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31

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

a l l o c a t i o n p r o b l e m s

R e n e V i c t o r V a lq u i V I D A L

Ins t t t u t e o f Mathemat i cal S ta t t s tws and Operat ions Research .

Techm cal Unn er st tv o f Denma rk L .vngt~r Denma rk

Received December 1982

Revised April 1983

This paper discusses the classical resource allocation prob-

lem. By very elementary arguments on Lagrangian duality it is

shown that this problem can be reduced to a single one-dimen-

sional maximization of a differentiable concave function.

Moreover, a simple graphical method is developed and applied

to a family of well-known problems from the literature.

pie i s the case

r , ( x , ) = p , ( 1 - e - ~ ' ) , p , , k , > O

t ha t w i l l be t h e or e t i c a l l y d i sc usse d i n S e c t i on 2 .

T h e r e a f t e r , i n S e c t io n 3 , a s i m p l e g r a p h i c a l m e t h o d

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

e xa m ple . I n Se c t i on 4 , e xa m ple s f r om the l i t e r a -

t u r e t h a t h a v e b e e n s o l v e d b y u n n e c e s s a r y

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

l a r p r ope r t i e s t o t he c a se d i sc usse d i n Se c t i on 2 .

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

Sec t ion 5.

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

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

b u d g e t i n g , s t r a t i f i e d s a m p l i n g , m a r k e t i n g a n d

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

f o r m u l a t e d :

N

F ( X ) = m ax ~ r , ( x , ) , (1)

sub j e c t t o

N

E x , = x . 2 1

t = l

a , ~ x , < < , b , , i = 1 . . . . . N , (3)

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

e f f o r t a l lo c a t e d t o t h e i t h a c t i v i ty a n d t h e r e t u rn s

r , ( x , ) a r e d i f f e r e n t i a b l e , s t r i c t l y c onc a ve a nd i n -

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

G e n e r a l a l g o r i t h m s t o s o l v e p r o b l e m ( 1 ) - ( 3 )

ha ve be e n de v e lope d by S r ika n t a n [ 9 ] , V ida l [ 10] .

S a n a t h a n a n [ 7 ], a n d B i t r a n a n d H a x [ 1 ].

He r e w e w i ll be de a l i ng w i th a spe c i al f a m i ly o f

p r o b l e m ( 1 ) - ( 3 ) , t h a t d u e t o t h e s p e c i a l s t r u c t u r e

o f t h e f u n c t i o n s

r , ( x , )

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

a v e r y e a s y s t r a i g h t f o r w a r d a p p r o a c h . O n e e x a m -

North-Holland

Euro pean J ourn al of Oper atio nal Research 17 (1984) 31 - 34

2 A n e x a m p l e

L e t u s f ir s t c o n s i d e r t h e f o l l o w i n g p r o b l e m :

N

F ( X ) = m a x ~ , p , (1 - e - ~ ' ~ ' ), ( 4)

t ]

s u b j e c t t o

x. 5)

O < ~ x , < ~ X

f o r i = 1 ,2 . . . . . N , ( 6 )

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

e f f o r t a l l o c at e d t o t h e i t h a c t i v i ty a n d p,, k, > O.

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

t th ac t ivi ty .

W i l k i n s o n a n d G u p t a [ 1 2 ] h a v e d e v e lo p e d a n

a l g o r i t h m t o s o l v e p r o b l e m ( 1 ) - ( 3 ) b a s e d o n d y -

n a m i c p r o g r a m m i n g . L a t e r , L u s s a n d G u p t a [ 5 ]

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

p r o b l e m b a s e d o n t h e K u h n - T u c k e r c o n d it io n s .

O u r a n a l y s i s w i ll b e b a s e d o n t h e w e l l -k n o w n

r e su l t s o f L a g r a ng ia n du a l i t y [ 11 ].

