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UNCUSSIViKn Security aimatflcaUoa
DOCUMENT CONTROL DATA • RAD
I. OUMMATINO ACTWITV 7SI
l «• •••Mil i»g»g j» «JM«jjj»g
Department of Operations Research Stanford University Stanford. California 9A305
!•■ •MU»
I. MPOIIT TITLI
On Optimal Assembly of Systems
4. DCtCRIPTIVt NOTtl Wm 3 SBS JS toOMlM SS3
1. AUTHOWtj AMI MM«, an« MM. BüiQ
Oerman, Cyrus, Lleberman, Gerald J., Ross, Sheldon M.
*■ MPONTOATC
August 25th, 1971 (• CONTNACT ON «HANT MO.
NOO014-67-A-0112-0052 k, )>>«OJ(CT NO.
(NR-OA2-002)
12 I» OMICINATaM-t ncrOKT NUMUMOi
No. 140
«27 ONR N000U-67-A-0112-i)0'j8
10. * VAILABILITV/LIMITATION NOTICC«
Distribution of this document Is unlimited.
II. (UPPLCMCNTARV N0TI1 It (PONSONINO MILITARV ACTIVITY
Statistics & Pvobablllty Program Office of Naval Research Arlington, Virginia 22217
' We are concerned with the following reliability problem: A system has k different types of components. Associated with each component Is a numerical value. Let (aJ) (J - l,...,k) denote the set of numerical values of the k components. Let R(a ,...,afc) denote the probability that the system will perform satisfactorily fl.e. RCa1 alc) is the reliability of the system) and assume R(a^,...,a ) has the properties of a Joint cumulative distribution function.
Now suppose a| £.. .£ a^ are n components ol .ype j (j - 1 k). Then n systems can be assembled from these components. Let N denote the number of systems that perform satisfactorily. N is a random variable whose distribution will depend on tlie way the n systems are assembled. Of all different ways in which the n systems can be assembled, the paper shows that EN is maximized if these n systems havr reliablliiy R(aj,...,a^) (1 - 1,..., The method used here is an extension of a well known resulc of Hardy, Littlewood and Polya on sums of products. Furthermore, under certain conditions, the same assembly that maximizes EN minimizes the variance of N.
Finally, for a similar problem in reliability, it is shown that for a series systems a construction can be found that not only maximizes the expected number of funrcloning modules but also possesses the etronger property of maximizing the probability that the number of functioning modules is at least r, for each 0 < r < n.
0
DD /Ä 1473 UNCLASSIFIED Security ClauiacatfM
■OMajMHAbHMMMBua.^ ^_ _—
'"■, ■'■-'■- '■■■- ■»»———— — —- ■!■ I !■ ■ I II- .1. !
'JHCLASBinKD
l«. KIV ■OMOI
AaslgnmenL
Hardy, Llttlewood, and Polya
Reliability
Series System
Joint Distribution Function
UMH« »on WT
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1». CROUPi AiKoaMlc downcrxMnc I« «pcclflad 1« OoO IH> nclW« S20a 10 Md Anwd rare*« Inductrl«! ÜMMal. Cat« tlM group nuii*»r. AUe, «hon •fpllcabto, »Horn Ihal opttoMl Mrfcln«* hi«« bMn Mod («f Orovp 3 and Ofoup 4 aa autlw lawl 3. REPORT TITLE: Emtf lh« coaplat« rapoil IM* la all capital UHora. Till»« In all erntet aheuld ba VbclaaatflaA If a KMaintfut ml« oannet ba aalaclad «ttbaut claaalfic» Hoik aho« Uli* clattificallao la «II capUala la paraalbaala laiiadlHaly (oUowtnc Iba Ulla.
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DD .Da0.".. 1473 (BACK) Unclassified Saeurity Claaaiflcallaa
■ -- — ---
• ——^——^ • ■*-, I-—--I ^,,,.,l..1„,„.,| ,., , !, ..„.,.,W
I ON OPTIMAL ASSEMBLY OF SYSTEMS
by
Cyrus Derman, Gerald J. Lieberman & Sheldon M. Ross
..
