On Optimal Reliability Design: A Review

ON OPTIMAL RELIABILITY DESIGN

A REVIEW

K. B. Misra, Electrical Engineering Deptt., Uni versi ty of Roorkee, Roorkee-247667, U.P. India.

constraint function fo(, 1 th type constraint and j th stage INTRODUCTION

ABSTRACT Once a system has been desi gne d to meet it s operational requirements, an analysis i s usually carried out to redesign the system such that reliability g oals are satisfied.The objective of this analysis is to work out a tradeoff between the design parameters oased on system reli a bility effectiveness, aJld the availability of resources. Efforts have been made to systematize r el i ability desi gn procedures,however much remains to be done in t h is area • Ai t hough each system is unique, a general design philos o phy and methodolo gy could be described tnat may be found useful for designing reliable systems. '£ he paper descri bes state of art on optimum design of systems based OH reliabili t,f effectiveness.

A sy s tem is characterised by its expected use . .iach system is unique and has a well defined job to perform. System definition therefore consists of stating what the system is required to do, what its subsystems are, functional inter-relationships of the constituent components and t he environment under which they operate. Once a hardware concept of a system has been developed a syste~ designer is faced with a probl~ of how best his s/ stem design matches with its projected performance during the intended period of i t s use. Th is leads to setting of some objectives and requirements based on certain s ys tem parameters and to check whether or no t these objectives would be met when the system is put into operation. If possible the design may have to be changed or improved to sati sfy these requirements. In other words,system effectiveness measures should be carefully examined in the final design of a 8'Jstem. As systems have grown to be complex, sophisticated and automation involved,one suc h measure of system effectiveness,that can not be sacrificed is reliability of the system. 'l 'he reliability design of a system t he refore has got to be carried out, i f the system is to yield its de sired performance. An optimal reliabilit y design is one which explores all pos~ ible means available to a designer to enhance the reliability of the system.Of cource,constraining thi s activity is the factor of restricted availability of resources.

NO'l'A'l' IO N system reli abili ty, 0 <. 11. <. 1 system unreliability,~ s= 1 - Hs numb er of subsystems ins a syst em, comoonent reliability of stage j, J <. p . d suosyJste m reliability of stage j, O<~.<.~

m.

J

'l

C

sub sy~s te~ unreliaoility,~j = 1 - R . J mi n imu:ll component s at st age j, operatiofl3.l r equirements , re d unda.ncy level at stabe j, total number of co:nuonents system inh ~ reilt availabili ty( steady st c:te), J~ A ~1 suosystem av;,Jrability,0 ~ Aj~' constant failure rate for the co mponents at stage j constant re pair r ate for t he compo nents at stage j mission time c 8 pi tal a vailable for ~he system desL;;n co 'a pone:lt cost f ClHction at s tage j nUiiluer of cOflstraints suos cri pt for t Il e constraint type, 1" 1 ( r reso~rces allocated to 1 th type constrain·t

Some of t he various means of increasing system reli ability are1 ) reducing the complexity of the system, 2) increasing the reliability of constituent components through a product improvement program, 3) using struct ural redundancy, 4) practi siI1i:; a planned mdntenance and repair schedule Curtailment of system complexity may yield in poor stability ani transient response of th e sy s tern and reduce ac curacy and

