172 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-4, NO. 2, MARCH 1974 Optimum Maintenance Policy for an Equipment Subject to Deterioration and Random Failure MANSOOR ALAM AND V. V. S. SARMA Abstract-Optimal maintenance policies for a machine with de- probability of machine failure is a function of maintenance gradation in performance with age and subject to failure are considered. and indicate the method of solution. Examples are worked The optimal policies are shown to be generally of a bang-coast nature, out at appropriate places to illustrate the results. Finally, except in the case where the probability of machine failure is a function out ma riae places oi the results.oFinaly of maintenance. It is also shown, in the case where machine failure is we summarize and comment on the results obtained. not taken into account, that a high depreciation rate tends to reverse this policy to coast-bang. The results obtained here extend and unify 1I. THOMPSON'S MODEL the recent results for this problem. The salvage value of the machine at time t, S(t), satisfies I. INTRODUCTION ~~~the differential equation r 1HE PROBLEM of machine' maintenance and re- placement has been the subject of considerable inves- dS(t)/dt = -d(t) + f(t)u(t) (1) tigation. McCall [4] gives a survey of maintenance policies where d(t) is the obsolescence function (in dollars) sub- for stochastically failing equipment. Recent advances in tracted from Sat time t;f(t) is the maintenance effectiveness optimal control theory led to modeling of this problem as function (in dollars) at time t added to S per dollar spent a control problem and application of Pontryagin's max- on maintenance; and u(t) is the maintenance function (in imum principle to obtain optimal maintenance policies. dollars) satisfying the constraint 0 < u(t) < U. Mainte- Two distinct models have been used in recent literature. nance here means money spent over and above the minimum Thompson [7] and Arora and Lele [1] consider the case spent on necessary repairs. The performance index that is when machine failure probability is not taken into account. to be optimized (maximized) is the discounted profit during They consider the degradation of a machine with time and the life of the machine plus the discounted salvage value at treat maintenance as a control variable which brings down time T, where T is the sale date of the machine; that is, this degradation. They show that the optimal maintenance policy is of a bang-coast nature. Kamien and Schwartz [3] V(T) = S(T)e-rT + F (p(t)S(t)-u(t))e-rt dt. (2) consider a model taking into account machine failure prob- Vp ability. They take the probability of machine failure as a state variable and maintenance as a control variable and In (2), p(t) denotes the production rate a.t time t per unit value of the machine at t, and r denotes the rate of interest. show that the optimal maintenance policy is nonincreasing. The problem is to choose a maintenance policy u(t) and a In the present paper, we incorporate both features in a sale date T so as to maximize the value V(T) of owning the single model. This represents, obviously, a more realistic siuation. In Section II, we briefly review Thompson's machine. A straightforward application of the maximu modelt[7]. an Sth IfI me by Arora and principle as in [2] gives the following optimal maintenance model [7 and the modified model suggested by Arora and poic U*( Lele [1], and we demonstrate the extreme sensitivity of the P u*(t) optimal maintenance policy to one parameter, namely, the u*(t f U, if f(t) > r/(p - (p - r) exp (-r(T - t))) rate of depreciation. The optimal maintenance policy is, -o, otherwise (3) normally, bang-coast (i.e., high-low, if there is a switching where p(t) = p was assumed to be constant. from high to low maintenance), but for high depreciation In p(t) is assing funconstant. rates the policy reverses to coast-bang (low-high). In Sec- In (1),f(t) is a nonincreasing function of t as considered tion III we incorporate the machine failure probability into by Thompson, and the optimal maintenance policy can the performance index and demonstrate the interesting fact only be of one of the followig the yp that the optimal maintenance policy is still bang-coast when 1) U - 0, if there is a switching from U to 0; the machine failure probability is independent of main- 2) U, for all t E [0, T]; tenance. In Section IV we consider the case where the 3) 0, for all t E [0, T]. On the other hand, we can generalize the model slightly by Manuscript received March 1, 1973; revised October 24, 1973. changin t8t M. Alam is with the School of Automation, Indian Institute of g (1 toJ Science, Bangalore, India. V. V. S. Sarma is with the Department of Electrical Communication dS(t)/dt - -a(t)- bS(t) + f(t)u(t) (4) Engineering, Indian Institute of Science, Bangalore, India. 1 The words "equipment" and "machine" are used synonymously where a(t) is the obsolescence function and b is a constant ijn this developfment. Equipment (or a machine) is bought for produc- depreciation rate. The justification of this choice is given condition. If the machine fails, it is "junked."byAoandLl[1.Basipetxsfrto,(4