e r v e d



M .K Malho tra e t a l. / Mo de l fo r schedu l ing pos ta l d i s t ribu t ion employees 375

e t a l . , 1 9 7 8 ) , p e r so n n e l s t a f f in g an d sch ed u l in g in

t h e h e a l t h c a r e i n d u s t r y ( W a r n e r a n d P r a w d a ,

1 9 7 2 ; Ab e r n a th y e t a l . , 1 9 7 3 ; an d Fr an z e t a l . ,

1 9 8 9 ) , an d c r ew p lan n in g in th e a i r l i n e in d u s t r y

( Mar s t en e t a l . , 1 9 7 9 ) . Th ese s tu d ie s h av e d emo n -

s t r a t e d t h e u s e f u l n e ss o f o p t im i z a t i o n m o d e l s a n dh e u r i s t i c m e t h o d s f o r m o d e l l i n g a c t u a l s e r v i c e

sy s t ems . Th e se t t i n g f o r o u r s tu d y i s t h e ma i l

d i s t r ib u t io n f u n c t io n o f p o s t o f f i ce s . I t d ea l s wi th

t h e d e t a i l e d s h o r t - t e r m e m p l o y e e a s s i g n m e n t a n d

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

o f d i s a g g r e g a t i o n , a n u n d e r a d d r e s s e d a r e a o f t h e

se r v ice sec to r sch ed u l in g l i t e r a tu r e .

T h e c o n s t r a i n ts a n d o b j ec t iv e f u n c t i o n o f o u r

m a t h e m a t i c a l p r o g r a m m i n g m o d e l r e p r e s e n t t h e

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

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

so n n e l i n th e U .S . Po s t a l Se r v ice . Th e s t r u c tu r e

o f t h e m o d e l i s s i m i l a r t o t h e l i n e a r p r o g r a m m i n g

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

H a n n s m a n a n d H e s s ( 1 9 6 0 ) , w h e r e b y c o s t s a r e

min imized su b jec t t o a i n v en to r y b a lan ce co n -

s t r a in t ( i n v en to r y o f ma i l i n t h i s ca se ) , av a i l ab le

cap ac i ty ( to t a l h o u r s th a t emp lo y ees can wo r k ) ,

a n d t h e c o n s t r a i n t o f w o r k g e t t i n g d o n e

( p r o cess in g o f ma i l) . Th e d ec i s io n v a r i ab le s a r e

u n d e r t i m e , r e g u l a r t i m e , a n d o v e r t i m e a l l o c a t i o n s

o f h o u r s to emp lo y ees f o r p r o cess in g th e g iv en

wo r k lo ad s . Ho wev er , t h i s mo d e l i s u sed f o r o b -t a in in g d e ta i l ed l ev e l d a i ly sch ed u l in g d ec i s io n s ,

a s o p p o s e d t o l o n g e r - t e r m a g g r e g a t e p r o d u c t i o n

p lan n in g d ec i s io n s .

Sec t io n 2 r ev iews th e p r o b lem se t t i n g in wh ich

th e mo d e l i s d ev e lo p ed , a s we l l a s t h e a s su mp -

t io n s mad e in sp ec i f y in g th e mo d e l co n s t r a in t s .

S e c t i o n 3 p r e s e n t s t h e m o d e I . T h e m o d e l is te s t e d

a n d v a l i d a t e d w i t h d a t a c o l l e c t e d f r o m a r e p r e -

sen ta t iv e s t a t io n in Sec t io n 4 an d th e mo d e l r e -

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

s io n s . Th e f in a l sec t io n d i scu sses th e p o ten t i a l

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

p o s ta l o r g an iza t io n s .

2 . P r o b l e m s e t t i n g

For a g iven c i ty , mai l i s de l ivered to f ina l

d es t in a t io n s f r o m sev e r a l d i f f e r en t s t a t io n s , an d

each s t a t io n i s sp l i t i n to a sp ec i f i c n u mb er o f

r o u te s . Fo r each r o u te , emp lo y ees f i r s t ' c a se ' t h e

m a i l ( a r r a n g e a c c o r d i n g t o d e l i v er y s e q u e n c e ) b e -

f o r e s t a r t i n g th e s t r ee t d e l iv e r y . ' Sch eme k n o wl -

ed g e ' i s r eq u i r ed f o r ca s in g th e ma i l f o r an y g iv en

r o u t e . T h i s m e a n s t h a t a n e m p l o y e e m u s t b e

t r a in ed b e f o r e b eco min g f u l ly p r o f i c i en t i n ca s in g

th e ma i l f o r an y sp ec i f i c r o u te . Th e wo r k f o r ce

c o n s i s t s o f p e r m a n e n t f u l l - t i m e ( F T ) e m p l o y e e s

wh o wo r k e ig h t h o u r s a d ay , an d p a r t - t ime f l ex i -b l e ( P T F ) e m p l o y e e s w h o c a n b e s c h e d u l e d f o r a

l i m i t e d a m o u n t o f t i m e e a c h w e e k , a n d n e e d n o t

b e ca l l ed in ev e r y d ay . Th e wo r k sh i f t f o r PTF

emp lo y ees can s t a r t a t an y t ime o f t h e d ay . Th e

m i x b e t w e e n t h e f u l l -t i m e a n d p a r t - t i m e w o r k e r s

d e te r min es th e sch ed u l in g f l ex ib i l i t y av a i l ab le f o r

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

u n f o r e s e e n c i r c u m s t a n c e s s u c h a s a b s e n t e e i s m ,

sick leave , and a t t r i t ion . Schedul ing f lex ib i l i ty i s

co n s t r a in ed b y a u n io n r u l e , wh ich cu r r en t ly s t i p -

u l a t e s t h a t a t l e a s t 9 0 % o f th e w o r k f o r c e s h o u l d

b e c o m p r i s e d o f F T e m p l o y e e s .

G e n e r a l l y , f u l l -t i m e e m p l o y e e s ' o w n t h e b i d ' t o

d e l iv e r ma i l o n a g iv en r o u te . Th e FT emp lo y ees

u su a l ly ca se th e ma i l o n th e i r o wn ed r o u te s , b u t

a l so can b e u sed f o r d e l iv e r in g ma i l o n a l t e r n a te

r o u t e s. A l t e r n a t e l y , F T e m p l o y e e s m a y b e u s e d

o n ly f o r d e l iv e r y p u r p o ses o n th e i r o wn r o u te s i f

t h e wo r k lo ad f o r t h e r o u te i s h eav y . I f a FT

emp lo y ee i s u n av a i l ab le d u e to v aca t io n o r s i ck

leav e , t h e r o u te i s t emp o r a r i ly a s s ig n ed d u r in g

t h e a b s e n c e p e r i o d t o a s e n i o r P T F e m p l o y e e

t r a i n e d f o r t h a t r o u t e . R e g u l a r d e l i v e r y f o r t h a tr o u t e i s a g a i n c a r r i e d o u t b y t h e F T e m p l o y e e o n

r e t u r n i n g f o r w o r k a f t e r t h e b r e a k . F T e m p l o y e e s

wo r k f o r f iv e d ay s o n ly in a week , wh e r eas th e

p o s t o f f i ce d e l iv e r s ma i l s ix d ay s a week . Th e

wo r k f o r t h e s ix th d ay o n each r o u te i s a s s ig n ed

t o a F T e m p l o y e e c a l l e d a ' T 6 ' w o r k e r . T h e T 6

emp lo y ee wo r k s o n f iv e d i f f e r en t r o u te s in a g iv en

w e e k , w i t h e a c h w o r k d a y c o r r e s p o n d i n g t o t h e

o f f d ay f o r t h e b id o wn er o f t h a t p a r t i cu la r r o u te .

