t h e o r e t ic a l p a p e r s
OPTIMIZATION OF BICRITERION QUASI-CONCAVE
FUNCTION SUBJECT TO L IN EA R CONSTRAINTS*
V. Rajendra Prasad
Indian Sta tistica l In stitu te
R.V. College Post
Bangalore 560 059
Y.P. Aneja
Faculty o f Business Administration
University o f W indsor
i) 'indsor. Ontario, C anada N 9B 3P4
and
K. P. K. N air
Faculty o f A dm inistration
University o f N ew Brunswick
Fredericton. New Brunsw ick. Canada B3B 5A3
A
bstra ct
In this paper we provide algorithms for maximisation and minimisation of
bicriterion quasi-concave function g
c^x) subjcct to linear constraints.
The algorithm for maximisation is based on bisection approach. The
algorithm for minimisation is an implicit enumeration method. W ith some
minor modifications, this algorithm also enumerates all efficient solutions of
bicriterion linear programs. M aximisation o f system reliability o f seriesparallel and parallel-series systems (with two subsystems) through optimal
assignment of components is treated as a special case.
1. Introduction
«
Suppose g (zu z 2) is a single valu e d function defined o n R %which
Satisfies the p r o p e r t ie s : (/) g (zu z 2) is quasi-concave a n d (//) g (~i, z.,)
"'creases with each argument. T h e p ro b lem s, we study in this paper, are
Maximisation an d minimisation o f g (cx x , c., x) subject to A x — b, x > 0
"here
and c.2 are 1 x n vectors, A is an m x n matrix a n d b an m x \
“This work is supported by the N a tu ra l S cien ces and E ngineering R esearch C ouncil
’r Canada, and is carried out during th e first au th or’s stay at U n iv ersity o f N ew
Brunswick, Canada.
vector. P ra c tic a l justification a n d several examples of maximisation
p ro b le m can be f o u n d in Gcofl'rion [ 6 ], Special cases of these two
problem s c a n be f o u n d in A n an d [1], A neja et al. [3], Bector and Dahl [5]
and S war up [1 0 ,1 1 ] , In all th e s e special cases, g (c, x, c2 x) is of the
form (cl x -|- «i) (c 2 A'+flo).
In Section 2 , we consider th e p roblem o f maximising g (c, .v, c2x)
subject t o .4.v = b, x > 0 . It is assum ed that g (z,, r 2) is continuously
differentiable on R 2 and stric tly increases with each argument. Wefiht
obtain som e p re lim in a ry results' a n d later develop an algorithm based on
these results. G eoffrion [ 6 ] co n sid e re d a more general maximisation
p ro b le m in w hich c1 x and c 2 x are replaced by two concave functions
o f x a n d gave algorithm s s e p a ra te ly for the general problem and the
p rob lem u n d e r consideration. T hese alg orithm s are based on para­
metric p ro g ra m m in g technique. W e arg ue th a t o u r algorithm is m ore
efficient t h a n a lg o rith m 2 o f G oeffrion [ 6 ], which was developed to solve
the p rob lem o f this section.
Fin ally we a p p ly the a lg o r ith m to a p r o b le m o f assigning components
optimally t o a series-parallel re lia b ility system so as to maximise the
system reliability.
In Section 3, we consider the m in im isa tio n p ro b le m . Here also, w
derive some p relim inary results a n d develop a n a lg o rith m to solve
the p r o b le m o n th e basis o f these results. If th e m atrix A is totally
u n im o d u la r a n d b an integral vector, the a lg o rith m yields an optimal
so lu tio n even w hen .v is restricted to be integral vector. Finally we apply
the a lg o rith m t o maximise the re lia b ility o f parallel-series reliability
system by o p tim a lly allo c a tin g t h e com ponents.
2.
M aximisation o f Bicriterion Quasi-Concave Function
In th is section, we consider th e p r o b le m :
Maximise
g (c1 x , c2 x),
subject to
,v € K =
{a :
A x = b, x > 0 }.
(Pi)
This is e q uiva le nt to the p ro b le m :
Maximise .
g (z u z 2),
subject to
(z l5 z2) e Z = {(zu z s) : (zlt z 3) =
(Cl x , e 2 x ) f o r sow*
a € K }.
(P;i
We assume th a t g (zv z,,) is c o n tin u o u sly differentiable o n R \ a n d strictly
increases w ith z^ a n d z2. In o u r n o ta tio n , (zx, z 2) ^
m e a n s zL ^
-2 > ) t and (z„ z.) ^
(.f„ r..,). A p o in t (?,, z.,) in Z is said to be efficient
if There doc.', not exist an o th e r p o in t ( i v vL>) in Z such t h a t (_yj, v ^ X z ^ r a).
The optimal solution ( ; f . ~f) o f p r o b le m 1% is necessarily an efficient
point due to property (//) of cr (,]. z.).
So. there exists a positive number
A* such that (;* ,:? ) maximises z, -• A* z, on Z.
O u r a p p r o a c h is to first
find A* and next ( rf, r?) and an v* € K such th at (rf, :* ) = {cl x*. c., .x*).
.Note rliat this ,v* is an o p j n u l s o l u t io n o f p r o b le m P j.
We shall tir^t present some p r e lim in a r y results a n d usin g these results
gives A*, (:*, -? ) an d ,v*.
vie develop an algorithm t h a t
T h e a lg o n tln n
starts with interval [ 0 , ^ ) -• nd p a r titio n s , in each ite ra tio n , a n interval
containing A* (selected in the p r e v io u s ite ra tio n ) into tw o subintcrvals. a n d
selects the one containing A * .
It yields A*. ( r f , r? ) a n d a *
in the final
iteration.
Preliminary Results
For the purpose of convenience, a p o in t (z^ r^) in Z is denoted by z
and g (-i. -•) bv g (;). Let us d e n o te by L (A ) the p r o b le m o f maximising
: r , A:, on Z. T h ro u g h o u t th is s e c t io n wc consider th e p ro b le m L (A) for
X > 0 only.
Let us d e n o te a n o ptim a l s o lu t io n o f L (A) by
:fA) = (z, (A), z, (A)j.
L emma
1. a , < A , and z ( \ ) ± : (A,) - - r , (A,) > z , (A,) a n d z , (A ,)< z a (A,).
Proof. Wc have r , (Al} ^ ^ (A.) a n d z, (X,) ^ z,. (A,) since z (A,) a n d
- (Aa) are optim al s olutio ns o f L (^j) a n d L (Aj) a n d z (Ai) 7^ - (Aa).