27

degradation in the quality of the product. The part improvement program demands the use of improved package, shielding techniques,derating etc. Although these techniques result in reduced failure rate of the component,but require more time for design and special state of art of production. Therefore,the cost of part improvement program is very high and may not be very practical in designing a system economically. On the other hand,employment of structural redundancy at subsystem level,keeping specific system topology,can provide a very effective means of improving the system reliability to any desired level. Structural redundancy may involve use of two or more identical components,such that when one fails, others are available in a way that the system is able to perform the specifhd task in presence of faulty components. Of course,depending upon the type of subsystem,various forms of redundancy schemes(vi~ active, standby, partial etc. )are available. The use of redundancy provides a quickest solution if time is main consideration,easiest solution if the component is already designed,cheapest method if the ?ost of redesigning a component is too h~gh and the only solution if the component reliability improvement is not possible. Thus much of the effort in designing system,is employed in allocating resources to incorporate structural redundancies at various subsystems which will eventUally provide a desired value of system reliability. Maintenance and repairs, wherever possible, undoubtedly boost the system reliability and should be employed in an optimal way. These facilities when combined with structural redundancy,may provide almost unity reliability for a system. Therefore,the basic problem in optimal design of a system is to explore t he exte~t of use of the above mentioned means of ~m­ proving system reliability within the resources available to a designer. Thus optimal desi gn of a system is an Operations Research problem. A proper formulation of problem is neede ~i to carry out such an analysis. The models used for such a formulation should be both practi cal and arnenable to known mathematical techniques of solution. On the other hand too much dependence on 'engineering judgment' as ag~ inst exact mathematical techniques may also mislead to an uneconomical or inaccurate designs. To illustrate tnis a simple example of a system, wi th two silb-systems. in series,is taken. It is desired to max~­ mize the system reliability subje?t to a cost constraint on system of 5 un~ts.Let us assume for simplicity that repairs are not carried out and the subsystems under consideration may employ active redundancy. The syste~ data is as follows.

28

subsystem 1 ~omponen t­

reliabili ty Componentcost

sUDsystem 2

0.60

0.65

2 units

1 unit

Obviously,through an engineering judgment, one would be tempted to improve upon the 'weakest link' in the system and arrive at a design: 2 units in parallel for subsystem 1 aad a nonredundant unit for subsystem 2 with a system reliability of 0.546.i3ut thi s figure of syste m reliability is not an optimum one. Through mathematical techniquffi o f Ooerations :tesearch one would obtain an opti;num design for the above mentioned sy fro t em as: non-redundant unit for subsystem 1 aIld three units in ;:Jarallel for subsystem 2 wi th an overall system reliability of O.5 743 ,which is undoubted ly superior to t t2 previ ous desi gn using engineering judgment. SYS'f EM EFFEC'. rIVENESS I HDI CSS A:'mrHEIR ASS ESS;i!:EN'i'

Generally, reliabili ty is considered to be a good criterion for system effecti veness and therefore as a system design parameter. However,depending upon the mission of the system some other criterion may be selected for reviewing alternate designs. Amongst the some other pa rameter s that mav be of interest are: 1. Availability,

2. Probability of mission success, 3. :'\ iean ti me to failure, 4. Duration of sing18 do wn time, 5. Operational readiness, etc.

Anyone of the aoove( or more ~h an. one can also be used) may form the cr~ tenon of optimal system design al1.r1 therefore, be tradei off with some of t ne co nstrained resources suc h as cost,weight,poVler consumption etc. J:<'or non-maintained systems, generally an index of reliabili ~y i s sufficient for the system reliability effectiven~ss. B~~.f~r maint ained systems, where repal.r facll. ~ t~e8 exist,it is not enough to base design on reliabili ty cri teri on. J<'or most of the processes or energy syst ems , the basic question is how much percentage of time, the system is available. This establishes a criterion for desi an based on system availability and thus oneOmay be interested in maximizing the system avail abili ty. Al ternate~y one may rather be interested in compar:ng the desian alternatives based on durat~on of single down time. ilhat ever may be the index of assessment, one should be able to build a mathe:natical mo del for the design problem that will fit into the present daytechniques of solution. techniques for the assessment of reliabili ty, from the knowledge o~ reliabilities or failure data of t ne const~tuent compo-

~he

-nents,has been very widely covered in the literature(1-36). The same is true for arw other index if chosen other than reliability. A wide variety of methods exists for reliabili ty calculation of general networks (i.e.,series-parallel or non-series parallel, maintained or non-maintained). Since for the sake of this paper,it is not necessary to go into details of the methods available for the calculation of reliability,availability or any other index,it is assumed that we have with us a procedure that calculates these parameters. A good review of the methods available for sys~em reliability calculations is provided in(33) The systems where components may have more than one type Of f~iture mode,have also been considered 6,~, 2-16). Also the maintained systems present no difficulty and have been dealt with nicely in (5,9,12,14, 15,16,19,21,23,24,26,27).

usually assign the number of identical uni ts at various subsystems in a way as to optimize the objective set. The objective could be maximizing reliability or some other parameter subject to a given budget limit or alternately minimize the spendings on a system subject to the condition that the system meets the reliability requirements. Since through redundancy allocation,we seek a set of parallel units for subsystems that can only be integers, the problem is that of non-linear integer programming and is di fficult to solve. It may be noted here that the ty~e of redundancy also plays an important part in arriving at an optimal system configuration. All subsystems may use similar type of redundancy or different types for different SUbsystems. rhe general types of redundancy that could be used for subsystems are i) standby, ii) active, iii) partial or fractional.

