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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 .