Optimal Maintenance, Repairmen, and Control Strategies for Systems with Breakdowms

Optimal preventive maintenance and repairmen policies for a system of machines subject to degradation with age and intermittent breakdowns and repairs are derived using optimal control theory. When this system of machines forms part of a dynamic process control system, a method of obtaining optimal closed-loop control law is indicated. I . INTRODUC~ON Many large industrial systems have two distinct kinds of problems, the problems of management and the problems of operation. The former are mostly economic decision problems such as resource allocation, maintenance, repair and replacement of equipment, personnel, etc. The second type deals with efficient t e c h c a l operation of the processes and is in the domain of automatic control. It is being increasingly realized that both these types of problems are amenable for solution via control theory and mathematical programming. There are also interactions between the two kinds of problems which need coordination at both levels for efficient operation of the integrated system. As an example, we might consider the management and operation of a large interconnected power system. The maintenance scheduling of generators, replacement of older machines, and man power planning form part of the economic decision aspect while the technical operation of the system involves maintaining appropriate voltage levels at various points; regulating the active and reactive power flows while maintaining the dynamic stability of the system in the face of machine shut down due to maintenance or breakdowns. The problem of optimal maintenance of equipment has been the subject of considerable research [I], [2]. Thompson [3] considered a machine's degradation with age and obtained optimal preventive maintenance (PM) policy via Pontryagin's maximum principle. His results were subsequently generalized in several recent papers [4H8]. Sworder and Kazangey [9] modeled a production process with several machines breaking down at random points in time as an LQG problem with jump parameters and derived an optimal feedback controller. In h s short paper, we consider a generalization of Thompson's model [3] for a system of machines taking into account intermittent breakdowns and repairs to obtain optimal PM and repairmen policies. Then a'e examine the implications of these in a process control situation in which this system of machines forms a part, using Sworder's approach [91. 11. O P ~ U L MAIMENANCE .~ND REPAIRMEN POLICIES FOR A NUMBER OF MACHIKES In this section, the various models are described in a form amenable for solution via optimal control theory. A . Model for Machine Deferioration Following Thompson [3], the model for the gradual deterioration of a system of machines with age, which can be partly offset via PM, is given by d S ( t ) / d r = d ( t ) + f ( t ) u ( t ) . (1) In (l), S ( t ) is the salvage value of the system of machines in dollars, d ( t ) is the inferiority gadient, f ( r ) is the maintenance effectiveness function, and u(t) , (0 G u ( t ) < U), is the number of dollars spent on PM, over and above necessary repairs. mended by B. Friedland, Chairman of the IEEE S-CS Applications, Systems Evaluation, Manuscript received December 11. 1974: revised November 13. 1975. Paper recomComponents Committee. V.V.S. Sarma is with the Depanment of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India. M. Alam was u,ith the School of Automation, Indian Institute of Science, Bangalore, si& of Liverpool, Liverpool, England. India. He is now with the Department of Electrical Engineering and Electronia, UniverRemark I : It may, however, be noted that the simple linear model (1) may not completely represent the observed economic behavior. In generd, an equation such as d S ( t ) / d t = d ( t ) + g ( S ( t ) , u ( t ) ) (2) may be more appropriate. Usually g ( . . . ) satisfies conditions such as 1) g(S,O)=O, implying that S ( f ) satisfies the linear relation when PM is zero and 2) g,,(S, u) > 0, indicating that a positive maintenance is always beneficial. Conditions on g,(S,u) and g,,(S,u) may also be used if PM is not uniformly effective at all conditions of the machine and at all times. The simpler model (1) satisfies conditions 1) and 2). Luenberger [IO] recently considered models analogous to (2). Arora and Lele's model [4] is a special case of (1) and (2) [6]. Remark 2: Extension of Thompson's model for a single machine to a system of machines involves the assumption that all machines are bought at the same time. If they are bought at different times, equations of the form (1) may be written for separate machines to determine individual PM policies. Otherwise, a policy of the form u(t)= 1 for all machines gives rise to the problem of maintenance allocation. B. Model for Intennittent Breakdowns and Repairs We assume that the machines normally require no human care and that there are N machines and m(m < A') repairmen in the system. At any time t , a machine may fail and call for service. Repair work is initiated immediately after breakdown if a repairman is free or else a broken down machine joins a queue. The system of machines, at time t , is characterized by state i, where i=O, 1,2;. . , N denotes the number of inoperative machines at f . If A and p characterize the breakdown and repair processes of each machine assumed exponential, the equations governing the probabilities Pim(t) of finding the system in state i are given by [ 111