Technical Report No. 140
August 25, 1971
Supported by the Army, Navy and Air Force Under Contract N00014-67-A-0112-0052 (NR-042-002)
with the Office of Naval Research
Gerald J. Lieberman, Project Director
Reproduction In whole or in Part is Permitted for any purpose of the United States Government
Department of Operations Research &
Department of Statistics Stanford University Stanford, California
r. KstiiibJncN oi'Air
Distribuiicu i'ü'Jmited
- ■ ■ Hi- *»■■. a ■■! _—. —J ■ i.f >■ 11. i - .
IpilllllllllJIU
I
^—— ll■l!",l*"' "■■",l1 '
MHBMMM ' M
10 0 i
;
[i
0
ID
ON OPTIMAL ASSEMBLY OF SYSTEMS
by
Cyrus Detman, Gerald J. Lieberman & Sheldon M, Ross
(0] Summary ——f-^ ^ {
We are concerned with the following reliability problem: A system
has k different types of components. Associated with each component
' ? 1 le a numerical value. Let (a } (j ■ l,,..,k) denote the set of Vl k ' numerical values of the k components, Let R(a ,,.,,& ) denote the
1 k probability that the system will perform satisfactorily (i.e R(a a )
1 k is the reliability of the system) and assume R(a ,. ,a ) has the
properties of a Joint cumulative distribution function.
1 1 Now suppose aJ. ^... •£ aJ are n components of type j (j • l,...,k).
Then n systems can be assembled from these components. Let N denote
the number of systems that perform satisfactorily. N is a random variable
whose distribution will depend on the way ehe n systems are assembled.
Of all different ways in which the n systems can be assembled, the paper
shows that EN is maximized If these n systems have reliability
1 k R(a.,...,a.) (i • 1, .,,n). The method used here is an extension of a
well known result of Hardy, Littlewood, and Polya on sums of products.
Furthermore, under certain conditions, the same assembly that maximizes
EN minimizes the variance of N.
Finally, for a similar problem in reliability, it is shown that
for a series systems a construction can be found that not only
mm - -' -
,—»**•
I
*" ' '
Ü maximizes ehe expected number of functioning modules but also possesses
the stronger property of maximizing the probability that the number of
functioning modules is at least r, for each 0 < r < n.
[1] An Optimal Assignment Theorem
A well-known result appearing in Hardy, Littlewood, and Polya [1]
asserts if a. < a. <...< a and if b. < b. <...< b . then 1— 2— — n 1— 2— — n'
n
1-1 lalb 1-1 *(1)
where i^ is any permutation of the Integers 1, 2,...,n.
Also in [1] is the generalization that for any k ^ 2 and if
0 < a, <-...< a, j - l,...,k then — 1 — — n' J ' '
n a j n V L - •-i 1-1 j-1
j n k
i-i j-i v
where ^,(1) - i (1 - l,...fn) and i|u,..,,tk are any k permutations
of 1, 2,...,a. Proof of both results follows from establishing the
inequalities for the special case of n - 2 and then resorting to a
standard argument regarding palrwise interchanges. First let n - 2
and k = 2. Then
1.
I
albl + a2b2 " alb2 " a2bl " al^bl~b2^ ~ a2^brb2^ " (a1-a2^bi"b2^ - 0
since a. <^ a« and b. f. b». Now assume the result is true for n - 2
MMMHMMimMHMMII UMitfMkMM ■«riiin».»! li In ^
linpppipiMPVfP«w)^VPqnpiMnpm«M*P«P*ffimr#i<Hn. iim ■"■ iiiniiii^wwi"«'^w^-
I I
and k «■ 2, 3 K-l. Then consider
ti
I n aJ (!)' 1-1 j-1 V
If for each J ■ 2,...,K, i|/ (1) - 2, * (2) - 1. Then
and
K J
j-2 *j(1) 2 (say)
.
11 ar. .«v - b, J.2 V2) 1
1 b2 •
(say)
.
and, hence.
K K n a^ + n a^
J-1 j-1 - a1b1 + a2b2
>^ a^- + a2b.
K K 2
11 a-^ -.v + 11 a. ,_v
::
If, for at least one J (say, j-2), «Ml) - l,i|», (2) - 2, then by
the Induction assumption
i s i
< a
2 K 1 a: n a^ •1-^(1)]
■ 1 2 J j a2 a2 ^3 ^(2)
K K n a^ ,.N + IT a; /().