Ma i l v o lu me can f lu c tu a te s ig n i f i can t ly b e -

tween d ay s o f a week an d ac r o ss d i f f e r en t r o u te s .W h e n t h e c a s i n g a n d d e l i ve r y w o r k l o a d f o r a

r o u t e e x c e e d s a n e i g h t - h o u r w o r k e q u i v a l e n t, e i-

th e r o v e r t ime i s i n cu r r ed , o r a s s i s t an ce mu s t b e

p r o v i d e d t o t h e F T e m p l o y e e w h o o w n s t h e b i d

f o r t h a t r o u te . Th i s a s s i s t an ce can b e in th e f o r m

o f p r o v id in g f u ll t ime ' r o u te r s ' t o t h a t r o u te f o r a

g iv en d ay . R o u te r s a s s i s t o n ly in ca s in g th e ma i l ,

t h e r e b y l e a v i ng t h e r e g u l a r o w n e r o f t h a t r o u t e

mo r e t ime f o r d e l iv e r y o n th e s t r ee t s b y r ed u c in g

th e ca s in g wo r k lo ad . Each r o u te r i s t y p ica l ly

t r a in ed f o r ca s in g ma i l o n ab o u t t en r o u te s . An -



376 M.K~ Ma lhotra et al. / Mo del fo r schedu ling postal distribu tion employees

o t h e r w a y t o h e l p o u t a n o v e r l o a d e d F T e m -

p l oye e on a pa r t i c u l a r rou t e i s t o a s s i gn a P TF

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

a ux i l i a ry de l i ve ry a s s i s t a nc e on t he s t r e e t s . O n

t h e o t h e r h a n d , F T e m p l o y e e s w h o s e r o u t e s a r e

u n d e r l o a d e d c a n a l s o b e u s e d f o r d e l i v e r i n g m a i l

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

o w n d e l iv e r y s c h e d u l e . S o m e F T e m p l o y e e s d o

no t ow n t he b i d t o de l i ve r on a spe c i f i c rou t e .

T h e y a r e k n o w n a s ' u n a s s i g n e d r e g u l a r s ' . C o r r e -

spond i ng l y , t he y a l so c a n be use d fo r de l i ve ry

p u r p o s e s o n a n y r o u t e w h i c h i s o v e r l o a d e d , a n d

f o r c a s i n g o n t h o s e r o u t e s f o r w h i c h t h e y p o s s e s s

s c h e m e k n o w l e d g e f r o m p a s t t r a i n i n g .

N o w t h e p r o b l e m a r i s e s o n a s s i g n i n g w o r k t o

t he a va i l a b l e e mpl oye e s i n suc h a w a y t ha t ove r -

t i me i s mi n i mi z e d . A c us t ome r s e rv i c e s t a nda rd

must a l so be sa t i s f ied: Al l f i rs t c lass mai l rece ived

a t a s t a t i on i n t he morn i ng mus t be de l i ve re d t he

sa me da y . The bu l k bus i ne s s c l a s s ma i l mus t be

de l i ve re d w i t h i n 48 hours , w h i c h me a ns t ha t a

l i m i t e d a m o u n t o f m a i I c a n b e i n v e n t o r ie d . E a c h

c a t e gory o f ma i l c a n be p hys i c a l ly i de n t i f i e d be -

fo re t he c a s i ng ope ra t i on be g i ns .

T h i s s c h e d u l i n g p r o b l e m i s a n i n v o l v e d o n e ,

b e c a u s e a s t a t i o n m a n a g e r d o e s n o t k n o w w h i c h

P T F e m p l o y e e s h o u l d b e u s e d i n w h a t c a p a c i ty t o

ob t a i n t he be s t r e su l t s . The op t i ons ope n a re t o

use a P TF fo r c a s i ng he l p , a ux i l i a ry de l i ve ry a s s i s -t a nc e on t he roa d , o r bo t h c a s i ng a nd de l i ve ry on

a spe c i f i c rou t e ( t o c ove r fo r t hose F T e mpl oye e s

w ho a re una va i l a b l e on a g i ve n da y due t o va c a -

t io n o r s i ck le a v e) . M o r e o v e r , e a c h P T F e m p l o y e e

i s t r a i ne d fo r c a s i ng ma i l ' on d i f fe re n t rou t e s .

W h i c h r o u t e m u s t h e / s h e b e a ss ig n e d to o n a

g i ve n da y so a s t o mi n i mi z e ove r t i me hours , a nd

a l so me e t t he s e rv i c e s t a nda rds? O t he r ob j e c t i ve s

a r e t o m i n i m i z e t h e i d l e t i m e f o r F T e m p l o y e e s ,

a n d t o e q u i t a b ly d i s t ri b u t e t h e o v e r t i m e h o u r s

a m o n g a l l F T e m p l o y e e s d e s i r i n g o v e r t i m e . T h ei mp or t a n c e o f t he se ob j e c t i ve s is c a p t u re d by

a s s i gn i ng d i f fe re n t r e l a t i ve w e i gh t s t o t he m i n t he

o b j ec t iv e f u n c t i o n , a n d b y a d d i n g a p p r o p r i a t e

c o n s t r ai n t s . T h i s p r o b l e m i s m o d e l e d a s a l i n e a r

p r o g r a m , a s p r e s e n t e d n e x t.

3 . S t a f f s c h e d u l i n g m o d e l

The c ons t ra i n t s e t a nd t he ob j e c t i ve func t i on

o f t h i s l i n e a r p r o g r a m m i n g m o d e l a r e r e p r e s e n t a -

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

pos t a l d i s t r i bu t i on sys t e m. I t c a n be so l ve d t o

y i e l d da i l y s c he du l e s , w h i c h i de n t i fy w ha t rou t e s

e a c h e m p l o y e e m u s t w o r k o n , i n w h a t c a p a c i t y

( o v e r t i m e o r r e g u l a r ) , a n d h o w m a n y h o u r s m u s t

b e s p e n t o n e a c h w o r k o p t i o n . F o r t h e P T Fe mpl oye e s , t he s c he du l e spe c i f i e s t he rou t e s on

w h i c h a P T F e m p l o y e e w o r k s f o r c a s i n g m a i l o r

re nde r i ng a ux i l i a ry de l i ve ry a s s i s t a nc e on t he

s t re e t s . I t a l so show s w h i c h P TF e mpl oye e s shou l d

be use d f rom t hose a va i l a b l e on a g i ve n da y . The

m o d e l o u t p u t s t h e a m o u n t o f u n c a s e d o r cu r -

t a i l e d ma i l on e a c h rou t e fo r t ha t da y . The s t a -

t i o n m a n a g e r c a n t h e n e i t h e r h o l d b a c k t h e b u l k -

bus i ne s s c l a s s ma i l on t ha t rou t e fo r t he ne x t da y ,

o r a u t h o r i z e m o r e o v e r t i m e . T h e c u r t a i l e d m a i l i s

g i ve n t he s a me p r i o r i t y a s f i r s t c l a s s ma i l a r r i v i ng

o n t h e f o l l o w i n g m o r n i n g . T h i s p r o c e d u r e p e r m i t s

t h e m o d e l t o m e e t t h e s e rv i ce s t a n d a r d s o f d el iv -

e r i ng t he bu l k bus i ne s s c l a s s ma i l a l so w i t h i n 48

hours , a nd i n f a c t is ide n t i c a l t o t he w a y t he

de c i s i ons a re a c t ua l l y ma de b y t he pos t a l supe rv i -

sors .

C u r r e n t l y t h e s c h e d u l e r a s s i g n s w o r k e r s f o r a

g i ve n da y t o t he i r r e spe c t i ve rou t e s on t he ba s i s

o f e mpl o ye e a va i la b i l it y . N o fo re c a s t i ng o f ma i l i s

done , s i nc e t he a c t ua l ma i l vo l ume f i gu re s a re

a va i l a b l e a t t he t i me t ha t s c he du l i ng de c i s i ons

m u s t b e m a d e . T h e P T F w o r k e r s a r e c a l l e d i n o nt h e b a s i s o f ac t u a l m a i l v o l u m e s e x p e r i e n c ed . T h e

b i d o w n e r s o f o v e r - l o a d e d r o u t e s r e q u e s t a d d i -

t i ona l a s s i s t a nc e o n a da i l y ba s is . Th e sup e rv i so r

c a n t h e n a s s i g n t h e m s o m e P T F w o r k e r f r o m t h e

p o o l a v a i l a b l e , a s s i g n a F T e m p l o y e e f r o m a n

u n d e r - l o a d e d r o u t e , o r a u t h o r i z e o v e r t i m e . A

c o m b i n a t i o n o f th e s e a l t e r n a t i v e s m a y a l s o b e

c o n s i d e r e d .

3 .1 . S t a f f s c h e d u l i n g m o d e l n o t a t i o n

T h e s t a ff s c h e d u l i n g m o d e l p a r a m e t e r s a n d

de c i s i on va r i a b l e s a re de f i ne d be l ow .

