Key Words-Multiple objectives, Interactive optimization, Reliability ... The method determines the most preferable reliability allocation are formulated as follows:.
264
IEEE TRANSACTIONS ON RELIABILITY, VOL. R-27, NO. 4, OCTOBER 1978
Interactive Optimization of System Reliability Under Multiple Objectives Toshiyuki Inagaki, Student Member IEEE Koichi Inoue, Member IEEE Hajime Akashi, Member IEEE Key Words-Multiple objectives, Interactive optimization, Reliability
allocation, Preventive maintenance Reader Aids-
Purpose: Present a derivation Special math needed for explanation: Theory of multiobjective optimization Special math needed for results: Nonlinear programming technique Results useful to: Reliability and maintainability theoreticians
wi(Xi)
weight of component i with Xi for given ,i
c,i T
Ej
installation cost for component i length of planning interval sum over i from I to m
PM
NLMIP NLP DM
Preventive Maintenance
NonLinear Mixed-Integer Programming NonLinear Programming Decision Maker Problem
PA
Analyst
3. MULTIOBJECTIVE OPTIMIZATION PROBLEM
Summary & Conclusions-This paper considers a multiobjective re3.1 Problem Definition liability allocation problem for a series system with time-dependent reliMultiobjective optimization problems with p objectives ability. The method determines the most preferable reliability allocation are formulated as follows: and preventive maintenance schedule. The problem is multiobjective nonlinear mixed-integer. The decision making procedure is based on interactive optimization and a nonlinear programming algorithm. The method is illustrated by a numerical example.
Problem 1
minimize
f(z)
f1(z),
...,
f,(z)}
subject to z E S where S is a feasible region determined by equality and inequality constraints. In general we have no solution which minimizes all of 1. INTRODUCTION the objectives simultaneously. Thus the concept of "Pareto optimum" becomes necessary to this kind of probIn many practical situations, decision making is com- lems. A solution is said to be Pareto optimum if one can plicated because of mutually conflicting goals. In the re- not decrease any one objective without increasing some liability design of systems, we are required to maximize other objectives. The set of Pareto optimum solutions reliability and minimize cost, weight, volume, etc. We forms a (p- 1)-dimensional surface, which is called a Parextend the optimal'reliability allocation problem preeto surface. sented in [1] into a multiobjective optimization problem. If the preference attitude of the decision maker (DM) Optimization methods for multiobjective problems are is globally known as an explicit utility function, a mulclassified into A) interactive methods (e.g. [2]), and B) tiobjective problem is reduced to a single objective opnon-interactive methods (e.g. [3]). The methods of A have timization problem where the utility function is maxian advantage over those of B in the sense that the decision mized on the Pareto surfae. However it is very difficult maker is not required to express his global preference to identify the utility function. structure analytically. This paper uses one of the methods Interactive optimization methods are attractive because of A. they do not require the identification of the global utility function. These methods search the most preferable so2. NOTATION AND NOMENCLATURE lution of the DM using the information of the following two quantities: a) marginal rate of substitution Muj, which m number of components in a system implies MUj units of decrease infj is exactly equal in value Weibull shape parameter of component i, to a marginal unit of increase infi, and b) trade-off ratio (fi > 1, for IFR),I3 (I3i, ...J,Im) Tij, which implies fj can be decreased by Tij units, by Weibull scale parameter of component i, increasing Jj for a marginal unit on the Pareto surface. X-(X1, ..., Xm) The most preferable solution of the DM is a decision ni ~~number of replacement carried out on com- vector z satisfying ponent i, n--(n1, ..., flm) XiL specified greatest lower bound of A i M1(z) = TiJ(z), i, j = I, ........., p XiU specified least upper bound of Xi g1(X1) production cost of component i with Xl for 3.2 Brief Summary of ICOM given /,S We adopt, in our reliability optimization problem, an
0018-9529/78/l1000-0264$OO.75 ©)
1978 IEEE
INAGAKI/INOUE/AKASHI: INTERACTIVE OPTIMIZATION OF SYSTEM RELIABILITY UNDER MULTIPLE OBJECTIVES
265
4. MULTIOBJECTIVE RELIABILITY interactive method called ICOM (Interactive Coordinatewise Optimization Method) [4] in extracting the DM's OPTIMIZATION most preferable solution from the set of Pareto optimum solutions. The DM is embedded within the algorithm and 4.1 Problem Formulation interactions are carried out between the DM and the prob- Assumptionslem analyst (PA) with the aid of a computer. 1) The system has m components with mutually s-inICOM consists of the following steps: dependent failure mechanisms. I. Problem transformation *2a) Each component is either up or down at any time Transform Problem I into a single objective Problem and has the Weibull failure distribution which is IFR. 2b) The reliability of component i over the time interval 2 with artificial constraints. [0, t] is given by (1) when maintenance is not performed.