Suppose n (A,) < z, (a ,). T his im plies za (A i)> r , (A,). W e can write
z . U . H A,
<-M > z, (A*)-!-*, zs (A^,
ic.. {r4 (A,) - z , <A1>r{za <A1) - z 2 <A ,)]< K < As,
i.e.. z, (A2)-!-Ao
z2
(A,) < z, (A,) l-Ao r , (A^,
which contradicts
the
o p tim ality
of
z (A2)
for
L (A»).
Therefoie
rj (A1) > z 1 ( a 2) an d co n se q u e n tly z., (Ai) < z ? (A2).
Lf.MM.v 2. L e t > .! < A ,.
A =,
Then A,
z
( Aj )
and
-Z iC ^ l/tz sC A s)-^ * !)]
(1)
Further i f
z, (A) + l z 2 (A) >
+
A
(Al)’
(2
(/)
<C A < Aa
(//) g (z ( A 0 ) > £ ( z (A)) => «• ( r (A)) > £ (■-) i f - € Z . z, < z, (A ) ami
/\
-s
_
> z%(A)
—
__
/A
(f/i) g (z (A,)) > g ( r (A)) -> g (z (A)) >
A
A
■*
<? ( r ) i f z € Z.
—
r, >
( A) <■»,/
_
r 2 < z 2 (A).
P roof. W e have A > 0 b y L e m m a 1. S u p p o s e A < Aj. Then
U i (Ai)“~-i (Ag)]/
(A;>)—z 2 (Ai)] < Ai, i.e., - i (A i)+ A j z., (A/) < z, (As)-fAx z 2 (Aa), which c o n tr a d i c ts th e o p tim a lity o f r (A,) f o r L (At ). Thus
A > A i a n d sim ilarly A
A2.
(i) In e q u a lity (2) m e a n s t h a t z (A) d o e s n o t lie o n the line passing
th r o u g h z (Ai) a n d z (A2). U sing (1) a n d (2). one can easily "ee
th a t At < A < A 2.
( n ) Assume th a t g (z (A]))
A
A
(z (A)) a n d (2) h o ld s .
A
__
C on sid e r a point
__
z € Z such th a t z x < z x (A) a n d z 2 > z 2 (A).
A
_
Since z t < Zi (Aj) < z 2 (Aj), we c an find a, 0 < a < l , such th a t zj (A) =
A
<■"-»
a z ! + ( l —a) Zi (Ai).
A
N o w con sid e r th e p o i n t z = a z - j - ( l —a) z (A!). It is
A lso za < z 2 (A) fo r otherwise opti-
ob v io u s th a t z ( Z a n d zx = zx (A).
_
rv
m a lity o f z (A) fo r L (X) w o u ld be v io la te d .
-
-
A
_
_
_
S u p p o se z 2 — r 2 (A).
A
Then
_
zi (A )+ Az2 (A) = a [ z j + A Z2]
+ ( l
—a) [z^Aj) -|- Az2 (Aj)]
< a [zi~l- A z 2] - | - ( l •—a) [zi ( a ) + A : 2 (Aj]
A
__ A
__
_
__
d u e to (2), i.e., z x+ A z 2 >Z!(A)-f- X z 2 (A) w hich c on trad icts th e opti_
___
r-J
___
m ality o f z (A) fo r L { A). T h e r e f o r e z 2< z 2 (a). N o w , by p rop erties (/') and
(/i) o f g{z), we can w rite
g ( z ( \ ) ) > g ( z ) > m in {g ( z ( \ t)), g (z)},
A
__
w hich implies g(z(A))> g (z) since g(z(Ai))>g(z(A)). (ii) also can be proved
nn the same lines.
L em m a 3 .
Let
h 'v X l i > : i
( A J ^ z (Aa) , A =
[ ^ ( A , ) - ^ (A2) ] / [ z a (X2) — ^ ( A ^ ]
f o r cm o p tim a l solution Z o f L ( \ ) .
( A j ) - M z 2 (A ,)
Then
(') £ ( - ( ai)) > g (Z ) => £(2) > g ( z ( \ ) ) f o r A > \
(ii) g(r(A2)) > g(?) ---> g (z ) > g ( z ( \ ) ) f o r A < A
Proof follo w s fro m Lem m as 1 a n d 2.
L em m a 4.
Let
a be
0J? / 3j6T
the value o f — — I ------- a t an optim al solution
OZ2 I
z
~
o f Z(A)
0Zt
Then
A
(i)
A
A
A
(//)
A
A
< a => g (-) < g (?) i f z € Z , zx > & and z 2 < g 3.
A >
a
=>
g
A
(:)
g
<
(i)
if
A
Z € Z ,
A
atul
Z !< ii
z 3> Z 2-
Proof.
A
A
A
(/) C onsider a i ( Z such th a t zj > Z i and z 2< z %■ Since z is optim al
for L (A), we have
[ i - 2 .]/[?a- l ]
< A- < a,
i.e.,
/I
A
'i+ a z 2 <
Zl~T<*Zi-
(3)
Since #(r) is c o n tin u o u sly differentiable quasi-concave, z i + a z 2 = ? i+ a 2 a
is a suppo rting hy perplane o f th e c lo s e d convex set S (Z) = ( z :g (z)^g (Z )}
A
at z ■= z a n d S(Z) £ { z :z 1+oLz2'^ Z 1-ra Z i }.
A
int S(z).
A
Suppose g (z ) > g ( Z )■ Then z €
/t
T h is implies z 1+ a z s > 2 1- f a ? a w hich co n tra d ic ts (3). Therefore
A
g(z)Kg(Z)-
(ii) also can be p r o v e d o n th e same lines.
Lem m a 5. F o r
X and
a o f Lem m a 4,
(>) £ « a)) < g(Z)for 0 < A < A j / A < a ,
(ii) g (z(A)) < ,g < z)fo r A > A i f A > a ,
(iii) g (z(A)) < g (z ) f o r A > 0 i f A == a.
P roof (/) a n d (ii) fo llo w f r o m L e m m a s 1 a n d 4.
(iii) follow s fro m (i)
(ii) and L em m a 4 since a n y o p ti m a l s o lu tio n z =?fc Z o f L (A) satisfy either
is/
rsj
rmJ
?i>Zi a n d z ^ < Z t o r zx< Zi a n d za > ? 2.
Lemma 6 . L e t K < A" < A. an d z ( \ x) e n d z(A..) lie on the lute {r/r,+
~XZit== Zi(X) + Az, (A)}.
T hen z (A,) is the unique o p tim a l solution of L (X\
fo r Ax <A < A"and z (X.,)
is
the
unique
o p tim a l solution
o f L (A) for
A < A < Aj.