P a r a m e t e r s :

D ; - - - D e m a n d o f m a i l i n f e e t t o b e d e l i v e r e d

on rou t e i .

p c i = N o r m a l p r o d u c t i v it y o f c a si n g , m e a s u r e d

a s f e e t o f ma i l c a se d pe r hour by a fu l l y

t r a i n e d e m p l o y e e o n r o u t e i .

j 1 = T h e o r d e r e d s e t o f F T e m p l o y e e s a t t h e

s t a t i on . Th i s i nc l ude s una s s i gne d re gu -



M.K. Ma lhotra et al. / M o d el fo r scheduling postal dis tr ibution employees 377

j 2

j3

jR

l u

Qi

Ci

U

q~.tj

q2tj

l a r s , a s w e l l a s t h e e m p l o y e e s w h o o w n

b i d s f o r t h e i r r o u t e s .

T h e o r d e r e d s e t o f P T F e m p l o y e e s .

T h e o r d e r e d s e t o f F T e m p l o y e e s w i ll in g

t o w o r k o v e r t i me . T h i s i s a s u b s e t o f j 1

T h e o r d e r e d s e t of e m p l o y e e s w h o o n l yd o r o u t e r w o r k o n s o m e s p e c i f i e d r o u t e s .

T h e s e t o f u n d e r l o a d e d r o u t e s ( h a vi n g

l e s s t h a n a n e i g h t - h o u r e q u i v a l e n t o f

w o r k ) o n a g i v e n d a y .

' O f f i c e h o u r s s t a n d a r d i z a t i o n ' f o r r o u t e i ,

d e f i n e d a s t h e n u m b e r o f ho u r s s p e n t f o r

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

n o r m a l e i g h t - h o u r d a y .

' S t r e e t h o u r s s t a n d a r d i z a t i o n ' f o r r o u t e i ,

d e f i n e d a s th e n u m b e r o f h o u r s s p e n t f o r

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

a n o r m a l e i g h t - h o u r d a y .

M a x i m u m a m o u n t o f m a i l in h o u r s e q ui v -

a l e n t w h i c h c a n b e c u r t a i l e d o n r o u t e i

o n a n y s i n g l e d a y .

A n u p p e r b o u n d o n t h e t o t a l m a i l c u r -

t a i l e d a c r o s s a l l t h e r o u t e s . T h i s p a r a m e -

t e r c a n b e s e t b y t h e m a n a g e m e n t , a n d

s e r v e s a s a s u r r o g a t e m e a s u r e o f c u s -

t o m e r s e r v i c e .

T h e s e t o f a l l t h e r o u t e s t o b e s e r v i c e d a t

a s t a t i o n .

1 i f a F T e m p l o y e e j o w n s t h e b i dt o d e l i v e r m a i l o n r o u t e i ;

0 o t h e r w i s e .

1 i f a F T e m p l o y e e j , w h o s e r v e s a s a

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

r o u t e i ;


1 i f a P T F e m p l o y e e j is t r a i n e d f o r

c a s i n g m a i l o n r o u t e i ;


1 i f a FT e m p l o y e e j is w i l l in g t o

b j = w o r k o v e r t i m e ;


10 i f t h e FT e m p l o y e e j is a v a i l a b l e

r ) - - f o r w o r k o n t h e g i v e n d a y ;

o t h e r w i s e .

i i f t h e F T e m p l o y e e j , w h o s e r v e s a s

t h e r o u t e r , i s a v a i l a b le f o r w o r k o n

t h e g i v e n d a y ;

o t h e r w i s e .

0

i f t h e P T F e m p l o y e e j i s a v a i l a b le

f o r w o r k o n t h e g i v e n d a y ;

o t h e r w i s e .

f = O v e r t i m e l im i t , i n h o u r s p e r d a y , f o r e a c h

e m p l o y e e . I t i s n o r m a l l y t w o h o u r s o f

o v e r t i m e , w h i c h r e p r e s e n t s t h e i n d u s t r y

s t a n d a r d o f a 2 5 % o v e r t i m e l i m i t p e r

e m p l o y e e .

p ; = R e l a t i v e c a s i n g e f f i c ie n c y o f a n u n t r a i n e d

P T F e m p l o y e e w o r k i n g a s a s u b s t i t u t e b i d

o w n e r o n a n u n t r a i n e d r o u t e . T h i s v a r i -

a b l e is a r e a l n u m b e r g r e a t e r t h a n 1 .0 f o r

a n u n t r a i n e d P T F e m p l o y e e , a n d e q u a l s

1 . 0 f o r a f u l l y t r a i n e d e mp l o y e e .

{1 / p i i f t h e r o u t e i i s c a r r i e d t e m p o -

r a r i ly b y a s u b s t i t u t e P T F e m -Ci

p l o y e e o n a g i ve n d a y ;


D e c i s i o n v a r i a b l e s:

D ~j = L a b o r h o u r s o f F T e m p l o y e e j a t r e g u l a r

w a g e r a t e a s s i g n e d f o r d e l i v e r y o n r o u t e i .

D O ~j = L a b o r h o u r s o f F T e m p l o y e e j a t o v e r -

t i m e w a g e r a t e a s s i g n e d f o r d e l i v e r y o n

r o u t e i .

D 2. = L a b o r h o u r s o f P T F e m p l o y e e j a t re g u -

l a r w a g e r a t e a s s i g n e d f o r d e l i v e r y o nr o u t e i .

D O 2 = L a b o r h o u r s o f P T F e m p l o y e e j a t o v e r -

t i m e w a g e r a t e a s s i g n e d f o r d e l i v e r y o n

r o u t e i .

C i~ = L a b o r h o u r s o f F T e m p l o y e e j at r e g u l a r

w a g e r a t e a s s i g n e d f o r c a s i n g m a i l o n

r o u t e i .

C O /lj = L a b o r h o u r s o f F T e m p l o y e e j a t o v e r -

t i m e w a g e r a t e a s s i g n e d f o r c a s i n g m a i l

o n r o u t e i .

C ~j = L a b o r h o u r s o f P T F e m p l o y e e j a t r e g u -l a r w a g e r a t e a s s i g n e d f o r c a s i n g m a i l o n

r o u t e i .

C O 2 -- L a b o r h o u r s o f P T F e m p l o y e e j a t o v e r -

t i m e w a g e r a t e a s s i g n e d f o r c a s i n g m a i l

o n r o u t e i .

R~s = L a b o r h o u r s o f F T e m p l o y e e j a t r e g u l a r

w a g e r a t e a s s i g n e d a s r o u t e r t i m e f o r

c a s i n g m a i l o n r o u t e i .

U j I = N u m b e r o f i d le l a b o r h o u r s f o r F T e m -

p l o y e e j .

U j R = N u m b e r o f id l e l a b o r h o u r s f o r r o u t e r F T

e m p l o y e e j .



3 7 8 M .K M alhotra et al. / Mo delfor schedulingpostal distribution employees

E ,

E"

S i

.~

d ; =

C u r t a i l e d m a i l o n r o u t e i in t h e c u r r e n t

p e r i o d , d e f i n e d a s h o u r s o f a d d i ti o n a l

c a s i n g w o r k r e q u i r e d .

C u r t a i l e d m a i l o n r o u t e i i n t h e p r e v i o u s

p e r i o d , d e f i n e d a s h o u r s o f a d d i ti o n a l

c a s i n g w o r k r e q u i r e d .

T h e u n p r o c e s s e d s u r p l u s m a i l o n r o u t e i

wh i c h e xc e e ds t he a l l owe d c u r t a i l e d l e v -

e l s . Th i s s u rp l u s m a y a r i s e due t o bus i -

n e s s m a i l in g s o f p r o m o t i o n a l a d v e r ti s e -

m e n t c a m p a i g n s .

O v e r - d e v i a t i o n o f t h e d i f f e r e n c e o f o v e r-

t i m e h o u r s a s s i g n e d t o th e j - t h a n d ( j +

1 ) - t h F T e m p l o y e e .

U n d e r - d e v i a t i o n o f d i f f e r e n c e o f o v e r t i m e

h o u r s a s s i g n e d t o j - t h a n d ( j + 1 ) -t h F T

e m p l o y e e .

3 .2 . S t a f f s c h e d u l in g m o d e l c o n s tr a i n ts

1) Ma i l a r i v i ng a t t he d i s t r i bu t i on s t a t i on a f t e r

s o r t i ng a t t he m a i n pos t o f f i c e m us t be de l i ve re d .