ALLOCATION PROCESS IN DESIGN The usual basic design procedure(8),a system designer has to follow, is to 1. define system reliability in terms of the operation requirements 2. develop an index of system reliability effectiveness and arrange the system into several non-interacting subsystems, 3. apply the mathematical techniques to evaluate alternate system configurations in terms of reliability (or any reliability index) and cost (or any other constraint on resources) , 4. specify a system configuration, maintenance policy and the relationship with other factors, 5. allocate failure and repair rates to each individual components so as to meet the required system reliabili ty goals.

'l 'he difference between (ii) and (iii) is that of capaci ty of each parallel unit and the subsystem loading requirement. For a detailed study of these types, wi th and without r~pairs,one could refer to(1-l8). References( 1 < 1 -16), in particular, present various cases 01' these redundancies under different conditions. Reli abili ty Allocation In allocating reliability,one i8 concerned wi th assigning of indi vidual component reliability so as to meet some prescribed level of system reliability subject to ce~ tain constraints on cost, weight etc. This may mean increasing the reliability of canponents from a currently achievable level and we assume here that the state of art permits this increase. This mR,Y also mean decreasing the failure rates of components at vario'-ls stages of a system to obtain an acceptable level of system reliability.Unlike redund ancy allocation, the rel iabili ty allocation is not an integer programming problem and is not so difficult to solve.

The above guidelines are by no means universal and the actual design procedure would depend entirely upon the system under study. However, it has establisned a fact, that allocation process either of reliability or redundancy to arrive at an optimal design, forms an integral part of the whole design procedure.

VALIDITl' OF A MODEL In most of the papers from the literature (37-90),a mathematical model for a system, constitutes of subsystems being in series. 'l 'his model is valid in most of the practical situations. A series model Signifies that for th e success of th~ whole syste~, success of each SUbsystem is important and necessary. Only those subsystems are considered in the model that have the above property. One should ,therefore,exercise restraint while prepari ng a mathe'llati cal model. Another important assumption that goes with a model of this type is,statistical independenc e of each subsystem,i.e.the failure

'He have already seen in an earlier section that use of either redundancy or high reliability components, are a few means of achieving a desired level of reliability of system, therefore, the designing includes an allocation of either redundancy or reliability in a system.

Redundancy allocation In allocatin g redundancy to a system we

29

or alternately minimize,either(-ln Rs)

of a unit in a subsystem does not cause or lead to the cause of failure of a unit in the other subsystem. Here,again care should be taken that a primary system and the secondary systems have either decoupling or are suitably represented in the ~odel dependin;; upon the degree of influence. The same is true for the other interacting systems.

n - " in R. (m.) L J J j =1 n or (-In Ass) = - ~ln Aj (m j ) J =1 subject to the constraints, n g (m.) = b gl j ( m j) ~ 0; 1 = 1 ,2, ••• , r 1

The independency of subsystems maices formulation of design problem little easier to attempt. In case of a maintained system,it may also nave to be asswned that tile repair facili ties do exist for each subsystem and tnat the repair on a subsystem could be carried out without affecting the repairs on other subsystems. This independence allows us to consider each SUbsystem separately as far as the expression for subsystem reliability is concerned. The same is true of nonmaintained syste~s. Any complicated situation Can be considered within a subsystem so long the independence of the subsystems is maintained.

, mj being integers

n

Problem 2 : Minimize

Z c. (m j

=1

J

). j

subject to the constraints, g1 (m j ):: n :f.j(m j ) -In Rs~O j= 1

Lln

ani gl (m j )

~

0 for 1=2,3, ••• ,r.

