Cost Benefit Analysis of Three Non-Identical Machine Model with Priority in Operation and Repair
Subject Areas : Machine learningNafeesa Bashir 1 * , Raeesa Bashir 2 , JP Singh Joorel 3 , Tariq Rashid Jan Jan 4
1 - Department of Statistics, University of Kashmir
2 - Mathematics and Quantitative Analysis, Amity University, Dubai, UAE
3 - Department of Statistics, University of Jammu
4 - Department of Statistics, University of Kashmir
Keywords: Reliability Measures, Mean Sojourn Time, Laplace Transformation, Laplace -Stieltjes transformation,
Abstract :
The paper proposes a new real life model and the main aim is to examine the cost benefit analysis of Textile Industry model subject to different failure and repair strategies. The reliability model comprises of three units i,e Spinning machine (S), Weaving machine (W), Colouring and Finishing machine(Cf). The working principal of the model starts with spinning machine (S) where in unit S is in operative state while as weaving machine, Colouring and Finishing machine are in ideal state. Complete failure of system is observed when all three units of system i.e. S,W and Cf are in down state. Repairperson is always available to carry out the repair activities in the system in which first priority in repair is given to Colouring and Finishing machine followed by Spinning and weaving machine. The proposed model attempts to maximize the reliability of a real life system. Reliability measures such as Mean Sojourn time, Mean time to system failure, Profit analysis of system are examined to define the performance of the reliability characteristics. For concluding the study of such model, different stochastic measures are analyzed in steady state using regenerative point technique. The tables are prepared for arbitrary values of the parameters to show the performance of some important reliability measures and to check the efficiency of the model under such situations.
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[12] S.C.Malik, M.S. Kadian, and J. Kumar , “Cost-Analysis of a System under Priority to Repair and Degradation”, International Journal of Statistics and Systems, Vol. 5,No.1, 2010, pp. 1 – 10.
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Cost Benefit Analysis of Three Non-Identical Machine Model with Priority in Operation and Repair
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Department of Statistics, University of Kashmir Raeesa Bashir Mathematics and Quantitative Analysis, Amity University, Dubai, UAE J P Singh Joorel Department of Statistics, University of Jammu T R Jan Department of Statistics, University of Kashmir
Received: 15/Aug/2020 Revised: 10/May/2021 Accepted: 07/June/2021 |
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Abstract
The paper proposes a new real life model and the main aim is to examine the cost benefit analysis of Textile Industry model subject to different failure and repair strategies. The reliability model comprises of three units i,e Spinning machine (S), Weaving machine (W), Colouring and Finishing machine(Cf). The working principal of the model starts with spinning machine (S) where in unit S is in operative state while as weaving machine, Colouring and Finishing machine are in ideal state. Complete failure of system is observed when all three units of system i.e. S,W and Cf are in down state. Repairperson is always available to carry out the repair activities in the system in which first priority in repair is given to Colouring and Finishing machine followed by Spinning and weaving machine. The proposed model attempts to maximize the reliability of a real life system. Reliability measures such as Mean Sojourn time, Mean time to system failure, Profit analysis of system are examined to define the performance of the reliability characteristics. For concluding the study of such model, different stochastic measures are analyzed in steady state using regenerative point technique. The tables are prepared for arbitrary values of the parameters to show the performance of some important reliability measures and to check the efficiency of the model under such situations.
Keywords: Reliability Measures; Mean Sojourn Time; Laplace Transformation; Laplace -Stieltjes transformation.
1- Introduction
In the present scenario of global viable market, the basic objectives of industries is to reduce the production cost and enhance the productivity of manufacturing systems of processing Industries. To meet the increasing demand of the society, the industries must offer long term production. The efficiency and accessibility of industry can be made profitable in terms of higher production and lower maintenance costs. Availability of complex systems can be improved by considering different techniques of reliability like maintenance, inspection, repairs, priority to repair of the failed unit and so on. In addition, these techniques in industrial design can substantially strengthen operational dependability through better system design. A very trivial change in the technique of reliability can cause a profound change in the operational dependability, that is why it is vital to specify model reliability targets before any strategy plan is taken into consideration .
There are systems where upon failure in a complex system the repair of a component is given preference over the other component depending upon the significance of the component. Such systems are called priority systems. When a unit of lower priority is under repair and a unit of higher priority arrives for repair, the lower priority unit is pre-empted and higher priority unit is repaired. After completion of the service of higher priority unit, the lower higher priority unit is given service. In reliability modeling, it is found that accessibility and performance of failed and repairable systems can be enriched using redundancy technique. But due to the high cost, it is not always possible to keep the unit as spare. Also, it sometimes becomes essential to prioritize one unit over the other to increase performance, availability and hence the efficiency of the system. The idea of priority in repair discipline in reliability modeling has been considered by various authors researchers in order to make the system available for required service with least possible operational cost.