T h e m a i l a r r i v e s f o r d i s t r i b u t i o n i n t h e m o r n i n g

( 5 - 7 a . m . ) , a n d b e f o r e n o o n . F o r t h e p u r p o s e o f

r u n n i n g t h e m o d e l , t h e a g g r e g a t e d m a i l a r r i v a l s

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

da y t o ge t t he t o t a l work l oa d fo r a g i ve n da y .

On l y a l i m i t e d i nve n t o ry o f bu l k bus i ne s s c l a s s

m a i l is a l l owe d . The de m a n d c o ns t r a i n t i s p re -

s e n t e d i n t wo pa r t s : one fo r m e e t i ng c a s i ng r e -

q u i r e m e n t s , a n d t h e o t h e r f o r m e e t i n g d e l i v e r y

r e q u i r e m e n t s . T h e t r a i n i n g p r o c e ss i s r e f l e c te d i n

t he c a s i ng c ons t r a i n t , a s i s m a i l c u r t a i l m e n t due

to insuff ic ient capac i ty .

1 . a) F o r c a s i ng r e qu i re m e n t s t o be s a t i sf i e d :

E qijRijl 1 + C i ( E w l j ( C l + b j C O l ) ) + S ij ~ j R j E j l

+ ~., q.2.{C.2. + C 0 2 . ) + E i = + ,

J EJ2 U \ tj ~ Ei

for a l l i ~ I . (1 a )

The r e l a t i ve c a s i ng e f f i c i e nc y o f a n un t r a i ne d

e m pl oye e (p~ ) a s s i gne d fo r c a s i ng a nd de l i ve r i ng

on rou t e i is r e p r e s e n t e d i n t he c i t e rm . I f a P T F

e m pl oye e i s c a r ry i ng m a i l fo r t he f i r s t t i m e a s t he

s u b s t it u t e b i d o w n e r , p ~ w o u l d n o r m a l l y b e g r e a t e r

t ha n one . T hus P i is a func t i on o f the P TF

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

g i v e n d a y , a n d w h e t h e r t h a t e m p l o y e e h a s h a d a

p r e v i o u s t r a i n i n g e x p o s u r e o n t h a t r o u t e . F o r

u n t r a i n e d e m p l o y e e s , a l o ss o f 3 0 - 5 0 % i n p r o -

duc t i v i t y c a n oc c u r on t he f i r s t da y . The l e a rn i ng

effec t i s s igni f icant , because i t en ta i l s a loss of

a v a i la b l e r e g u l a r t i m e h o u r s . A n e m p l o y e e is g e n -

e ra l l y fu l ly t r a i ne d i n a we e k ' s t i m e , a nd r e t a i n s

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

w h e n n o t w o r k i n g o n i t f o r e x t e n d e d p e r i o d s o f

t i m e . T he d i s c us s i on on t he E i a n d S i t e rm s i n

t h e c a s i ng r e q u i r e m e n t s c o n s t r a i n t is c o v e r e d l a t e r

on in th i s sec t ion .

1 . b. i) F o r de l i ve ry r e qu i re m e n t s t o be s a t is -

f ied :

E w ~ ( D l j + bs D O l j ) + E ( D ~ + D O ~ ) = S T i]Cj l jEj2

for a l l i • I ,

1.b.ii)

w l . ( D ] j ) = STi for a l l i • I , . ( l b)

P a r a m e t e r S T ,. , t h e s t r e e t h o u r s s t a n d a r d i z a t i o n

fo r rou t e i , i s f a i r l y i n s e ns i t i ve t o w i de f l uc t ua -

t i ons i n t he vo l um e t o be de l i ve re d . De v i a t i ons i n

m a i l v o l u m e s f r o m r e f e r e n c e v o l u m e s a r e c a p -

t u r e d a s i n c r e a s e d o r d e c r e a s e d w o r k l o a d s f o r

c a s i ng on l y . S om e t i m e s t he re a re a f e w rou t e s

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

m a i l . T h e s e r e g u l a r b i d o w n e r s a r e t h e n f r e e d u p

t o d e l i v e r m a i l o n o v e r l o a d e d r o u t e s , a p r o c e d u r e

c a l l e d ' p i vo t i ng ' . I t i s ha nd l e d i n t he m ode l by

s e t t i ng t he wlj . pa r a m e t e r e qua l t o one fo r t hos e

F T e m p l o y e e s w h o o w n t h e b i d o n u n d e r l o a d e d

r o u t e s . T h i s e q u a l i t y i s m a d e p r i o r t o r u n n i n g t h e

m o d e l b y i n s p e c ti n g t h e d e m a n d d a t a f o r a gi v en

da y . The re a re ge ne ra l l y l e s s t ha n f i ve rou t e s on

a n a v e r a g e d a y w h i c h a r e u n d e r l o a d e d , a n d a s

s u c h c a n b e p i v o t e d . T h e n t h e ( 0 , 1 ) p a r a m e t e r

e n s u r e s t h a t o n l y t h e b i d o w n e r , o r F T e m p l o y e e s

s e rv i ng a s p i vo t s , c a n de l i ve r on a g i ve n rou t e . No

s uc h r e s t r i c t i on i s p l a c e d on t he P TF e m pl oye e s .

T h e y c a n d e l i v e r o n a n y r o u t e w i t h t h e s a m e l e v e lo f p ro f i c i e nc y . The s e c ond de l i ve ry c ons t r a i n t

s how n a s 1 . b .i i) e n s u re s t ha t b i d owne rs o f und e r -

l oa de d rou t e s f i r s t f i n i s h de l i ve r i ng on t he i r rou t e s

b e f o r e p i v o t i n g o n t o o t h e r o v e r l o a d e d r o u t e s .

2 ) O v e r t i m e h o u r s a s s i g n e d s h o u l d n o t e x c e e d

a c e r t a i n f r a c t i on o f r e gu l a r t i m e hou r s .

2 ( D O / l j + C O l j) < f r ] for a l l j C J 1 ,i e l

E ( D o 2 + C 0 2 ) <_fr} f o r a l l j ~ j 2 . ( 2 )

i ~ l



M.K. M alhot ra e t a l , / Mo de l for s c he dul ing pos ta l d i s tr ibu t ion e m ploy e e s 379

T h e p r e s e n c e o f t h e a p p r o p r i a t e r v a r i a b l e

e n s u r e s t h a t a n e m p l o y e e i s a s s i g n e d o v e r t im e

on l y if p re s e n t i n t he s t a t i on on a g i ve n day . S om e

o f t h e P T F e m p l o y e e s a r e a s s i g n e d t o th e r o u t e s

whos e i nd i v i dua l owne rs a re no t a va i l a b l e . The n

t h e r e g u l a r h o u r s f o r t h o s e r o u t e s r e m a i n e q u a lt o e i gh t , bu t t he a va i l a b l e hou r s fo r t ha t P TF

e m pl oye e a re s e t e qua l t o z e ro . S i m i l a r l y , t he T6

e m p l o y e e s d o n o t h a v e a u n i q u e i d e n t i t y , b u t a r e

s l o t t e d i n t o d i f f e re n t rou t e s on d i f f e re n t da ys o f

t h e w e e k .

3 ) T h e r e g u l a r t i m e c o n s t r a i n t s t a t e s t h a t e a c h

e m p l o y e e c a n n o t w o r k m o r e t h a n a s p e c i f i e d

n u m b e r o f h o u r s o n a r e g u l a r w a g e r a t e . T h e F T

e m p l o y e e s a r e g u a r a n t e e d e i g h t h o u r s o f w o r k

e v e ry d a y. H e n c e u n d e r t i m e i s p e n a l i z e d i n t h e

ob j e c t i ve func t i on .

3 . a ) F o r F T e m p l o y e e s :

E ( D ] ] + C ] j) + U , .= S r ) f o r a l l j • J ' ,i E I

ER Ij+ Uj~=8r] f o r a l l j • j I~ ( 3 a )

i ~ I

3 . b ) T h e P T F e m p l o y e e s c a n b e u s e d f o r l e s s

t h a n e i g h t h o u r s o f w o r k o n a n y d a y . T h i s c o n -

s t r a i n t e n s u r e s t h a t P T F e m p l o y e e s a r e a s s i g n e d

r e g u l a r w o r k h o u r s o n l y i f t h e y a r e p r e s e n t i n t h es t a ti o n o n t h a t d a y . M o r e o v e r , r e g u l a r w a g e r a t e s

a r e o n l y a p p l i c a b l e t o e i g h t o r f e w e r w o r k h o u r s

p e r d a y .