.j-2*j(1) j-2 V2>
K k ■ n a^ + n a^
Lj-2 1 j-2 ^
,11, + (a^aj^)
/ 1 lv
4.Vv».
y:2av«
li|Mtt|HM«ftlMriiMMMJHHMM*Mi*M»itei^ft^MMM ^M^M^Mi.^.-..^,..,....- .,
^-«-r--» ""»» r*"*"^ """" ' ..„^.„;.-. jiniiiiiu mi >IIIIIIIPIIII mil n i ■MMMMMm
7r h
11 I i
..
..
;
i
..
i-i K
K K • n aj + n a^
J-2 J-2 + (a^ - ah n a^
1 j-2
naj J-l
K + n a
j-l
j
Hence the inequality holds for n - 2, and all Integer values of k.
Returning to arbitrary n, If ^.....ij/ are not all equal to
tj;. then there exist two values of 1 (say 1 ■• 1, and 1-2) such
that Ml) <^ iM2) does not hold for all j-2 k. However, by
considering the permutations i|)! (j - 2t...(k) which are the same as
f., (j - 2,...,k) except that t|;' (1) ±$'(2) for every j - 2,...,k
It follows from the above result for n - 2 that (keeping iK (1) - 1)
n k . n k .
I " «}.(!) 1 I n aJ (1) 1-1 j-l V1; i-1 j-l V
Hence, the original permutations could not be optimal.
Now let R(x.,...,x ) be any real-valued joint cumulative
probability distribution function. For our purposes we want to prove
the following extension.
Theorem 1: If a^ £.. .^ a^ (j - 1 k) , then
n 1 k n 1 2 k Ji^i ai) - J/^CD'^d) Xd)^
where ^(1) - 1 (1 - l,...,n) and ij» (J-2 k) are any
permutations of 1, 2 n.
1. This extension turns out to be a rediscovery of a special case of a result obtained by Lorentz [2].
I UMMawo. —toMMmf- -
„„,„..,,,„.., .....< »••< „i..,,!,.^ „ „^.,.^.„,.^.|....JI ,1) i . ""''—''"•'"-""
I i.
..
h
1.
Ü ! i
Proof: As in the above proof we need only prove the theorem for
n ■ 2. Also, without loss of generality we can then take a^ - 0,
a^ ■ 1, j = 1 k, and we can take R to be a discrete
distribution function with mass only at the points X. XJc where
X (m ■ l,...^ ) are all the points consisting of the k coordinates
which are 0 or 1. Let C denote the probability mass at the
point X . For any iK»"'»^!, we can w1'"'-16
ok
R(ai|'1(i) \(i) m=l m J-l mi *j(i)'
where g (a) = 1, if a > x mj — mj
= 0, if a < x mj
,th where x , is the j coordinate of X . mj ■" m
This is so since
k i i *! gmj(ai:j(i)
) = 0* " ^(i) ' Xmj f0r at leaSt one j
= 1, if a , /.v > x . for all j = l,...,k. i|/.(i) - mj j » .
i.
But since g (a) are non-decreasing functions it follows from the
Hardy, Littlewood, and Polya result for products that
k k k k n gmAah + n gm.(a
j„)i n gm.(aj ,..) + n g.(^ n.) j«! mJ 1 j=i mJ 2 j=i mJ ^y' j=i J *j( ^
for any permutations i|;-,... ,i(). . Then, since c ^ 0 (m»l,...,2)
5
wmmmrn ■^Mito»—^aMMfcfMIIUiMjmil^tirii ■ ■■'■'■ - ■■■ - ■ ■ ■ i«nf»-- ' - ■ ■■' ■ ■ • ■
"■""" ■'" """"I"
_„ ,>,,<rf»<WW'»W|VlW —«IM
—TT— MMPHiiii J
I I
i
i 1 1
it follows that
1 k
Ü
R(a. , • • • »a, ) + RCa» a») ^ '*'^aj. Q\ »• • • »^ /-j.)'
+ R<aii»1a)M",\<2))
for every ij^» \ as was to be shown.
2. Application to Reliability Theory
We are concerned with a type of system that has k components.
Associated with each component Is a numerical value. Let {a }
(J m 1 k) denote the set of numerical values of the k components.