If availability is of interest, R. in abOlTe is replaced by A. and R by A . J 'f he first constraint inJabove rs of ~&urse on reliabili ty. In problem 1 , the r-constraints may involve a cost-constraint also. 'rhe cost-constraint has been considered linear and nonlinearboth, in the literature. It may be noted here that a series-model offers both obje~ ti ve function 'm d constraints of separable forms and are easy to handle. Also it is wortn mentioning that in case of maintained systems, the subsystem reliability R. would be computed with re[.lairs conside- J red,Le.R j = f(m j , '\j and ~j) for a given repair facility and exponential distributions taken for failure and repair times. :lhereas for nonrnaintained systems Rj = f(m j , Aj ), of course all ).j known, R j is basi cally a function of m .. However, if "'j or rather Pj are also tiLcen 2.S variables, then the problem becomes that of mixed-integer programming. In that case, constraints may also be expressed in terms of Pj and mj •

Amongst the various design problems(37-9 0 ) that have been considered in the literature tne following problems are widely covered. 'r hey could be broadly classified in two groups-one covering redundancy allocation and the other,reliability allocation. Redundancy allocation

The first problem can be stated as;max~m~­ ze system reliaoility or availability subject to given r-constraints,linear or nonlinear,i.e. glj (m j ) may be linear or nonlinear in m.. 'r he system has a seriesmodel of nJsub-systems. n Pro blem·l : illaximize Rs 11 Rj(m j ) j=l n or Maximize Ass= Aj (m j ) j=1

(b)

n-

Reliability allocation

For a series-model of n subsystems, the proble:n Clay be easily formulated as

n

L

LJ=1

Sometimes i t may also be of interest to minimize cost of a system subject to the condition that the system atleast has a figure of reli abili ty or availability greater or equal to a given value.

DESIGN PROBLEMS

subject to the constraints,

1

-

mj >--.\

For a series-system model,repairs have ef~ ect as far as the subsystem repairs are concerned since once a sUb-system fails, the whole syste!J1 fails. For the reliabiliw of such a system,one has to improve the weakest subsystem in the system but not overlooking the ~vailability of resources. In certain casest47) a system model may not be series,however the type of model depends entirely upon the functional interactiDn and topology of subsystems or components. The assumptions in preparing a mathematical formulation of design may also thereby change with the type of model.

( a)

J

PrOblem 3 : maxi!llize rl s

glj(m j ) ~ bl ;1=1,2 •.• ,r.

j =1

s. t.

30

witl, R. <: Jmin ...

~

R. J

system cost. They derived maximum reliabili ty for a fixed system cost and therefore,solved an unconstrained problem,using a variational method. Gordon(38) also tried a problem of a single constraint using standby redundancy. The method of (37 )was extended to include any number of constraints by Misra(40),but this is again an approximate method of solution. This required an estimate of system reliability. References(9,15,38) describe a trial and error procedure for reliability optimization under constraints.

1 ,2, ••• , r.

1 R. ¥ J max

j.

In case the model is of non-series Parallel type, the reliability expression of such a system is of non-separable form and thus can not be analysed as multistage process. '1'he reliability of such a system will have got to De found using techniques described in (1-36). The problem then can be stated as Problem 4 maximize Rs == f (R 1

'

R2 , ... ,Rn)

n

s.t'?=lglj(Rj)~bl; wi th (c)

1, 2 , •••

1

,r.

(R . -V j. J.... J max

~ R.

R.

J min"

AVailabilitYallocatio!,l

In designi~ systems for reliability and repairability, one may like to trade-off between system mean time to failure MTBF, and syste m meantime t o repair, MTTR. The trade-off is based on cost and availability; A system designer,for instance may be interested in determinill6 the pair (:ll'1'B F,MTTR) for which availability is maximum subject to a cost constraint. Hence the proble~ of allocation for a series-model will be Problem 5