E ( D ~ + C ~ ) - < 8 4 f o r a l I j • j 2 . ( 3 b )i ~ l

3 .c ) T h e P T F e m p l o y e e s c a n n o t d e l i v er f o r

m o r e t h a n s ix h o u r s , i n o r d e r t o e n s u r e t h a t

a l m os t a l l m a i l i s de l i ve re d by m i d -a f t e rnoon .

T h i s m a n a g e m e n t p r e f e r e n c e w a s i n c o r p o r a t e d

i n t o t he m ode l i n the fo rm o f c ons t r a i n t (3c ) . Th i s

c ons t r a i n t i s e s pe c i a l l y i m por t a n t fo r t he ne w

P T F e m p l o y e e s w h o d o n o t p o s s es s s c h e m e

k n o w l e d g e f o r m a n y r o u t e s , a n d s o m u s t b e m a d e

t o g i ve a ux i l i a ry a s s i s t a nc e on t he s t r e e t s .

E ( D 2 + D O / ~ ) < 6 4 for a l l j • J 2 . (3c )i E l

4) The e qu i t y c ons t r a i n t r e qu i re s t ha t a l l t he

F T e m p l o y e e s w h o a r e w i l l in g t o w o r k o v e r t i m e

m us t r e c e i ve i t i n a n e qu i t a b l e f a s h i on . The d i f -

f e r e n c e s i n o v e r t im e a s s ig n e d a r e c o n s i d e r e d b e -

t w e e n c o n s e c u t i v e p a i r o f e m p l o y e e s b e l o n g i n g t o

t h e s e t o f F T e m p l o y e e s d e s i r in g o v e r t i m e . N o

ove r t i m e e qu i t y c ons t r a i n t s e x i s t s fo r t he rou t e r s

o r P T F w o r k e r s .

E ( D O l j' + C O ] j ) - E ( D O ] ( ] + , , + C O 1 ~/+l) )i ~ l i ~ l

= d f - d 7 f o r a l l j • j 3 a n d j4~lJ31. (4 )

5 ) Th e c u r t a i l e d m a i l c ons t r a i n t r e qu i re s t h a t

m a i l wh i c h c a nno t ge t c a s e d on a g i ve n da y ,

t e r m e d a s t h e ' c u r t a i l e d m a i l ' , i s a d d e d t o t h e

n e x t d a y 's p r o c e ss i n g r e q u i r e m e n t s . T h e t o t a l c u r-

t a i l e d m a i l c o m e s f r o m t w o s o u r c e s - f r o m t h e E i

t e rm , a n d t he S~ t e rm i n t he c a s ing c ons t r a i n t .

T h e E~ t e r m c o m e s f r o m t h e t r a d e - o f f b e t w e e n

c o s t a n d c u s t o m e r s e r v i c e , w h e r e b y t h e m a n a g e -

m e n t f e e l s t ha t i t i s be t t e r t o c u r t a i l m a i l a c ro s s

a l l rou t e s up t o a c e r t a i n l i m i t U , be fo re ove r t i m e

i s u s e d t o p roc e s s i t. The S i t e rm i s t he s u rp l u s

c u r t a i l e d m a i l ove r a nd a bove t he c u r t a i l e d m a i l

E~, a nd on l y a r i s e s wh e n no t e nou gh r e s ou rc e s

a re a va i l a b l e t o p roc e s s t he g i ve n wo rk l oa ds .

S uc h ove r l o a d s i t ua t i ons c a n oc c u r i f a s u rge o f

p ro m o t i ona l m a i l h it s a bus i ne s s o r r e s i de n t i a l

rou t e on a ny g i ve n da y . The d i f f e re n t i a t i on be -

t we e n t he s e t wo c a t e go r i e s o f c u r t a i l e d m a i l is

r e q u i r e d i n o r d e r t o e n f o r ce t h e m a n a g e m e n t

p r i o r i t i e s i n t he ob j e c t i ve func t i on . The re i s a n

up pe r l i m i t on i nd i v i dua l E i t e rm s , a s we l l a s t he

s um o f t he E , t e rm s fo r a ll t he rou t e s i n t he

s t a t i on on a g i ve n da y . No s uc h uppe r l i m i t e x i s t s

on the S~ te rm .

The c u r t a i l e d m a i l c ons t r a i n t r e f l e c t s t he fa c t

t h a t t h e a m o u n t o f c u r ta i l e d m a i l o n a n y g iv e n

d a y w i ll d e p e n d u p o n t h e a v a i la b l e w o r k f o r c e

s iz e , a nd t h e de m a nd o f m a i l t o be p roc e s s e d . In

o r d e r t o a v o id s o l u t io n s w h e r e b y e x ce s si v e a m o u n to f m a i l i s l e f t unp roc e s s e d on a ny i nd i v i dua l

r o u t e , w e c o n s t r a i n t h e m a x i m u m d e s i r e d a m o u n t

o f u n c a s e d m a i l o n a n y r o u t e . A t t h e s a m e t i m e ,

t h e t o t a l a m o u n t o f m a i l w h i c h c a n b e c u r t a i le d

a c ros s a l l t he rou t e s i n t he s t a t i on a l s o ha s a n

u p p e r b o u n d .

E ; < C i f or a l l i • I ,

}~Ei<U.i ~ l

(5 )



380 M.K. M alho tra e t a l. / Mod e l fo r schedu l ing pos ta l d is t r ibu tion employees

T h e c o e f fi c i e nt o f o n e p a r t o f c u r ta i l e d m a i l

( E i ) i n t h e o b jec t iv e f u n c t io n i s l e s s t h an th e

co e f f i c i en t f o r o v e r t im e . T h en th e so lu t io n to t h e

m o d e l w o u l d a lw a y s tr y t o u s e u p t h a t p a r t o f t h e

cu r t a i l ed m a i l ( E i ) u p to t h e sp ec i f i ed v a lu e o f

t h e u p p e r b o u n d . T h e m a n a g e m e n t c a n t h u s c o n -t r o l t h e t o t a l cu r t a i l ed m a i l t h a t t h e so lu t io n m u s t

ref lec t . In th is way , they a lso ind i rec t ly con t ro l

t h e n u m b e r o f o v e r ti m e h o u r s i n c u r r e d o n a g i v en

d ay . T h e su r p lu s p a r t o f cu r t a i l ed m a i l ( S~) i s p u t

in t h e o b jec t iv e f u n c t io n w i th a v e r y h ig h r e l a t i v e

v a lu e f o r i t s co e f f ic i en t . T h i s h ig h co e f f i c i en t v a lu e

in d ica t e s t h a t t h e su r p lu s m a i l h e ld b ack i s n o t a

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

ad d i t i o n a l r e so u r ce s a r e e i t h e r n o t av a i l ab l e , o r

c a n n o t b e a u t h o r i z e d d u e t o b u d g e t c o n s t r a i n t s .

T h e c u s t o m e r s e r v i c e s t a n d a r d s w o u l d d e c l i n e i n

th a t c a se .

3 . 3 . O b j e c t i v e f u n c t io n

T h e o b jec t iv e f u n c t io n i s a we ig h ted su m o f

the fo l lowing ob jec t ives :

1 ) M in im ize th e t o t a l o v e r t im e in cu r r ed , f o r

b o th th e f u l l - t im e , a s we l l a s t h e p a r t - t im e em -

p loyees .

2 ) M i n i m i z e t h e u n d e r t i m e i n c u r r e d f o r t h eF T e m p l o y e e s .

3 ) M i n i m i z e t h e t o t al a m o u n t o f u n p r o c e s s e d

m a i l o v e r t h e p l an n in g h o r i zo n ( d a i ly i n t h i s c a se ) .

T h i s t e r m e n s u r e s t h a t t h e m o d e l d o e s n o t y i e l d a

t r i v i a l so lu t io n , wh e r eb y a l l t h e m a i l t o b e d e l iv -

e r ed i s se t a s id e a s t h e cu r t a i l ed v o lu m e .

4 ) M i n i m i z e t h e s u m o f t h e d e v i a t i o n s f r o m

e q u i t a b l e o v e r t i m e a s s i g n m e n t t o t h e F T e m p l o y -

ee s .

