256 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-20, NO. 4, NOVEMBER 1971

probability gA /(JA + .s) corresponds to A failing before S back to item A. In such a case, a different mathematical modeland the probability P2,US/(,UA + /s) corresponds to an should be used. There are other models possible, e.g., a modelunnecessary switch (A to B) occurring during the time that A that allows one of the failure modes for S to be independentis operative. of time.

Concluding Remarks Acknowledgment

The derived unreliability formulas for two items with The author is indebted to Dr. J. H. Bennett and Dr. G.sensing and switching are more meaningful than the Sogliero for many valuable discussions and to the referees forcorresponding ones derived in [1 ] using a less complete model. their helpful suggestions.Before applying the formulas in this paper, one shouldexamine carefully the assumptions made in this model, e.g., Referencethe six mutually exclusive events in which the device fails, andcheck against real situations. In this model, we assume that 1] L.A. Aroian, "The reliability of items in sequence with sensing and

after the device is s d tswitching," in Redundancy Techniques for Computing Systems,after the device is switched to item B, no more switching iS R.H. Wilcox and W.C. Mann, Eds. Washington, D.C.: Spartan,allowed. In some practical stuation, the device may switch 1962, pp. 318-327.

A Direct Method for Maximizingthe System Reliability

JAYDEV SHARMA and K.V. VENKATESWARAN

Abstract-A simple computational procedure has been developed for lI. Previous Literatureallocating redundancy among subsystems so as to achieve maximumreliability of a multistage system subject to multiple constraints which Various papers are available on the parallel redundancy caseneed not be linear. The computational time is quite short. Two with a specific constraint. Moskowitz and McLean [1]examples are shown.r

considered the problem of maximizing reliability with a costReader Aids: constraint using a variational method. Proschan and Bray [2]

Purpose: Helpful hints extended Kettelle's [3] computational method for maximizingSpecial math needed for explanations: Elementary probability reliability subject to a cost constraint, to multiple constraints.Special mathl needed for resullts: SameResults useful to: Design and reliability engineers A dynamic programming approach was suggested by Bellman

and Dreyfus [4]. Fan et al. [5] used the discrete maximum1. Introduction principle for maximizing reliability. Tillman and Liittschwager

[6] developed a method for maximizing reliability or minimiz-Reliability of an overall system can be increased by ing cost subject to several constraints by using an integer

introducing redundancies in subsystems. In order to ensure inggramiing formulation strai by used a intexthat factors such as cost, weight, and volume remain within progra prormulation. Mizukami [7] used a convex

integer programming method for maximizing reliability withresources available, system reliability is optimized with respect multiple linear constraints. Federowicz and Mazumdar [8]totesecontrants formulated the redundancy allocation problem to obtainFor solving this problem (or optimizing system reliability) app roximate solutions. Ghare and Taylor [9] maximized the

many authors have used cumbersome approaches. This paper, reiblt of paale reudn.ytmsb rnhhowever, provides a simple computational approach. It has adbudpoeue ir 1]ue iayagrtmtbeen tried on many problems, with satisfactory results . opimz syte reiblt orcs.ujc t utpecn

Manucrit rceiedMrch15,197; rvise Juy 2, 171.straints. All these procedures use time consuming and rigorousThe authors are graduate students in the Department of Electrical methods. Some procedures may not converge to a solution;

Engineering, University of Roorkee, Roorkee, India. some methods provide only approximate results.

SHARMA AND VENKATESWARAN: MAXIMIZING SYSTEM RELIABILITY 257

111. Problem Formulation and Computational Procedure to see that no constraints are violated, i.e., that the system as

Let there be k stages (subsystems) connected in cascade stated is possible.Step 2: Find the stage (of those under consideration) that(series) in a system. Stage i consists of ni redundant (parallel) has the highest unreliabilitY Add a redundant component to

elements each having probability of failure qi. All the elements that stage.in a stage must fail before the stage fails. All stages must Step 3: Check constraints.operate for the system to operate. a)3f any constraintsv

Assuming that failure of each individual element occurs b) If no constraint has been reached, go to Step 2.independently and that the probability of failure is qi : q1 for c) If any constraint is exactly satisfied, stop. Thei # j, the system unreliability is current n is the optimum number of redundant components.,

k Step 4: Remove the redundant component added in Step 2.Q(n) = 1 q-n(1-q1fi) (1) The resulting number is optimum for that stage. Remove this

stage from further consideration.Step 5: If all stages have been removed from consideration,

where n is a vector of nonnegative integers such that n (nl, stop; the current n is optimum. Otherwise, go to Step 2.

n2, * * n*n) represents the number of elements in all stages.The constraints (which need not be linear) on the allocation of IV. Numerical Exampleredundancies are

Consider the following two problems.k 1) Maximize the reliability subject to two linear constraints.

G11(n) 6 b1; j = 1, 2,* , r. (2) The system is shown in Table 1. The problem is to find nl, n2,

=1d n3, n4, n5 to maximize system reliability.Solution: Starting with n = (1, 1, 1, 1, 1) add one element

The problem, therefore, can be stated as the selection of n at a time as shown in Table II.such that Q(n) is minimized subject to the constraints of (2). Hence we obtain the optimum number of redundantIt is easy to show that (for qi small enough) components

kn = (3, 4, 5, 4, 3).

