IEEE TRANSACTIONS ON RELIABILITY, VOL. R-26, NO. 4,-OCTOBER 1977 261 Decision Making in Reliability V.S. Srinivasan K.V. Ramachandra index for equipment type; 0 = A, B b index for tests T number of tests Key Words-Distribution-free methods, Hypotheses, Type I error, Zg, ,, sum or product over all ;; 4' is an index Power function, Wilcoxon-Rank-Sum test, Mann-Whitney test, Markov ° type I error chain, Availability. power of the test Reader Aids- Purpose: Report of derivations NOTATION FOR MODELS I & II Special math needed: Elementary Statistics Results useful to: Reliability theoreticians f,.,Fo9(.), R,5(.) pdf, Cdf, Sf of time-to-failure of 0 Summary & Conclusions - Distribution-free methods are evolved for tA maximum failure time of A units in a test 4 models of evaluating time-to-failure. Two decision procedures for rb rank of failure time of B among all units choosing the better of two equipments are given in models I & II. in the test, for test b; rb = 1, ..., N Decision procedures for whether two equipments have the same avail- expf (), expfc (.) Cdf, Sf for the standard exponential dis- ability or reliability are given in models III & IV; in these models, it is tribution; expfc (x) = exp (- x) assumed that the mission time is a r.v. in the interval (0, To). nber suc (x) I ex ( N The method of model I is superior to existing tests like the K a number such that 1 . K . N Wilcoxon-Rank-Sum test and Mann-Whitney test in a particular case. Z(b, K) 1 if rb > K, 0 otherwise; for test b Model II is based on the time taken to return to the initial state by a Xm number of steps needed for Z(b, K) to Markov chain and is optimum for large sample sizes. In model III, even return to Z( 1, K), at repetition m of step though the distributions of time to failure and time to repair-completion 1, in model II. are negative exponential, the method is general, distribution-free and E i optimal. Hence the type I error is constant even if the distributions of E {n; I} s-expected value of n under Hi time to failure and repair-completion differ from the assumed distribu- csqfc(; v) Sf of a chisquare r.v. with v degress-of- tions. Model IV is also distribution-free and optimum. freedom. INTRODUCTION NOTATION FOR MODELS III & IV Suppose that there are two types of equipments, say A and X(¢, b), Y(4', b) time-to-failure or repair-com- B which are meant to serve a common purpose. A manu- pletion of 4, for test b facturer or a customer might wish to decide which of the two 7r(4, b) X(¢), b)/ Y(O, b) equipments is superior. Without loss of generality assume that X¢,, ,p0 constant failure or repair rate the manufacturer claims that B is superior to A and the of 4' customer wants to test his claim. 0¢ a steady state availability B can be said to be superior to A if i) the Cdf of B is always Tm mission time, a r.v.; Tm < To below the Cdf of A, or ii) the steady-state availability of B is V(O, b) 1 if 4 has not survived upto higher than that ofA, or iii) the reliability of B is higher than time To, 0 otherwise; for test b that of A. in model IV There is no prior knowledge about either the value of b total number of tests for which summarizing statistics such as mean time-to-failure or the V(O, b) = 1, for model IV functional form of the failure or repair distribution. Non- n* mn {b,} parametric decision procedures can profitably solve such CO index for recorded observation problems because of their invariant property of type I error on time-to-failure of 4 for under situations that can widely vary from the assumed condi- model IV; Cr1, = 1,. ,. b, tions on the distribution of time between failures. X'(4', C<g) time-to-failure of 4' at recorded observation C,, for model IV I represents 17 or X' ; I is an index COMMON NOTATION FOR ALL MODELS b represents C¢, for model IV un- less "test b" is specified H0,H1 null hypothesis, alternate hypotesis 4¢(.;41,Fp(.;JI),R¢(.;I1) pmf, Cdf, Sf oflI(4, b) i index forH;i=0,1 K¢, (1 -0¢/
Transcript
IEEE TRANSACTIONS ON RELIABILITY, VOL. R-26, NO. 4,-OCTOBER 1977 261
Decision Making in Reliability
V.S. SrinivasanK.V. Ramachandra index for equipment type; 0 = A, B
b index for testsT number of tests
Key Words-Distribution-free methods, Hypotheses, Type I error, Zg, ,, sum or product over all ;; 4' is an indexPower function, Wilcoxon-Rank-Sum test, Mann-Whitney test, Markov ° type I error
chain, Availability. power of the test
