Operations Research Games a Survey

8/9/2019 Operations Research Games a Survey

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S o c i e d a d d e E s t a d i s t i c a e I n v e s t i g a c i d n O p e r a t i va

T o p

2001) Vol. 9, No. 2, pp. 139 216

O peration s Research Gam es A Su rvey

P e t e r B o r m H e r b e r t H a m e r s a n d R u u d H e n d r i ck x

C e n t ER and De p ar t m e nt o f Ec on om e t r i c s and Ope ra ti ons Re se arch

Ti lburg Univers i ty , P .O. Bo x 90153, 5000 L E Ti lburg, The Nether land s

e -m a il: P . E . M . B o r m @ k u b . n l H . J . M . H a m e r s k u b . n l R . L . P . H e n d r i c k x k u b . n l

A b s t r a c t

This paper surveys the research area of cooperative games associated with several

types of operations research problems in which various decision makers players) are

involved. Cooperating players not only face a joint optimisation problem in trying,

e.g., to minimise total joint costs, but also face an additional allocation problem in

how to distribute these joint costs back to the individual players. This interplay

between optimisation and allocation is the main subject of the area of operations

research games. It is surveyed on the basis of a distinction between the nature of

the underlying optimisation problem: connection, routing, scheduling, production

and inventory.

Key Words: Optimization problems, allocation problems, cooperative games.

AMS

s u b j e c t c l a s s i f i c a t i o n :

91-02, 91A12, 90B30, 90B35, 90B10, 90B05.

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

Typically, operations research analyses situations in which one decision

maker, guided by an objective function, faces an optimisa tion problem. The

theo ry concen trates on the question of how to act in an optima l way and,

in particular, on the issues of computational complexity and the design of

efl3cient algo rithms. Game t heor y on the oth er han d analyses situatio ns

involving at least two inte ract ing decision makers called players), with

possibly diverging interests. Roughly speaking, it deals with mathematical

models of competition and cooperation.

Competitive or noncooperative game theory studies situations in which

the players can negotiate about what to do i.e., pre-play communica tion is

allowed), but enfbrceable binding agreements are assumed not to be possi-

ble. Therefore strategic analysis and individ ual incentives play a prominent

role here. In cooperative game the ory enfbrceable binding agreements are

possible and also side payments may be allowed. Now the main issue is fair

allocation, eit her of joint costs or joint revenues.


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P. Borm H. Hamers and R. Hendrickx

S i n ce t h e e a r l y d e v e l o p m e n t s o f o p e r a t i o n s r e s e a rc h a n d g a m e t h e o r y

t h e r e h a s b e e n a s t r o n g i n t e r p la y b e t w e e n t h e t w o d i sc ip l in e s . E s p e c i a l l y t h e

i n t e r re l a t io n b e t w e e n o p e r a t i o n s r e se a r c h a n d n o n c o o p e r a t i v e g a m e th e o r y

is w e l l- k n o w n : b e t w e e n d u a l it y re s u l t s i n m a t h e m a t i c a l p r o g r a m m i n g a n d

m i n i m a x r e s u l t s f b r z e r o - s u m g a m e s , b e t w e e n li n e ar c o m p l e m e n t a r i t y a n d

b i m a t r i x g a m e s , b e t w e e n M a r k o v d e c is i o n p r o c e s s e s a n d s t o c h a s t i c g a m e s ,

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

T h e i n te r r e l a ti o n b e t w e e n o p e r a t i o n s r e s e a r c h a n d c o o p e r a t i v e g a m e t h e -

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

t i o n s re s e a r c h g a m e s . O n e c a n s a y t h a t a n i m p o r t a: n t p a r t o f t h e i n t e r p l a y

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

( d i sc r e te ) s t r u c t u r e o f a g r a p h , n e t w o r k o r s y s t e m t h a t u n d e r l i es v a r i o u s

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

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

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

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

p l a y e r s c a n p o s s i b l y c r e a t e e x t r a g a i n s o r sa v e c o s ts c o m p a r e d t o t h e s i t u -

a t i o n i n w h i c h e v e r y b o d y o p t i m i s e s i n d i v id u a l ly . H e n c e th e q u e s t i o n a r is e s

h o w t o s h a r e t h e e x t r a r e v e n u e s o r c o s t s a vi n g s.

O n e w a y to a n a l y s e t h i s q u e s t i o n i s t o s t u d y t h e g e n e rM p r o p e r t i e s ( e. g.,

b a l a n c e d n e s s o r c o n v e x i ty ) o f a ll g a m e s a r is i n g f r o m t h a t s p e ci fi c t y p e o f

o p e r a t i o n s r e s e a rc h p r o b l e m a n d t o a p p l y a s u i t a b le e x i s ti n g g a m e t h e o r e t i c

s o l u t i o n c o n c e p t ( e .g . , c o r e o r S h a p l e y v a l u e ) t o t h i s c la s s . A n o t h e r w a y i s

t o c r e a t e a c o n t e x t s p e c if i c a l l o c a t i o n r u l e . S u c h a ru l e c a n b e b a s e d e i t h e r

o n d e s i r a b l e p r o p e r t i e s i n t h i s s p e c i f ic c o n t e x t o r o n a k i n d o f d e c e n t r a l i s e d

m e c h a n i s m t h a t ; p re s c ri b e s a n a l l o c a t i o n o n t h e b a s i s o f t h e a l g o r i t h m i c

p r o c e s s a l o n g w h i c h a j o i n t l y o p t i m a l c o m b i n a t o r i a l s t r u c t u r e is e s t a b l i s h e d

( e. g. , f o l lo w i n g a n a l g o r i t h m t o c r e a t e a m i n i m a l c o s t s p a n n i n g t r e e ) . A g en -

e r a l r e f e r e n c e o n c o o p e r a t i v e g a m e s i s D r i e s s e n ( 1 9 8 8 ) a n d v a r i o u s s p e c i f i c

o p e r a t i o n s r e s e a r c h g a m e s a r e t r e a t e d i n C u r i e l ( 1 9 9 7 ) .

T h e o r i g in a l r e q u e s t o f T O P t o t h e a u t h o r s o f t h i s p a p e r w a s t o w r i t e a

complete s u r v e y o n o p e r a t i o n s r e s ea r c h g a m e s . G i v e n t h e a b u n d a n c e o f p a -

p e r s o n t h i s t o p i c , s t a r t i n g m o r e o r le s s f r o m t h e b e g i n n i n g o f t h e s e v e n ti e s,

a n d t h e v a s t i n c r e a s e d u r i n g t h e n i n e t ie s , t h i s c o n s t i t u t e s a m i s s io n i m p o s -

s ib le . M o r e o v e r , t h e d e f i n i t io n o f o p e r a t i o n s r e s e a r c h g a m e s i s n o t s o s t r i c t

t h a t a u n i q u e s e l e ct i o n o f r e s e a rc h s t r e a m s i s p r e s c r i b e d . C o n s e q u e n t l y , t h e

c h o ic e o f t o p i c s t r e a t e d i n t h is s u r v e y is s o m e w h a t b i a s e d t o w a r d s o u r o w n


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Operations Research Gam es A Survey 141

e x p e r t i s e , k n o w l e d g e a n d i n t e r e s t s . A f ir s t g e n e r a l a i m o f t h i s s u r v e y is t o

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

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

d a t e s t a t e - o f t h e - a r t . L a s t b u t n o t l e a s t w e h o p e i t s t i m u l a t e s r e s e a rc h e r s t o

e n t e r t h i s f ie ld . T h e r e a r e s ti ll m a n y q u e s t i o n s t o b e a s k e d , g a p s t o b e f il le d

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

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

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

w o r d s t o p r o v o k e p o s s i b l y d if fh r en t i n d i v i d u a l a s s o c i a t i o n s a n d t o e n t e r n e w

r e s e a r c h t r a c k s .

T o b e t t e r s t r u c t u r e t h e s u r v e y , w e h a v e c h o s e n t o m a k e a d i v i si o n o f

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

o f t h e u n d e r l y i n g o p t i m i s a t i o n p r o b l e m . I n o u r v i e w , t h is c a t e g o r i s a t i o n a ls o

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

t e c h n i q u e s a n d r e s u l t s a c r o s s t h e d i f fe r e n t c l a ss e s o f o p e r a t i o n s r e s e a r c h

g a m e s . W e d i s t in g u i s h b e t w e e n :

(i) Connect ion f i x e d t r e e , s p a n n i n g t r e e

(ii) Routing C h i n e s e p o s t m a n , t r av e l li n g s a l e s m a n

(iii) Scheduling s e q u e n c in g , p e r m u t a t i o n , a s s i g n m e n t

( iv) Product ion l i n e a r p r o d u c t i o n , f l o w

(v) i n v e n t o r y

E a c h c a t e g o r y w i ll b e t r e a t e d i n a s e p a r a t e s e c ti o n . W i t h i n e a c h c a t e g o r y

w e h a v e c h o s e n o n e r e p r e s e n t a t i v e c l a ss w h i c h i s d i s c u s s e d in s o m e d e t a il ,

s t a r t i n g f r o m t h e l e v e l o f a p e r s o n w h o is n o t f a m i l ia r w i t h t h e t o p i c . F o r

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

h o w t o g o f r o m o p e r a t i o n s r e s e a rc h t o g a m e t h e o r y . A f t e r d i sc u s s i n g th e

m a i n r e s u l t s f r o m t h e l it e r a t u r e f o r t h e r e p r e s e n t a t i v e c l as s , t h e o t h e r c l a ss e s

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

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

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

l e a s t c l o s e ly r e l a t e d , b u t w h i c h w i ll n o t b e d i s c u s s e d i n th i s s u r v e y . F o r t h o s e

i n t e r e s t e d w e h a v e a d d e d a s e l e c t e d li s t o f r e f e re n c e s . W i t h i n a f i n a n c ia l

c o n t e x t w e m e n t i o n b a n k r u p t c y g a m e s , cf. O N e i l l (1 9 8 2) , A u m a n n a n d


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P. Borm , H . H am ers and R . H endr i ckx

M a s c h l e r ( 1 9 8 5 ), C u r i e l e t a l. ( 1 9 8 7 ), Y o u n g ( 1 9 8 8 ), K a m i n s k i ( 2 0 00 ) a n d

C a l l e j a e t a l. ( 2 0 0 1 b ) , d e p o s i t g a m e s , c f . I z q u i e r d o a n d R a f e l s ( 19 9 6 ) a n d

B o r m e t a l. ( 2 0 0 1 ) a n d s h o r t e s t p a t h g a m e s , c f. F r a g n e l l i e t a l . ( 2 0 0 0 ) ,

V o o r n e v e l d a n d G r a h n ( 20 0 0) a n d G r a h n ( 20 0 1 ). A n i n t e re s t i n g c l as s w i t h

c l e a r o p t i m i s a t i o n f > a t u r e s d e a ls w i t h c o s t s h a r i n g i s s u e s , c f. M o u l i n a n d

S h e n k e r ( 1 9 9 2 a , b ) a n d S p r u m o n t ( 1 9 9 8 ). A n i c e s u r v e y o f t h i s l i t e r a t u r e c a n

b e f o u n d i n K o s t e r ( 1 99 9 ). A n o t h e r t y p e o f g a m e s w h i c h d i r e c tl y i n v ol v e s

c o m b i n a t o r i a l s t r u c t u r e s is t h e c l a ss o f c o m m u n i c a t i o n g a m e s a n d a ll i ts

v a r ia n t s . S u r v e y s c an b e f o u n d i n S l ik k e r a n d N o u w e l a n d ( 2 00 1 ) a n d B i l b a o

( 2 0 0 0 ). A n i n t e r e s t i n g r e c e n t c o n t r i b u t i o n w h i c h u s e s t e c h n i q u e s f r o m li n e a r

p r o d u c t i o n i s S u i j s e t a l . ( 2 0 0 1 ) .