1 k We assume that R(a , ...,a ) is the probability that the system will
1 k perform satisfactorily (i.e. R(a ,...,a ) is the reliability of the
system) where R(a ,... ,a ) as a function of a a has the
properties of a joint cumulative distribution function. For example, if
the system's performance depends on values of k random variables,
Y.,...,Y. , to the extent that the system performs satisfactorily, if
and only if, Y < a-', J - l,2,...,k, then if Y.,...^. have joint
Ik Ik distribution FCy.,...,y.), then R(a ,...,a ) » F(a ,...,a ) will
have the assumed interpretation and properties.
Now suppose a^ <^. .£ a-' are n components of type j
(j = 1 k). Then n systems can be assembled from these components.
Let N denote the number of systems that perform satisfactorily. N
is a random variable whose distribution will depend on the way the n
systems are assembled. A direct application of theorem 1 asserts:
k-1 Theorem 2: Of all the (nl) ' different ways in which the n systems
can be assembled, EN is maximized if these n systems have reliabilities
R(a.,...,a.) (i ■ l,...,n).
lllllllM'M ! ! I - ■
.^^^^jaimimaaj^i^».^,^^.. ... ... -.,..,..,.. , ;.t.,m..,..a..«..-,,u-.,. , - ■ J.....v..<...... ,.vi, ,,- ■,...
^ PHIHMI «imwm, UN i. rnn^i—■n«fiiiwiiM..JiwiNWW .1 imm»mmw ■■■u.ii i. 1 .ii^p^ ■pl|liiiilim)lliwi|,>JW.'iwi'"»»Pt»wiW!Mi|H
I [
.
1; I fl Proof. For any assembly defined by i<i.(i) ■ 1, iK (i) (j - 2,...,n)
^ EN- I *bl nV *l (4S K «0-
1:
i-i »i») ^d) >k<t)'
The previous result holds.
We can assert, also, the following result.
1 k Theorem 3: If Rte^ /iv*"'8!!) (±0 1. ^2 for every 1 and
^2''"'^Ic t^en t^e same assembly that maximizes EN minimizes the
variance of N.
n . . . . Proof. Since variance N - ^ RCa^ ^y" ^ (i))(1"R(% (i) % (i)'»
XXX K X K
the truth of the theorem need only be established for n ■ 2. That It
Is true for n > 2 follows from the following lemma:
Lemma: If 1/2 1 P1 1 q1 1 ?£ 1 1. Pi 1 I2 — p2* fnd Pl+P2 - ql+q2
then pj^d-pj^) + p2(l-p2) <^ q^^d^) + q2(l-q2).
Proof: On subtracting the right hand member from the left we get
P1(I-P1) + P2(I-P2) - qj/i-qi) - q2(1-q2)
2 2 2 2 pl + p2 ' pl " p2 " ql ' q2 + ql + q2
ql ' Pl + q2 ' P2 + (pl ' ql) + (p2 " q2)
l|l^^j||ai^„. :.. . .■...■„....„,.„>..,:..^„^.—...^-^■.■.AJ.^-. JJJ.^-,~...,..-^.-..t.J........^Jai^tokiiMtwivMiii*****^^»«^*^^
„***.*&& «ff*^**-.':'.
l
i.
mmmmmmmmiimmmmmmmmmiMmmmP**mmmiM\*i iuww.^-w -.■,i.',-r ■',•-., j'.','i';-v. 4 ■;'.■: ■V/f ■:
(q-L " Pi^^i + Vi> + ^2 " p2^q2 + p2^ ~ (ql " V " (q2 " P2)
(^ - P1)[q1 + ?!"!] + (q2 - P2)(q2 + P2 ~ ^-^
< (p, - qoHqi + PT - 1) + (q9 ~ P,)(q, + p, - 1) '2 V12,VV11 " ^l 2 l'2/VM2 ' t'2
1 1
(P2 " ^^l + Pi " i) " (P2 " q2^q2 + P2 " -^
(P2 - q2)(ql + pl ' q2 " P2)
■(P2 - q2^q2 " qi + P2" Pp - 0•
Some remarks are In order:
1. Theorems 2 and 3 will be applicable to special coherent
structures, namely, those where the reliability function Is not only
monotone but also a distribution function.
2. It should be stressed that the exact form of the distribution
function need not be known, only that the reliability function can be
assumed to be In the form of some distribution function.