P r o o f N o te tha t z (A,) a n d z (A.) arc a lso o p t i m a l f o r L (A). Consider a
A s u c h th a t Aj < X < A. S u p p o s e z (A) 9 ^z (A,). T h e n we h a \ c z x (A)--,.j(A,)
a n d z 2 ( a ) > z 2 (A^ by L e m m a 1. W e also have
- i (A) + X z ,{ X ) > --i OM + A z 2 (A,).
i.e.,
[zj (A O -Z x (A)]'[z 2(A)—z 2 (A1) ] < A < I
i.e.,
Zj (A,) -f- X z 2 (Aj) <^ z 1 (A)
A ~2 (A),
w hich c o ntradicts th e o p tim a l ity o f z ( A i ) f o r L (A). Therefore z (A) =
z (Aj) a n d z (Aj) is th e u n i q u e op tim a l s o lu ti o n ofZ-(X). Similarly, z(A2)
is the u niqu e o p tim a l s o lu tio n
o f L (A) f o r A < A < A2.
T he follow in g a l g o r i t h m d evelop ed on th e basis o f above results
yields optim al so lu tio n s o f p ro b le m s Pj a n d P 2 s im u ltan e o u s ly .
A lg o rith m 1
S te p 0: F in d a point zO) t h a t maximises z l c o o r d i n a t e o n Z. This can be
done by m a x im isin g c xx on K. Let .v(l > b e th e p o i n t th a t maximises
exx o n K. T h e n
= (cxx (1), c 2x(1)). Set x* — x (‘>, x* ----- z(*> and
Q* — g (zC1) ) . S im ila rly o b ta in x(2M h a t m axim ises c.,x on K and
find th e image z<2' o f x^K I f q* < g ( : '2)), set x* = x<s>, z* = z,3>
a n d q* = g(zW ).
S te p 1: F in d A = [ z ^ — z ^ ]/'[ z ^ — z ^ ] a n d set F 0 — z\^ -f A zi'- .
M aximise zx+ A za o n Z by solving L P : m axim ise (cj-|- A c2) ,v on
K. Let x be th e o p tim a l so lu tio n o f th is L P a n d z ----- (c{x, c.,x) and
P = 2 i+ A 2 2. I f F " = F0, set X* = A. a n d go t o Step 4. I f g (l ) > q *,
go t o Step 3. O th erw ise go to Step 2 .
S tep 2 : I f x* = xW, set a'(2) = * an d z<2> — z ■ O therw ise set a (1) = x
a n d z(J) = z a n d go to Step 1.
S te p 3 : Set x* = x, z* = z a n d q* = g (z ).
E v a lu a te a — ~
d z 3
z= z-
I f a < A, set a:® = % a n d z& = z a n d go to
r)iT
at
dzi
Step 1.
I f a > A, set .vW = x , z*1) = z a n d go t o Step 1. I f a — A,
stop, .x* a n d z* a re o p tim a l solutions o f th e problem s P x a n d P 2,
respectively.
Step 4 : Define the f u n c tio n 0 (0) = g (z(U+fl (ZV ) - Z0))) a n d maximise
>'i (6) over O < 0 < 1 . Let 0* m a x im ise ^ (0) subject t o
1.
Set :*
r O ) H - 0 * a n d x * = .*(*}+6* (XW - XW) a n d stop.
x* and z* arc o ptim a l s o lu t io n s o f th e p ro blem s Pj a n d P 2,
respectively.
Validity o f A lgorithm
T h e optimal s o lu tio n o f the p r o b l e m P a is an efficient p o in t z* a n d
therefore there exists a finite p o sitiv e A* such th a t z* is a n op tim a l
s o lu tio n of L(A*). T he a b ov e a lg o r ith m s ta r ts with th e interval [0, °o) in
search o f A* a n d in each ite ra tio n it d iv id e s a n interval c o n ta in in g a A*
into t w o subintervals a n d selects th e su b in te rv a l th a t co n ta in s A*. The
a lg o r i th m obtains \* in Step 3 o r Step 4 o f th e final iteration.
Suppose
and z (2>o f S te p 1 o f the algorithm are obtained
as optimal solutions o f L ( A^ a n d L (\.z), resp ectively. Then Ai <
and the
interval [Alt AJ contains at least one value o f A*. Further,
L e m m a 7-
O') g (-) < g (-(Ai)) i f z € Z , rj > z 1 (Aj) a n d z 2< z , (Ax).
(H) g (-) < g (= (A.)) i f z € Z.
< - i (A2) a n d z 2> ~2 (A,).
Proof. The p r o o f is based o n i n d u c t io n o n th e n um b e r o f iteration s.
In the first iteration A, = 0 a n d Aa is ta k e n t o be oo as a c o nv entio n.
S u p p o se the lem m a h o l d s fo r r l h ite r a tio n a n d let zO) a n d z(2) o f Step 1
of /th iteration be s o lu tio n s o f £(Ai) a n d jL(A2) and z an optim al solu tion
ofi(A) where A is as described in Step 1.
2’ K < A <A2.
If F ^
F 0, we have, by L e m m a
S uppose < /*>£ (?) a n d z * = rW.
Then g (z (1))
A
and d u e to Lemmas 2 a n d 3, g (r(A))
■C A
A
(z*) for A > A and g ( z ) < g(z)
A
—
.
Z, Zj < Zi a n d z 2> g 2. It m eans t h a t th e interval [A1; A] c o n ta in s
a value of A* a nd th e lem m a ho ld s f o r (/’+ ! ) t h ite ra tio n when A2 a n d z(2)
are updated as A, =
A~and z<2) = z •
S im ila rly t h e cases (a)
(2) a n d
:* = z<*> (b) q* < g (i) a n d "A < a, (c) q* < g iz ) a n d 7 > a where
«=
~ — at z — g can also be p r o v e d using L em m as 2, 3, 4 a n d 5.
0^21 0-1
Theorem 1. I f F = F 0 in S tep 1, the o p tim a l solution o f the problem P 2
lies on the line segm ent jo in in g :W and z (2) o f S te p 1.
Pl'oof If F = F 0, zW a n d z(2> are o n the line ^ + A za = z i + A
Suppose z<4 and z<2) are o btained as o p ti m a l s o lu tio n s o f i(Aj) a n d Z(A,),
respectively.
By L em m a s 6 a n d 7, it is e n o u g h to consider the optimal
s o lu tio n s o f L(Xi), L (A) a n d
C o n s id e r an optim al s o l u t i o n : ef
L (AO which do es
z t -\- Aj.
z, <
not
z ? a n d --2> 4 ° o r b)
o p tim a l fo r L(Aj).
«
m axim ise
A» <
o p tim a l fo r Z(A).
Z.
We have either (oj
a n d r . < 4 “ since b oth r and .-"arc
Suppose
A, i.e., = ? +
on
(a) h o ld s .