P r e l i m i n a r i e s

I n th i s s e c t io n w e i n t r o d u c e s o m e b a s ic n o t a t i o n a n d d e f in e a n u m b e r o f

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

t h e p a p e r .

T h e s e t o f r e a l n u m b e r s i s d e n o t e d b y IR. F o r a f i n it e se t N w e d e n o t e b y

IRN t h e s e t o f r e a l v e c t o r s o f l e n g t h I N I, w h e r e t h e c o o r d i n a t e s c o r r e s p o n d

t o t h e e l e m e n t s o f N . IR ~ is t h e s e t o f al l e l e m e n t s o f R N i n w h i c h a ll

c o o r d i n a t e s a r e n o n n e g a t i v e a n d R ~ + d e n o t e s t h e s e t o f a ll v e c t o r s in w h i c h

a ll c o o r d i n a t e s a r e p o s i t i v e . T h e s e t o f s u b s e t s o f N i s d e n o t e d b y 2 N . F o r

S C N , e s

d e n o t e s t h e v e c t o r i n I RN w i th e~ = 1 for a l l i E S a nd e~ = 0

for a l l i E N \ S .

A coopera t ive gam e w i t h t rans f e rab l e u t i l it y , o r T U g a m e , i s d e s c r i b e d

b y a p a ir ( N , v ) , w h e r e N = { 1 , . . . , n } d e n o t e s t h e s e t o f p l a y er s a n d

v

: 2 N --

R i s t h e

c h a r a c t e r i s t i c ] h n c t i o n ,

a s s i g n i n g t o e v e r y c o a l i ti o n

S C N o f p l a y e r s a v a l u e

v S ) ,

r e p r es e n t in g t h e m a x i m a l t o t a l m o n e t a r y

r e w a r d t h e m e m b e r s o f t h is g r o u p c a n o b t a in w h e n t h e y c o o p e r a t e . B y

c o n v e n t i o n , v ( 0 ) = 0 .

T h e

i m p u t a t i o n s e t I v )

o f a g a m e ( N , v ) i s d e f i n e d a s t h e s e t o f i n d i -

v i d u a l l y r a t i o n a l a l l o c a ti o n s o f v ( N ) :

I v ) - - { . : _> v { i } ) }

i N


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Operations Research Games: A Survey

143

and the

core

is defined as

C v) = { x C R N I ~ T x e = v N ) , V s c N : ~ z ~

> v S ) } .

i N i S

So the core consists of all allocations of v N ) such that no coalition S

has an incentive to part company with

N \ S

and establish cooperation on

its own. A TU game N,v ) is called

balanced

if it has a non emp ty core

and totally balanced if the core of every subgame is nonempty, where the

subgame corresponding to some coalition T C N, T 5k 0 is the game

T, v T)

with

v T s ) = v S )

fbr all S C T.

Every game can be uniquely decomposed as a linear combination of

una nim ity games. For T C

N ,T ~ O,

the

unan imity game N, uT)

is

defined by

1 if T c S,

UT S)

= 0 otherwise

for all S C N.

An order on the players is a bijection a : N -+ {1 , . .. , n}, where a i) = j

means th at player i is at posi tion j. The set of all orders on N is den oted

by [IN. For every order cr E FIN, we define the

marginal vector m~ v)

recursively by

~ T t ; _ , k ) v ) = v { o - - 1 1 ) , . . . , o - - 1 k ) } ) - v { o - - 1 1 ) , . . . , o - - 1 k - 1 ) } )

for all k E {1,..., n}. The Shapley value of N, v) is defined as cf. Shap ley

1953))

1

< v ) - Z ~ ~ ( v ).

I v l ~

~EII N

The Shapley value is a linear operator on the class of all TU games and for

the una nim ity game N, UT) it equals

( ~ a r ) = l r .

I I

A game N, v) is called

superadditive

if for all coali tions S, T C N wit h

S T = 0 we have

~ s ) + ~ T ) < v S u T )

and

convex

if fbr all i 6 N and all S C T C

N \ { i }

we have

v S u { i } ) - v S ) < v T u { i } ) - v T ) .


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144 P . B o r m , H . H a m e r s a n d R . H e n d r i c k x

In a superadditive game, it will always be beneficial for two disjoint coali-

tions to cooperate and form a larger coalition. In a convex game, a player's

marginal contribution to a large coalition is larger than his marginal con-

tribution to a smaller coalition, which is stronger than superadditivity. A

game is convex if and only if its core is the convex hull of all marginal

vectors. Furthermore, every convex game is totally balanced.

The

e x c e s s

of coalition S C N with respect to an impu ta ti on 0~ E

I ( v )

is defined by

E(S, x) = v(S) - ]~ xi.

i S

The e x c e s s v e c t o r with respect to x, denoted by O(x), is the vector in ]R2 r~

containing the excesses of all coalitions in (weakly) decreasing order.

Fox a game (N, v) with

[ ( v ) O ,

the

n u c l e o l u s

is defined (cf. $chmeidler

(1969)) as the unique imputation n u ( v ) such that O (n u (v ) )


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145

In a similar way, the definitions of imputation set, nucleolus and plnas are

altered. In this cost setting, the n atu ral coun terpa rt of convexity, as defined

for reward games, is concavity. A cost game N, c) is called concave if for

all i E N an d all S C T C N \ { i } we have

u { i } ) - 4 s ) >_ u { i } ) - c T ) .

Similarly, the counterpart of superadditivity is subadditivity.

C o n n e c t i o n

In this section we consider operations research problems which involve

connection networks in an interactive cooperative setting. We look at two

such problems in particular : mai nte nan ce problems, which involve a fixed

tree network, and minimum cost spanning tree problems, in which the con-

nection network is still to be decided upon.

First, we look at main ten anc e problems, which form a special class of fixed

tree problems. Our expositio n is mai nly based on tile overview given in

Koster 1999).

The idea behin d a maint enan ce problem is the fbllowing. A group of

players is connected by some fixed network to a certain service provider,

e.g., by a road ne twork to a comm unit y centre. This network is a tree in

which the service provider is situa ted at the root. Each road in this network

has some maintenance costs associated with it and the question is how the

main tena nce costs of the entire network should be divided in a fair way

among all users.

Formally, a m a i n t en a n ce p r o b l em is a triple G, t, N) where

G = t7, E) is a tree with vertex set V and edge set E; th e root r has

only one adjacent edge.

t : E -+ ~ is a nonnegativ e cost functi on on the edges of the tree.

N = { 1 , . . . , n} is a finite player set; each player i E N is loca ted at

some vertex

v i )

E V and every vertex in V except the root corre-

sponds to exactly one player.


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P. Borm, H. Hamers and R. Hendrickx

In order to analyse maintenance problems, we introduce solne more nota-

tion. First no te th at every vertex v C V is connected to the root of the tree

by a unique path Pv including v itself). We denote the edge in P v that is

incident on v by ev. The p recedence rela tion 4 on V is defined by

v ~ 4 v ~z v r is on the path Pv.

A t r u n k of G is a set of vertices R C V which is closed under the rela tion

4, i.e., i f v E R a n d v ~ 4 v, then v ~ C R. The set of Jbllowers of player

i E N is given by

F i ) = { j E N I v i ) ~

v j)} and the set of

predecessors

b y P i ) = { j E N I v j ) ~ v i)}. The to tal costs of a trunk R equal

T R)= Z t e )

Wit h each mainten ance problem F = G, t, N) we associate a

m a i n t en a n ce

g a m e N , c r ,) defined by

c o S ) = m i n { T R ) l v i ) E R for all i E S and R is a tru nk} 3.1)

for all S C N, S 0 and co 0) = 0. By non neg ati vit y of the cost function,

the tru nk R th at minimises total costs in 3.1) is the smallest trun k

R s

containing all the vertices at which the members of S are located, i.e.,

Z t e )

v~Rs \ {r}

The d u al u n a n i m i t y g a m e N , u } ) wit h T C N, T ~ is defined by

1. i f T n S ~ ) ,

u~r S) = 0 othe rwise

for all S C N. The coaliti on T in u~r can be seen as having some veto

control: if no memb er of T is present in a coalition, this par ticu lar coalition

has value 0. Note that u} is a concave game.

Pr opos i t i on 3 .1 . L e t F = G , t , N ) be a m a i n t en a n ce p r o bl em . T h en t h e

assoc ia t ed cos t gam e N , co) can be deco mp osed in the .fol lowing wa y:

co Z

i N


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Operat ions Research Games: A Survey 147

The decompositio n of cr in terms of dual u nan imi ty games is interp reted

as tbllows. In order to determine the costs of a coalition S, we have to find

the smallest trun k containing all members of S. Edge

ev(i)

is present in

this smallest tr un k whenever a member of S is a tbllower of player i, i.e.,

s n F i )

Because all coefficients of the dual una nim ity games in P roposit ion 3.1

are nonnegative, every mainte nanc e game is concave. As a consequence, the

core of such a game is no nem pty and has a nice structure. The lit erature

offers a large numbe r of characte risations of the core of maint enan ce games,

two of which will be pr esent ed here. The first one is in terms of trunks.