Kumar et al. [2] obtained reliability analysis of a warm standby model performing in normal/abnormal environment under certain assumptions where the distribution of failure time has been taken as negative exponential and the repair, inspection and replacement time distribution has been taken as arbitrary. Chander [11] discussed the two non-identical reliability models in which the failure and repair time of each unit were independent and uncorrelated random variables. The author obtained the various reliability measures using Regenerative Point Technique (R.P.T ). Also, Zhang Wang [15] studies a cold standby system with priority in use and repair. Malik et al. [12] studied cost- benefit analysis of a model under and gave priority to repair and degradation. The numerical and graphical behaviour of important reliability measures has been obtained for arbitrary values of costs and parameter. Further, Kadyan et al. [6] considered reliability performance of model under priority to repair and degradation. The authors obtained the availability and profit of the system using Regenerative point technique (RPT). Sureria and Malik [5], Malik [13] analyzed computer system with priority to software replacement over hardware replacement and obtained the reliability and economic measures of computer system under certain assumptions. The model has been analyzed stochastically by using R.P.T and the results of the model has been compared with the existing model. Chib et al. [3] assumed two types of repair facilities for stochastic analysis of two unit priority system. Rathee and Chander [9] carried out the reliability analysis of parallel system in which he gave priority to repair over maintenance. Munday and Malik[14] obtained profit analysis of a Computer System with Priority for hardware redundancy and repair. Recently Kumar et al. [4] studied profit analysis of computer systems with priority and maximum operation time. The author gave priority to software upgradation over hardware repair and replacement and hence obtained the profit of the system. The profit of the system has been compared with some existing model in literature. Also Bashir et al. [7] used the technique of inspection of degraded unit and obtained cost analysis of two unit model. Gorjian Jolfaei [8] carried out reliability analysis of power generation engines in a wastewater treatment plant. Kumar and Malik [1] examines the weathering server system with identical element in warm standby and used the concept of first come first serve repair policy for redundant systems in weather conditions. The various reliability measures of the system has been obtained using R.P.T. The author proved that the concept of first come first serve repair policy proved to be economically beneficial. Sharma and joorel [10] developed a model for fruit juice manufacturing system for its reliability analysis, which consists of four non identical units viz. Fruit cutter, Fruit pulper, Sterlizer and Homogenizer. The author used the concept of emergency repair in which the team of experts is available on the failure of Sterlizer or Homogenizer, to carry out the failure activities. [16] Obtained the reliability of two unit cold standby model under the assumption that the system is operating under different weather conditions and gave priority to preventive maintenance over inspection. To determine the nature of the system, various system effectiveness has been obtained using regenerative point technique and semi markov process. [17] Analysed the concrete mixture plant which comprises of seven units subject to the coverage factor by considering profust reliability approach which is based on probability assumption as well as fuzzy state assumption. The performance analysis of the system has been done using Markov birth–death model and differential equations solved by Runge–Kutta method. The results of the proposed model have proved beneficial for system designers. [18] have made an effort to obtain the system performance of sewage treatment plant by using markovian birth-death process. Various system measures like reliability, availability, maintainability and obtainability of the system has been obtained. [19] explored the generators in steam turbine power plant and obtained the availability and profit of generators. Using supplementary variable technique amd markov birth-death process different system measures has been evaluated. Results of the system have been obtained for availability and profit that would be useful for system designers to enrich the reliability and thus performance of plant
The basic aim of the paper is to obtain the profit evaluation of textile industry with exponential failure time distribution subject to different repair strategies. The System consists of three non-identical components Spinning (S), Weaving (W), Coloring and Finishing (Cf). Initially unit S is in working mode whereas unit W and Cf are in idle state. Failure of the system occurs only if all the unit fails. Single server is available in the system to carry out repair activities and Cf gets first priority over repair followed by S and W. The failure time (FT) ~ exp. (
). The repair time distribution is taken as general.
2- Notations and Symbols
· 
, 
= failure rates of unit 
respectively.
· 
= activation rates of unit 
and 
from idle to operation respectively
· 
cdf of repair time of the failed unit 
respectively
· * = Laplace transform i.e. 
TRANSITION DIAGRAM
~ = Laplace-Stieltjes transform i.e. 
.
· 
are normal and operative.
· 
are under repair.
· 
are waiting for repair.
· 
is in ideal state.
· 
Total possible states of the system, 
3- Transition Probabilities
Let 
represents the transition probability from state 
to 
in transient state. Then the various transition probabilities, can be written as follows:
=
= 
= 
In transition probabilities by taking 
we get the steady state probabilities i.e 
and it can be easily verified that 
for all possible values of 
, i.e
+
+
+
The MST 
is the expected time taken by a system in a given state before transiting to any other state. If 
is the sojourn time of 
then MST is given by:
4- Mean Time to System Failure
We define as the time to system failure and 
be the c.d.f. of the time to system failure for the first time when the system starts operation from state 
We find the equations of for various values of i, by using regenerative point technique.