Problem 2 minimize f1(z)
R1(t)
=
exp[-(Xi t)13z]
(1)
3) Technological development can decrease Xi but has no influence on /i. subject to z E S and f, (z) S E3 (I = 2, .......... .., p). 4) Every fresh component starts from a perfect state, viz. repair is to like-new. II. Initial Pareto Optimum Solution 5) gi(Xi) and w1(Xi) are monotonically decreasing cona) Set up initial values for Ej (j = 2, ..., p) and solve vex functions of C2 in the half-open interval (XiL, Xiu]. As Problem 2. If every inequality constraints is active, the Xi -* XiL, gi(Xi) - +x and wi(Xi) -> +o±. solution of Problem 2 is Pareto optimum. If we use mul6) The PM policy is preventive replacement. tiplier methods [5] in solving Problem 2, Tj1's (j = 2, 7) Replacements are perfect, e.g. instantaneous, never p) at z are obtained as Lagrange multipliers. do any damage, and there is no queue. We take the following three quantities as objectives for b) Ask the DM to judge whether or not Mj,(z) = TjI(z) holds for allj. If he says YES, stop the algorithm because our reliability problem. a) mission reliability, viz, the probability that the the mst preferale solutio for him.If NO, go to step system operates successfully throughout the planning inIII. terval [0, T] under PM schedule. Under the above asIII. Coordinatewise Optimization sumptions, the optimum replacement schedule is periodic for each component [1]. The mission reliability f1 is a) Selectfk (k = 2, ..., p) which does not satisfy Mkl(z) = Tkl(z) at z. We callfk a coordinate axis in p-dimensional f1(X, n) = exp[ - 1i XAi(ni+ 1)1-31i Ti]; space of objectives. b) total cost b) Change Ek and solve Problem 2 where Ej'S (j = 2, f2(X, n) = X (n1+ 1) [g1(X1) ± C1i]; . p, j k) are fixed at the same values as in the previous c) system weight step. c) Ask the DM to judge whether Mkl(z) = Tkl(Z). If he f3(X) = Ei wi (Xi). says YES, go to step III-d: otherwise return to step III- If the weight of spare components must be included in b. the system weight,f3(X) is replaced byf3(X, n) = Ei (ni+ 1) d) Select another coordinate axis and iterate the same wi (A>). The multiobjective reliability optimization problem is procedures as III-a to c until Mj1(z) = Tj1(z) holds for all stated as follows for a given set of shape parameters 83. j (j = 2, .. ., p). If the above equality holds for all j, stop the algorithm. The resulting solution z is the most prefProblem 3 erable solution of the DM. Remark 1. Coordinatewise optimization is carried out by maximize n) the bisectioning method, for example, based on the inminimize f2(A, n) formation whether Mkl(z) is more or less than Tkl(z). mi z f3(X) Remark 2. If Problem 2 is not very difficult to solve, the PA had better make the DM see the Pareto surface off1 subject to ~ ,.. vs.fk which is obtained by solving Problem 2 for various 0.f*(4; G) >f G) :L)
i(4
500
viz.
400
Decision Making Process T21(10-3)t f3 f,
the
f*(4'';
global optimum
of
a
f*(4t';
NLP Problem 4'
gives
an
upper bound on that of a NLMIP Problem 4. The global ptmiato method has not yet been established for nn 0.9600 linear nonconvex problems and we merely can obtain the 'best' local optimum by solving the problem for various _/ 2 0 .... | / .............initial values of decision variables. / 0.9550_ .......................... / ...............The solution of Problem 4' in step 6, for example, is the best local optimum that we could obtain. Thus we can 65........ expect that the 'best' local optimum is practically the Figure 1 Decision Making Process on Pareto Surfaces global optimum and that the following relation holds: |
;
|30
INAGAKI/INOUE/AKASHI: INTERACTIVE OPTIMIZATION OF SYSTEM RELIABILITY UNDER MULTIPLE OBJECTIVES
0.9664 ¢ ft (4; G) 0.9663 = ft (4"; L) The difference between 0.9663 and 0.9664 is about 0.01% which is negligible. Thus our method can give a good solution to a NLMIP Problem 4 which has no exact -
method of solution.
[3] [4]
[5]
267
R. L Keeney, "Multiplicative utility functions", Operations
Research, vol 22, 1974, pp 22-34. Y. Sawaragi, K. Inoue, H. Nakayama, "A decision making model
for environmental management systems", New Trends in Sys1976 Dec, tems Analysis, International Symposium, Versailles, edited by A. Bensoussan, J. L. Lions, Springer, 1977. D. A. Pierre, M. J. Lowe, Mathematical Programming via Augmented Lagrangians-An Introduction with Computer Programs, Addison-Wesley, 1975.
ACKNOWLEDGMENT T. Inagaki; Dept of Precision Mechanics; Faculty of Engineering;
The authors express their thanks to Associate Prof. H. Nakayama of Konan University for his discussion on
Kyoto University; Kyoto 606 JAPAN.
ICOM.
For the biography of T. Inagaki see vol R-27, 1978 Apr, p 40. K. Inoue; Dept of Precision Mechanics; Faculty of Engineering; Kyoto University; Kyoto 606 JAPAN.
REFERENCES
[1] [2]
T. Inagaki, K. Inoue, H. Akashi, "Optimal reliability allocation under preventive maintenance schedule", IEEE Trans. Reliability, vol R-27, 1978 Apr, pp 39-40. S. Zionts, J. Wallenius, "An interactive programming method for solving the multiple criteria problem", Management Science, vol 22, 1976, pp 652-663.
For the biography of K. Inoue see vol R-23, 1974 Apr, p 33. H. Akashi; Dept of Precision Mechanics; Faculty of Engineering; Kyoto University; Kyoto 606 JAPAN. For the biography of H. Akashi see vol R-27, 1978 Apr, p 40.
Manuscript received 1977 Nov 26; revised 1978 May 4.
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