A r ‘° <
r,+ A
fl)
/1 A '1'
T h e n we have [-*, - - r,] [ r , - : ; 1
w hich contradicts that ri>no;
T h e re f o r e (b) h o l d s a n d d u e to Lemma 7, .?(r)<s(r'i.
A
_
Similarly, if a n o p ti m a l so lu tio n j o f L (X .>) d o e s not maximise rj - .< .
on Z , th e n g ( r ) < g ( 2 ® ).
N o w it is e n o u g h t o consider only the optima
solutiens o f L { \) in o r d e r t o find the o p t i m a l s o l u t i o n o f P..
Suppose z is a n o p tim a l s o lu tio n ofZ-(A) b u t is not on the line segmeu
A
jo in in g zW a n d z&K
A
A
A
T h e n either z \ > z ,J a n d r 2 <
A
z z > z f a n d c o n s e q u e n tly by L em m a 7, g{z) <
or
(-i
< r (‘ as:
A
£(~(1>) or g(zj <
H ence the L em m a.
I f the a lg o r ith m te rm in a te s in Step 3, th e n b y Lemma 5, A of lins
iteration is a v a lu e o f X* a n d z* is th e o p tim a l s o lu tio n o f P.,. If tbs
a lg o rith m te rm in a te s in Step 4, th en b y T h e o r e m 1 th e best point :* ®
th e line segment j o i n i n g zW a n d z<2> is th e o p tim a l so lu tio n o f P, and;
p o in t x * $ K su ch t h a t z* = (Ci**, c2x * ) is a n op tim a l s o lu tio n o f Pj.
D iscrete Case
C onsider th e p r o b l e m Pi w ith .v r e s tr ic te d t o be an integral vectw
L e t / b e th e set o f a ll integral p o in ts o f K . A p o in t .v in r is said t o t
efficient w ith respect to I if th e re d o e s n o t exist a y in / such th:
(Ci-Y,^*) ^ ( ci )a c-j}’)- N o t e th a t a p o i n t w hich is efficient w .r.t. / nee
n o t be efficient in K. T o maximise g i c ^ ^ x ) on F, it is enough!
c onsider p o in ts w h ic h a re efficient w .r.t. /.
I f A is to ta l ly u n im o d u ia r a n d b is a n in te g ra l v ector, a lgo rithm
be made use o f t o solve th e above p r o b le m . I n th is case, a p p ly algorithm'
1 ignoring the in te g ra lity restriction. I f it te rm in a te s in Step 3, x* is tfc
required o p tim a l s o lu tio n . I f it te r m in a te s in Step 4, enumerate#
points which are efficient w.r.t. / a n d satisfy c 1x > z [ l) a n d c .A ^ r ^ a n d ta l
the point a m o n g th e m w hich gives m a x im u m value o f g. I f th is poin t is
better than x*, t h e n it is t h e r e q u i r e d s o lu tio n . O th e rw ise x* is the
required so lu tio n . S om etim es t h e s t r u c tu r e o f th e m a trix A enables us t o
develop im plicit e n u m e ra tio n t e c h n iq u e s such as b r a n c h a n d boun d
method to carry o u t th e a bo ve m e n t i o n e d e n u m e ra tio n , as illu strated in
the following special case.
Special Case : M axim isation o f R e lia b ility o f Series-P arallel S y ste m
Consider a series-parallel r e l i a b i l i t y system c o n sis tin g o f tw o parallel
systems Cx a n d C 2 in series. S u p p o s e C x (C 2) consists o f n fji? ) positions
and there are n ( = n 1+ n 2) c o m p o n e n t s a n y one o f w h ic h c a n be assigned
to any position. Let p o s itio n s o f C x be d e n o te d b y 1 ,2 ,...,n 1 an d those
of C., denoted by «A+ 1,.. ,n. A s s u m e th a t re lia bility o f c o m pon e nt j is
Pu when it is assigned to p o s it io n i. W e represent an assignm ent by ir x 1
vector A-=(.vn , ..x ln,.Y21,..., x Vl, .. ,x„u . . . , x lJ,)T where x u = 1 if com ponent
j is assigned to p o s itio n i a n d z e ro o th e rw ise . F o r a n assignm ent x , the
system reliability can be w ritte n as
n,
n
R s{x) = [ 1 — ir
tt
n
n
qx,J ] [ 1 — 7T
7t q x ,J ],
,'=1 / = 1
0
/=«! + 1 / = 1 ''
where qij= l —p i).
This can be r e w r itte n as
^ ( - Y) = [ 1 - e x p (-s'x-v)] [ 1— exp ( —avy)],
where
5ai , . . -,Sln, •• ->Snil , ••■
5'i = (.S'iI,
and
s .,= ( 0 , 0 , ■
0,.. .,0)
> - A u,
are two 1 X /;2 vectors
log qu .
Now consider th e p ro b le m (R.,) o f assigning these n c om po ne n ts to n
positions o f th e system so as t o m axim ise th e system reliability. This
and
problem is, in m athem atical te rm s,
Maximise
subject to
c2x ),
x ( K a n d x bein g integral,
where g i ^ x ^ x ) = [1 - e x p ( - « i * ) ]
[ 1 —e x p (- -c2x)], C j= su
K = { x : S X jj — 1 fo r / = 1 t o it,
*/; = 1 f o r j = 1 t o n and x,v > 0 }.
S
c2= s 2 and
Consider a f u n c t i o n f ( x , y ) = 11—expC—-v")] [ 1 cxp( v)] f r o m ^ t0
We now p resent a re su lt c o n c e rn in g f ( x , y ) w hich enables us to use
algorithm 1 fo r solving t h e p r o b le m R s.
Lem m a 8 . f ( x , y )
is q u a si-c o n c a w on R\ — \(x .y ) | a > 0 . i><>}.
See A p p e n d i x f o r p r o o f .
Since g ( z ! ,z t) is q u a s i - c o n c a v e on R 1
by
L .m m a
S
and
strictly
increases w ith Zl a n d z 2, we can make use o f A lg o rith m 1 as suggested
ea rlie r f o r t h e d is c r e te case. Tf A lgorithm 1 term inates in Step 4,
A l g o r i t h m 2 given b e lo w c a n be used to enu m erate all assignments that
satisfy
a n d c a. v > 4 " .