Q(n) _ q(n) - jj q/ni. (3) The system reliability is 0.985. This solution is the same as thei= 1 optimum calculated by other methods.

2) Maximize the reliability with three nonlinear constraints.The problem of maximizing the system reliability reduces The system is shown in the following table.

approximately to minimizing

k Stage 1 2 3 4 5

q(n) qfnli (4) ElementReliability 0.80 0.85 0.90 0.65 0.75

subject toThe constraints are

k

E G,jin).b1; j=l1,2,* * *,r. (5) g1(n)=n,2 +2n22 +3n32 +4n42 +2n26110i=l

g2 (n) = 7(n1 + en 1 /4) + 7(n2 + en2 /4) + 5(n3 + en3 /4From (4), it is apparent that a good way to reduce q(n) is

to pick out the maximum term on the right-hand side and + 9(n4 + en4 /4) + 4(n5 + en5 /4) < 175reduce it by adding a redundant unit to the correspondingstage. The objective is to reduce q(n) in successive steps. The g3 (n) = 7n, en1/4 + 8n2 en2 /4 + 8n3 e'3 /4algorithm to be used is to add one redundant element to thestage with the highest q1fli (assuming no constraints are + 6n4en4I4 +9n5ens /4 S200.violated). This algorithm has intuitive and empirical appeal,but has not been rigorously proved to give the optimum The procedure is the sane as the previous one and is given insolution. The constraints become active only in the neighbor. Table III. When n = (2, 2, 3, 4)-8th row-te fourth stage hashood of the boundary of the feasible region of n. Therefore, the highest unreliability, and one element "should" be addedthe sequential steps involved in solving the problem are as

follows.~~~ ~ ~ ~~~~~~~~~~hr unushuald ctonditiion2s,f this substep may not be apprpiae

system there must be at least one element in each stage. Check if one is violated.

258 IEEE TRANSACTIONS ON RELIABILITY, NOVEMBER 1971

TABLE I

Stage 1 2 3 4 5

ElementReliability 0.9 0.75 0.65 0.8 0.85

Cost 5 4 9 7 7

Weight 8 9 6 7 8

Constraints: Cost-< 132; Weight< 142

TABLE II

Number of Elements Unreliability of StageTotal Total

n, n2 n3 n4 n5 1 2 3 4 5 Weight Cost

1 1 1 1 1 0.1 0.25 0.35a 0.20 0.15 38 321 1 2 1 1 0.1 0.25a 0.1225 0.20 0.15 44 411 2 2 1 1 0.1 0.0625 0.1225 0.20a 0.15 53 451 2 2 2 1 0.1 0.0625 0.1225 0.04 0. 15a 60 521 2 2 2 2 0.1 0.0625 0.1225a 0.04 0.0225 68 591 2 3 2 2 0.1a 0.0625 0.042875 0.04 0.0225 74 682 2 3 2 2 0.01 0.0625a 0.042875 0.04 0.0225 82 732 3 3 2 2 0.01 0.015625 0.042875a 0.04 0.0225 91 772 3 4 2 2 0.01 0.015625 0.015006 0.04a 0.0225 97 862 3 4 3 2 0.01 0.015625 0.015006 0.008 0.0225a 104 932 3 4 3 3 0.01 0.015625a 0.015006 0.008 0.003375 112 1002 4 4 3 3 0.01 0.003906 0.015006a 0.008 0.003375 121 1042 4 5 3 3 0.01a 0.003906 0.005252 0.008 0.003375 127 1133 4 5 3 3 0.001 0.003906 0.005252 0.008a 0.003375 135 1183 4 5 4 3 0.001 0.003906 0.005252 0.0016 0.003375 142 125

aThis is the stage to which a redundant component is to be added.

TABLE III

Number of Componentsin Stage Unreliability of Stage

nl n2 n3 n4 n5 1 2 3 4 5 g, (n) g2(n) g3(n)

1 1 1 1 1 0.2 0.15 0.1 0.35a 0.25 12 73.1 48.81 1 1 2 1 0.2 0.15 0.1 0.1225 0.25 24 85.4 60.81 1 1 2 2 0.2a 0.15 0.1 0.1225 0.0625 30 90.8 79.02 1 1 2 2 0.04 0.15a 0.1 0.1225 0.0625 33 100.4 93.12 2 1 2 2 0.04 0.0225 0.1 0.1225a 0.0625 39 109.9 109.22 2 1 3 2 0.04 0.0225 0.la 0.042875 0.0625 59 123.1 127.52 2 2 3 2 0.04 0.0225 0.01 0.042875 0.0625a 68 130.0 143.62 2 2 3 3 0.04 0.0225 0.01 0.042875b 0.015625 78 136.0 171.13 3 3 3 3 0.008 0.0225 0.01 0.042875b 0.015625 83 146.1 192.5

alhis is the stage to which a redundant component is to be added.bThis indicates that the stage has been removed from further consideration.

