396 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, MAY 1975 application are not always ideal (in the sense that in many cases can be avoided and the second stage clustering made easier at it generates clusters in which "natural" boundaries, although practically no additional computational cost. well defined on the given data set, are violated by the resulting One of the referees brought to the attention of the author the partition). This, however, is not a serious obstacle since the fact that the proposed algorithm is related to that described by method is intended as the first stage in a two-stage clustering Fu [2]. approach: global clustering followed by a local one performed on preliminary clusters obtained in the first stage. In an n- REFERENCES dimensional space containing N vectors Xi = (xi1, .,xi.) the [1] D. J. Eigen et al., "Cluster analysis based on dimensional information with applications to feature selection and classification," IEEE Trans. proposed algorithm works essentially as follows. Syst., Man, Cybern., vol. SMC-4, pp. 284-294, May 1974. In each dimension a window size wd is determined and histo- [2] K. S. Fu, Adaptive, Learning and Pattern Recognition Systems Theory and Applications, J. M. Mendel, K. S. Fu, Eds. New York: Academic, grams calculated. A threshold td is then used to determine valid 1970, pp. 72-75. clusters in each dimension (via comparison of histograms and thresholds). In this way each dimension is subdivided into si, i 1, n sections (we will assume for simplicity that si > 0, for i = 1,n). A two-dimensional illustration of the obtained grid is shown in Fig. 1. It is clear that this method will create at most Optimal Maintenance Policies for Machines Subject to C = ]J.1 si, i = 1, n clusters. In many cases the number of Deterioration and Intermittent Breakdowns clusters will be smaller since some of the obtained hypercubes V. V. S. SARMA AND MANSOOR ALAM will be empty, nevertheless, the number of obtained clusters grows very fast (essentially exponentially with the number of dimensions), and obviously many of the obtained clusters will Asrc-pia rvniemineac oiis o ahn dimensions) and obviously many of the obtained clusters will subject to deterioration with age and intermittent breakdowns and be artificial and will only hamper local clustering in the second repairs, are derived using optimal control theory. The optimal policies stage. are shown to be of bang-bang nature. The extension to the case when An example of this is shown in Fig. 2 where three clusters of there are a large number of identical machines and several repairmen in normally distributed samples were generated in a two-dimensional the system is considered next. This model takes into account the waiting space and the preceding algorithm resulted in the generation of line formed at the repair facility and establishes a link between this nine clusters most of which were artificial. We are going to problem and the classical "repairmen problem." propose a simple modification of [1] that improves its perfor- mance considerably in a number of cases without decreasing its I. INTRODUCTION computational efficiency. Optimal control theory has been applied recently by several investigators to obtain optimal maintenance and replacement policies for equipment. Thompson [1], Arora and Lele [2], and A look at Fig. 2 suggests that if clustering were performed Sethi [3 ] consider the degradation of a machine's capability with dimensionwise (coordinate by coordinate) rather than globally, age and assume that the deterioration can be partly offset via the resulting clustering could be much more acceptable in a preventive maintena ce. Alam and Sarma [4] consider the effect number of cases. We will define the modification more formally as of random catastrophic failure on the maintenance policies. follows: 1) m = n,M = 1, and S, = (Xi,.-..,XN); 2) apply This correspondence considers the intermittent breakdowns algorithm [1] to coordinates xim of vectors Xi e Sk,k = 1,. M, and associated repairs of the equipment and the stochastic obtaining M clusters Si, i = 1, * - -,M, with a new value of M nature of these processes. The approach leads to unification of (determined by the nature of data); 3) m = m - 1; 4) if m = 0, this work with the now classical "repairmen problem" [5 ], when then the algorithm ends, otherwise go to step 2). Here m is an there are large number of machines in the system. iteration count and M is the number of clusters. The performance of this algorithm could be further improved II. PROBLEM FORMULATION-SINGLE MACHINE CASE if coordinates were not taken sequentially but rather in the order A. Thompson's Modelfor Machine Deterioration of quality of generated clusters (e.g., in Fig. 2(a) the second coordinate would be better to start with). Ordering could be In order to define the model the following notation is used: made in a number of ways. A natural method using information T sale date of the machine, obtained during the calculation of histograms is to assign each V(T) discounted profit during the life of the machine plus dimension a number qi, which measures the prominence of the discounted salvage value at time T, in dollars, generated clusters, S(t) salvage value of the machine at time t, in dollars, qi = E g(h), along the histogram (1) r rate of interest, u(t) number of dollars spent on preventive maintenance at with g(h) given, for example, by time t satisfying the constraint 0 < u(t) . U (preven- g(h) = h2, for h > ti, g(h) = 0, otherwise. (2) tive maintenance here means money spent over and above the minimum spent on necessary repairs because CONCLUSION ~~~~~~~of intermittent breakdowns), CONCLUSION ~~~~~~f(f) maintenance effectiveness function at time t, in dollars The proposed modification improves the global clustering in a added to S per dollar spent on maintenance, number of cases such as that in Fig. 2 where three clusters exist, the algorithm [1] generates nine, but the proposed modified algorithm generates the correct three clusters. The presented Manuscript received Aphril17,D 197a4;revsteedfNoEmberia 19, 1974. i° example is artificial and favorable to the proposed method. Engineering, Indian Institute of Science, Bangalore, India. . . . . . . ~~~~~~~~~~~~~M. Alam iS with the School of Automation, Indian Institute of Science, It l lustrates, however, thnat in a number of cases artificil results Bangalore, India.