If an assignment ,v is represented by a
p e r m u t a t i o n ( V j . V o , o f 1, 2 ,
write c ,x
=
then \ jv^
ni
n
S siv a n d <\>.v S
v„
«=1
'
"
X- 17,-f 1
1 l or/
1 to n and we can
F o r a partial
permutation
(i{1,u 2,...,U k )o £ 1,2,.. ,/ 7, le t Cj(»j... ,«*•) represent the m axim u m value of Si,-,..
i- 1 '
o n th e set o f all p e r m u t a t i o n s generated fro m ( m, , S
i mi l arl y let
n
c2(itx,...,ui<) represent t h e m a x im u m value o f S
.v,-,.. on the same set of
I=
1
‘
p e rm u ta tio n s . d (u x..... Uk), i — 1,2, can be obtain ed u sin g H ungarian method
fo r t h e assig nm ent p r o b l e m . Let N denote the set {1.2, ...//}.
A lg orith m 2
Step 0 : Set G ={(1), (
2
a n d a x= r p a n d a.,
:'2"
p a r tia l p e r m u ta tio n
fr o m G and set G = G \
I f k = n — 1, go to Step 3. Otherwise evaluate
£ , - ( « ! , f or 7 = 1,2. I f Ci(uh .. ,itk) > c/j fo r / = 1,2. then set
G = U {(llL% ,.,U k,y)} u G
y € N - { u 1, . . . , u k}
Step 1: Select a
Step 2: I f G=<f>, stop, x * is o p tim a l.
O therw ise g o t o Step 1.
Step 3: I f (uL,....,u„ ) is th e p e r m u ta tio n
>h
ev a lu a te S
sit, — a x a n d
*=1
S
Sj
/= « !+ 1 '
a*. I f o n e o f these tw o values
go to Step 2.
O th erw ise ev a lu a te g( S ,v(V, ,
/ —I
'
I f il; is g re a ter th a n
A*> c 2-v*), set X* = 1 if j = i ti
is non-positive,
n
+1 V -
g e n e ra tio n fr o m (»,, ,.,i
n
a n d 0 otherwise f o r 7= 1 to n. G o to Step 2.
3. M inimisation o f Bicriterion Quasi-Concave Function
In this section, we co n sid e r th e p r o b l e m :
Minimise
"( c^. v.CoA),
subject to
a- $ K = {a :/1a-=Z>, . y > 0 }.
This is equivalent t o the p ro b le m :
Minimise
subject to
{zu z , ) ^ Z ,
(p4)
whcreZ is as described in Section 1 .
Note that Z is a convex p o ly h e d r o n . In this section, a p o in t (z1,z2)
in Z is said to be efficient if a n d o n ly if there does n o t exist an other
point (vi.j'j) in Z such th a t
< (zx,z 2). By p rop erties (z) and
Jj»
||(
(") of g ( ri,r 2), there exists an efficient extrem e p o in t ( z j , z2 ) th a t minimises
£(“i»-2) on Z.
* *
Since ( z Jf r2 ) is efficient, th e re exists a positive num ber X*
* *
such that (zl , r2 ) minimises z 1Jr \ * z 2 o n Z.
*
for >* and find
O u r a p p ro a c h is to search
*
* *
, z%) and x * $ K such (z 1, z 2 ) =
r zx*) d u rin g the
search. In this section also, a p o in t ( z ^ z ,) in Z is d en oted b y z and
g(zlt
by g(z). W e den ote by Jl/(A) th e p ro b lem o f minim ising Zi+Az,.
on Z and denote by z ( \ ) an optim al s o l u t i o n o f M(A). T h r o u g h o u t this
section we consider the p ro blem M (A) f o r A > 0 only.
Preliminary R esults
Lemma 9. A1<A., and z(X1)^ z(A .i )=>z1(A1) < z 1(A.:,) a n d z 2 (A1) > z 2(A 2)-
Proof is similar to th a t o f Lem m a 1.
L emma 10. L e t Ax< X 2, z ( A j ) : j £ z ( A 2) and A = [z 1(A,)—z 1(AJ)]j[z2(A1) —z 2(A2)]
Then A ^ A ^ A2, Further, i f zJ(X)+Az 2(A )< z 1(A1)+ A z 2(A1), then A1 < A < A 2.
Proof is similar to t h a t o f Lemma 2.
Suppose Aj<A 2 and z(Aj) an d z(A2) a r e op tim a l solutions o f M(Ax) and
^(A2) such th a t z(Aj)96 z(A2). Let A—[zxfAg)—z 1(A1)]/[za(A1) —z 2(A2)] a n d h
be the point o f intersection o f th e lines i 1 :z14 -A1z 2= z 1(A1) + A 1z2(A1') and
L.,:Zi-f-Atz ,= z 1(A2‘)-f A2z 2(A2j .
Let c = z ^ A ^ - f A z a ^ ) .
Zi+Az2= c intersectsL t a t z(Aj) a n d L.-, a t z(Aa).
N o te
th a t th e line
I f A1 < a < A 2, th e region
A z2< c, z 1 + A 1z 2> z x(A1) -j- A1z 2(A1) a n d zx -f- A2z 2^Zx(A 2) + (A 2za(A»)}
v . x m a t o M m m j x ■s'v’*-
A'VJ> K J,-K- SA!#
i% a w s g f c f o r iW %
f?< ^ ' 4 V X -i'V>vmi ft. Let mdc»oteite
triangfc b y d ( * xJ » ). T h is tr> »x& t.\ssW *m \« l-<£. 1 a> M'adcd tegsoa.
T h e p K r o w w tf )m e:ir c u n v ) m r iw & t^ V y w v i ;(.'*) >0 tfic
th e b o u n d a r y o f Z -
V"m_ \. ' ^ n ^ e
VV >>->
g { h # f vr
i*EMMA I / . g t . ^ » >
. . S
%J J ^ c o n c ^ e
*U> W ^ ^ t Q T A ^ x - c K . . f t
** ParS
o j ^ it k
«wvt
c h o u g h t(7 S)>ow tha
zJM + A tS ziA } > Z j t e t f t A j Z j f r j ,
aitd
•->^>H
v ,^ o s e ^
> 2A W > V K ,z ^
fov « { 0 ,w j.
r. I 2^ > , f 0j. Soj3j fc ^ X }< I{ ^ a iuid som e o p t h m i aoiuti
t a ) 4 . j - / n <* ? - if
* i ( i ) + ^ * ( h ) w h ic h c a a (m < iic tt o \ c ^ c i m a l k y
T herefore z(A ){M *i,> s) & r \ <<>c<\.
> I > A Vi.e., z t$ )4 -X zJ X
o f p (fy ta r
'
M( X )
^
V
L emma
12.
L e t A i < , \ < A 2. Then
O' "(-)
£(-(A»)), g{h)} fo r any optim al solution z o f M (AJ
sa tisfyin g r, > r,(A,) and
i?(r(A2)), g(A)} fo r any optim al solution 7 ofM(*«)
(//) .?(-)
satisfying -jC m O V ) am / ~2> r.,( A 2).