Reader Aids-Purpose: Report of derivations NOTATION FOR MODELS I & IISpecial math needed: Elementary StatisticsResults useful to: Reliability theoreticians f,.,Fo9(.), R,5(.) pdf, Cdf, Sf of time-to-failure of 0
Summary & Conclusions - Distribution-free methods are evolved for tA maximum failure time of A units in a test4 models of evaluating time-to-failure. Two decision procedures for rb rank of failure time of B among all unitschoosing the better of two equipments are given in models I & II. in the test, for test b; rb = 1, ..., NDecision procedures for whether two equipments have the same avail- expf (), expfc (.) Cdf, Sf for the standard exponential dis-ability or reliability are given in models III & IV; in these models, it is tribution; expfc (x) = exp (- x)assumed that the mission time is a r.v. in the interval (0, To). nber suc (x)
Iex (
N
The method of model I is superior to existing tests like the K a number such that 1 . K . NWilcoxon-Rank-Sum test and Mann-Whitney test in a particular case. Z(b, K) 1 if rb > K, 0 otherwise; for test bModel II is based on the time taken to return to the initial state by a Xm number of steps needed for Z(b, K) toMarkov chain and is optimum for large sample sizes. In model III, even return to Z( 1, K), at repetition m of stepthough the distributions of time to failure and time to repair-completion 1, in model II.are negative exponential, the method is general, distribution-free and E ioptimal. Hence the type I error is constant even if the distributions of E {n; I} s-expected value of n under Hitime to failure and repair-completion differ from the assumed distribu- csqfc(; v) Sf of a chisquare r.v. with v degress-of-tions. Model IV is also distribution-free and optimum. freedom.
INTRODUCTION NOTATION FOR MODELS III & IV
Suppose that there are two types of equipments, say A and X(¢, b), Y(4', b) time-to-failure or repair-com-B which are meant to serve a common purpose. A manu- pletion of 4, for test bfacturer or a customer might wish to decide which of the two 7r(4, b) X(¢), b)/ Y(O, b)equipments is superior. Without loss of generality assume that X¢,, ,p0 constant failure or repair ratethe manufacturer claims that B is superior to A and the of 4'customer wants to test his claim. 0¢ a steady state availability
B can be said to be superior to A if i) the Cdf of B is always Tm mission time, a r.v.; Tm < Tobelow the Cdf of A, or ii) the steady-state availability of B is V(O, b) 1 if 4 has not survived uptohigher than that ofA, or iii) the reliability of B is higher than time To, 0 otherwise; for test bthat of A. in model IV
There is no prior knowledge about either the value of b total number of tests for whichsummarizing statistics such as mean time-to-failure or the V(O, b) = 1, for model IVfunctional form of the failure or repair distribution. Non- n* mn {b,}parametric decision procedures can profitably solve such CO index for recorded observationproblems because of their invariant property of type I error on time-to-failure of 4 forunder situations that can widely vary from the assumed condi- model IV; Cr1, = 1,. ,. b,tions on the distribution of time between failures. X'(4', C<g) time-to-failure of 4' at recorded
observation C,, for model IVI represents 17 or X' ; I is an index
COMMON NOTATION FOR ALL MODELS b represents C¢, for model IV un-less "test b" is specified
H0,H1 null hypothesis, alternate hypotesis 4¢(.;41,Fp(.;JI),R¢(.;I1) pmf, Cdf, Sf oflI(4, b)i index forH;i=0,1 K¢, (1 -0¢/
262 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-26, NO. 4, OCTOBER 1977
rb(l) rank of I(B, b) among {I(B, b), The results are
I(A, b}Z'(b, 1) I if rb(1) = 2, 0 otherwise a = 1/32 - 3.1%pi(f), qi(J) Pr {Z'(b, I) = 1} under Hi;
qi-1 -pi(I) = [1 -(l/2) expfc (A)] 5
S(I) ZbZr(b, I)binm (.; p, N), binfc (.;p, N) pmf, Sf of binomial distribu- The power function m is plotted in Fig 1 against A. The ( is
tion. compared with "Rank-Sum-Test" of Wilcoxon [3] for a = 5%.The decision procedure (1) is superior to the Wilcoxon Test(the Mann-Whitney Test is equivalent for no ties [3]) even
Models I & II when the type I error is smaller for decision procedure (1).
1. One B unit and N - 1 A units are put on test (a total of Nunits). Record the rank of the time-to-failure of the B unit. 100 -
2. Test 1 is run a total of T times. In model I, T is fixed inadvance. In model II, Tis a r.v. 90
3. All times-to-failure for A units are i.i.d. All times-to-failure WRSfor B units are i.i.d. Failures of A and B units are s- 80independent.