n jIJ1 [ lil'rBF/iVlTBF+luTTR] j

maximize A -

subject to a constraint n

on cost of

L

c j (Wl'3F, MTTR) ~ C

j=1 also, (MTBF)j (Ml'T R). J or alternately

Everett(41)attempted to solve redundancy optimization problem through the use of Lagrangian multipliers but hao considered only one constraint. Misra( 42 )described an approximate method for any number of constraints yet keeping the computational effort minimum. MessingerJet al.(43) provide a good review of earlier methods,and have considered approximate methods of allocating spare units based on incremental reliability per pound and Lagrangian mul ti plier algori thIns. They have also considered standby redundancy, and certain series-parallel systems. Banerjee and Rajamani(44) used an analytic approach after formulating the redundancy optimization as non-linear programming problem through a parametric transformation and finally using concept of u~dominating sequence as described in(9). Tillman(45-47)provides two powerful and general methods for reliability and redundancy optimizations using Discrete Maxi.Im.l!n Princi ple and Sequential Unconstrained Minimization Technique (SUMT),and considers both series-parallel and nonseriesparallel systems. The constraints are also allowed to be nonlinear. However, in( 45 and 46) the solution of nonlinear transcendental equations may not always be easily possible. Convergence and accuracy of solution are not guaranteed. Federowicz and Mazumdar(48)solve for optimal redundancy using Geometric programming fo cmul at ion. They again provi de an aDProximate solution and round-off to the nearest integers. Geometric programmingis fairly simple for a problem of single constraint but not so attractive for higher number of constraints. Misra and Sharm a(49) provide a simpler Geometric progra:n:ning formulation than (48) and allow consideration of switching redundancy al so.

0 ~ 0 ; ¥ j.

;>;

minimize co st, s.t. an aVailability constraint. The cost functi on above is gene ralJy a nonlinear function of design variables. A SHO:\T :lEVIE;{ OF !vl ATHSI.1 A'l'ICAL TEC:INIQUES iJlany authors( 37-9 0 )have considered above problems using various formulations ann computatio n al techIliques. The methods available are: gradient; variational;dynamic programming;integer programming,including b ranch and bound method; geomet ri c programming,direct searcn, etc.

A least-s~uare f ormulation is presented in (50,51) which is justified as a simpler formulation, especially at the design stage. :£ he solution is obtained in (51) throue;h a direct search procedure requiring only functional evaluations and the~ by allowing a vari ety of problem al terations or modifi cations.

i.1 oscowitz and i,: c:uean(37)perhaps were th e firs1; to formulate mathematically t h e optimization of sy s tem reliability subject to

J sing generalized relationships between

31

reliabili ty and other measures of sy stem effectiveness s~ch as cost,weight,~~d performance,reso~rces are allocated to various parts of the system(52). l~euer,et al. (52) basically used Lagrangian multiplier in trade-off analysis. Laut(53) presents novel effectiveness ideas in trade-off procedures for subsystem design improvement based on the concept of subsystem perform· ance,reliability,availability and cost effectiveness. The variational method,Least square formulation, and the discrete maximum ,., .... inciple,althou6h being versatile,offer only an approximate solution. Geometric programming also provides an approximate solution after many simplifying assumptions. In most oftthe aPDroxima~e methods. the basic assump ion rema1ns OIle same:tne oec1s10n Va rlables are treated as beinc continuous and the final integer solution is obtained by rounding off the real solution to the nearest integers. Usually this procedure is sati sfactory and fortunately provides a true optimum solution. 'l 'his is due to the objecti ve function oeil16 well-behaved. '£ herefore,inpractice,the ap~roximate methods are preferred t o obtain an economic al solutio li. Dyna~ic programming, Inte ge r progrwnming, BranCh and Bound and the di rect searc h techniques(54-8 1) have also been sug5ested for reliability design of a system. lhese teChniques fall under the category of exact methods but are generally timeconsuming or requi re a big chunk of computer memory. An article by Bellman and Dreyfus(54)gives a dynamic programming solution to be optimal redundancy problem, and a orocedure is suggested for considerini!: alterna te desic;ns at each stage. Xe ttelle( 55) ,again usine dyn::lffii-c progra.'nming (D.P) developed a simple algorithm and a computer program for the problem arli illustrated wide aP tJli cations for t he model. Another dynwnic programming model by Liitschwager( 56 ) solves the proble'll of maximiziIl<'; reliability per unit of power used,subject to R cost constraint. 'r he model also permits selecti on of desi Gn alternatives for e a c h stage. Proschan and 3ra.;( 5'7) consider a g ener alization of Kettelle' s( 55) dyn am ic programming algorithm to include m~ltiple constraints. Fyffe et al. (58 ) also describe a method similar to tue aoove ones.However, Burton and i-1 oward(5 9) attempT; to generalize th e D.P. techniques for apylication to networks wi th elements in series, parallel or combination of these. Jensen(60) describes a similar aP9roach. 11ie se t echniques are good for a sm a ll sy s tem and C311 be effectively used onl y with one or two con straints. Misra( 61) describes a summation form of functional equation wi th a view to overcome the computatiunal ila~grds and memory requirements of dynamic programming formulation. iloodehouse(62) suggest s how to solve three constraints problem using .::l. P.