P r opos i t i on 3 .2 . A vec t or o f cos t share s x E R N i s ar t e l e m en t o f C cr)

i f and on l y i f x >_ 0 and } -~ i ER x i


10/61


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Opera t ions Research Games : A S urvey

4

4

1 ; o 1 1 1

4

2 1 3:43

4

0,4,0,3t ~

( 4 , 9 , 5 , 8 )

149

F i g u r e 2 : Home down pa i l~ t i ng so lu t i on

p a i n t i n g s o l u t i o n o f t h i s m a i n t e n a n c e p r o b l e m e q u a l s ( 4, 4 , 4 , 4 ) + ( 0 , 1 , 1 , 1 ) +

0,4,0,3) ( 4 , 9 , 5 , 8 ) .

T h e h o m e - d o w n p a i n t i n g s o l u ti o n P ( F ) t u r n s o u t t o b e t h e n u cl e ol u s o f

t h e c o r r e s p o n d i n g m a i n t e n a n c e g a m e ( N , c r ) ( cf . M a s c h l e r e t a l. ( 1 9 9 5 )) .

Theorem .1. e t r = O , t , N ) be a m a i n t e n a n c e p r vb l e m . T h e n P ( P ) =

~ c~)

C o n s i s t e n c y o f t h e h o m e - d o w n p a i n t i n g s o l u t i o n is s tu d i e d i n G r a n o t

e t a l. ( 1 99 6 ), G r a n o t a n d M a s c h l e r ( 19 9 8 ) a n d v a n G e l l e k o m a n d P o t t e r s

(1999) .

A n a l t e r n a t i v e p a i n t i n g s o l u t i o n i s g i v e n b y d r o p p i n g c o n d i t i o n ( i v ) a n d

r e p l a c i n g c o n d i t i o n ( v ) b y

( v ) E v e r y w o r k e r i s d o i n g h is j o b a s c lo s e t o t h e c o m m u n i t y c e n t r e a s

cond i t ions ( i ) - ( i i i ) a l low.


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150

P. Borm, H. f lamers and R. Hendrickx

R u l e s ( i) -( ii i) a n d ( v ') d e t e r m i n e t h e down-home pa in t ing so lu t ion f o r m a i n -

t e n a n c e p r o b l e m s , w h i c h w e d e n o t e b y P ~, a n d w h i c h is g iv e n b y

P ' ( P ) - - ~ iF j ) l t e v j ) ) . (3.2)

jEP i)

E x a m p l e 3 , 2 , C o n s i d e r t h e m a i n t e n a n c e p r o b l em in E x a m p l e 3 .1 . T h e

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

1 1 1 1 1 ,

P I ( p ) = 1 2 ( , 4 , 4 , ~ ) + 5 ( 0 , 1 , 0 , 0 ) + 6 ( 0 , 0 , ~ , ~ ) + 3 ( 0 , 0 , 0 , 1 ) = ( 3 , 8 , 6 , 9 )

T h i s d o w n - h o m e p a i n t i n g s o l u ti o n i s t h e S h a p l e y v a l u e o f t h e c o r re -

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

T h e o r e m 3 . 2 .

Let P = G, t , N ) be a m ain tena nce prvb lem. The n

P ' ( P ) =

B e c a u s e c r is c o n c a v e , t h e S h a p l e y v a l u e li es in t h e c o r e o f t h e g a m e .

T h i s a l so fo ll o w s i m m e d i a t e l y f r o m P r o p o s i t i o n 3 .3 a n d e q u a t i o n ( 3 .2 ) .

I n o r d e r t o d e f in e a t h i r d p a i n t i n g s o l u t i o n , w e n e e d t o i n t r o d u c e s o m e

m o r e c o n c e p t s . A

pseudo subtree

o f a t r e e G = (V , E ) i s a c o n n e c t e d s u b -

g r a p h G 1 = ( V r, E I) s u c h t h a t t h e r e e x i s t s a n / E V r w h i c h i s r n i n i m a l i n

G ~ w i t h r e s p e c t t o 4 a n d w h i c h h a s o n l y o n e a d j a c e n t e d g e i n E ~. A weigh t

s y s t e m f o r m a i n t e n a n c e p r o b l e m P = ( G , t , N ) is a p a i r f l = ( T , w ) , w h e r e

7 - = ( G I , . . . , G p ) is a p a r t i t i o n o f G i n to p s e u d o s u b t r e e s a n d w E R { is a

w e i g h t v e c t o r s u c h t h a t f o r a l l i E N w i t h t ev i)) > 0 t h e r e i s a j E F i ) ,

w h o i s i n t h e s a m e p s e u d o s u b t r e e a s i , w i t h j ~> 0 . T h e s e t o f a l l we i gh t

s y s t e m s f o r F i s d e n o t e d b y B ( P ) .

N o w w e d e f i ne t h e weigh ted down-home pa in t ing so lu t ion P~ c o r r e s p o n d -

i n g t o s o m e w e i g h t s y s t e m fl E B ( F ) . I n th i s c o n t e x t , e v e r y p s e u d o s u b t r e e

G k h a s i ts o w n l o c a l c o m m u n i t y c e n t r e , w h i c h is s i t u a t e d a t t h e r o o t o f G k .

T h e s o l u t i o n i s d e t e r m i n e d b y t h e f o l lo w i n g r u le s :

(i ) E v e r y w o r k e r w o r k s w i t h s p e e d w i .

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

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

(iii ) E v e r y w o r k e r d o e s h i s j o b o n a n u n f i n i s h e d r o a d s e g r n e n t b e t w e e n t h e

l o ca l c o m n m n i t y c e n t r e a n d h i s h o m e .


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O p e r a t io n s R e s e a r c h G a m e s : A S u r v e y

151

( iv ) I f t h e r o a d b e t w e e n a w o r k e r ' s p r e d e c e s s o r i n t h e t r e e a n d t h e l o c a l

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

p a r t o f t h e n e t w o r k .

(v ) E v e r y w o r k e r is d o i n g h i s j o b a s c lo s e t o h i s r e s i d e n c e a s c o n d i t i o n s

( i ) - ( iv ) a l low.

F r o m P r o p o s i t i o n 3 .3 i t f ol lo w s th a t e v e r y w e i g h t e d d o w n - h o m e a l lo c a -

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

c h a r a c t e r i s a t i o n o f t h e c o r e ( B j o r n d a ] e t a l . ( 1 9 9 9 )) .

T h e o r e m 3 . 3 . L e t F = ( G , t , N ) b e a m a i n t e n a n c e p r o b l e m . T h e n ] o r e v e r y

x E C ( c r )

t h e r e e z i s t s a w e i g h t s y s t e m [3 E

B ( F )

s u c h t h a t

x = P , ~ ( F ) .

A s i m i l a r re s u l t i n t h e c o n t e x t o f i r r ig a t i o n n e t w o r k s c a n b e f o u n d i n

K o s t e r e t al. ( 1 99 8 ). R e l a t e d p r o b l e m s w i t h a p p l i c a t i o n s o f f i x e d t r e e p r o b -

l e m s a r e d i s c u s s e d i n M e g i d d o ( 19 7 8) a n d G a l i l ( 1 98 0 ). S o m e c o m p u t a t i o n a l

i s s u e s a r e a d d r e s s e d i n G r a n o t a n d G r a n o t ( 1 9 9 2 a ) .

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

a i r p o r t p r o b l e m . A i r p o r t p r o b l e m s w e r e i n t r o d u c e d b y L i t tl e c h il d a n d O w e n

( 1 9 7 3 ) a n d i t s S h a p l e y v a l u e a n d n u c l e o l u s a s w e l l a s t h e i r p r o p e r t i e s w e r e

s t u d i e d i n L i t tl e c h i l d ( 1 9 7 4) , L R t l e c h i l d a n d O w e n ( 19 7 7 ), L R t l e c h i ld a n d

T h o m p s o n ( 19 77 ), D u b e y (1 98 2) a n d P o t t e r s a n d S u d h o R e r (1 99 9 ).

T h e d e s c r i p t i o n o f a n a i r p o r t p r o b l e m c a n b e s h o r t e n e d t o a p a i r ( N , k ),

w h e r e N = { 1 , . . . , n } is t h e p l a y e r s e t a n d k ff JR ) i s a v e c t o r o f m a r g i n a l

c o s ts , w h i c h a r e i n t e r p r e t e d a s f o ll ow s . E v e r y p l a y e r o w n s a n a i r p l a n e w i t h

c e r t a i n c h a r a c t e ri s ti c s , w h i c h d e t e r m i n e t h e m i n i m a l l e n g t h o f a l a n d i n g

s t ri p t h i s p l a n e c a n u s e . A s s u m i n g t h a t t h e p l a y e r s a re o r d e r e d i n i n c re a s -

i n g l e n g t h o f t h i s s t r i p ( i. e ., / ~ 1 ~ -~ ' ~ - -- ~ ' n ) a n d m a i n t e n a n c e c o s ts a r e l in e a r

i

i n s t r i p le n g t h , p l a y e r i ' s t o t a l c o s t s e q u a l ~ j = l k j , w h e r e k j r e p r e s e n t s t h e

e x t r a c o s t s o f m a i n t a i n i n g t h e l o n g e r s t ri p o f p l a y e r j i n r e l a t i o n t o t h e

s h o r t e r s tr i p o f p l ~ e r j - 1. T h e p r o b l e m i s h o w t o d iv i d e t h e m a i n t e -

n a n c e c o s ts o f a s tr i p t h a t a c c o m m o d a t e s a ll a ir p la n e s , Y ] i cN ki , a m o n g t h e

p l a y e r s .

W i t h e a c h a i rp o r t p r o b l e m ( N , k ) w e a ss o c i a t e a n

a i r p o r t g a m e ( N , c )

w i t h c o s t f u n c t i o n c ( S ) = 2 i k j , w h e r e i = m a x { j l j E S } . S i n c e t h i s

j = l

g a m e i s a s p e c i a l c a s e o f a m a i n t e n a n c e g a m e , i t is c o n c a v e a n d w e h a v e a


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152 P. Borm, H. Hamers and R. Hendrickx

n i ce e x p r es s i o n fo r t h e S h a p l e y v a lu e . F i r s t , n o t e t h a t o n e c a n d e c o m p o s e

t h e c o s t f u n c t i o n c a s f o l l o w s :

c = klu~v + k2u{~,.. . ,~d + + knu{n } .