T h e L a n g r a n g i a n p r o b l e m i s

N

L ( u ) = m a x Y'~

{ p , ( 1 - e - ~ , ' , ) - u x , }

N

= E m ax { p , ( 1 - e ' , ) - u x ,

037 7-22 17/8 4/$ 3.00 '- ' 1984, Elsevier Science Publish ers B.V. (Nor th- Holl and)

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3 2 R. K V. Vidal / A graphical method to solve allocanon problems

s u b j e c t t o

O < ~ x i ~ X f o r i = 1 ,2 . . . . . N ,

wh e r e u ~ : 0 , i s s c a l a r, de no m ina t e d m u l t i p l i e r i n

E v e r e t t ' s s e n s e . T h e s u f f i c i e n t c o n d i t i o n s s a y t h a t

i f w e f i n d u * a n d

x,* u*),

t h e s o l u t i o n t o L ( u * ) , s o

t h a t

N

~ . x t u * ) = X , (7)

i I

t he n , t he x,* u*) a r e t he so lu t i on t o t he o r ig ina l

p r o b l e m ( 4 ) - ( 6 ) . M o r e o v e r , s i n c e o u r p r o b l e m

F ( X ) i s c o n c a v e s u c h a u * w i ll a lw a y s e x i st .

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

N s u b p r o b l e m s s o t h a t

L , u ) - ~ m a x { p , l

- e - * , ' ) - u x , } , ( 8)

sub j e c t t o

O ~ x , < ~ X .

T h e s u b p r o b l e m h a s a l w a y s a s o l u t i o n f o r u ~ 0

( We ie r s t r a s s ' t he or e m ) . T he op t im a l so lu t i on , x ,* i s

e i t h e r a b o u n d a r y p o i n t o r a n i n t e r i o r p o i n t . I f i t i s

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

i s d i f f e r e n t i a b l e , i t a lso ha s t o be a s t a t i on a r y po in t

s o t h a t

d r, = d { p , ( l - e - ' , ' , ) } = u

d x , d x ,

o r

l o g , ( p , k , ) l o g , ( u )

x = k , k , ( 9 )

If x,* < 0 the n reset x,* -- 0; if x,* > X the n reset

X,* = X . F igu r e 1 i l l us t r a t e s t he o p t im a l f unc t i on

x*, u). T h e n i t is su f f i c i e n t t o f i nd u* , so t h a t ( 7 ) i s

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

pr ob l e m ( 4 ) - ( 6 ) . Hor s t [ 2 ] g ive s a t he or e t i c a l d i s -

c u s s i o n o f p r o b l e m ( 1 ) - ( 3 ) a l o n g t h e a b o v e m e n -

t i o n e d L a g r a n g i a n d u a l i t y p r i n c i p l e s .

Wha t i s spe c i a l i n p r ob l e m ( 4 ) - ( 6 ) i s t he f a c t

t h a t w e a r e a b l e t o o b t a i n e x p l i c i t l y x*, u), give n

b y ( 9 ) . M o r e o v e r , i n t r o d u c i n g t h e o n e - t o - o n e

t r a n s f o r m a t i o n

0 = logo u ) , 1 0 )

( 9 ) be c om e s

x , * = A , + B , a , (11)

w h e r e

log , (

pik i )

A , =

k

( 1 2 )

I

s , = ( 1 3 )

I t i s obv ious t ha t t he g r a ph o f x ,* ( 0 ) i s e a sy t o

d r a w , d u e t o t h e l i n e a r f u n c t i o n ( I I ) . F u r t h e r m o r e ,

( 7 ) be c om e s

N

Ex* O)=X

(14)

t ]

a nd the f un c t i on ( 9 (X) ob t a ine d f r o m ( 14) i s p i e c e -

w i se l i ne a r . T he g r a ph o f ( 9 ( X) i s f ound by i de n t i -

f y i n g t h e p o i n t s a t w h i c h 9 X) c ha nge s i t s s l ope .

O n c e w e h a v e f o u n d 0 ( X ) , g i v e n X w e c a n f i n d

0 , t ha t r e p l a c e d i n t he g r a ph o f x,* 9) wil l give

t h e o p t i m a l v a l u e s o f x , * . T h i s s t r a i g h t f o r w a r d

a ppr oa c h i s i l l us t r a t e d i n t he ne x t s e c t i on by so lv -

i n g a n u m e r i c a l ex a m p l e .

Px k

F i g . I . T h e f u n c t i o n

x,* u).

X ~

3 A s im p l e g r a p h i c a l a p p r o a c h

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

T a b le 1 [ 12] . Pa r a m e te r s A , (12) a n d B , ( 13) a r e

g ive n i n T a b le 2 .

F i g u r e 2 s h o w s t h e g r a p h f o r f u n c t i o n s x,* O) as

g i v e n b y ( 1 1 ) a n d t h e c o n d i t i o n x , * ~ [ 0 , X ] . T h e s e

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

p a p e r .