Kulshrestha arld Gupta(63) attempt reliabili ty allocation in a fl.Jn-redundant serie s systen wi til a given budge t, assuming a cost reU ';bili ty relationship for ee.ch comp o nent type. They use D.P.to achieve thi s allocation but with the assum p tion of high reliability of components. Another approach is to formulate the redun dancv problem as an integer linear program-ing model. A knapsack formulation fo r the optimal redundancy problem is given by v.Hees and v.d.Y.eerendonk(64). T ill~an and Liittschwager(65) formulate the problem as an integer linear program. Their model permi ts inclusion of multi ple constraints for a system. Furthermore,their model can be used to select the best alternate design at each stage. Kolesar( 66) gi ves an inte ge r linear programming model incorporating multiple constraints and considers various failure modes. Using the saIDe general approach, iYii zukami (67) ap p roximates the reli abili ty conCave o bjecti ve fUIlction by piecewise linear functions. Tillrnan(68)formulated the problem as an integer programming problem a nd consi dered "the subsystems whim can fail in s everal modes and are subject to linear or nonlinear constraints. An ap~ lication to life support system is given in (71). Ghare and 'raylor( 70) provide another procedure known as3ranch and Bound in the literature. lVli sra and Sharma(74) solve reliability problem USill6 zero-one programming and a nonbinary tree search procedure.This is sh0wn to be an effective method on a computer. Bodin(69) considers parallelseries and series-parallel problem, and a mixed problem an(l presents an alternate method for solving such problems for more restricti ve condi tions. Misra(72) describes a 0-1 type of formulation and an algorithm (L-B) for three different cases of optimization problem. This approach can be used for solution of reliability problems wit h any type of constr8ints. ~e f e rence ( 73) overcomes the size-problem of a system by mod ifyinG the ?pproach of (72). '1'he algorithm of (72) i s vers9.tile 9.nd solves several types of design problems efficiently,exactly an r:l ecnnomic :l.lly. 'r he method :naj prove to be quite adv an.tageous if a good starting point is chosen otherwise the number of vari able s increases (al t holJ.;h not too rapidly as with others) with the size of the systeT. an.,l \\"i th low 'C'J 3t coeffi ciR!lts' of subsys tems in relation to permissible r es·J urces fOT constraints. Inoue, et al. (75) depart fr om the conve ntion 31 proOle Ei ?..nd present all optimal reliabili t~' de~i g n of process systems t:l.ki~ the c a03c it ; of units into account. The suosys~ex configarat i ons studied i n(75) are 0:'- ac"ti ve an .-] stan dby partial redundarwy form. l'be algori tll'li used is t he sa.'!le as sUi.:;.:;ested in (72).