H e n c e , t h e S h a p l e y v a l u e o f ( N , c ) e q u a l s

O C ) k l L , . L ) 1 1

= . . , + k s 0 , - -

n n - - l ' ' ' n - - 1 ) + ' + k n ( 0 ' ' ' ' 0 ' 1)

O f c o u r s e , t h e r e s u l t s fo r m a i n t e n a n c e g a m e s w . r .t , t h e c o r e, t h e n u c l e o lu s

a n d t h e w e i g h t e d S h a p l e y v a l u e s i n d u c e e a s y e x p r e s s io n s f o r a i r p o r t g a m e s

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

a n d K o l p i n ( 1 9 9 8 ), w h o l o o k a t ir r i g a t i o n n e t w o r k s . I n B r a n z e i e t al . (2 0 0 1 ) ,

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

o f s i d e p a y m e n t s t h a t a r e fe a s i b le .

N e x t , w e c o n s i d e r a c l as s o f p r o b l e m s t h a t i s c l o s e l y r e l a t e d t o m a i n t e -

n a n c e p r o b l e m s :

minimum cost spannin 9 tree

o r

mcst problems.

C o n t r a r y

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

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

b a s e d o n F e l t k a m p ( 1 99 5 ).

C o n s i d e r a g r o u p o f v i ll ag e s , e a c h o f w h i c h n e e d s t o b e c o n n e c t e d t o

s o m e s o u r c e , e i t h e r d i r e c t l y o r v i a o t h e r v i ll a g e s. E v e r y p o s s i b l e c o n n e c t i o n

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

c o n n e c t e v e r y v il la g e t o t h e s o u r c e s u c h t h a t t h e t o t a l c o s t s o f c r e a t i n g t h e

n e t w o r k a r e m i n i m a l . K r u s k a l ( 19 5 6) a n d P r i m ( 19 5 7) p r o v i d e t w o g r e e d y

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

p r o b l e m s c a n b e f o u n d i n G r a h a m a n d H e l l ( 19 8 5 ).

C o n s t r u c t i n g a n m c s t , h o w e v e r , is o n l y p a r t o f t h e p r o b l e m . I n a d d i t io n

t o m i n i m i s i n g t o t a l c o s t s , a c o s t a ll o c a t io n p r o b l e m h a s t o b e a d d r e s s e d a s

w e ll . C l a u s a n d K l e i t m a n ( 1 9 73 ) i n t r o d u c e d t h i s c o s t a l l o c a t i o n p r o b l e m ,

w h e r e u p o n B i r d ( 1 9 7 6 ) t r e a t e d t h i s p r o b l e m w i t h g a m e t h e o r e t i c m e t h o d s

a n d p r o p o s e d a c o s t a l l o c a t i o n r u l e , k n o w n a s t h e B i r d r u l e .

F o r m a l l y , a n m s c t p r o b l e m is a t r i p l e 7 - = ( N , , , t ) , w h e r e N = { 1 , . . . n }

i s t h e p l a y e r s e t , * i s t h e s o u r c e a n d t : E N . --+ R + i s t h e n o n n e g a t i v e c o s t

f u n c t i o n .

Es

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

S C N * , s o t h a t S, Es) is t h e c o m p l e t e g r a p h o n S :

E s = { { i j } l i j c s i j } .


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Ope rat ions Resea~Y~h Gam es: A Su rve y

153

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

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

t h e n a m e o f t h e p r o b l e m .

G i v e n an m c s t p r o b l e m T = N , , , t ) a n d a n m c s t N * , R ) f o r t h e g r a n d

c o a l i t i o n ,

B i r d s t re e a l l o c a t io n , [ ~ R ( T )

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

p l a y e r i C N t h e c o s t o f t h e f ir s t e d g e o n t h e u n i q u e p a t h i n N * , R ) f r o m

p l a y e r i to t h e s o u r c e , . T h e c o m p u t a t i o n o f t h is a l l o c a ti o n c a n b e in t e -

g r a t e d i n to t h e P r i m a l g o r i th m , w h i c h , s t a r t i n g f r o m a fi x ed r o o t , c o n s t r u c t s

a n m c s t b y c o n s e c u t i v e l y a d d i n g e d g e s w i t h t h e l ow e s t c os t, w i t h o u t i n tr o -

d u c i n g c y c l e s .

A l g o r i t h m 3 . 4 ( B i r d s r u l e ) .

i n p u t : a n m c s t p , v b l em ( N , , , t )

o u t p u t : a n e dg e s e t R C E N . o f a n m c s t a n d c o r r e s p o n d i n g B i r d a l l o c a t io n

/~ R T )

t . C h o o s e t h e s o u r c e a s r o o t.

2 . h t i t i a l i s e R = ~ .

. F i n d a m i n i r a a l c o s t e dg e e = { i , j } E E N . \ R i n c i d e n t o n , o r a n y o f

t h e v e r t ic e s p r e s e n t i n o n e o f t h e e d g es i n R i n s u c h a w a y t h a t j o i n i n g

e to R d o e s n o t i n t r o d u c e a c y c l e .

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

o t h e r v e r t e x i is a p l a y e r w h o w a s n o t y e t c o n n e c t e d t o th e s o u r c e .

A s s i g n t h e c o s t

~ . ~ T ) = t e )

t o a g e n t i .

5 . J o i n e t o R .

6 . I f n o t a l l v e r t i c e s a r e c o n n e c t e d t o t h e r o o t i n t h e g r a p h

N * ,

R ) , g o

b a ck t o s t ep 3 .

N o t e t h a t t h e B i r d a l lo c a t i o n d e p e n d s o n t h e a c t u a l m c s t t h e a l g o r i t h m

a r r iv e s a t , w h i c h is d e t e r m i n e d b y t h e c h o i c e s m a d e i n s t e p 3 o f t h e a lg o -

r i t h m .

T h e f o l lo w i n g e x a m p l e i l lu s t r a te s t h e a l g o r i th m .

E x a m p l e 3 . 3 . C o n s i d e r t h e m c s t p r o b l e m T w i t h N = {1, 2, 3} as p r e -

s e n t e d i n F i g u r e 3 , w h e r e t h e n u m b e r s o n t h e e d g e s r e p r e s e n t t h e c o st s.


16/61

1 5 4

P . B o r m , H . H a m e r s a n d R . H e n d r i c k x

3

Figure : A m i n i m u m c o s t s p an n in g t re e p r o b l e m T

W h e n w e a p p l y A l g o r i t h m 3 .4 to t h i s p r o b l e m , t h e f ir s t e d g e w e j o i n

t o R i s e i t h e r { , , 1 } or { * , 3 } . S u p p o s e w e c h o o s e t h e f i r s t o n e , t h e n w e

s e t / 3 ~ T ) = 1 0. S u b s e q u e n t l y , w e a d d { 1 , 2 } t o R , s e t / 3 ~ T ) = 6 , a d d

{ 2 , 3 } a n d s e t / ? 3 R T ) = 5 . T h i s g i v e s u s a c o s t a l l o c a t i o n o f 1 0 , 6 , 5 ) . O n

t h e o t h e r h a n d , s u p p o s e w e s t a r t w i t h { , , 3 } . T h e n w e e n d u p w i t h c o s t

a l l o c a t i o n

f i n T )

= 6 , 5 , 10) .

T h e t w o m i n i m u m c o s t s p a n n i n g t re e s a re d r a w n i n F i g u r e 4 .

2 2

6 ~ 6 5

1 10 3

3

Figure 4:

T w o m i n i m u m c o st sp a n n i n g t r ee s

W i t h e a c h m c s t p r o b l e m T = N , , , t ) w e a s s o c i a t e a

m c s t g a m e N , c 7-),

w h e r e

c 7 - S )

r e p r e s e n t s t h e m i n i m a l c o s t s o f a t re e o n S * = S U { * }:

c T S ) = m i n { ~ t ~ ) I n C E s , a n d S *, R ) i s a tre e}

eER

f o r a ll S C N , S 0 a n d cT ~ )) = 0 . T h e f o l l o w i n g t h e o r e m c o m e s f r o m

G r a n o t a n d H u b e r m a n 1 98 1 ).


17/61


155

T h e o r e m 3 .5 . L e t T = ( N , , , t ) be a m i n i m u m c o s t s p a n n i n g tr ee p r o b le m .

T h e n f o r e v er y m i n i m u m c o st s p a n n i n g tr ee ( N * , R ) , B i r d s a ll o ca ti o n ru le

f i n ( T ) i s a n e z t r e m e p o i n t o f t h e c or e o f t h e co r r e sp o n d i n g m i n i m u m c o s t

s p a n n i n g t re e g a m e ( N , c T ) .

I t i m m e d i a t e l y f o l lo w s t h a t e v e r y m c s t is b a l a n c e d . A n a l t e r n a t i v e p r o o f

f o r n o n e m p t i n e s s o f t h e c o r e i s g i v en in G r a n o t a n d H u b e r m a n ( 1 98 2 ).

A t h r t h e r o v e r v i e w o f m c s t p r o b l e m s i s g i v en i n A a r t s ( 19 9 4 ) a n d t h e

c o r e a n d n u c l e o l u s a r e s t u d i e d i n G r a n o t a n d H u b e r m a n ( 1 98 4 ) a n d S o ly -

m o s i e t al . ( 1 9 9 8 ). A a r t s a n d D r i e s s e n ( 1 9 9 3 ) s t u d y t h e i r r e d u c i b l e c o r e o f

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

t e r m i n e t h i s s e t. I n M o r e t t i e t al . (2 0 0 1 ) a n d N o r d e e t a l. ( 2 0 0 1 ), e x i s t e n c e

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

I n N o u w e l a n d e t a l. ( 1 9 93 ) i t is s h o w n t h a t e v e r y n o n n e g a t i v e m o n o t o n i c

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

t h e v e r t i c e s a s w e l l a s w i t h t h e e d g e s .

T h e r e a r e a l a rg e n u m b e r o f v a r i a t io n s o n t h e m c s t p r o b l e m a s p r e s e n t e d

a b o v e . I n F e l t k a m p e t al. ( 1 99 4 ), m i n i n m m c o s t s p a n n i n g e x t e n s i o n p r o b -

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

i n s u c h a w a y t h a t t o t a l e x t e n s i o n c o s t s a r e m i n i m a l . I n t h is f r a m e w o r k ,

t w o a l l o c a ti o n r u l es a r e p r e s e n t e d t h a t a r e i n s p i re d b y K r u s k a l ' s a l g o r i th m

f o r f i n d i n g m i n i m u m c o s t s p a n n i n g t r e e s . I n S u i j s ( 2 0 0 1 ), m c s t p r o b l e m s

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

c o s t s a n d m a i n t e n a n c e c o s ts . S i nc e t h e l a t t e r c o s t s a r e u n k n o w n e x a n t e ,

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

t e r m i n e a n o p t i m a l n e t w o r k is p r e s e n t e d a n d a t w o s t a g e B i r d ~ l lo c a t io n is

d e f i n ed a n d s h o w n t o b e a co r e a ll o c a t io n o f t h e c o r r e s p o n d i n g c o o p e r a t i v e

s t o c h a s t i c m i n i m u m s p a n n i n g t r e e g a m e ( cf . S u i js (2 0 0 0 )) .