We ha ve se e n t ha t t he g r a ph f o r t he p i e c e wi se

l i n e a r f u n c t i o n 0 ( X ) , t h a t s p e c i fi e s t h e o p t i m a l 0 *

f o r a g iv e n v a l u e o f X , c a n b e d r a w n b y i d e n t i f y i n g

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R . V . V . V i d a l A g r a p h i c a l m e t h o d to s o l v e a l l oc a t io n p r o b l e m s 33

Table 1

A num erica l exa mple; X = l0 b. N = 4

Activity

2 3 4

p, 3-106 4 .106 2.106 106

k, 3-10 -~ 2.10 -6 10 -6 10 -6

p k 9 8 2 1

Table 2

A and B parameters

Activity

2 3 4

A, 0,73.106 1,04.106 0,69.106 0

- B, 0,33.106 0,5 .106 106 100

.-. 1

- 1

t h e p o in t s 0 , w h e r e t h e r e i s a c h a n g e o f s l o p e .

T h e s e a r e t h e p o i n t s

( 0 ~ ,0 2 . . . . .

0s) shown in F ig . 2 ,

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

T a b l e 2 a n d a r e g i v e n i n T a b l e 3 t o g e t h e r w i t h t h e

c o r r e s p o n d i n g v a l u es o f X .

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

O ( X ) ,

t h i s i s sh o w n in F ig . 3 . T h a t i s , g iv e n a v a lu e o f X ,

f r o m F i g . 3 w e c a n o b t a i n t h e v a l u e o f

0 ,

a n d ,

w i t h t h i s v a l u e , f r o m F i g . 2 w e c a n o b t a i n t h e

va lues of x ,* . Thus for X = 106 , we ge t 0* = 0 .93 ,

a n d t h e o p t i m a l s o l u t i o n w i l l b e

x ~ = 4 . 2 4 . 1 0 5 , x ~ = 5 . 7 6 - 1 0 5 , x ~ = x ~ = 0 .

Fig. 3. Graph of O X).

Table 3

The function 0 X)

~ X , / l O ~

1 2.21 0

2 2.08 0.0436

3 0.69 1.1973

4 0.08 2.3136

5 0 2.42

6 - 0.31 3.142

7 -0.818 3.818

8 -1 4

: < I 0 ; :

2

l

0

- 1

x l i

Fig. 2. Graph of x,* 0).

N o t i c e t h a t w e a r e a b l e t o g e n e r a t e e a s i l y t h e

o p t i m a l s o l u t i o n f o r a l l p e r m i s s i b l e v a l u e s o f X ,

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

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

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

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

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

t h e p r o b l e m t h a t a r e e a s y to u n d e r s t a n d a n d t h e re -

f o r e e as y t o i m p l e m e n t .

4 . A f a m i l y o f p r o b l e m s

T h e a p p r o a c h d e s c r i b e d i n S e c t i o n 3 t o s o l v e

p r o b l e m ( 4 ) - ( 5 ) c a n b e a p p l i e d t o o t h e r p r o b l e m s

o f t h e t y p e ( 1 ) - ( 3 ) a s f a r a s w e a r e a b l e t o f i n d

s i m i l a r s u i t a b l e t r a n s f o r m a t i o n s a s ( 1 0 ) a n d 1 1 ) .

T a b l e 4 g i v e s o t h e r e x a m p l e s f r o m t h e l i t e r a t u r e

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

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3 4

T a b l e 4

O t h e r e x a m p l e s

R . V . V . V i d a l / A g r a p h i c a l m e t h o d t o s o l v e al l o c at i o n p r o b l e m s

r , x , ) 0 A, B, Refere nce

p, 1 - k , / x , ) _ _ 1 0 v / p ~ , S r i k a n t a n [ 9]

p , . k , > 0 ~[ff

- p , x , 1 Klein {4

f xl/k I

, > O , k > 2 u I / * - 1 0 - p , x )

p , / a , - x , ) _ _ 1 a , - V ~ , Jense n [31

p , , a , > 0 Ilrff

p, logc 1 + m , x i ) L u s s a n d

p , . m , > 0 1 1 p , G u p t a [ 5 ]