Therefore, MTSF for different states is given by:
Taking L.S.T of above equations, and simplifying for 
,
On taking 
in above equation and 
, as
we have
implies that 
is a proper cdf.
Therefore, MTSF is given by
Where,
(1-
)+
+
5- Availability Analysis
We define availability, 
as the probability that the system is in upstate and is available to serve its required purpose when it initially start from regenerative state 
The availability equations for different states is given as:
+
+
+
Taking the LT of above equations, we get the set of linear
equations in 
and solving for 
, we get
The steady state availability is given as:
where,
(1)
(2)
6- Busy Period Analysis (BPA)
It is the probability that the repairman is occupied in repairing the failed component at time ‘
’ given that the system has started from regenerative state at 
. The BPA for different states is given by
Taking the LT of above equations and solving for 
, 
In steady state, BPA is given by
+
And 
given by (2)
7- Expected No. of Visits by Repairman
We denote 
as the expected no. of visits by the repairperson to repair the failed unit. The expected no of visits for different states is given as:
+
Taking the LT of above equations and solving for 
,
In steady state, the no. of visits per unit time is given by
Where,
And 
given in (2)
8- Profit Analysis
The profit encountered to the model in steady state by considering different costs and revenues, is calculated as:
Where
Revenue per unit up time of the system,
Cost per unit time for which the repair is busy,
Cost per unit visits by the repairman.
9- Behavior of Reliability Measures:
In this section, for practical illustration, the behavior of some important reliability measures has been examined for some specific values of parameters.
Table 1.Performance of 
and fixed parameters
,
, 
, 
,
and 
on MTSF.
| MTSF | ||
|
|
| |
0.1 | 3.9904 | 5.2570 | 2.8029 |
0.2 | 2.9559 | 3.9690 | 2.3256 |
0.3 | 2.3585 | 3.2605 | 2.0005 |
0.4 | 1.9677 | 2.8004 | 1.7637 |
0.5 | 1.6917 | 2.4719 | 1.5834 |
0.6 | 1.4863 | 2.2227 | 1.4416 |
0.7 | 1.3276 | 2.0256 | 1.3271 |
0.8 | 1.2013 | 1.8648 | 1.2328 |
0.9 | 1.0985 | 1.7307 | 1.1537 |
1.0 | 1.0065 | 1.6612 | 1.0653 |
| MTSF | ||
|
|
| |
0.1 | 1.8571 | 1.6701 | 2.1038 |
0.2 | 2.0716 | 1.8292 | 2.4664 |
0.3 | 2.1878 | 1.9227 | 2.6476 |
0.4 | 2.2646 | 1.9869 | 2.7667 |
0.5 | 2.3221 | 2.0350 | 2.8596 |
0.6 | 2.3680 | 2.0727 | 2.9397 |
0.7 | 2.4054 | 2.1019 | 3.0123 |
0.8 | 2.4351 | 2.1230 | 3.0792 |
0.9 | 2.4563 | 2.1352 | 3.1401 |
1.0 | 2.4677 | 2.1369 | 3.1938 |
| Profit | ||
|
|
| |
0.1 | 2.5482 | 2.4729 | 2.8455 |
0.2 | 2.1945 | 2.2126 | 2.4681 |
0.3 | 1.8936 | 2.0004 | 2.1495 |
0.4 | 1.6221 | 1.8157 | 1.8639 |
0.5 | 1.3656 | 1.6459 | 1.5954 |
0.6 | 1.1151 | 1.4828 | 1.3339 |
0.7 | 0.8643 | 1.3212 | 1.0724 |
0.8 | 0.6091 | 1.1574 | 0.8061 |
0.9 | 0.3463 | 0.9888 | 0.5318 |
1.0 | 0.0737 | 0.8134 | 0.2468 |
| Profit | ||
|
|
| |
0.1 | 3.59827 | 4.17309 | 3.80843 |
0.2 | 3.67789 | 4.37047 | 3.85116 |
0.3 | 3.83764 | 4.68783 | 3.99023 |
0.4 | 4.08536 | 5.15737 | 4.23971 |
0.5 | 4.44261 | 5.84607 | 4.63524 |
0.6 | 4.95729 | 6.89995 | 5.2584 |
0.7 | 5.739 | 8.68624 | 6.31344 |
0.8 | 7.08058 | 12.4277 | 8.45213 |
0.9 | 10.0689 | 25.9994 | 15.3405 |
1.0 | 11.1245 | 30.3421 | 20.4563 |