Ro u t i n g

I n t h i s s e c t io n w e d i sc u s s c l as s e s o f o p e r a t i o n s r e s e a rc h p r o b l e m s i n

w h i c h t h e o b j e c t i v e is t o f in d a r o u t e o f m i n i m a l c o s t s w i th i n a g r a p h . F i r s t ,

w e d i s c u s s t h e c l a ss o f C h i n e s e p o s t m a n g a m e s a s i n t r o d u c e d i n H a m e r s e t

a l. ( 1 9 9 9 a ) . S e c o n d , w e d i s c u s s tr a v e l l i n g s a l e s m a n g a m e s a s i n t r o d u c e d i n

P o t t e r s e t al . ( 1 9 9 2 ). W e d i s c u s s tw o v a r i a n t s o f t h e t r a v e l l in g s a l e s m a n

p r o b l e m : t h e fi x e d r o u t i n g p r o b l e m a n d t h e S t e i n e r tr a v e ll in g s a l e s m a n


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156

P. Borm, H. Hamers and R . Hendr ickx

p r o b l e m .

I n t h e C h i n e s e p o s t m a n p r o b l e m , w h irl1 is i n t r o d u c e d i n M e i - K o K w a n

( 1 96 2 ), o n e c o n s i d e rs a s i t u a t i o n i n w h i d l a p o s t m a n h a s t o d e l iv e r m a i l to

e a c h s t re e t, o f a c e r t a i n c i ty . H e h a s t o s t a r t a n d f in i sh a t t h e p o s t o t ~ c e .

F o r e a c h s t re e t c o s t s a r e i n v ol v e d e a ch t i m e t h e p o s t m a n v i s it s t h i s s t r e e t.

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

c o s t s a re m i n i m i s e d . T h e m a i n d i ff e re n c e b e t w e e n s e v e r a l c l as s e s o f C h i n e s e

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

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

g r a p h is u n d i r e c t e d , E d m o n d s a n d J o h n s o n ( 19 7 3 ) p r e s e n t a p o l y n o l n i a l l y

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

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

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

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

p o s t o tY ic e a n d v i s i t a ll e d g e s ( p l a y e r s) b e f o r e r e t u r n i n g t o t h e p o s t o ff ic e ,

t h e c o s t a l lo c a t i o n p r o b l e m i s c o n c e r n e d w i t h a f a ir a l lo c a t i o n o f t h e c o s t

o f a c h e a p e s t C h i n e s e p o s t m a n l ;o u r i n t h e g r a p h . T h a t i s, t h e c o s t o f a

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

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

F o r m a l l y , a C h i n e s e p o s t m a n o r C P p r v b l e m i s a t up l e F = (N , G , v0 , g , t ) ,

w h e r e N = { 1 , . . . , n } i s t h e s e t o f p l a y e r s , G = (V , E ) i s a c o n n e c t e d u n d i -

r e c t e d g r a p h w i t h v e r t e x s e t V a n d e d g e s e t E , v0 C V r e p r e s e n t s t h e

p o s t o r ific e, g : E --+ N i s a b i j e c t i o n r e l a t i n g t h e p l a y e r s t o t h e e d g e s a n d

t : E - -+ F' i s a n o n n e g a t i v e c o s t f u n c t i o n a s s i g n i n g c o s t s t o t h e e d g e s . A n

E - t o u r w i t h r e s p e c t t o v0 a s s o c i a t e d w i t h c o a l i t i o n S C N i s a c l o s e d w a l k

v O , e l , . . . , e k , v o ) t h a t s t a r t s a t t h e p o s t o t fi c e v 0 , v i s i t s e a ch p l a y e r i n S a t

l e a s t o n c e a n d r e t u r n s t o v 0, i .e ., S C { g ( e j ) I j C { 1 , . . . , k } } . N o t e t h a t a n

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

a ll S - t o u r s i s d e n o t e d b y D S ) .

S u p p o s e a c o a l i t i o n S is s e r v e d a c c o r d i n g t o t h e S - t o u r ( v0 , e l , . . . , e k , v 0 )

E D ( S ) t h e n t h e t o t a l c o s t s o f t h i s t o u r a r e y ~ k t e j ) W e w i l l a s s u m e

j = l

t h a t e a c h p l a y e r i E S p a y s t , h e c o s t s t ( g - l ( i ) ) h i m s el f. I n t h i s w a y w e

a l r e a d y a l l o c a t e t h e

separable cos t s Y~ . i~s t g - l i ) )

o f a n S - t o u r . N o t e t h a t

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

nonseparab le cos t s f o r c o a l i t i o n S , y ~k t e j) - ~ i ~ s t g - l i ) ) , h a v e t o b e

j l

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

C h i n e se p o s t m a n


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Operat ions Research Games: A Survey

157

o r U P g a m e ( N , c ) c o r r e s p o n d i n g t o P = N , G , v 0, g , t ) , d e f i n e d b y

e S ) =

k

m i n [ ~ - ~ t ( e j ) - E t ( g - l ( i ) ) ]

D ~- *

vo ,e l , . . . , ek , vo )E S ) j = l . i C S

f o r a ll S C N . I n t h e fo l lo w i n g e x a m p l e , w e s h o w t h a t a C P g a m e n e e d n o t

b e b a l a n c e d .

E x a m p l e 4 . 1 . C o n s i d e r t h e C P p r o b l e m N , G , v 0, g , t ) w i t h N = { 1 , . . . , 5 },

g r a p h G = V , E ) a s d e p i c t e d i n F i g u r e 5, t ( e j ) = 1 a n d g ( e j ) = j f o r a l l

j { 1 . . . 5 } .

v 1

e 1

/

e

5 \

v 2

A

e4 ~ / .................... e 3

v 3

F i g u r e 5 :

A C*Jinesep o s t m a n p r o b l e m

D e n o t e t h e c o r r e sp o n d i n g C P g a m e b y N , c ) . T h e n c ( N ) = 1 a n d

c ( S ) = 0 f o r S A = { { 1 , 2 , 5 } , { a , 4 , 5 } , { 1 , 2 , 3 , 4 } } . L e t x R x a n d

s u p p o s e

x C(c ) .

T h e n

2 = 2c N )= Z Z xi _< Z c s)= 0.

S E A i C S S E A

C o n t r a d i c t i o n , s o N , c ) is n o t b a l a n c e d .

I n s p it e o f t h is r e s u l t, b a l a n c e d n e s s , t o t a l b a l a n c e d n e s s a n d c o n c a v i t y

h a v e b e e n e s t a b l i s h e d f o r C P g a m e s t h a t a r i se f r o m s o m e s p e ci fi c c l a ss e s

o f g r a p h s . A g r a p h G = V , E ) is s a i d t o b e global ly UP balanced ( to tal ly

balanced, concave)

i f t h e i n d u c e d C P g a m e is b a l a n c e d t o t a l ly b a la n c e d ,

c o n c a v e ) f b r a l l p o s s i b l e v 0 E V a n d a l l n o n n e g a t i v e c o s t f u n c t i o n s o n t h e


20/61

1 5 8

P . B o r m , H . H a m e r s a n d R . H e n d r i c k z

e d g e s . G i s c a l l e d l o c a ll y C P b a l a n c e d ( t o t a l l y b a l a n c e d ~ c o n c a v e ) i f t h e

i n d u c e d C P g a m e is b a l a n c e d t o t a l l y b a l a n c e d , c o n c a v e ) fo r s o m e v0 E V

a n d a l l c o s t f u n c t i o n s .

I n T h e o r e m s 4.1 - 4 .4 s o m e r e s u l t s a r e s t a t e d f r o m H a m e r s 1 9 9 7 ), G r a -

n o t e t a l. 1 9 9 9) a n d G r a n o t a n d H a m e r s 2 0 0 0 ).

T h e o r e m 4 . 1 .

L e t G b e a c o n n e c t e d u n d i r e c t e d g r a p h . T h e n t h e f o l l o w i n g

t h r e e a s s e r t i o n s a r e e q u i v a l e n t :

( i ) G i s w e a k l y E n l e r .

( i i ) G i s g lobal ly C P balanced.

( i i i ) G i s loca l ly CP balanced .

A g r a p h is c a l le d w e a k l y E u l e r i f e a c h b i c o n n e c t e d c o m p o n e n t 1 i n G is

E u l e r i a n i .e . , th e d e g r e e o f e v e r y v e r t e x is e v e n ) .

T h e o r e m 4 . 2 .

L e t G b e a c o n n e c t e d u n d i r e c t e d g r a p h . T h e n t h e f o l l o w i n g

f i v e a s s e r t i o n s a r e e q u i v a l e n t :

( i ) G i s w e a k l y c y c l i c .

( i i ) G i s g l o b a l ly C P c o n c a v e .

( i i i ) G i s g lobal ly C P to ta l ly ba lanced .

( i v) G i s loca l ly C P co ncav e .

(v ) G i s loca l ly C P to ta l ly ba lanced.

A g r a p h G i s c a l le d w e a k l y c y c l i c i f e a c h b i c o n n e c t e d c o m p o n e n t is a

c i r c u i t .

T h e C h i n e s e p o s t m a n p r o b l e m i n w h i c h th e u n d e r l y i n g g r a p h is d i r e c t e d

h a s a l so b e e n s t u d i e d i n t h e l i t e r a t u r e . A l l d e f i n i t i o n s fo r t h e u n d i r e c t e d c a s e

a s p r e s e n t e d a b o v e c a n b e e x t e n d e d t o t h e d i r e c t e d c a s e in a s t r a i g h t f o r w a r d

w a y .

1 A b i co n n ec t ed co m p o n en t o f a g r ap h G i s a m ax i m a l s u b g r ap h o f G i n w h i ch each

pai r o f ver t ices is conne c ted by a t l eas t two edge d i s jo in t pa ths .


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O p e r a t io n s R e s e a r c h G a m e s : A S u r v e y 159

Theorem 4 .3 . L e t G b e a s t r o n g l y c o n n e c t e d d i r e c t e d g r a p h . T h e n G i s

g l o b a l l y C P b a l a n c e d .