5 ) M i n i m i z e t h e s u m o f t h e r e g u l a r h o u r s

w o r k e d b y t h e P T F e m p l o y e e s . T h i s d e c i si o n v ar i-

a b l e is o n e w h i c h t h e m a n a g e m e n t c a n c o n t ro l ,

b e c a u s e a P T F e m p l o y e e n e e d n o t b e u t i l i z e d f o r

fu l l e igh t hours .

6 ) M in im ize th e su r p lu s cu r t a i l ed m a i l .

M in Z = 1 . 5 { Y ' . ~ ] ( D O ~ j + C O ~ ,.)i ~ l j ~ j 3

+ E E ( D O / + C O ) }i ~ I j E J 2

+ 1 .25 ( ]~] U j+ ~ U j } + ~ 7 ~ E ij E J l j c J R i ~ l

E ( d 7 + d ] -)j E J 3

j~=lJ3[

i ~ l j ~ j 2 i ~ l

( 6 )

A se r i e s o f sen s i t iv i t y r u n s we r e m ad e in o r d e r

to o b ta in so m e in s ig h t s i n to t h e r e l a t i v e we ig h t s

o n d i f f e r en t t e r m s in t h e o b jec t iv e f u n c t io n . W h en

th e to t a l o v e r t im e in cu r r ed i s g iv en th e sam e

w e i g h t a s t h a t o f c u r t a i l e d m a i l , t h e m o d e l u s e su p l a r g e r a m o u n t s o f o v e r ti m e t h a n c u r t a i l e d

m a i l . T h i s is u n d e s i r a b l e f r o m t h e m a n a g e m e n t ' s

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

we ig h ted ev en s l i g h t ly l e s s t h an to t a l o v e r t im e ,

v e r y s m a l l a m o u n t s o f o v e r t i m e a r e i n c u r r e d .

S in ce th e co e f f i c i en t o f t o t a l cu r t a i l ed m a i l i n t h e

o b jec t iv e f u n c t io n i s l e s s t h an th e co e f f i c i en t o f

to t a l o v e r t im e , t h e m o d e l w i l l a lway s a t t em p t t o

cu r t a i l a s m u ch m a i l a s p o ss ib l e b e f o r e a s s ig n in g

o v e r t i m e . R e d u c i n g t h e u p p e r b o u n d o n t h e t o t a l

cu r t a i l ed m a i l j u s t r e su l t s i n an eq u iv a l en t n u m -

b e r o f h o u r s sh i f t i n g in to t o t a l o v e r t im e . T h u s f o r

a g iv en se t o f p r o d u c t iv i ty v a lu e s , t h e m an ag e -

m en t c an co n t r o l t h e d a i ly o v e r t im e b y sp ec i f y in g

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

day .

T h e co e f f i c i en t f o r t h e t o t a l o v e r t im e t e r m i s

1 .5 , i n o r d e r t o r e f l e c t t h e r e l a t i v e d o l l a r co s t f o r

o v e r t i m e . T h e c o e f fi c i en t s o f o t h e r t e r m s i n t h e

o b jec t iv e f u n c t io n a r e n o t a l l eq u a l t o u n i ty , i n d i -

ca t in g th a t t h ey h av e d i f f e r en t r e l a t i v e im p o r -

t a n c e t o m a n a g e m e n t . W i t h e q u a l w e i g h t s o n

u n d e r u t i l i z e d f u l l t i m e h o u r s , P T F w o r k e r h o u r s ,a n d t h e c u r t a i l e d m a i l ( E l ) , t h e m o d e l m a y n o t

u se u p a l l av a i l ab l e f u l l t im e h o u r s b e f o r e cu r t a i l -

i n g m a i l o r a s s i g n in g P T F w o r k e r s o n t h a t r o u t e .

T h u s a h i e r a r ch y o f p r io r i t i e s ex is t s , wh e r eb y th e

h e l d b a c k s u r p l u s m a i l ( S i ) i s t h e l e a s t d e s i r ab l e ,

an d a s su ch i s a s s ig n ed a v e r y h ig h co e f f i c i en t

v a lu e o f 2 0 . Ov e r t im e i s t h e n ex t l e a s t d e s i r ab l e

o p t io n w i th a co e f f i c i en t v a lu e o f 1 .5 . T h e FT

u n d e r u t i l i z a t i o n s h o u l d b e d r i v e n d o w n t o z e r o

b e f o r e m a i l i s cu r t a i l ed o n an y r o u te , o r ad d i -

t i o n a l h e l p a s s i g n e d i n t h e f o r m o f a P T F e m -



M.K, M atho tra e t a l. / Mo de l fo r schedu l ing pos ta l d is t r ibu t ion em ployees 381

p l o y e e . T h e mo d e l a c h i e v e s t h i s b y g i v i n g a r e l a -

t iv e l y h i g h e r w e i g h t o f 1 .2 5 t o t h e u n d e r u t i l i z e d

h o u r s. P T F w o r k e r h o u r s h a v e t h e s a m e w e i g h t in

t h e o b j e c t i v e f u n c t i o n a s t h e c u r t a i l e d m a i l E i , s o

t h a t a P T F w o r k e r m a y n o t b e u s e d f o r e ig h t

h o u r s i f m a i l c a n b e c u r t a i l e d o n r o u t e s u p t o t h es p e c i f i e d l i m i t U . T h i s m e a n s t h a t t h e m o d e l i s

i n d i f f e r e n t b e t w e e n c u r t a i l in g m a i l ( E i ) o r u s i n g

m o r e P T F h o u r s o n a n y r o u t e , a s l o n g a s t h e

o t h e r c o n s t r a i n t s o f t h e m o d e l a r e s a t is f i ed . F i -

n a l ly , t h e e q u i t a b l e a s s i g n m e n t o f o v e r t i m e b e -

t w e e n e m p l o y e e s d e s i r i n g o v e r t i m e h o u r s h a s t h e

l o w e s t r e l a t i v e p r i o r i t y , a n d a s s u c h i s a s s i g n e d a

w e i g h t o f u n i t y i n t h e o b j e c t i v e f u n c t i o n .

4 . M o d e l v a l i d a t i o n

T h e m o d e l d e v e l o p e d i n t h e p r e v i o u s s e c t i o n i s

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

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

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

T o v a l i d a t e t h e m o d e l , a n d s h o w i ts v a l u e a s a

d e c i s i o n s u p p o r t s y s t e m , it w a s a p p l i e d t o a n

a c t u a l c a s e s i t u a t i o n .

A l l ma j o r U . S . c i t i e s h a v e a l a r g e ma i n p o s t

o f f i c e , w h i c h s e r v e s a s t h e c e n t r a l c l e a r i n g h o u s e

o f m a i l t o s m a l l e r d e s t r i b u t i o n s t a t io n s . T h e m a i l

i s s o r t e d a c c o r d i n g t o t h e z i p c o d e s , a n d d i s -

p a t c h e d f o r d i s t r i b u t i o n t o t h e s t a t i o n s t w i c e a

d a y . T y p i c a l l y e a c h z i p c o d e c o r r e s p o n d s t o a

d e l i v e r y s t a t io n . W i t h i n e a c h z i p c o d e , t h e m a i l

c a n b e d i s t r i b u t e d a l o n g s e v e r a l d i f f e r e n t r o u t e s .

T h e s o r t i n g o f m a i l a l o n g d i f f e r e n t r o u t e s f o r a

g i v e n z i p c o d e i s a l s o d o n e a t t h e c e n t r a l p o s t

o f f i c e . I n o r d e r t o c a p t u r e t h e s c h e d u l i n g c o m -

p l e x i t y a t a s t a t i o n , t h e l a r g e s t o n e i n a ma j o r

m i d w e s t e r n c i ty w a s s e l e c t e d . T h i s s t a t i o n c o n -

s is t s o f t w o z i p c o d e s , t o g e t h e r t o t a l li n g m o r et h a n f i f ty r o u t e s . T h e n u m b e r o f ro u t e s a t m o s t

o t h e r s t a t i o n s u s u a l ly d o e s n o t e x c e e d t h i rt y .