Transcript
396 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, MAY 1975
application are not always ideal (in the sense that in many cases can be avoided and the second stage clustering made easier atit generates clusters in which "natural" boundaries, although practically no additional computational cost.well defined on the given data set, are violated by the resulting One of the referees brought to the attention of the author thepartition). This, however, is not a serious obstacle since the fact that the proposed algorithm is related to that described bymethod is intended as the first stage in a two-stage clustering Fu [2].approach: global clustering followed by a local one performedon preliminary clusters obtained in the first stage. In an n- REFERENCESdimensional space containing N vectors Xi = (xi1, .,xi.) the [1] D. J. Eigen et al., "Cluster analysis based on dimensional informationwith applications to feature selection and classification," IEEE Trans.proposed algorithm works essentially as follows. Syst., Man, Cybern., vol. SMC-4, pp. 284-294, May 1974.
In each dimension a window size wd is determined and histo- [2] K. S. Fu, Adaptive, Learning and Pattern Recognition Systems Theory andApplications, J. M. Mendel, K. S. Fu, Eds. New York: Academic,grams calculated. A threshold td is then used to determine valid 1970, pp. 72-75.
clusters in each dimension (via comparison of histograms andthresholds). In this way each dimension is subdivided intosi, i 1, n sections (we will assume for simplicity that si > 0,for i = 1,n). A two-dimensional illustration of the obtained gridis shown in Fig. 1. It is clear that this method will create at most Optimal Maintenance Policies for Machines Subject toC = ]J.1 si, i = 1, n clusters. In many cases the number of Deterioration and Intermittent Breakdownsclusters will be smaller since some of the obtained hypercubes V. V. S. SARMA AND MANSOOR ALAMwill be empty, nevertheless, the number of obtained clustersgrows very fast (essentially exponentially with the number ofdimensions), and obviously many of the obtained clusters will Asrc-pia rvniemineac oiis o ahndimensions) and obviously many of the obtained clusters will subject to deterioration with age and intermittent breakdowns andbe artificial and will only hamper local clustering in the second repairs, are derived using optimal control theory. The optimal policiesstage. are shown to be of bang-bang nature. The extension to the case whenAn example of this is shown in Fig. 2 where three clusters of there are a large number of identical machines and several repairmen in
normally distributed samples were generated in a two-dimensional the system is considered next. This model takes into account the waitingspace and the preceding algorithm resulted in the generation of line formed at the repair facility and establishes a link between thisnine clusters most of which were artificial. We are going to problem and the classical "repairmen problem."propose a simple modification of [1] that improves its perfor-mance considerably in a number of cases without decreasing its I. INTRODUCTIONcomputational efficiency. Optimal control theory has been applied recently by several
investigators to obtain optimal maintenance and replacementpolicies for equipment. Thompson [1], Arora and Lele [2], and