P ro o f,
(/) We have
a
,)]
•[r2(A1i~ ^ .,] = A1 < X,
i.e., r,4-Ar, < c, which implies ; ( i (A^A,). Now, statem ent (/) follows
from quasi-concave p ro p e rty o f g,
(ii) also can be proved on th e sam e lines.
L e m m a 1 3 . Suppose r(Aj) and z(\.,) lie on the line r ^ - A v - r , ( A ) + A ~ r 2(A).
Then r(Aj) is the unique optim al solution o f M(X) fo r Ax < A < a~ a n d z(A,)
is the unique o ptim al solid ion o f M( A) f o r A < A < A2.
This lemma can be proved on th e same lines as L em m a 6 n o tin g that
-(Ai), r(Aj a n d r(A2) o f this
ri';-A2r 2 on Z . respectively.
le m m a
minimise
+ A! z2, zx+ Ar, and
The following a lg o rith m d e v e lo p e d on the basis o f abo ve results
yields optim al s o lu tio n s o f p ro b le m s P 3 and P 4 sim ultaneously. The
algorithm initially obtains r (l) a n d z^2> which minimise Zj a n d z 2 on Z,
respectively.
If z (,0
4'-’ o r r!.1' = z 9 \ th e n
th e
a lg o rith m
stops giving
one of z(‘> and r |2> as an op tim a l s o lu ti o n o f problem P 4. Otherwise the
algorithm perform s several ite ra tio n s giving a n efficient p o in t z<r+2> in /t h
( r ^ 1) iteration.
I n r t h ( r ^ l ) iteratio n, th e a l g o r t h m starts with a collec tio n IF of
pairs of indices. An element ( i , j ) in W represents th e p a ir o f efficient
points z(o a n d zV). T he a lg o r ith m selects a p a ir (/,_/') in W a n d obtains,
using z<» a n d zM, a new efficient p o i n t z(r+-'>which results in th e inclusion
of pairs ( / , r + 2 ) a n d ( r + 2 , j ) in W a n d e lim in a tio n o f (i,j) fro m W. A
point x(r r2> in K t h a t c o rre s p o n d s t o z(r+2) is sim ultaneously obtained
along with ~(r+2>. P o in t z* given in th is ite ratio n is a p o in t in { z V \ ...z ^ V )
such th a t g (z* ) =
m in
g ( z {0)-
Some
of
the
clcmcnt!> >s
are
l< /< r+ 2
e lim in ate d by a c r ite r io n in v olving th e v a lu e o f a- a t this z*. The algorithm
sto p s w hen W=<t>. z* o f final ite ra tio n is a n o p tim a l solution of problem
P 4 a n d th e c o r r e s p o n d in g p o in t v* in K is o p tim a l ( o r problem P3.
A lgorithm 3
S te p 0: O btain
(a-^)
th a t
m inim ises ^ .v ^ v y ) on K and set
c 2.v(D) a n d Z'.a>= (c 1A-'2', c., a ' 2’)- T a k e A, 0 and A,= z
and, as a c o n v e n tio n , ta k e O.zx-'r-* as th e objective function of
M (c c ). T h e n zW a n d z|2> are o p tim a l s o lu tio n s o f -U(AJ and iW(A.j,
.vO
respectively.
= z'i'K th e n z'2; a n d .v,2>are optimal solutions
If
o f the p r o b le m s P 4 a n d P 3.
th en r " 1 and .v'1^ me
If ziIJ -- - 1/' ,
optim al s o l u t i o n s o f P 4 a n d P 3.
I f z^1) a n d z ,2) do not coincide
in an y c o o r d in a te , set W-— {(1,2)}. /■= 2. </1 = r J i1) and </2=Zo2\
S tep 1: If
^ g ( z ( 2->), set z * = z U), q* = g ( z lU) a n d a * = a i".
set z * = z ^ , q * = g ( z (2)) a n d .v*= a',:!|.
S tep 2■ C hoose a n y
Evaluate A, =
( i , j ) f r o m W.
Set
W =
Otherwise
^ \ { ( / . 7 j} and r = r -f 1.
— z ^ J / l z ^ — zf^J a n d set d = z [ °
A,
z^
and
find ,\'(r>t h a t m in im ises (^-fArCo)^ o n A'. Set z1' >— (c1.v1'’', r 2.v(,J).
T h e n z ^ i s a n op tim a l so lu tion o f M (A r).
Set dr = z\r>
If dr~ d , go t o S tep 7. O therwise find th e s o lu tio n
^i+A/z2= (// a n d r 1+ A ,z2= </,■ a n d
z i Jr \ z 2-=dr a n d z 1-\-\jZ2= d J.
q{r, j ) = g (h([ ’J), / $ ’» ).
th e
so lu tio n
Ar
■
h{-i'r>) of
(h{l ' j>, h(r Ji) of
Set q ( i,r ) = g (li\nr\ h\!'r)) an d
I f £ ( z ( ') ) > 9* go to Step 4.
S tep 3: Set z * — z (-r\ x * = x ( r\ q * = g (z(r'>).
S te p 4: I f q (i,r )< q * , set W = W \ j {/,/•)}.
S te p 5: I f q(r, j) < q * , set W = W \j { ( r , j) } .
S te p 6: D elete fro m W each (u,v) if q(u, v) > q*.
S te p 7: I f W^<j>, go t o Step 1. O therw ise s to p ,
solutions o f th e p ro b le m s P 4 a n d P 3.
z* a n d .v* are optim al
I 'alidily o f A lgorithm
Consider an elem ent ( i , j ) o f W. L et
E i} -= { z:z is optim al fo r M( X) f o r some A, Aj<A< A;}
U { - i s optim al fo r M(Xf) a n d 2 1> z (p , z2< z (<p}
L ' { - i s optim al f o r M( Xj ) and r 1 < r (1J), z2> - 2 )}Let L ;j be the line segment jo in in g zW and : 0 \ Defining Eu as La
without end p o in ts z <n and z (i>w h e n A, = Ay-, we can write
E r, = E ir u E rJ U { - (r))
(4)
for an o ptim a l so lu tio n z (r) o f M ( Ar) when X j ^ X r^ X j .
Lem m a
14.
c
J.0 i f =1 (A)+A
(A)
=
: (/ ' + I z f where
A=
[z{p ~
4 ' ] / [4 ° ~ - 2 }JP ro o f Ciise ( i ) : A, — A < \ j
The equality -i(A )+ Az2 (A) = r (1')+ A z ^ implies th a t
z(A)andz®
optimal f o r M(X) a n d lie on th e line z! + Aza= z 1(A) +Az 2 (A). In this case,
we have {z:z is optim al for M(A) f o r some A, A,< A <Ay}=zW by Lemma 12.