4 .1 . D ata co l l ec t ion

T h e f i rs t s t e p i n d a t a c o l l e c t i o n w a s s e t t i n g t h e

m o d e l p a r a m e t e r s . T h e s e p a r a m e t e r s a r e: 1 ) t h e

r e f e r e n c e v o l u m e o f m a i l o n e a c h r o u t e , 2 ) t h e

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

t h e ma i l ( o f f i c e t i me ) , a n d 3 ) t h e t i me w h i c h

s h o u l d b e s p e n t o n t h e s t r e e t s f o r d e l i v e r i n g t h e

r e f e r e n c e v o l u m e i n a n e i g h t - h o u r d a y . T h i s i n -

f o r m a t i o n w a s o b t a i n e d f r o m t h e s t a n d a r d s d a t a

m a i n t a i n e d a t t h e s t a t i o n . T h e c a s i n g p r o d u c t i v i t y

w a s o b t a i n e d a s t h e f e e t o f m a i l w h i c h c a n b ec a s e d i n a n h o u r b y a w o r k e r t r a i n e d f o r c a s i n g

o n t h a t p a r t i c u l a r r o u t e . T h i s p r o d u c t i v i t y i s t h e

p c ~ p a r a m e t e r u s e d i n t h e d e m a n d c o n s t r a i n t .

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

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

a d d r e s s e s a l o n g d i f f e r e n t r o u t e s , a n d t h e v a r i a -

t i o n o f t h e m i x b e t w e e n l e t t e r m a i l a n d f l a t m a i l

a l o n g d i f f e r e n t r o u t e s . S o m e r o u t e s h a v e m o r e

b u s i n e s s a d d r e s s e s t h a n r e s i d e n t i a l o n e s . T h e a v -

e r a g e i s a b o u t 4 f e e t o f m a i l p e r h o u r , w h i c h

t r a n s l a t e s i n t o c a s i n g 1 8 l e t t e r s o r 8 f l a t s p e r

m i n u t e .

T h e a c t u a l m a i l a r ri v a l v o l u m e d a t a f o r e a c h

r o u t e w a s c o ll e c t e d n e x t . I t w a s d e c i d e d t h a t d a t a

w o u l d b e c o l l e c t e d f o r o n e w h o l e w e e k . T h e m a i l

a r r i v e s a t t h e s t a t i o n a t 5 - 7 a . m . i n t h e m o r n i n g ,

a n d t h e n a g a i n a t 1 0 a . m . T h e m a i l v o l u m e s a r e

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

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

S i m i l a r ly t h e a f t e r n o o n v o l u m e i n c l u d e s th e m a i l

c u r t a i l e d i n t h e m o r n i n g . T o o b t a i n t h e t o t a l w o r k

l o a d f o r t h e d a y , t h e m o r n i n g m a i l v o l u m e f r o m

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

f r o m t h i s t o t a l t o a v o i d d o u b l e c o u n t i n g . T h e n

t h e m o d e l r e c e i v e s th e s a m e v o l u m e o f t o t a l m a i l

a s t h a t a c t u a l l y a r r i v i n g a t t h e s t a t i o n , a n d t h e

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

e v e r y d a y t h e m o d e l s t a r t s a f r e s h , a f t e r a c c o m o -

d a t in g t h e r e s u lt s o f m a n a g e m e n t ' s d e c is i on s o n

c u r t a i l e d v o l u m e s o n t h e p r e v i o u s d a y . T h i s t r e a t -

m e n t m a k e s p e r f o r m a n c e c o m p a r i s o n s e a s i e r . T h e

c a s i n g w o r k l o a d i s t h e r e b y o b t a i n e d b y d i v i d i n g

t h i s t o t a l v o l u m e b y t h e c a s i n g p r o d u c t i v i t y . T h ed e l i v e r y w o r k l o a d i s a l w a y s t h e s a m e a s t h e

s t r e e t t i me , i r r e s p e c t i v e o f t h e ma i l t o b e d e l i v -

e r e d .

I n o r d e r t o s c h e d u l e t h e e m p l o y e e s , t h e i r p r o -

f i l e s o f t r a i n i n g a s w e i 1 a s b i d a s s i g n me n t s mu s t

b e k n o w n . T h i s d a t a w a s c o l l e c t e d f o r a l l t h e F T

( i n c l u d i n g t h e r o u t e r s ) a n d P T F e m p l o y e e s w h o

a r e i n v o lv e d i n t h e d e l i v e r y p r o c e s s . F o r e a c h

P T F e m p l o y e e , t h e l i s t o f r o u t e s o n w h i c h t h e y

a r e t r a i n e d w a s a l s o o b t a i n e d . S i mi l a r l y , t h e l i s t

w a s o b t a i n e d f o r t h e f iv e F T e m p l o y e e s w h o



382 M.K. M alhotra et al. / Mod el fo r scheduling pos tal distribution employees

w o r k e d a s t h e r o u t e r s . T h e o v e r t i m e d e s i r e d l i s t

c ons i s ts o f a bou t 20 e mp l oye e s , a n d i nc l ude s a

fe w T6 e mpl oye e s a l so . Th i s l i s t i s u se d t o c on -

s t r u ct t h e o r d e r e d s e t j 3 , r e q u i r e d f o r e n f o r c i n g

t he e qu i t y c ons t ra i n t . T h i s pha se o f da t a c o l l e c -

t i on a l so s e t t he va l ue s o f a l l (0 , 1 ) pa ra me t e r s i n

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

sh i p , a nd t o r e s t r i c t c a s i ng by P TF w orke rs t o

o n l y t h o s e r o u t e s f o r w h i c h t h e y h a v e b e e n p r e v i -

ous l y t r a i ne d .

T h e n e x t p h a s e o f d a t a c o l l e c ti o n f o r v a l i d at -

i n g t h e m o d e l w a s t o o b t a i n t h e a t t e n d a n c e

r e c o r d s o f a l l t h e e m p l o y e e s f o r t h e g i v e n w e e k .

S o m e e m p l o y e e s o n l y w o r k p a r t o f th e d a y d u e t o

t e m po ra ry j ob - re l a t e d d i s a b i l i ti e s , a nd so t he i r

w o r k d a y is s h o r t e n e d b e c a u s e t h e y c a n n o t d e -

l i ve r on t he s t r e e t s . S ome e mpl oye e s a re a w a y on

t h e i r a n n u a l v a ca t i o n , so t h a t t h e r p a r a m e t e r

c o r r e s p o n d i n g t o th e i r p r e s e n c e i n t h e s t a t i o n is

s e t e qua l t o z e ro fo r t he w ho l e w e e k . The c o l l e c -

t i o n o f t h e s e a t t e n d a n c e r e c o r d s e n s u r e s t h a t o n l y

t hose e mpl oye e s w ho a re a c t ua l l y a va i l a b l e a t t he

s t a t i on a re s c he d u l e d fo r t he g i ven w e e k .

4 .2 . S o m e a d j u s t m en t s

A n a l y s is o f m a n a g e m e n t s c h e d u l e s s h o w e d t h a t

b i d o w n e r s a r e a s s i g n e d o v e r t i m e o n t h e i r o w n

r o u t e , i r r e s p e ct i v e o f w h e t h e r t h e y a r e o n t h e

o v e r t i m e d e s i r e d l i st o r n o t . D e v e l o p m e n t o f f e a -s i bl e s c h e d u l e s i s c o n s i d e r e d m o r e i m p o r t a n t . T h e

e q u i t y c o n s t r a i n t w a s t h u s n o t a b i n d i n g o n e f r o m

t h e m a n a g e m e n t ' s p e r s p e c ti v e , a n d s o w a s s ub s e -

q u e n t l y d r o p p e d f r o m f u r t h e r r u n s o f t h e m o d e l .

I t c a n a l w a y s b e i n c l u d e d w h e n e v e r a n e q u i t a b l e

a s s i g n m e n t o f o v e r t i m e h o u r s b e t w e e n a se l e ct

g r o u p o f F T e m p l o y e e s b e c o m e s a n i m p o r t a n t

c o n s i d e r a t i o n .

F i n a l l y , a s e t o f e m p l o y e e a s s i g n m e n t s w e r e

de ve l ope d a nd show n t o t he pos t a l supe rv i so r ,

w h o g e n e r a l l y c o n s i d e r e d t h e m q u i t e f e a si b le .

O n e o f o u r e a r ly co n c e r n s w a s w h e t h e r t h e m o d e l

m a d e a s s i g n m e n t s t h a t w e r e t o o s h o r t i n d u r a -

t i on . The supe rv i so r s a s su re d us t ha t t h i s c onc e rn

w a s u n w a r r a n t e d f o r c a s i n g w o r k . A s s i g n m e n t s a s

sho r t a s f i ve mi n u t e s a re c on s i de re d re a sona bIe .

