Mar 11, 2016 - To cite this article: Irfan Ali, Yashpal Singh Raghav & Abdul Bari (2012) Optimal Maintenance for a System with Probabilistic Time Constraint, ...
Safety and Reliability
ISSN: 0961-7353 (Print) 2469-4126 (Online) Journal homepage: http://www.tandfonline.com/loi/tsar20
Optimal Maintenance for a System with Probabilistic Time Constraint Irfan Ali, Yashpal Singh Raghav & Abdul Bari To cite this article: Irfan Ali, Yashpal Singh Raghav & Abdul Bari (2012) Optimal Maintenance for a System with Probabilistic Time Constraint, Safety and Reliability, 32:3, 51-59, DOI: 10.1080/09617353.2012.11690962 To link to this article: http://dx.doi.org/10.1080/09617353.2012.11690962
Published online: 11 Mar 2016.
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Date: 12 April 2017, At: 02:01
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OPTIMAL MAINTENANCE FOR A SYSTEM WITH PROBABILISTIC TIME CONSTRAINT lrfan Ali , Yashpal Singh Raghav and Abdul Bari Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India
Abstract In this paper, we repair an optimum number of components in a system, repair time taken as a random variable in the constraint function. By applying probabilistic programming, constraints function is converted into an equivalent deterministic form. A numerical example is presented to illustrate the computational procedure for random time constraints, and the problem is solved by using LINGO software. Keywords: Selective maintenance, reliability, chance constraint, probabilistic programming.
1. INTRODUCTION We consider a system which requires performing a sequence of identical production runs after every given (fixed) period. A production run in the system consists of several subsystems where each subsystem can work properly if at least one of its components is operational. The following assumptions are also made: (i) (ii) (iii) (iv)
all the components can be repaired if deteriorated or failed all component states are independent reliability of each component in a subsystem is the same cost spent and time taken on repairing each component within a subsystem are the same
The reliability of a system can be increased by proper selection of the number of repairable components to be repaired from its subsystems. The system is virtually repaired under the limitations on some parameters such as cost spent and time taken in repairing the failed components. A repairable system is a system in which the failed or deteriorated components can be repaired to operate normally. Some
52 examples of repairable systems are a computer network, a manufacturing system, a power plant or a fire prevention system. Many authors have discussed the allocation problem of repairable components. Among them are Rice et al. (1998), Cassady et al. (2001a, b), Schneider and Cassady (2004), Rajaopalan and Cassady (2006), lyoob et al. (2006), Schneider et al. (2009), Ali et al. (2011a, b), Bakhshi et al. (2011); and many others. Consider a system in which m subsystems are connected in series, the i- th subsystem consisting of ni components connected in parallel ( i = 1,A , m) . Let r, be the reliability of each component in the i- th subsystem.
"'
.......
Figure 1: Parallel-Series Systems
In fig.1 the system for a fixed period is a series arrangement of the subsystems (subsystem 1, subsystem 2 ... subsystem m), its reliability can be defined as R
=IT {1-(1-r;)n;}
(1.1)
i=l
At the completion of a particular mission, each component in the system is either functioning or failed. Suppose that failed components (if there are any) can be repaired to a functioning condition if there is a possibility that after repair this component work properly otherwise it is replaced before the next mission starts. Ideally all failed components are repaired prior to the beginning of the next mission. However, it may not be possible to repair all failed components. Let ti denote the amount of time required to repair a component of subsystem i , and let ai denote the number of failed components in subsystem i at the end of the mission. The total time required to repair all failed components in the system prior to the next mission is given by:
53
(1.2) i=l
Suppose that total amount of time allotted to perform repair of failed components between mission is To time units. If T0 < T , then all failed components may not be repaired prior to beginning of the next mission. In such a case, a method is needed to decide which failed components should be repaired and replaced prior to the next mission and which components should be left in a failed condition. This process is referred to as selective maintenance. Rice et al (1998) define the following mathematical programming model for this selective maintenance problem. Let d; denote the number of components in subsystem i repaired prior to the next mission. Thus for the next mission we define selective maintenance for repaired components as:
(1.3) The resulting mathematical programming model is given as: m
Max R= IJ1-(1-r1 )(n;-a;)+d;
(i)
i=l
Subject to
(1.4) (ii)
i'=l
0 ~ d 1 ~a andd1 integer, i =l,2, ... ,m
(iii)
2. NOTATIONS R(t) T Po r; t;
n; m; t0
T;
a,,
reliability of the system total repair time of the system specified probability of the system reliability of every component available for subsystem repair time of every component available for subsystem optimum number of redundant components number of subsystems set up time for designing a system expected repair time for component of subsystem standard deviation of repair time for component of a subsystem
54
3. SOLUTION USING CHANCE CONSTRAINED PROGRAMMING The problem at hand addresses the issue of maximizing the total system reliability, where the repair time of each component is randomly variable. Let us assume that component repair times t;, i = 1,2, ... m in many practical situations are not fixed. Let us assume that tj, i = l, ... ,m are independently normally distributed random variables, where n; is the redundant component in the ith subsystem. So we write the above problem in the following chance constrained programming form: m
Max R= Til-(1-rdn;-a;)+d;
(i)
i=l
Subject to P[
~
(t;d;)+t 0
(3.1)
s ]~Po
(ii)
T
0 s d; sa and d; integer, i =1,2, ... ,m
(iii)
Where t; repair time function is a random variable. T is a known non-negative constant and Po, o s Po s 1 is a specified probability. Let us assume that the time t;, i = 1, ... ,m in the constraint function are assumed to be independently and normally distributed random variables. Let t' = (t1, ... tm) and r£ = (dl, ... dm) . Then the constraint function (t' n +to) I will also be normally distributed random variables with mean E (t' d + t0 ) and variance (t' d + t0 ) Then, the mean is obtained as follows:
E(t' d
+t0 )=E(~tA J+ t = ~d;E(t;)+ t = ~d;,U; +t 0
0
0
(3.2)
where .U; = E(t;), i = l, ... ,m. and the variance as follows: (3.3) i=I
i=l
i=l
where a;2 = V(t;). now, we consider the probabilistic constraint {3.1, (ii)} which can be expressed as:
55 P{J(t) :