70-The decision problem is formulated as- 160
H:A andB are i.i.d.50-
H1: RB(X) > RA (X) for every x, viz, B is superior to A.40 I
MODEL I 30- WRS-WILCOXON RANK-SUM TEST;4-5%(I)-DECISION PROCEDURE(M);
Decision Procedure 20- 1/32 - 3 1%
RejectHoi.f.f. rb=Nforallb (1) to
Type I Error o2 3 4 5 6 7
From the theory of extreme value statistics [1, 2] it is A
known that - Fig. 1.
a = (1IN)T (2)Case 2
Power functionFB(x) = FA(lX),r?Kl
The performance of decision rule (1) is evaluated for twocases. The power function is- FA(x) = expf(x)
( = HbE{RB(tA)}. (3) N = 2,T = 5
Eq. (3) is evaluated for each case by putting the expression for The results are-FA into RB.
az = 1/32 3.1%o,Case 1
(3 = (71 + 1-FB(x) = FA(x -/),/\> 0
The power function (3is plotted in Fig 2 against ri. The (3isFA(x) = expf (x) compared with "Rank-Sum-Test" of Wilcoxon [3] for ae = 5%.
WhenN =20 and T -1 50 that al = 5%, (1) and the WilcoxonN = 2, T =5 Rank Sum Test are equivalent and the power curves coincide.
SRINIVASAN/RAMACHANDRA: DECISION MAKING IN RELIABILITY 263
Hi forms a Markov chain [4] with the transition matrix
0 i Var {n;i} = [2/[ppiqi] - 6 (8)01 02 03 04 05 06 07 08 09 l0
rl From the central limit theorem, it follows that for large M,Fig. 2. the distribution of S (defined in step 3) under HO approaches
a chisquare distribution with v = 1. IfK is chosen such thatMODEL II y < 0.5 then it can be proved that
Decision Procedure Var {n; 1} > Var {n;O}. (9)
Step 1 Choose a value of K, say k. Then for each test b (see Hence this method is optimum for large M.assumption 1), calculate Z(b, k) and form the sequence{Z(1, k), Z(2, k) .... . }. Stop running tests when Z(b*, k) THE POWER FUNCTION= Z(l, k) for b* > 1; calculate Xm = b* - 1. For example, if{Z(b, k)} = {O, 1, 1, 1, 1, 0}, then b* = 6 and Xm = 5. Since for large M, the distribution of S Var {n; 0}/ Var
{n; 1} also approaches a chisquare distribution with v = 1, theStep 2 Do step 1 a total ofM times. The repetitions are power function iss-independent.
P1 = (1/3) expfc (2Ai),SteP S Reject H0 i.f.f. S > S (4)
= 1 - (1/3) expfc (2A/).Analysis
The power function ,B is plotted in Fig 3 against A\. The ,B isIt is easy to verify that the stochastic process Z(b, k) under compared with "Rank-Sum-Test" of Wilcoxon [3 ] for az = 5%7.
264 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-26, NO. 4, OCTOBER 1977
Decision Procedure
Step 1'. For each test b, i) calculate 77(Q, b), ii) compute the
90 / rank of r(B, b) and iii) calculate Z'(b,77). Then compute S(r?).808 / Step 2'. Determine S* such that binfc (S*; 1/2, ) =
30 1 / The values of pi(71) can be determined from
20 / pi(71) f0 FA (X; 7)fB(X; 7l)dx.I10 Thus-
O0 Po(1) =21 2 3 4 5 6 7 2
KAKB log (KA/KB)Fig. 3. P,(71) I + [KA/(KA -KB)] (13)
(KA -KB)The decision procedure (4) is superior to the Wilcoxon test(the Mann-Whitney Test is equivalent for no ties) WhenM is pf 6B -p6A, then KB -+ KA From (13)lt follows thatlarge. P1(??) -oP0(r?) asKB -+KA, as it should be. It can also be
proved from (13) that when OB > Pp1(71)> 1/2.
MODELS III & IVPr{S(1) = r; HO is true} = binm (r; 1/2, T) (14)
1'. One B unit and one unit A are put on test (a total of 2units). For models III & IV, record the times-to-failure ofA & B units. For model III, record the repair-completion Pr {S(rq) = r; H, is true} = binm {r; p1 (a), T} (15)times of A & B units.
2. (Same as models I & II). In model III, T is fixed in Since p I (71) > 1/2, when H1 is true, it follows from theadvance. In model IV, T is a r.v. Neyman-Pearson lemma that the decision procedure (1 1) is
3. (Same as models I &II) optimum.4'. All times to repair-completion for A are i.i.d. All times to The power function f is-
repair-completion for B are i.i.d. Failures and repairs ofA & B are s-independent. = binfc (S;pI(jq), 7). (16)
MODEL III MODEL IV
The steady state availability of b is given by the well-known The decision problem is formulted as-asymptotic renewal theorem.