A look at Fig. 2 suggests that if clustering were performed Sethi [3 ] consider the degradation of a machine's capability withdimensionwise (coordinate by coordinate) rather than globally, age and assume that the deterioration can be partly offset viathe resulting clustering could be much more acceptable in a preventive maintena ce. Alam and Sarma [4] consider the effectnumber of cases. We will define the modification more formally as of random catastrophic failure on the maintenance policies.follows: 1) m = n,M = 1, and S, = (Xi,.-..,XN); 2) apply This correspondence considers the intermittent breakdownsalgorithm [1] to coordinates xim of vectors Xi e Sk,k = 1,. M, and associated repairs of the equipment and the stochasticobtaining M clusters Si, i = 1, * - -,M, with a new value of M nature of these processes. The approach leads to unification of(determined by the nature of data); 3) m = m - 1; 4) if m = 0, this work with the now classical "repairmen problem" [5 ], whenthen the algorithm ends, otherwise go to step 2). Here m is an there are large number of machines in the system.iteration count and M is the number of clusters.The performance of this algorithm could be further improved II. PROBLEM FORMULATION-SINGLE MACHINE CASE
if coordinates were not taken sequentially but rather in the order A. Thompson's Modelfor Machine Deteriorationof quality of generated clusters (e.g., in Fig. 2(a) the secondcoordinate would be better to start with). Ordering could be In order to define the model the following notation is used:made in a number of ways. A natural method using information T sale date of the machine,obtained during the calculation of histograms is to assign each V(T) discounted profit during the life of the machine plusdimension a number qi, which measures the prominence of the discounted salvage value at time T, in dollars,generated clusters, S(t) salvage value of the machine at time t, in dollars,
qi = E g(h), along the histogram (1) r rate of interest,u(t) number of dollars spent on preventive maintenance at
with g(h) given, for example, by time t satisfying the constraint 0 < u(t) . U (preven-
g(h) = h2, for h > ti, g(h) = 0, otherwise. (2) tive maintenance here means money spent over andabove the minimum spent on necessary repairs because
CONCLUSION ~~~~~~~ofintermittent breakdowns),CONCLUSION ~~~~~~f(f)maintenance effectiveness function at time t, in dollarsThe proposed modification improves the global clustering in a added to S per dollar spent on maintenance,
number of cases such as that in Fig. 2 where three clusters exist,the algorithm [1] generates nine, but the proposed modifiedalgorithm generates the correct three clusters. The presented Manuscript received Aphril17,D197a4;revsteedfNoEmberia 19, 1974. i°example is artificial and favorable to the proposed method. Engineering, Indian Institute of Science, Bangalore, India.
. . . . . . ~~~~~~~~~~~~~M.Alam iS with the School of Automation, Indian Institute of Science,It l lustrates, however, thnat in a number of cases artificil results Bangalore, India.
CORRESPONDENCE 397
TABLE I where PO(t) and P1(t) are the solutions of (3) and (4), respectively,OPTIMAL MAINTENANCE POLICIES FOR A MACHINE WITH INTERMITTENT a .c
BREAKDOWNS AND REPAIRS and ki characterizes the additional expenditure when the machineis being repaired. Equation (5) is derived from (2) on the basis
Parameters of optimal maintenance that the output from a machine is obtained only when it isDifferent Cases policy u*(t) working and repair expenditure is incurred when it is in state 1.
T' (Stop main- T A straightforward application of maximum principle as in [7]tenance) (Sael date)
1-Tmo oti(Be w gives the following optimal maintenance policy u*(t):1. Thompson' s solution (Breakdow,ns 9not considered) 10.6 34.8
2. a 0.05, " =0.5 0.0 25.0 U*(t) = u, if f(t) > e-rt[ PP (e-t - e-rT)3. X = 0.05, " =1.0 0.7 28.0 ' ~ ' IG + pi)r4. A = 0.05, u = 1.5 4.8 30.0
+ AP {e-(+?u+r)t e-(e++r)T} + e-rTd(t) inferiority gradient at time t, in dollars subtracted from (A + iu)(G + i + r) J
S at time t, (6)p production rate at time t, output value at t/salvage
value at time t, and u*(t) = 0, otherwise. The sale date T may be obtained byy labor rate, in dollars per unit time. adopting Thompson's procedure [1].