The proof of Theorem 4.3 translates the problem to a linear program-

ming problem and applies a balancedness result established in Owen 1975).

Theorem 4 .4 . L e t G b e a s t r o n g l y c o n n e c t e d d i r e c t e d g r a p h . T h e n G i s

d i r e c t e d w e a k l y c y c l i c i f a n d o n l y i f G i s g l o b a l l y C P c o n c a v e .

A di rected weakly cyclic grap h is a 1-sum 2 of directed circuits.

We conclude the discussion on CP games by considering an allocation

rule for the class of problems in which the underl ying graph is an un direc ted

weakl y Euler graph. This class of CP p roblems with player set N is denote d

by W E N .

In order to i ntro duce a rule th at divides for each P = N, G, v0, g, t) E

W E N

the costs of a minimal N-tour among the players, we need the notion

of followers of a bridge wit h respect to v0. An edge of G is called a b r i d g e

if removal of this edge leads to a disconnec ted graph. We denote th e set of

bridges in G by B G ) . Edge e E E is called a f o l l o w e r of b with respect

to v0 if each path th at conta ins bot h v0 and e also contains b. T he set of

followers of b will be denoted by F b G , v o ) . Note that b E F ~ G , v o ) and

that the set of followers depends on the location of v0 in the graph.

Let b E B G ) . The n th e p ost man needs to cross this bridge twice if he

is to make a tour containing some edge in F b G , v o ) . It seems reasonable

that each player in F b G , v0) will pay an equal share of the costs of crossing

b for the second time. So, if a tour th at visits a certain player contai ns

bridges, he has to contribute a fair share in the nonseparable costs of all

these bridges. Formally, the division rule 7 : W E N --+ R N is defined for all

F = N , G , v o , g , t ) E W E N

by

for all e E E.

t b )

I F a , 0 ) l

The following example illustrates the 7 rule.

2The 1-sum of graphs G and H is defined as the graph derived from G and H by

coalescing one vertex in G with another vertex in H.


22/61

160 P. Borm, H. Hamers and R. Hendrickx

E x a m p l e 4 . 2 . C o n s i d er t h e C P p r o b l em N , G , vo , g , t ) , w h e r e t h e g r a p h

G is d e p i c t e d i n F i g u r e 6 l e f t) ,

t b l )

= 52 ,

t b2)

= 44 and t b3 ) = 33 .

E

go vo

E3

F i g u r e 6 : W eakly Euler graph G left); the components o f G a f t e r removal of

bridges right)

O b s e r v e t h a t G is i n d e e d a w e a k l y E u l e r g r a p h , b e c a u s e t h e r e m o v a l o f

t h e b r i d g e s b l , b2 a n d b3 l e a d s t o t h e c o m p o n e n t s E 0 , El , E2 a n d E a , w h i c h

a r e a l l E u l e r i a n . B e c a u s e I F G , b l ) l = 2 6 , I F G , b 2) l = 1 1 a n d I F G , b 3)l =

1 1 , a c c o r d i n g t o t h e ~, r u l e e a c h p l a y e r i n F G , b l ) \ F G , b2) U F G , b3))

5 5 44

p a y s ~ ; = 2 , e a c h p l a y e r i n F G , b2) p a y s 5 g + 1~ = 6 a n d e a c h p l a y e r i n

5 33

F G,

b3 pa ys ~ + ~ = 5 .

T h e 7 r u l e c a n b e c h a r a c t e r i s e d b y t w o d i f fe r e nt s e t s o f p r o p e r t i e s . T h e

f i rs t c h a r a c t e r i s a t i o n u s e s fi v e p r o p e r t i e s c a l l e d ef fi c ie n c y , s t a n d a r d , n u ll ,

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

a C P p r o b l e m i n to a n u m b e r o f s i m p l e s u b p r o b l e m s a n d u s e s th e a d d i t i v e

s t r u c t u r e o f t h e 7 r u l e .

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

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

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

w i t h r e s p e c t t o a n e x t r e m e b r id g e . A

bridge cluster

i s a m a x i m a l s e t o f

e d g e s th a t n e e d t h e s a m e s e t o f b r i d g e s t o b e c o n n e c t e d t o t h e p o s t o f fic e.

S o f o r P = N , G , v o , g , t) E W E N a n d B G ) = { b l , . . . , b q } w e h a v e t h e

b r i dg e c lu s t e r s { CN G , v0 )} j~{0 ,..., q} , w he re

Co G , vo ) = E \ U be . c ) F b G , vo )


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Operat ions Research Games: A Survey

161

is t h e s e t o f e d g e s t h a t d o n o t n e e d a n y b r i d g e t o b e c o n n e c t e d to v 0 a n d

f o r a l l j c { 1 , . . . , q}

C j G , v o ) = G , v o ) \

Ub B a)nFv a ,vo) ,b bj F G, v0)

is t h e c l u s t e r o f e d g e s t h a t n e e d t h e b r i d g e s { b E

B G ) ] b j E F b G ,

v0) } t o

b e c o n n e c t e d t o v0 . A b r i d g e

b j E B G )

i s c a ll e d a n

ez t r em e br i dge

of G i f i t

h a s n o o t h e r b r i d g e a s a t b ll o w e r , o r e q u i v a l e n t l y , i f C j G , v 0) - - Fb~ G , v 0 ).

T h e f o ll ow i n g e x a m p l e i l lu s t r a te s t h e n o t i o n s o f b r i d g e c l u s te r a n d e x t r e m e

b r i d g e .

E x a m p l e 4 . 3 . C o n s i d e r t h e g r a p h G in E x a m p l e 4 .2 . T h e n

C o G , v o ) = E o

a n d C j G , vo ) = E j U { b j } f o r j E { 1 , 2 , 3 } . T h e e x t r e m e b r i d g e s a r e b2 a n d

b3.

N e x t , w e d e sc r i b e a p r o c e d u r e t o c o n s t r u c t t h e c o n d e n s e d g r a p h o f a

w e a k l y E u l e r g r a p h G w i t h r e s p e c t t o a n e x t r e m e b r id g e . L e t v0 E V

a n d l e t b E

B G )

b e a n e x t r e m e b r id g e o f G . L e t v{ b e i n c i d e n t o n b

s u c h t h a t t h e r e e x i st s a p a t h b e t w e e n v0 a n d v { i n t h e g r a p h V , E \ { b } ) .

L e t

V F b G , v o) )

b e t h e s e t o f v e r t i c e s in c i d e n t o n t h e e d g e s i n

Fa G , vo ).

T h e g r a p h __G a r i se s f r o m G b y r e m o v i n g a ll e d g e s

Fb G , vo )

a n d v e r t i c e s

V F b G , v o ) )\ { v ~ } .

L e t

IFb G, vo) l = m ,

t h en t h e g r ap h G* a r i s e s f r om __G

b y c o n n e c t i n g a c i r c u i t o f l e n g t h

m

t o t h e v e r t e x v~ . T h e g r a p h G * is c a l le d

t h e

c o n d e n s e d g r a p h

o f G w i t h r e s p e c t to t h e e x t r e m e b r id g e b. N o t e t h a t

G * is a l so a w e a k l y E u l e r g r a p h . M o r e o v e r , t h e n u m b e r o f e d g e s i n G a n d

G * c o i n c i d e .

E x a m p l e 4 . 4 . C o n s i d er t h e g ra p h G in E x a m p l e 4.2 . F i g u r e 7 s h ow s t h e

g r a p h G * t h a t a r is e s f r o m G b y c o n d e n s a t i o n w i t h r e s p e c t t o th e e x t r e m e

br i dge ba .

T h e c o n d e n s e d C P p r o b l e m o f

P = N , G , v o, t , g ) E W E N

w i t h r e s p e c t

t o t h e e x t r e m e b r i d g e b E

B G )

is

P b = N , G * , v o , t * , g * ) ,

w h e r e G * =

V * , E * ) is t h e c o n d e n s e d g r a p h o f G w i t h r e s p e c t to b , g * : E * + N is

a b i j e c t i o n s u c h t h a t

g * e ) = g e )

for a l l e E

E \ F b G , v o )

a n d t h e c o s t

f u nc t i o n t * : E* ~ IlK_ i s de f i ned b y

S t e )

i f e E

E \ F b G , v o ) ,

t * e )

0 o t h e r w i s e .

L e t

P = N , G , v o , g , t ) E W E N .

C o n s i d e r t h e f o l l o w i n g t h r e e p r o p e r t i e s f o r

a d i v i s i on r u l e f :

W E N --+

R N :


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V

v 4 v 3

V

V5 V V5

F i g u r e 7 :

The condensed

g r a p h G * f r o m

G wi th respec t to

b

E f f ic i e n cy : ~ i e N

k P ) = ~ b e ,~ c :)

t (b) .

B r i d g e c lu s t e r s y m m e t r y :

L e t B G ) = { b l , . . . , b q}, t h e n f v ~ , ) P ) =

f ~ ( ~ . ,) ( r) f o r a n < , ~ 2 e c j a , v o ) , j e { 0 , . . . , q } .

C o n d e n s a t i o n p r o p e r t y : L e t b b e a n e x t r e m e b r i d g e o f G a n d l e t P b =

( N , G * , v o , g * , t *)

b e t h e c o n d e n s e d p r o b l e m w i t h r e s p e c t t o b, t h e n

f ~ ( ~ )( r ) = . ~ ( r b ) f o r a n ~ e E \ u d a , v 0 ).

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

s a m e s e t o f b r i d g e s t o b e c o n n e c t e d t o t h e p o s t o f fi ce w i ll c o n t r i b u t e t h e

s a m e s h a r e in t h e n o n s e p a r a b l e c o s ts . T i l e c o n d e n s a t i o n p r o p e r t y i s a k i n d

o f c o n s i s t e n c y p r o p e r t y . A l l p l a y e r s w h o a r e n o t i n t h e b r i d g e c l u s t e r c o r -

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

g r a p h a s i n t h e o r i g i n a l g r a p h . N o w , a r u l e is c a l le d c o n s i s t e n t i f i n b o t h

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

T h e o r e m 4 .5 . The a l l oca t ion ru l e 7 : W E N --+ R N i s t he un i que ru l e t ha t

s a t i sf i e s e f fi c ie n c y , b r id g e c lu st er s y m m e t r y a n d t h e c o n d e n s a t i o n p r o p e r t y .