= ~~ + ii,~~~,) (10)H RB(Tm;X) =RA(Tm;X')foreveryTmH1: RB(Tm;X')> RA (Tm;X') for every Tm
The decision problem is formulated as-Decision Procedure
Ho: 6A =BStep 1" For each test b, record X(¢, b) i.f.f. V(Qb, b) = 1 and
H1: 0B > 0A the corresponding times-to-failure which are recorded are
SRINIVASAN/RAMACHANDRA: DECISION MAKING IN RELIABILITY 265
{X(¢, C,d,), C<, = 1 ....... b; b¢. T}. Stop running tests REFERENCESwhen n * = N.
[1] P.G. Hoel, Introduction to Mathematical Statistics, Asia Publish-Step 2" Compute S(X'). ing House, Bombay, 1957.
[2] E.S. Keeping, Introduction to Statistical Inference, D. Van3,, Nostrand Company Inc., New York, 1962.Step 3" Determine S* such that binfc (S*; 1/2, ) [3] M. Hollander, D.A. Wolfe, Nonparametric Statistical Methods,
John Wiley and Sons, New York, 1973.Step 4 Reject Ho i.f.f. S(xI) > S* (17) [4] W. Feller, An Introduction to Probability Theory and its Appli-
cations, Vol. I, John Wiley & Sons, New York, 1957.
P {S(X') = r;H is true} = binm {r; P, (X[') N1 /9A Dr. V.S. Srinivasan (born 1940 Jan 9) is a Sr. Scientific Officer in1r{S(t) = r;HI '} = im}(1)Electronics & Radar Development Establishment, Bangalore India. His
interests are in the area of signal detection, statistics, and reliability. Hewhere received his BSc and MSc degrees in Statistics from the University of
Madras and the PhD in Operations Research from the University of
p1 (X') -f RB(x; X')fA (X ; X')dX. (20) Delhi. He has several publications to his credit in the theory of0 reliability and signal detection.
Since p1() > 1/2, when H1 is true, it follows from the K.V. Ramachandra; Computer Group; LRDE; Bangalore-560001Neyman-Pearson lemma that the decision procedure (17) is INDIA.Optimum.
The power function ,B is- Mr. K.V. Ramachandra (born 1939 Mar 30) received his BSc and MScdegrees from Mysore University and Karnataka University. He is em-ployed in Electronics & Radar Development Establishment, Bangalore
binfc (S*;,p1(X), N). (21) India. He has registered with Bangalore University, India for his PhD
degree in Mathematics. He has published several papers in Mathematics
ACKNOWLEDGMENT and Physics. His interests are in the area of signal detection and estima-tion in Radar and reliability theory.
The authors are extremely thankful to the Editor for his Manuscript received 1975 November 13; revised 1976 April 22, 1976valuable comments and constructive suggestions. Thanks are June 10, 1976 July 10, 1977 January 8, 1977 February 10.also due to Dr. R.P. Shenoy, Director, LRDE, Bangalore, Indiafor his encouragement and permission to publish this paper. nun
Man u scri pts Rece ived For information, write to the author at the address listed; do NOT write to the Editor.
"Human reliability", H. Kragt; Dept. of Industrial Engineering; "Correspondence on 'A method for calculation of networkEindhoven University of Technology; Eindhoven, the reliability"', "Sequential technique for system reliability eval-NETHERLANDS. uation", S.K. Banerjee; Dept. of Electrical Engineering; Indian
Institute of Technology; Powai, Bombay 400 076 INDIA.
"Enumeration of the total number of redundant structures of "Computing reliability and repair time distribution of ann-identical components", Ivan B. Ram; Bhabha Atomic Re- almost hierarchical complex system using reduced statesearch Centre; Reliability Evaluation Laboratory; Trombay, enumeration", Stefan Arnborg; Forsvarets Forskningsanstalt;Bombay 400 085 INDIA. National Defense, Research Institute; S-104 50 Stockholm 80
SWEDEN."A note on 'Redundancy optimization in general systems" ', H.Sivaramakrishnan; Transmission Research Department; Indian "Estimation in a Pareto distribution derived from stochasticTelephone Industries Ltd.; Dooravani Nagar; Bangalore-560 hazard functions", Dallas R. Wingo; 5020 South Lakeshore016 INDIA. Drive #2711; Chicago, IL 60615 USA.