-■'-'■■—-^ ' ' :.;».■■.-..,.—.,.......■ ""---^■'^■"■"'itMmiiililriiT 1 •' 1 1' '1 r-'"i ' inii r'iiniiiWirrriniii 1 r
.~**m W*I|>',M"",~'—~'™*~-1—-■—-■—■—-—--—-———————*—---———-—-—^——-TT" n . i _____
i.... „ lH..||.l|..|.l
I I I [3] Another Application of the Hardy. Llttlewood and Polya Result
To Reliability
Consider the following model: A stockpile consists of n units
of each of k different types of Items. Associated with each Item
Is a probability that the Item will perform effectively. We denote
by ?i, j - l,...,k, 1 - l,...,n, the probability that the 1
unit of the type j will perform effectively. (This item will be
referred to as the P^ item.) These probabilities are assumed to be
independent and it is supposed that
?! 1 p2 i ••• 1 pi» j - l.-.-.k.
From these nk items we must construct n modules, where a module
consists of one of each of the k types of items. We say that a
module functions if all of its k elements perform effectively. The
problem is to construct the modules in such a way so as to maximize
the probability of attaining at least 0 £ r £ n functioning modules.
Let us define the 1 module to be that module containing the
item P.. If our objective was to maximize the expected number of
functioning modules, then the Hardy, Llttlewood and Polya result
yields the solution. Namely, that the i module consists of the
12 k items P., P, P.. We now show that this construction not only
maximizes the expected number of functioning modules but it also
possesses the stronger property of maximizing the probability that
the number of functioning modules is at least r, for each
0 < r < n.
mmm mmmm Lmta mm .■■.,..^..^.L«.ia..i.aM ' '■■■--' -"■■
mmm • • '■ ■'■ — ■'"-■'■" p n ' ^ —-- ' im i—— ■ ■ ■■ ■ •
I I ::
ü
i.
Theorem A; The probability of obtaining at least r functioning
modules is maximized, for each r - l,...tn, by letting the 1
module consist of the items
P., P.,.#.,P., 1 ■ lf«a>ln»
Proof: Consider any arbitrary construction of modules - call it
C. - whose 1st module consists of the items
pi p2 p3 k
•L 12 13 1k
where i2 / 1, and suppose that the sth module of C. consists of
the items
pi p2 p3 k
Now consider a different construction - call it C. - whose 1st module
consists of the items
pi P2 P3 pk
1» 1» 0 »• ••»"0 i i ä3 Äk
and whose sth module consists of the items
pi p2 p3 pk
where
1.
£r » min(ir,jr), mr - max(ir,jr), r - 3,...,k.
Suppose further that the remaining modules at CL are identical to
those at C.. The probability that either the 1st or the sth module
10
äauult 1 -
mmmmmmm wmmm mmmii IM iwtmw^r^^i^m» — -^■^^WIIHWI I
I
I I I :
i.
of C. functions Is
l 1« 1^ If. 1 10 1, 1. 8 1 j« j. "2 3 '2 *3
while the corresponding probability under C„ Is
1 1 A3 Äk 1 1 £3 Äk s 12 1113 i^
As the number of the other n - 2 modules which function is
stochastically equal under C, or C, It follows by the Hardy,
Llttlewood and Polya result that the number of functioning modules
is stochastically larger under C„ than It Is under C.. Hence we
need only consider constructions whose first module contains both F^
2 and P.. Similarly we can show that we need only consider constructions
12 k whose first module Is P., P. P-. Repeating this argument on the
other modules completes the proof.
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We make the following remarks:
1. The main reason we have considered this problem is that
often the "total system" works if, and only if, at least r of the
modules function.
2. If we suppose that k ■ 2 and that a module functions
if at least one of the items in the module performs effectively then
a proof similar to the above shows that the construction stochastically
maximizing the number of functioning components Is the one whose 1
1 2 module consists of the items P., P .. .. 1' n+1-1
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REFERENCES
[1] Hardy, G.H., Littlewood, J.E., and Polya, G., Inequalities
Combridge University Press, 1934.
[2] Lorentz, G.G., An Inequality for Rearrangements. American
Mathematics Monthly, 60, March 1953, pp. 176-179.
12
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