Example 1: Consider Thompson's example [1] with inter-Assumptions: d, f, u are piecewise continuous, d is non- mittent breakdowns and associated repairs taken into account.
decreasing and f is nonincreasing, and p is assumed to be Suppose S(O) = 100, d(t) = 2, p = 0.1, r = 0.05, and f(t) =constant. 2/(1 + t)012. The results for this example are shown in Table I.The model is now given by the differential equation
dS(t) = -d(t) + f(t)u(t) (1) III. PROBLEM FORMULATION-MULTIMACHINE CASEdt In an establishment there are usually a large number, say, N,
and the performance index to be maximized is given by of identical machines bought for production purposes. Eachmachine can be analyzed separately by the procedure indicated
V(T) = S(T)e-rT + Y(e-rT - 1) in the previous section, and optimal preventive maintenancer policy can be arrived at. Yet it is rather unusual to have a
T separate repairman for each machine, in view of "operative+ {pS(t) - u(t)}e-rt dt. (2) efficiency," i.e., the repair crew may be idle for most of the
time. Also, if the number of repairmen is too small, the assump-
B. Modelfor Intermittent Breakdowns and Repairs tion that the machine is repaired immediately after a breakdownis no longer valid. If all the repairmen are engaged, a new
It is assumed, following Feller [6], that the machine normally arrival forms a waiting line. This problem is termed "machinerequires no human care. At any time t, the machine may break interference." The steady-state behavior of queueing processesdown and call for service. The breakdown time of the machine involved has been a subject of considerable investigation (see,and the repair time are taken as random variables with for example, Benson and Cox [8], Morse [9], and Barlow [10]).exponential distributions. Thus a machine is characterized by The deterioration of each machine of the system can betwo parameters, ). and u, corresponding to breakdown and described by a differential equation of the form (1). The state ofrepair. We say that the machine is in state 0 if it is working and the system can be characterized by the probability Pim(t) wherein state 1 if it is being repaired. The differential equations for the i = 0,1,2,. . *,N, denotes the number of machines that are notprobabilities P5(t), i = 0 or 1, of finding the machine in state i working at any time t and the superscript m denotes the numberat time t, are given by [6] of repairmen. The state of the system is represented by index i.
dP0(t) _ If A and , are parameters characterizing the breakdown anddt ) po(t) + iPi(t), PO(O) = 1 (3) repair processes of each machine, it is straightforward to write
the differential equations for P5m(t) analogous to (3) and (4)dP1(t) = Apo_(t) - pP1(t), P1(0) = 0. (4) depending on the number m(l . m < N) of repairmen.
dt The objective is to obtain the optimal preventive maintenancepolicy u*(t) for the system of N machines and to obtain the
C. Solution by Maximum Principle optimum number of repairmen m*. The equation governing the
Thompson [1] has demonstrated via Pontryagin's maximum deterioration of N machines of the system is againprinciple that for the model described by (1) and (2), the optimal dS(t)maintenance policy u*(t), is bang-bang. When the intermittent )d(t) + f(t)u(t)breakdowns are taken into account, (3) and (4) are also to be dtconsidered. The problem of finding optimal maintenance policy whrSt)inoiteptdasheoalavgeauefteu*(t) now reduces to maximization of the expected profit during sytmomahnsttie ndlar,d)isheytmthe life of the machine given by obsolescence function at time t, also in dollars, u(t) is the total
y -1 nmber of dollars spent on preventive maintenance at time t,E{V(T)} = S(T)er'T + -(e-rT-1 satisfying the constraint 0 . u(t) < U, and f(t) is the cumula-
tive maintenance effectiveness function.
+ fT [pS(t)P0(t) u(t) - kiP(t)]erIt dt (5) The corresponding objective function to be maximized is theJO expected profit over the life of the system of N machines. This
398 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, MAY 1975
can be obtained by a generalization of (5) and is given by OPTIMAL MAINTENANCE TALIEI 1OPTMALMAITENNCEPOLICIES FOR THE TWO-MACHINE CASE
E{V(T)} S(T)e-rT + M (e-rT 1)-r
Parameters of optimal Optimal net Optimal profitrT Number of maintenance policy u*(t) profit per unit time
+ f[{ps(t, po (t) + aipS(t)Pim(t) repairmen V*(T)+ 1ff \I1\I m ~~~~~~~~~~~StopMain- Sale date V(Ttenance, T' T
+ k2P2m(t) + + kNPNm(t)}l erc dt (7) obtained in each case. Example 2 gives the results in a two-machine case.