32

Misra (76)further advances the use of the algorithm of (72) for considering optimal design of a system of sUb-systems which may employ any general type of redundancy,i.e. standby partial or active. This is described as a mixed-red undancy system. In( 76) the original L-B algo ri thm of (72) is modified to suit reliability design problem. Several steps, for reducing variables and computer time,are suggested. Misra(77)uses this modified algorithm(76) for the design of a maintained system with any general type of redundancy and repair facility. Availabili ty and reli abili ty optimizations are considered in (77) with a non-linear cost function. Thus this algorithm allows solution of more practical design problems Another series of papers(78-8l,84) describe techniques that can be used for a system with many subsystems and are simple, economical and provide helpful tips for quick redundancy allocation. These algorithms often provide optimal solution. A new aspect of desi~n is considered in( 82~ Misra and Ljubojevic(82) allow consideration of component reliability improvement and redundancy both in arri ving at optimal design. A typical cost-curve is considere~ This design problem, basically a mixed-integer programming problem,is solved by a simple technique in(82). Lientz's(83)method for maximizing system reliability is based on stochastic approach wherein probability distributions are attached to family of allocations. This allows a user to reflect his experience with the components and the considerations of fac-tors not easily accountable in mathematical formulation. Reference(85)presents a summary of organization,a~plication,feasibility and results of CROS (Computer Reliability Optimization System) besides providin~ many early references. Hansler et al. (86)concern themselves with optimizing reliability of computer systems. References(87-9 0 )describe use of linear programming and dynamic program ~ ing for ~va­ ilability, failure and repair rates allocation in the design of maintained systems. Particularly, Lambert (90) et a1. consider redundancy also as design variable. CONCLUSIONS From the foregoing sections,it is amply indicated that a wide variety of formulations and mathematical techniques have been put forward for the reliability design of a system. The type of systems considered are mainly series,series-parallel,parallel-series and non-series parallel(what some of the authors call as 'complex system'). The

33

design selection criterion in majority of Cases is reliability and as such a large number of approaches have been presented by sever al authors fo r the solution of Problem 1 . Availability and failure and repair rate allocations have also been studied. Steady state availability or inherent availability is chosen to identify optim al desi gn in many papers. In one case consideration to component reliability improvement program and redundancy is also given as means for system reliability improvement whereas in the other Case design of a maintained system with a given repair facili ty h as been attempted. In future perhaps repai r facility and maintenance schedules may also be incorporated in the formulation. I!l several papers, the type of redundancy considered for improving reliability has been active redundancy, however, in other papers standby ann partial redundancies have been studied. In one case a mixed redundancy system is also designed. Turning to the techniques for the problems mentioned above, practically all Operations Research methods have been used by now, vi z, Gradient,variation a l,~namic programming, Integer pro gramming including Branch and Bound and 0-1 type of programming,Linear Integer programming, Piecewise Lineari zation. Geometri c programming and direct search etc. Some methods provide p-xact and the other approximate solutions. "Sxact methods are usually time consuming and may sometimes be come comput ati onally unwieldy for a large system or with more than two or three constraints. The type of constraints is immaterial with some approaches, for instance( 72). In the opini on of the author, the al go ri thm of (72) perhaps the only answer to complicated reliabili ty design problem, as this algorithm allows any arbitrary form of objective and constraint functions. The size of the system may poss! bly be restri cted on small computers but with large and fast computers,this may not be problematic(although memory requirements do not increase very much in (72) with the size of the system). Possibly a combination of Branch and Bound and this search al go ri thm may yield a practical solution for reducing computer time. In this way,we observe that attempts have been made to systematize reliability desi gn of a system ,yet much remains to be done in preparing proper system models, design formulation and in developing mathematical techniques for arriving at an optimal design suc h th at maximum benefits accrue from the system when put in operation. A schematic procedure is needed to be conceived of for these designs.

(17) 13 • .1.0. Arnst ad ter, rteli a bility Mathematics, :~l cGraw l1ill, l~ew 'fork,197l. (18) A.E.Green and A.J.Bourne,-1eliability Technolo gy , hley-Interscience,London, 1972. Assessment technique s

REFEltENCES The references that follow, by no means cover the entire literature that is available on t h e topic,however, CRre has been taken to include all possiole representative references which reflect the philosophy of design tech niques and are easily accessible in the form of publisned work. A reader is also reco~nended to look for the cross-references ill the list provided oelow. fhe listing has been done in chron010 6 ical order under subtitles. Books

.J • .2.Gaver, "rime to Failure and Avail-

ability of Paralleled Systems with Hepair' ,IEEErrans. Rel., '101. R-12, pp.30-38, June 1963.

( 1 ) I. J a zovsky, 1eli a bility ';'neory and Pr a-

( 2)

(3) (4) (5)

(6)

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34

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35

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36