We also have {z : z is op tim a l f o r M(Xj) and 2 1< - \J\ za> z ^ } = <j>.
Otherwise, we arrive at a c o n trad ictio n th a t zf/> is n o t o ptim a l for M(X).
The set {z:z is optim al fo r M ( A,) a n d z 1> z i ,'), 2.,< z ^ } c ontains z('> due to
is optim al for M(X). W e claim that
/i
/i
there does not exist an optimal s o lu tio n z o f M(A,) such th a t Z j> z /
and
Lemma 9 and since A/ = A and
/i
z2 <Zg^ .
for M(Xj).
Otherwise we arrive at a c o n tra d ic tio n th a t z <J> is n o t optimal
Therefore, the set {z:z is optim al fo r M(Xf) a n d r L> -V
is contained in L u a n d c o n se q u e n tly E i} C LaCase (ii): A,- < a = Aj.
This case can be verified on th e sam e lines as case (/).
Case (iii): A,•< \ < Xj.
W e c a n easily see, f o l l o w i n g L e m m a 13 an d th e a rg u m e n ts o f case (/),
th at
{z:z is op tim a l fo r M( A) f o r some A, A,<>.< A«}
{ - ' " . - '‘‘JL *- - is
o p tim a l fo r M(X)},
{z:z is o ptim a l fo r A/(A,) a n d
a n d { z . z is optim al fo r M( Xj ) a n d
^
r : < 4''}
<!>,
< z \J). r 2> 4 ”} —</..
( 6)
(?)
W e also have
{z:z is optim al fo r M ( A)} = i-,/.
O therw ise, we arrive a t a c o n tr a d ic a tio n th a t e ith e r r 1' 1 is not optimal
fo r M { A,) o r z(j) is n o t o p tim a l f o r M(Xj). F r o m E q u a t i o n s ^ ) - ^ ) we can
n o w c o n c lu d e E u —Lij.
A t the e n d o f each iteration o f the algorithm , the optimal
Solution o f the problem P t belongs to
l/
I 'a U{-*}
0 ,/y lV
T
heorem
2.
P r o o f W'e prove the t h e o r e m by in d uc tio n o n th e n u m b e r o f iterations
S up p o se the theorem is tr u e f o r / th iteratio n. Let H *'1 a n d H'1' " represent
W in Step 7 o f I th a n d (/-}-1) th iterations. A ss u m e that (/,y)€
is
chosen in Step 2 o f (/-[-I) th ite ra tion a n d s u p p o se d , ~ d in that step. We
have Eij £ L tj by L em m a 14 a n d g ( z ) > g ( z * ) f o r r b y
quasi concave
o f g. T h u s g ( z ) ^ g ( z * ) f o r z$.Esl a n d the o p tim a l s o lu tio n o f Px belongs to
U
Eu U [z*} since W (/ " = ^/(/)\ { ( ' ./)} w h e n d, — d.
ijywm ^
Suppose dr < d. C o n s id e r th e case q(i, r) ~^q* a n d q { r , j ) ^ q * . W'e have
g ( - (r))>& (z*) fo r the revised z* a n d du e to L e m m a s 11 a n d 12, # ( - ) > g(=*)
f o r z$ E ir U E rJ, that is, g ( z ) ^ g ( z * ) for z ^ E ^ by E q u a tio n (4). We also have
g (z)> g (.z*) fo r z$ E m if (u ,v )sW « > \{(i, j)) a n d q ( ii,r ) ^ q * . N ow , an optimal
so lu tion o f Pi belongs to
U
E Uv
b y in d u c tio n hypothesis
M c o ' d +o
since
{( m, v)}. S im ila rly other three cases can
(u,v)e WO)
also be verified.
F o llo w in g the above a rg u m e n ts , one can e a sily see tha t the theorem
h o ld s for the first ite r a tio n also .
T h e o re m 2 implies th a t final z* o f the A lg o r ith m 3 is o p t ;mal fo r the
p ro b le m P 4 since final IV is empty.
T h u s final x* th at satisfies
(ci* ,c2x * ) ^ z * (final) is a n o p tim a l so lu tio n o f th e p r o b le m P 3. In Step 2,
any L P technique can be u s e d to find
th a t m in im ises (Ci + A,C2).v on K.
Discrete Case
Consider the p ro b le m P 3 with .v restricted to be a n integral vector.
Algorithm 3 yields an optim al s o lu tio n f o r this discrete case also i f A is
totally u n im o d u la r a n d b is an integral v e c to r pro vided th a t we use in
Step 2 simplex m e th o d or any L P te c h n iq u e tha t yields a n extrem e point
o f K as an op tim a l so lu tio n . T his is because all extreme p o in ts o f K are
integral when ,4 is totally u n im o d u la r a n d b is an integral v e cto r a n d x*
o f Algorithm 3 is always a n integral v e c to r if we use in Step 2 any LP
technique as m en tio n ed above.
Special Case: M a x im isa tio n o f R e lia b ility o f Parallel-Series S y ste m .
Consider a parallel-series r e lia b ility system consisting o f two series
systems S y an d S 2 in p a ra lle l. S u p p o se S ^ S z ) consists o f «i(«2) positions
and there are n{ = n1+ it2) c o m p o n e n ts a n y o n e o f which can be assigned to
any position. A ssum e tha t re lia b ility o f c o m p o n e n t j is p a if it is assigned
to position th a t is, reliability o f a c o m p o n e n t depends also o n the posi­
tion in which it is fixed. N o w the p r o b le m is how to assign n com ponents
to n positions o f th e system in o r d e r t o m aximise the system reliability.
Denote the positio ns o f S i by l,2,...,/?i a n d those o f S 2 by ^ + 1,
We represent a n assignm ent by a n n2 x 1 vector . v = ( a " u . . . . , . v 1„ , x 21, . . . , x 2„ , . . . ,
where .v ,;= l if co m p o n e n t j is assigned to p o s itio n / a n d zero
otherwise. F o r an assignm ent x , th e system reliability can be written as
nx
y?p(.Y) =
l — [ 1 —
n
n
i r p x ‘J ] [ l -
it
,= 1
j= 1
u
n
n
7T
i^ n , + l 7 = 1
p Xi i ].
,]
In other words,
RP{x) = 1 —[ 1 —exp ( —
— e x p ( —a,.*)],
where
a l ~ (a l l ) • ' ■>a i«> °U1 >• • • >a 2 " > - ■•a / ; 1 1> ■■• 5 a /(1
and a., = ( 0 ,..., 0 ,
are two
1
0 ,...,0 )
+
x n vectors
and <ijj = — log p i}.