where p is now the total production rate. Also, Example 2: When there are two machines, the three possiblestates of the system are 0,1,2, denoting the number of machines
0 < N- 1 < XN-2 < ... < a2 < a, < 1 that are not working at any time t. The probability expressions
and are to be developed for the two-machine problem with onek, < k2 < ..*< kN. repairman and with two repairmen. The parameters a and k are
chosen to be oto = 1, al = 2, k1 = 1, k2 = 2.The parameters a and k characterize the decreased productivity The following numerical values of various parameters were
and the increase in additional repair expenditure, respectively, as employed for computation of optimal maintenance policies:the number of machines not working increases. Equation (7) canbe rewritten as S(0)=200 d(t)=4 p=0.2 r=0.05
rT + m T f(t) = 4/(1 + t)1"2 -= 0.05 ,I = 1.0.E{V(T)} =S(T)e-r + (e-'T 1)r The results are shown in Table II. The complete computational
T details are given in [11]. The optimum number of repairman+ J {i(t)S(t) u(t) - k(t)}eTt dt (8) is m* = 1.
where IV. CONCLUDING REMARKS
N-1 In this correspondence the general problem of machine
( =o maintenance and replacement has been examined, taking intoN account the inherent breakdowns and repairs. If there are a
k-(t) = kjPjm(t). large number of identical machines in the system, considerationj= l of the breakdowns and repairs leads this problem to the domain
Remark 1: p3(t) is the equivalent production rate for the of queueing theory. In the problem considered in this paper,multimachine problem, and p3(t)S(t) is the total production in there are still various questions to be resolved. For example,dollars at time t. when there are several different types of machines in a system, aRemark 2: If one assumes that the output at any time t is model describing the deterioration with age can easily be
proportional to the number of machines working at that time, obtained, but the characterization of system state when thethen the parameters aci are given by breakdowns and repairs are taken into account becomes quite
involved.
N= i = 0,1,2,..*,(N- 1). REFERENCESN[1] G. L. Thompson, "Optimal maintenance policy and sale date of a
Hence machine," Management Sci., vol. 14, no. 9, pp. 543-550, 1968.[2] S. P. Arora and P. T. Lele, "A note on optimal maintenance policy
N-1 _- N-1 and sale date of a machine," Management Sci., vol. 17, no. 3, pp.
(t) =mP N N-
P [3]S. P. Sethi, "Simultaneous optimization of preventive maintenance andi=O N N i=O replacement policy for machines: A modern control theory approach,"
AIIE Trans., vol. 5, no. 2, pp. 156-163, 1973.NJJ(t)/p is thus the production function x(t) defined by Arrow [4] M. Alam and V. V. S. Sarma, "Optimum maintenance policy for an
equipment subject to deterioration and random failure," IEEE Trans.et al. [5]. The optimal preventive maintenance policy can be Syst., Man, Cybern., vol. SMC-4, pp. 72-75, Mar. 1974.shown to be [5] K. J. Arrow, D. Levhari, and E. Sheshinski, "A production function
for the repairman problem," Rev. Econ. Stud., vol. 39, no. 3, pp.-rt 241-249, 1972.
't'__-_U_f______>_e [6] W. Feller, An Introduction to Probability Theory and Its Applications,U*(t) = u, if f(t) >t N- vol. I. New York: Wiley, 1957.
e-rT - P Exipx
(t) e-rt
dt [7] A. E. Bryson and Y. Ho, Applied Optimal Control. Waltham, Mass.:oi=O []Blaisdell, 1969.
J L o F. Benson and D. R. Cox, "The productivity of machines requiringattention at random intervals," J. Roy. Stat. Soc., vol. 13, no. 1,
and u*(t) = 0, otherwise. pp. 65-82, 1951.The number of repairmen m(1 . m . N) is obviously an [91 P. M. Morse, Queues, In7entories and Maintenance. New York:
integer. To determine the optimum number of repairmen m*, N [10] R. B. Barlow, "Repairman prohlems," in Studies in Applied Probabilitywit fie m, m 12...,Nmsbeold. This will. ................and Management Science, K. J. Arrow, S. Karlin, and H. Scarf, eds.problems wtfiem,m=12 *,msbesle.TsW11 Stanford, Calif.: Stanford Univ. Press, 1962.
result in a different set of differential equations for PLm(t) for [111 M. Alam, "Optimal preventive maintenance, replacement and repair-. . . ~~~~~~~menstrategies for stochastically failing systems: A modern control
each m. m* iS to be chosen to maximize the optimum E{V(t)} theory approach," Ph.D. dissertation, Indian Inst. Sci., Aug. 1974.