W h e r e a s in t h e C h i n e se p o s t m a n p r o b l e m e a ch e d g e i n t h e g r a p h h a s t o b e

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


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Operations Research Games: A Survey 163

t o u r t h a t v i s i t s a ll t h e v e r t i c e s i n t h e g r a p h e x a c t l y o n c e . F o r e x a m p l e , a

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

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

t o h is h o m e u n i v e r s i ty . T h e p r o b l e m i s t o s e l e c t a r o u t e i n w h i c h t o t a l t r a v e l

c o s t s a re ln i n i m i s e d . I t i s w e l l k n o w n t h a t f i n d i n g s u c h a r o u t e i s a n N P - h a r d

p r o b l e m . N e v e r t h e l e s s , m a n y r e a l liI> p r o b l e m s a r e r e l a t e d t o t h e t r a v e l li n g

s a l e s m a n p r o b l e m . T h i s h a s r e s u lt e d i n m a n y h e u r i s t ic a p p r o a c h e s t o f in d

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

t r a v e l l in g s a l e s m a n p r o b l e m w e r e f e r t o L a w l e r e t a l. 1 9 8 5 ) .

F i s h b u r n a n d P o l la c k 1 9 83 ) i n t r o d u c e t h e c o s t a l l o c a ti o n p r o b l e m t h a t

a r is e s i f i n t h e u n d e r l y i n g g r a p h e a c h v e rt e x , e x c e p t t h e o n e t h a t c o r r e s p o n d s

t o t h e h o m e l o c a t i o n , c o r r e s p o n d s t o a d i ll > re n t p l a y e r . T h e c o s t a l lo c a t i o n

i s c o n c e r n e d w i t h a f a ir a l l o c a t i o n o f t h e c o s t o f a c h e a p e s t H a l n i l t o n i a n

c i rc u i t in t h e g r a p h . T h a t is t h e c h e a p e s t t o u r t h a t s t a r t s i n t h e v e r t e x t h a t

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

r e t u r n s h o m e .

F o r m a l l y , a

t ravel l ing salesman

o r

TS problem

i s a t u p l e N , , , t ) , w h e r e

N = { 1 , . . . , n } is t h e s e t o f p l a y e rs , r e p r e s e n t s t h e h o m e l o c a t i o n a n d

t : E N * --+ IR + i s t h e c o s t t h n c t i o n a s s i g n i n g c o s t s t o t h e e d g e s c o n n e c t i n g

t h e v e r t i c e s in N * = N tO { , } . W e a s s u m e t h a t t s a t is f i e s t h e t r i a n g l e

i n e q u a l i t y .

E s

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

S , s o t h a t S , E s ) is t h e c o m p l e t e g r a p h o n S :

E s = { { i , j } l i , j E S , i j} .

B y d e f i n i n g t h e w o r t h o f a c o a l i t i o n S a s t h e m i n i m a l c o s t s o f a H a l n i l t o n i a n

c i rc u i t in t h e g r a p h S t o { * }, E s u { , } ) , w e o b t a i n t h e c o r r e s p o n d i n g

travelling

s a l e s m an o r T S gam e .

T h e f o l lo w i n g e x a m p l e , d u e t o T a m i r 1 9 8 9) , i l lu s t r a te s t h a t T S g a m e s

n e e d n o t b e b a l a n c e d .

E x a m p l e 4 . 5 . C o n s i d e r t h e T S p r o b l e m N , , , t ) w i t h p l a y er s e t N =

{ 1 , . . . , 6 } ,

t { i , j } )

= 1 fo r a ll edge s

{ i , j }

d e p i c t e d i n F i g u r e 8 a n d t b r a l l

o t h e r e d g e s { i, j } ,

t { i , j } )

e q u a l s t h e m i n i m a l c o s t s o f a p a t h c o n n e c t i n g i

t o j u s i n g t h e d e p i c t e d e d g e s .

W e d e n o t e th e c o rr e s p o n d i n g T S g a m e b y N , c ). T h e n

c N )

= 8 w i t h

op t im a l t o u r , , 4 , 5 , 6 , 1 , 2 , 3 , * ) ) , c { 1 , 2 , 4 , 5} ) = 5 , c { 3 , 4 , 5 , 6} ) = 5 a nd


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P . B o r m H . H a m e r s a n d R . H e n d r i c k x

5 1

Figure 8:

A t r a v e ll in g s a le s m a n p r o b l e m

4 { 1 , 2 , 3 , 6 } ) = 5 . L e t x ~ R N a n d s u p p o s e x c C ~ ) , t h e n

16=2 N)= Z xi+ Z xi+ E

i6{1 4,5} i6{3,4 ,5 ,6} i6{1,2,3,6}

Contradiction, so (N, c) is not balanced.

In case there are less than six players some results with respect to bal-

ancedness are established . Pot ter s et al. (1992) show th at 3-person TS

games have a no ne mp ty core. Tami r (1989) shows th at each 4-person TS

game has a nonempty core and provides Example 4.5 showing that a 6-

person TS game can have an empt y core. Finally, Kuipers (1993) proves

that 5-person TS games are balanced.

The travelling salesman model can be extended to the case in which the

costs depend on the direction in which the salesman travels through the

edges. In this contex t, Po tte rs et al. (1992) provide a 4-person TS game

with an empty core.

Pot ter s et al. (1992) also introduce the class of fixed rout ing games.

The idea of a fixed routing game is that the salesman decides about the

Hamiltonian circuit he will use to visit all the players. Then the value of a

coalition S in a fixed routing game is defined as the costs of the restricted

tour that visits the players in S in the same order as prescribed by the orig-

inal Hamil ton ian circuit and skips all othe r players. Pot ter s et al. (1992)


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

t o n i a n c i r c u i t is a n o p t i m a l r o u t e tb r t h e r e l a t e d T S p r o b l e m . D e r k s a n d

K u i p e r s 1 9 97 ) g iv e a t i m e ef f ic i en t a l g o r i t h m t h a t c a l c u l a t e s c o r e e l e m e n t s

o f f i x e d r o u t i n g g a m e s . I n K u i p e r s e t al . 2 0 0 0) a n d S o l y m o s i e t al . 1 9 9 8)

O n 4) a l g o r i t h m s a r e p r o v i d e d t h a t c a l c u l a t e t h e n u c l e o l u s o f f ix e d r o u t i n g

g a m e s .

F i n a ll y , w e m e n t i o n S t e i n e r T S g a m e s . T h e s e g a m e s a r is e f r o m s it u a -

t io n s i n w h i c h s o m e o f t h e e d g e s b e t w e e n p a i rs o f p l a y e rs m a y b e a b s e n t .

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

t h e c h e a p e s t S t e i n e r t o u r . A S t e i n e r t o u r i s a c lo s e d t r a i l t h a t s t a r t s i n t h e

h o m e l o c a t i o n a n d v i s i ts e a c h v e r t e x o f S a t l e a s t o n c e . F o r t h e s e g a m e s

H e r e r a n d P e n n 1 9 95 ), G r a n o t e t a l. 2 00 0) a n d G r a n o t a n d H a m e r s 2 00 0)

h a v e c h a r a c t e r i s e d c o n c a v i t y b y t h e s t r u c t u r e o f t h e a v a il a b le e d g e s .

S c h e d u l i n g

I n th i s s e c t i o n w e d i s cu s s c la s s es o f o p e r a t i o n s r e s e a r c h g a m e s t h a t a r e

r e l a t e d t o s c h e d u l i n g p r o b l e m s . F i r s t , w e d is c u s s v a r io u s c l a s se s o f s e q u e n c -

i n g g a m e s as i n i t i a t e d b y C u r i e l e t a l. 1 9 8 9) . W e f o c u s o n b a l a n c e d n e s s

a n d c o n v e x i t y a n d d i s c u s s t w o c o n t e x t s p e ci fi c s o l u t i o n c o n c e p t s : t h e e q u a l

g a i n s p l i t t i n g ru l e a n d t h e s p li t c o r e . S e c o n d , w e c o n s i d e r p e r m u t a t i o n

g a m e s , i n t r o d u c e d i n T i j s e t a l. 1 9 8 4 ), w h e r e w e f o c u s o n t o t a l b a l a n c e d -

n e ss . F i n a ll y , w e d i sc u s s a s s i g n m e n t g a m e s , i n t r o d u c e d i n S h a p l e y a n d

S h u b i k 1 9 71 ), w h i c h f o r m a s p e c ia l c la s s o f p e r m u t a t i o n g a m e s a n d h a v e

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

T h e m a i n c h a r a c t e r i s t ic o f a s e q u e n c i n g s i t u a t io n is t h a t a n u m b e r o f j o b s

t a s k s , o p e r a t i o n s ) h a v e t o b e p r o c e s s e d in s o m e o r d e r o n a n u m b e r o f )

m a c h i n e s ) in s u c h a w a y t h a t s o m e c o s t c r i te r i o n is m i n i m i s e d . I n s p i te o f

t h i s c o m m o n c h a r a c t e r i s ti c , s e q u e n c i n g s i tu a t i o n s c a n b e c la s si fi ed o n t h e

b a s is o f m a n y f e a t u re s . W e m e n t i o n t h e n u m b e r o f m a c h i n e s , t h e s p e c if ic

p r o p e r t i e s o f m a c h i n e s e .g . , p a r a l le l , s e ri a l) , t h e c h o s e n c o s t c r i t e r io n e .g . ,

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

j o b s e . g. , r e a d y t i m e s , d u e d a t e s ) a n d p o s s i b l y t h e s p e c if ic o r d e r in w h i c h

t h e jo b s h a v e t o b e p r o c e s s e d o n t h e m a c h i n e s e .g . , j o b - s h o p , f lo w - j o b ).

O b v i o u s l y , s e q u e n c i n g s i tu a t i o n s a r is e i n m a n y a p p l i c a t io n s : t h e p r o c e s s o f

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


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a i r p la n e s , e t c. F o r a r e v i e w o f s c h e d u l i n g t h e o r y w e r e c o m m e n d L a w l e r e t

a l . ( 1993 ) .

A s a s p e c if ic e x a m p l e w e d e s c r i b e t h e c la s s o f o n e - m a i n e s e q u e n c i n g

s i t u a t i o n s a s i n t r o d u c e d i n C u r i e l e t a l. ( 1 9 89 ) . I n a o n e - m a c h i n e s e q u e n c i n g

s i t u a t i o n t h e r e i s a q u e u e o f p l a y e r s , e a c h w i t h o n e j o b , i n f r o n t o f a m a c h i n e .