IEEE TRANSACTIONS ON RELIABILITY, NOVEMBER 1971 259

to the fourth stage. But this violates the third constraint; Referencestherefore, the fourth stage has n4 = 3 and is eliminated fromfthere csefrat stage has th

3

e highes [11 F. Moskowitz and J.B. McLean, "Some reliability aspects offurther consideration. Now the first stage has the highest systems design," IRE Trans. Rel. Qual. Contr., vol. RQC-8, pp.unreliability. One proceeds as before. The optimum solution is 7-35, Sept. 1956.n is (3, 2, 2, 3, 3). The system reliability is 0.9045. [2] F. Proschan and T.H. Bray, "Optimum redundancy under

multiple constraints," Oper. Res., vol. 13, pp. 800-814, Sept.-Oct. 1965.

[3] J.D. Kettelle, Jr., "Least cost allocation of reliability invest-V. Conclusion ment," Oper. Res., vol. 10, pp. 229-265, Mar.-Apr. 1967.

[41 R.E. Bellman and S.E. Dreyfus, "Dynamic programming and the

This paper provides simple computations for the solutions reliability of multi-component devices," Oper. Res., vol. 6, pp.200-206, Mar.-Apr. 1958.

of redundancy allocation problems. The method is not [5] L.T. Fan, C.S. Wang, F.A. Tillman, and C.L. Hwang, "Optimiza-affected by the kind or number of constraints (except in the tion of systems reliability," IEEE Trans. Rel., vol. R-16, pp.tediousness of calculations). A computer program would be 81-86, Sept. 1967.[61 F.A. Tillman and Liittschwager, "Integer programming formula-simple and computing would be small for many systems. The tion on constrainted reliability problems," Management Sci., vol.simplicity of the method and its suitability for computer 13, pp. 887-899, July 1967.solution make this method highy useful to reliability engi- [7] K. Mizukami, "Optimum redundancy for maximum systemreliability by the method of convex and integer programming,"neers. It has not been rigorously proved that this method is Oper. Res., vol. 16, pp. 392408, Mar.-Apr. 1968.exact, but no faulty solution has yet been found. Even if the [8] Federowicz and Mazumdar, "Use of geometric programming tomethod is not exact, it appears to be very close, maximize reliability achieved by redundancy," Oper. Res., vol.

16, pp. 948-954, Sept. 1968.[9] P. Ghare and R. Taylor, "Optimal redundancy for reliability in

series systems," Oper. Res., vol. 17, pp. 838-847, Sept. 1969.Acknowledgment [101 K.B. Misra, "A method of solving redundancy optimization

problems," IEEE Trans. Rel., vol. R-20, pp. 117-120, Aug. 1971.[111 J. Sharma, "Optimization of reliability function of a system,"

The authors are grateful to Dr. K.B. Misra and to the M.S. degree project, Univ. Roorkee, Roorkee, India, 1971.referees for their helpful comments. Available from the author.

CorrespondenceCorrespondence items are not refereed.

The Digital Computer in Medical Electronics 1) Periodically calibrate the associated instrumentation with respectto gain, zero, and linearity, either fully automatically or under the

H. WILLIAM PERLIS manual control of the operater.2) Guide the medical staff in the proper operation of the

The issues of reliability and maintainability with regard to medical instrumentation via graphical displays.electronic equipment have not, in the past, been given much attention 3) Maintain records of operator errors for subsequent humanby either the users or manufacturers of the equipment. With the factors analysissimultaneous advent of a serious shortage of trained medical personnel, 4) Maintain calibration and maintenance records for guidance in thethe growing demand for high-quality medical care, and the increasing future purchase, modification, or specification of instrumentation.use of digital techniques in a heretofore analog oriented environment, 5) Aid in the trouble-shooting of the existing equipment.the digital computer must be considered as an integral part of a medical 6) Aid in the education of new personnel in proper equipmentelectronic complex rather than the end link in a data gathering and operation and maintenance through the use of "programmed learning"reduction chain. displays.

In reaching a diagnosis or in ascertaining the effects of a course of Many of the procedures and techniques now commonplace in thetreatment, the physician is performing a pattern recognition task aerospace industry could be applied in the medical environment toinvolving many input parameters. The data presented by the electronic increase systems reliability if the system designer is cognizant of thedevices must be sufficiently accurate to aid in this discrimination, special problems inherent in the medical environment.

With the goals of increasing overall system reliability in mind, the Digital computers are being used in greater numbers every year indigital computer can be integrated into a medical electronic system to medical environments to collect and analyze data, to presentperform the following functions. conclusions drawn from this data and even, in some cases, to interact

with the patient in a closed-loop mode in accord with preprogrammed

Manuscript received April 21, 1971; revised August 7, 1971. This algorithms [1]. Therefore it is important to examine the issues ofwork was supported in part by U.S. Public Health Service Contract equipment reliability and maintainability with respect to the currentPH 43-67-144 1. environment of operation and to project to future environments.

The author is with the University of Alabama Medical Center, Whereas five to ten years ago an intensive coronary care unit containedBirmingham, Ala. 35233. several beds with their associated electrocardiographic displays watched