Ca s i ng fo r a l l rou t e s i s done i n one b i g ha l l , a nd

Table 1Representative assignment schedule for a typical day

Route Re gu la r/ T6 /PT F Time Overtime Additional Duty Time Total

carrie r hours hours assistance hours hours

1 XY Z 8.00 0.52 Rout er3 Case 0.18 8.70

2 PKE 8.00 0.00 PTF1 Carry 0.53 8.53

3 SKD 8.00 0.00 PTF2 Carry 0.75 8.75

4 RTW 8.00 0.00 Rou ter3 Case 1.57 9.57

5 JKL 8.00 0.00 PTF1 Carry 2.38 10.38

6 NR F 8.00 0.00 PTF1 Carry 0.52 8.52

7 MKM 8.00 0.00 PTF3 Case 1.05

PTF5 Carry 0.50 9.55

51 DAR 8.00 1.00 PTF2 Case 0.27 9.27

52 GTK 8.00 0.00 PTF2 Case 2.05 10.05

53 MMD 8.00 1.10 Non e 9.10

54 QU T 8.00 1.69 Non e 9.69

55 BIN 8.00 0.00 Rou ter2 Case 1.17

PTF5 Case 0.26 9.43

56 LCV 8.00 0.60 Rou ter5 Case 1.63 10.23

Notes

1) The assignment schedule is shown only for a few routes. The identity of the workers has been protected by giving them code

initials.

2) The PTF employees who are trai ned on more routes t end to case more fr equently than carry. Here PTF1 is a relatively new

employee.



M.I'L Malhotra et al. /M od el for scheduling postal distribution employees 383

there is virtually no travel or setup time involved

when changing from one work center to another.

Furthermore, experience with the model (see next

section) shows that only about 4% of all casing

assignments are less than 20 minutes. In terms of

total work hours per week, these assignments arenegligible.

As for delivery work, the supervisor did not

think that a delivery assignment of less than half

an hour on any route was profitable, because of

the travel times involved between different routes.

No specific constraint was introduced to handle a

delivery assignment of less than 0.5 hour, again

because a very small number of variables give

assignments of this nature. For the scheduling

problem based on this case station data, there are

a total of about 2500 variables and 330 con-

straints in the model. For several test runs of the

model, only six variables on the average give

fractional assignments of less than 0.5 hour, which

are then manually allocated to the regular bid

owners as overtime work. This adjustment is re-

flected in the schedules and results presented

next.

4.3. Mo del results

A detailed representative schedule developedby the model for the first day of the week is

partially shown in Table 1. While the model does

not give the start and finish times for these as-

signments, a lot of casing activity occurs in the

afternoon when the bid owners are away on their

delivery routes. This frees up the work centers for

casing work, and hence no interference is caused.

Similarly, the PTF workers can finish delivering

on some routes, and then return to the station forcollecting the cased mail for the remaining routes

that they have been assigned for the day. The

central idea is that they either help deliver or

case on overloaded routes, and overtime is as-

signed only after all the regular time hours have

been used up. Sometimes it may become neces-

sary to assign overtime to even the PTF employ-

ees in order to get the desired amount of mail

processed.

The schedules were similarly developed for the

remaining days of the week, and their summa-

rized results are shown in Table 2, along with the

results of the schedules actually developed by the

management (two supervisors developed the

schedules on different days of the week for which

the data was collected). The table shows the

number of regular hours, overtime hours, and

total hours incurred, along with the curtailed mail

held back on a daily basis. The upper bound on

the total curtailed mail was made equal to the

mail actually left uncased by the workers on that

particular day. This equality allows cleaner com-

parisons between the model and actual results.Table 2 shows that the model cuts overtime

usage by 61%, from 118.8 hours to only 68.1

hours. The model consistently incurs a lower

Table 2

Comparison of the model with the actual decisions

Day of week ' Regular hours Overtime hours

Model Actual Model Actual

Total hours Curtailed hours

Model Actual Model Actual

Monday 509.23 501.20 23.47 31.50Tuesda y 509.63 498.80 3.57 14.40

Wednesday 493.28 484.60 15.72 24.40

Thu rsd ay 498.67 490.20 11.73 20.20

Fri day 500.94 489.50 13.06 24.50

Saturday 456.93 453.70 0.57 3.80

Total 2968.68 2918.00 68.12 118.80

Overt ime% Model = 2.24

Cost at $10. 65/h our Model --- $ 32705

532.70 532.70 34.10 34.10513.20 513.20 29.15 29.15

509.00 509.00 27.73 27.73

510.40 510.40 56.45 56.45

514.00 514.00 0.00 0.00

457.50 457.50 36.72 36.72

30 36 .8 0 3036.80 184.15 184.15

Actual = 3.91

Actual = $ 32 975

Note. The perc enta ge savings in dollars for the whole week is computed as follows:

(Actual - Model)100% 270-100%A = 0.82%.

Actual 32975



40

30

20

10

MODEL

Mon Tue Wed Thu Fr i Sa t

384 M .~ Malhotra et al. / Model for scheduling postal distribution employees

Day of week

Figure 1. Comparison of model and actual overtime hours

overtime percentage on each day than achieved

by the actual decisions. This improvement is ac-

complished due to better utilization of the PTF

employees, who are assigned complete eight hours

of work before overtime allocation decision is

made for FT employees. Thus the model does amuch better job of using up all the available

regular time hours, since it can slot all the avail-

able employees optimally in an effort to incur as

little overtime as possible. The difference be-

tween the overtime hours incurred by the model

and actual decisions is graphically shown in Fig-

ure 1.

In terms of to ta l costs, the results are less

dramatic. Table 2 also shows that the total hours

worked in the post office are equal to those

worked by the model. This result is expected,since both the model as well as the actual deci-

sions are based on equivalent work loads. On the

average, the model outperforms the actual deci-

sions by 0.82% in total dollar costs incurred.

The model therefore develops feasible sched-

ules, and also gives results which are close to

those obtained by the management. The compu-

tational times for the model are also quite attrac-

tive. The model was solved using LINDO

(Schrage, 1987), and reached an optimal solution

in 112 seconds and 400 iterations on an average

on a PRI ME 9955 computer. The model solu-

tions exhibit multiple optima, whereby the objec-

tive function values are identical on different

optimizers (LINDO, SCICONIC and MPSX), but

there are differences in some specific assign-

ments. The report generation is done by means ofa Statistical Analysis of Systems (SAS) program,

which converts the model output into the form

which can be readily interpreted.

5 . C o n c l u s i o n

In this study we have developed and validated

a staff scheduling model for the postal distribu-

tion system. The primary purpose of this research

was to construct a representative staff schedulingmodel. Even though actual implementation has

not been carried out so far, our experience sug-

gests that the model can be used in several differ-

ent ways by the postal management. In the plan-

ning mode, the model can be used for developing

weekly schedules based on forecasted demand.

The model can also be used for constructing

detailed daily schedules. The mail volume for the

pre-noon arrival is usually a certain percentage of

the early morning mail arrival. All the scheduler

needs to do is to input the mail volumes and



M.K~ Malhotra et al . /M od el for scheduling postal distribution employees 38 5

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

s e q u e n t l y p l a n t h e a c t u a l a s s i g n m e n t s o n t h e b a -

s is o f th e s c h e d u l e g e n e r a t e d . H u m a n i n t e r v e n -

t i o n c a n a d j u s t f o r s u c h f a c t o r s a s p r o d u c t i v i t y

d i f f e r e n t i a l s a n d t h e e l i m i n a t i o n o f t h e f e w d e l i v -

e r y a s s i g n m e n t s w h i c h a r e l e s s t h a n 0 . 5 h o u r . T h i s

m a n - m a c h i n e a p p r o a c h c a n s a v e s c h e d u l e r s '

t i m e , a s w e l l a s h e l p t h e m t o c o n s t r u c t n e a r

o p t i m a l s c h e d u l e s . F i n a l l y , t h e m o d e l c a n b e u s e d

t o e v a l u a t e t h e i m p a c t o f c e r t a i n l o n g - t e r m p o l i c y

i s su e s o n s y s t e m p e r f o r m a n c e . T h e s e i s s u e s , w h i c h

d e a l w i th t h e m i x b e t w e e n p a r t - t im e a n d f u l l- t i m e

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

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

r e s e a r c h .

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