The problem (Rp) o f maximissing R P(x ) over the set o f all assignments
can be posed as
Minimise
subject to
g(cr v,c 2x),
xgAT a n d x being integral,
where
and
g(CiX,czx ) = [ \ — exp(— Cj.y)] [ I — exp(—c2.v)], c L= ai, c2= a,,
K —i x :
n
S
*,; = 1 f o r / = l t o / ; ,
/= l
n
ii *,/ = 1 for /' = 1 to n and
/= i
x a > 0).
Since g (z u z $ ' s quasi-co ncave on R~ b y L e m m a 8 and the constraint
m a trix A is to ta lly u n i m o d u l a r in this case, A lg o rith m 3 yields an optimal
so lu tio n o f the p r o b l e m R p. A n y a ssig n m e n t te chnique can be used to
solve the L P in Step 2 o f the algorith m .
4. Discussion
A lg o rith m 1 o f Section 2 yields an o p tim a l solu tion o f problem Pj.
T his a lg o rith m is b a se d on q ua si-c on c a ve p ro p e r ty an d monotonic
p r o p e r ty (with resp ect to each arg u m e n t) o f g (z 1,~2). Geoffrion [ 6 ] also
gave a n a lg o rith m to solve the p r o b le m Pj. H o w e v e r, he exploited the
two properties o f g i n a diiferent m an n e r. H e show ed th a t as we move
f r o m one end o f efficient f r o n tie r / '( th e set o f efficient points) o f Z to the
o th e r end, g(zu z z) is no n -d e c reasin g u p to s o m e p o i n t a n d non-increasing
f r o m th a t p o in t o n w a rd s. G e o ffrio n ’s a l g o r i t h m starts a t one end of F
a n d m oves a lo n g F b y p a r a m e tr ic p r o g r a m m i n g te c h n iq u e until it reaches
a n edge o f Z th a t c o n ta in s o p tim a l z*. O u r a lg o r it h m divides in each
ite ra tio n a p a rt ox F c o n ta in in g optim al z* (selected in previous iteration)
in to tw o parts a n d select the one th a t c o n ta in s =*. This is done by
solving a n LP p r o b le m . I f the a lg o r ith m e n d s in Step 4 , it maximises
g (zl t z 2) on the edge o f Z th a t c on tain s ■*. I f it end s in Step 3. it directly
gives z*.
I f the simplex m e t h o d is used to solve L P o f each iteration, the
op tim a l basic feasible s o l u t i o n o f L P in a n i te r a tio n can be used as initial
basic feasible s o lu t io n o f L P in th e next i te r a t io n o f th e a lg orithm . We
c a n see that no ba sic feasible s o lu tio n is visited m o r e th a n once
th r o u g h o u t the a lg o rith m . Since o u r a l g o r i t h m is based on bisection
ap proach, we claim t h a t o u r a lg o rith m w o u ld p e r f o r m b e tte r t h a n that of
G eoffrion [ 6 ].
A lgo rithm 3 o b ta in s o p tim a l s o lu tio n s o f p ro b le m s P , a n d P 4. It
im plicitly enumerates a ll efficient extrem e p o in t s o f K a n d finds a n optimal
one. The a lg o rith m yield s efficient f ro n tie r, i f th e fo llo w in g modifications
are d o n e :
(1) Ignore the p o in ts W * ) a n d h0 '’) o f S tep 2.
( 2) I n S te p 3, u p d a te W also as W = W \ j { ( i yr), ( r , j )}
(3) Delete Steps 4,5, and 6 .
(4) Store :<~r> o f each ite ra tio n in c lu d in g : « and z(2>. Eliminate
- 0 )(r (2)) if there exists a
such th a t r(r,< r ( 1>(z<>><z<2>). Arrange
all the remaining r ^ ’s in th e increasing o rder o f z x coordinate
and take convex linear c o m b in a tio n s o f each p a i r o f successive
points in th a t order.
To generate all efficient p o in ts o f Z o f Section 2, we can further
modify the A lgorithm 3 by re p la c in g m inim isation by maximisation
and eliminating initial
and z(2) w hen they are not efficient.
An im portant advantage o f A lg o rith m 3 is that it gives a n optim al
solution of problem P 3 even if .v is re stric te d to be integral p ro v id e d that
the matrix A is to ta lly u n im o d u la r a n d b is integral. W e have taken
this advantage t o solve the p ro b le m o f assigning com ponents op tim a lly to
a series-parallel reliab ility system so as t o maximise the system reliability.
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APPENDIX
P r o o f o f L em m a 8
W e prove the lem m a by show ing that for any tw o
d istin c t
a rb itra ry
p o in ts
C'j.v'i)
and ( x 2,y t ) in
/ ( . V i + A ^ - f A y ) > m in { /( .V ij 'j ) , /(.*■>,>',)},
for 0 ^
m in
A^
1 where x = x 2— x t and j = y 3 — y v
S u p p o se
/(^C j + Ax, J 'x + A j)) = / ( a - j + A ^ . J i + A0 ))) <
for som e A", 0 < A ° < 1.
m in { / (aY -’V )’ J (■''2-.vi )} ,
(9)
L et (A'0,v0) = (x 1,7 l )-f-A 0(^,_j>) an d
K A ) = / ( . v 1+ ( A ° - l - A ) i , }’1 + (A 0 +A )jp),
for A € [ — A0, 1 — A0].
/f(0 ) =
W e can write
m in
h(X) < m in { /7 (— A0) , h(l — A0) } ,
(10)
—A ° < A < 1 —A0
w e have
/;' ( 0) = y ( l - S ) * + $ ( 1 — y ) j ,
and
h \0 ) = —y x 2—Sj>2+ y 8 (* -|-J>)2,
w here y =
exp ( — x 0) and S =
exp ( — y n).
W e k n ow that (x ° ,y ° )
( 0 ,0 ) . A ssum e ( / , / ) > ( 0 ,0 ). T h e n 0 < y < l an d 0 < g < l .
S in ce h(A) and h \ A) are con tin u ou s, equation ( 10) im p lie s >i'(0 )= = 0 w h ich in turn
im plies on e o f x and y is p o sitiv e and the other is n e g a tiv e . N o w o n e ca n ea sily see
that
yS(x+J *)2 < max (yx2, Sjj>2)
i.e ., A"’(0 ) < 0 . This co n tra d icts the equality in ( 10) .
I f o n e o f x ° an d i" is 0 h \ Oj^iO
w hich also contradicts th e eq u ality in ( 10). T h er efo re th e r e c a n n o t exist A°,0 < A “ <1
such th at ( 9) holds and h en ce th e lem m a.