m~

p , x , / a , + x , ) __1 - m, ~ Lue nbe rger 161

p , . a , > 0

2 p , ~ , S i v a zl ia n a n d

p > 0 1 0

p2 Stanfe l [ 81

U 2

5 Co n c l u s i o n s

B y ve ry e l e me nt a ry a rgume nt s on La gra ng i a n

dual i ty i t i s show n tha t a fami ly of the c lass ica l

re sourc e a l loc a t i on p rob l e m t ha t ha s be e n so l ve d

i n t he l i t e ra t u re by unne c e ssa ry c umbe rsome a l go-

r i thms posse sse s som e ge ome t r i c a l p rope r t i e s t ha t

pe rmi t s t he de ve l opme nt o f a ve ry s i mpl e

s t ra i gh t fo rwa rd a pproa c h . The s i mpl i c i t y o f t he

me t hod ha s be e n shown by so l v i ng a we l l -known

e xa mpl e f rom t he l i t e ra t u re . More ove r . we ha ve

be e n a b l e t o i de n t i fy se ve ra l e xa mpl e s f rom t he

a va i l a b l e li t e ra t u re wh e re a s i mi la r a ppro a c h c ou l d

e a s il y be de ve l ope d .

Referen ces

[ 1] G . R . B i t r a n a n d A . C . H a x , O n t h e s o l u t io n o f c o n v e x

k n a p s a c k p r o b l e m s w i t h b o u n d e d v a r ia b l e s, Pr o c . 9 th I n -

t e rn a ti o na l M a t h e m a t i c a l P r o g r a m m i n g S y m p o s i u m ,

B u d a p e s t 1 9 7 6 ) 3 5 7 - 3 6 7 .

[ 2] R . H o r s t , O n r e d u c i n g a r e s o u r c e a ll o c a t i o n p r o b l e m t o a

s i n g l e o n e - d i m e n s i o n a l m i n i m i z a t i o n o f a d i f f e r e n t i a b l e

c o n v e x f u n c t i o n , O p e r a t i o n a l R e s . 3 2 1981 ) 82 1-82 4.

[3] A. J ensen ,

A D i s t r i b u t i o n M o d e l A p p l i c a b l e t o E c o n o m i c s

M u n k s g a a r d , C o p e n h a g e n , 1 9 5 4 ) .

[4 1 M . K l e i n , S o m e p r o d u c t i o n p l a n n i n g p ro b l e m s , N a v a l R e s .

L o g i s t . Qu a r t . 4 1 9 5 7 ) 2 6 9 - 2 8 6 .

[ 5] H . L u s s a n d S .K . G u p t a , A l l o c a t i o n o f e f f o rt r e s o u r c e s

a m o n g c o m p e t i n g a c t i v i t i e s , O p e r a n o n s R e s . 2 8 1975)

3 6 0 - 3 6 6 .

[ 6 ] D . G . L u e n b e r g e r , O p t i m i z a t i o n b y V e c to r S p a c e M e t h o d s

Wiley, New York , 1969) .

[ 7] L . S a n a t h a n a n , O n a n a l l o c a t i o n p r o b l e m w i t h m u l t i s ta g e

c o n s t r a i n t s ,

O p e r a t i o n s R e s . 1 9

1 9 7 1 ) 1 6 4 7 - 1 6 6 3 .

[8] B .D. Siva zl i an an d L.E. S t anfe l ,

O p n m t z a t i o n T e c h n i q u e s

i n O p e r a t i o n s R e s e a r c h

Prent i ce-Hal l . Englewood Cl i f f s .

N J , 1975) .

[ 9] K . S. S r i k a n t a n , A p r o b l e m i n o p t i m u m a l l o c a t i o n . O p e r a -

t io n s Res . 1 1 1 9 6 3 ) 2 6 5 - 2 7 3 .

[ 10 ] R . V .V . V i d a l , O p e r a t i o n s r e s e a r c h i n p r o d u c t i o n p l a n n i n g ,

P h . D . t h e s i s, I M S O R , T h e T e c h n i c a l U n i v e r s i t y o f D e n -

mark 1970}.

[ 11 ] R . V .V . V i d a l , N o t e s o n s t a t i c a n d d y n a m i c o p t i m i z a t i o n ,

I M S O R , T e c h n i c a l U n i v e r s i ty o f D e n m a r k 1 9 7 6) .

[ 12 ] C . W i l k i n s o n a n d S .K . G u p t a , A l l o c a t i n g p r o m o t i o n a l

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

a p p r o a c h , P r o c . 5 t h I n t e r n a t i o n a l C o n f e r e n c e o n O p e r a -

t io n a l Res ea r ch , V e n i c e 1 9 6 9 ) 4 1 9 - 4 3 2 .