E a c h p l a y e r m u s t h a v e h is j o b p r o c e s s e d o n t h is m a c h i n e . T h e f in i te s e t o f

p l a y e r s is d e n o t e d b y N = { 1, . . . ,n } . T h e p o s i t i o n s o f t h e p l a y e r s i n t h e

q u e u e a r e d e s c r i b e d b y a b i j e c t i o n c r C [ I N . W e. a s s u m e t h a t t h e r e i s a n

i n i t i a l o r d e r a 0 C

FIN

o n t h e j o b s b e f o r e t h e p r o c e s s i n g o f t h e m a c h i n e

s t a r t s . T h e p r o c e s s i n g t i m e P i o t h e j o b o f p l a y e r i is t h e t i m e t h e m a c h i n e

t a k e s t o h a n d l e t h i s j o b . F o r e a c h p la y e r i C N t h e c o s t s o f s p e n d i n g t i m e i n

t h e s y s t e m c a n b e d e s c r i b e d b y a li n e ar c o s t f h n c t i o n

ci

: R + - + R d e f i n e d

b y c i t ) = c~i t w i t h a i > 0 . A s e q u e n c i n g s i t u a t i o n a s d e s c r i b e d a b o v e i s

d e n o t e d b y

N , c r o , p , a )

w i t h p , a C R ~ + .

T h e c o m p l e t i o n t i m e C ( a , i ) o f t h e j o b o f p l a y e r i if t h e j o b s a r e p r o c e s s e d

( in a s e m i - a c t i v e w a y ) a c c o r d i n g t o t h e o r d e r a E F IN i s g i v e n b y

= Z PJ

{ j E N [ 7 j ) < _ ~ ) }

A p r o c e s s i n g o r d e r i s c a l le d s e m i - a c t i v e i f t h e r e d o e s n o t e x i s t a j o b w h i c h

c o u l d b e p r o c e s s e d e a r l i er w i t h o u t a l t e r i n g t h e p r o c e s s i n g o r d e r , i .e ., i f t h e r e

a r e n o u n n e c e s s a r y d e l a y s . T h e t o t a l c o s t s o f a ll p l a y e r s i f t h e j o b s a r e

p r o c e s s e d a c c o r d i n g t o t h e o r d e r a e q u a l Y '~ i~ N

t r i G a , i ) .

C l e a r l y , b e c a u s e

I I N i s f i ni te , t h e r e e x i s t s a n o r d e r f o r w h i c h t o t a l c o s t s a r e m i n i m i s e d . A

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

s a v in g s i s a n o r d e r i n w h i ch t h e j o b s a r e p r o c e s s e d i n d e c r e a s i n g o r d e r w i t h

r e s p e c t t o t h e u r g e n c y in d e x u i d e f i n e d b y u i = ~ ( cf . S m i t h ( 1 9 5 6 ) ).

P i

E x a m p l e 5 . 1 . C o n s i d e r a o n e - m a c h i n e s e q u e n c i n g s i t u a t i o n ( N , c r0 ,p , c~),

w h e r e N = { 1 , 2 , 3 } , = 1 , 2 , 3 ) , p = 2 , 2 , 1 ) a n d = 4 , 6 , 5 ) . T h e n t h e

u r g e n c i e s f o r t h e p l a y e r s a r e u l = 2 , u 2 = 3 a n d u 3 = 5 , r e s p e c t i v e l y . H e n c e ,

t h e o p t i m a l p r o c e s s i n g o r d e r is (3 , 2 , 1 ) w i t h t o t a l c o s t s 5 . 1 + 6 . 3 + 4 . 5 = 4 3 .

N o t e t h a t a n o p t i m a l o r d e r c a n b e o b t a i n e d f r o m t h e i n i ti a l o rd e r b y

c o n s e c u t i v e s w i t c h e s o f n e i g h b o u r s i , j w i t h i d i r e c t l y i n fr o n t o f j a n d

rui < uj.

T h i s p r o c e s s w i ll b e r e f e r re d t o a s t h e S m i t h a l g o r it h m .

B y r e a r r a n g i n g f r o m t h e i n it ia l o r d e r t o a n o p t i m a l o r d e r , a n a l l o c a t i o n

p r o b l e m a ri se s : h o w s h o u l d t h e m a x i m a l t o t a l c o s t s a v in g s t h e p l a y e rs c a n


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Operations Research Games: A Su~vey 167

o b t a i n b e d i v i d e d a m o n g t h e p la s, e r s ? A g a i n , t h i s p r o b l e m is ta c k l e d u s i n g

c o o p e r a t i v e g a m e t h e o r y b y a n a l y s in g c o r r e sp o n d i n g s e q u e n c i n g g a m e s .

F o r a se q u e n c i n g si t u a t io n ( N , a 0 , p , a ) t h e c o s ts C s ( a ) o f c o a l i t i o n S

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

C s ( a ) = ~ i ~ s c ~ iC ( a, i) . W e

w a n t t o d e t e r m i n e t h e m a x i m a l c o s t sa v i n g s o f a c o a li t i o n S w h e n i t s m e m -

b e r s d e c i d e t o c o o p e r a t e . F o r th i s , w e h a v e t o d e f i n e w h i c h r e a r r a n g e m e n t s

o f t h e c o a l i t i o n S a r e a d m i s s i b le w i t h r e s p e c t t o t h e i n it i a l o r d e r . A b i j e c t i o n

cr E I IN i s ca l l ed admissib le f o r S i f i t s a t is f i e s t h e f o l l o w i n g c o n d i t i o n :

P ( a , j ) = P ( a 0 , j )

for a l l j E N \ S , w h e r e f o r a n y T E

N

t h e s e t o f p r e d e c e s s o r s o f a p l a y e r

j E N w i t h r e s p e c t t o ~- i s d e f i n e d a s P ( T , j ) = { k E N I T ( k ) < T ( j ) } .

T h i s c o n d i t i o n i m p l ie s , in p a r t i c u l a r , t h a t t h e s t a r t i n g t i m e o f e a c h

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

o r d e r a n d t h e p l a y e r s o f S a r e n o t a l lo w e d to j u m p o v e r p l a y e r s o u t s i d e

S . T h e s e t o f a d m i s s ib l e o r d e r s f o r a c o a l i ti o n S is d e n o t e d b y A ( S ) .

B y d e f i n i n g t h e v a l u e o f a c o a li t io n S a s t h e m a x i m u m c o s t s a v in g s

c o a l it io n S c a n a c h i ev e b y m e a n s o f a n a d m i s si b le r e a r r a n g e m e n t w e o b t a i n

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

s eq u enc i ng g a m e ( N , v ) ,

w h i c h i s d e f i n e d b y

v s ) = m a x i ) - 5 . 1 )

f o r a l l S C N.

E x p r e s s i o n (5 .1 ) c a n b e r e w r i t t e n i n t e r m s o f

g.ij

= m a x { 0 ,

o~jp

- - o z i P j }

w h i c h e q u M s t h e c o s t s a ~ in g s a t t a i n a b l e b y p l a ~ e r i a n d j w h e n i is d i r e c t l y

i n fr o n t o f j , r e g a r d l e s s o f t h e e x a c t p o s i t i o n i n t h e o r d e r . F o r t h i s w e n e e d

t h e n o t i o n o f c o n n e c t e d c o a li t io n . A c o a l it i o n S is c a l le d connec ted w i t h

r e s p ec t t o cr i f f o r a l l i , j E S an d k C N s uch t h a t o r( i) < o r (k ) < o r ( j )

i t h o l d s t h a t k E S . T h e S m i t h a l g o r i t h m a n d ( 5. 1) i m p l y t h e fo l l o w i n g

p r o p o s i t i o n .

P r o p o s i t i o n 5 . 1 . Le t ( N , ~ro,p, c~) be a sequencing s i tuat ion and let ( N ,v )

be the corresponding sequenc ing gam e. Th en ] or any coali tion S that i s

connected wi th respect to ao we have

v s ) = ) _ 2

i,j~S:~o(i)


30/61

168

P . B o r m , H . H a m e r s a n d R . H e n d r i ck x

F o r a c o a l i ti o n T t h a t i s n o t c o n n e c t e d w i t h r e s p e c t t o or0 t h e d e f i n i ti o n

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

v T ) =

v S) ,

S T\~o

w h e r e T \ c r 0 i s t h e s e t o f c o m p o n e n l , s o f T , a c o m p o n e n t o f T b e i n g a m a x -

i m a l l y c o n n e c t e d s u b s e t o f T .

E x a m p l e 5 . 2 . L e t N = { 1 , 2 , 3 } , a 0 = 1 , 2 , 3 ) , p = 2 , 2 , 1 ) a n d c~ =

4 , 6 , 5 ) . I t i s r e a d i l y ve r i f i ed t ha t g12 = gu3 = 4 an d g13 = 6 T h e n v { i } ) =

0 f o r al l i N , v { 1 , 2 } ) - - v { 2 , 3 } ) - - 4 , v { 1 , 3 } ) - - v { 1 } ) + v { 3 } ) - - 0

a n d v N ) = 14.

T h e fo l l o w i n g t h e o r e m , d u e t o C u r i e l et a l. 1 9 8 9 ) , s h o w s t h a t s e q u e n c i n g

g a m e s a r e c o n v e x g a m e s .

T h e o r e m 5 . 1 .

L e t N , c ro , c ~ ,p ) b e a s e q u e n c i n g s i t u a t i o n . T h e n t h e c o r r e -

s p o n d i n g s e q u e n c i n g g a m e N , v ) is c o n v e x .

I n p a r t i c u l a r, T h e o r e m 5 .1 i m p l ie s t h a t s e q u e n c i n g g a m e s a r e t o t a l ly )

b a l a n c e d .

A n o t h e r w a y o f p r o v i n g b a l a n c e d n e s s o f s e q u e n c i n g g a m e s is b y e x p li c-

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

s p l i t t i n g r u l e , i n t r o d u c e d i n C u r i e l e t a l . 1 9 8 9 ) , a n d t h e s p l i t c o r e , i n t r o -

d u c e d in H a m e r s e t al . 1 9 9 6 ) , a r e r u l e s t h a t y i e l d a l l o c a t i o n s t h a t a r e i n

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

R e c a l l t h a t t h e s e t o f p r e d e c e s s o r s o f p l a y e r i E N w i t h r e s p e c t t o t h e

p r o c e s s i n g o r d e r cr i s g i v e n b y

P cr , i )

= { j E N l a j ) < a i ) } . W e d e fi n e

t h e s e t o f t b l lo w e r s o f i E N w i t h r e s p e c t t o cr t o b e F a , i ) = { j E N I

o r j ) > o r

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