COST-BENEFIT ANALYSIS OF A SINGLE UNIT SYSTEM WITH SCHEDULED MAINTENANCE AND VARIATION IN DEMAND

The present paper analyses the reliability and cost -benefit for a single unit system with scheduled maintenance and variation in demand. As Change in d emand affects the production of system also, hence sometimes, the system needs to be shut down when th e umber of produces are in excess as compared to those demanded. Revenue in case of both types of up states i.e., when demand is greater than or equal to production and when demand is less than production have also been taken under consideration while carrying out the cost-benefit analysis. The loss in curred to the system when it is kept shut down due to l ss demand has also been taken into account. Optimized reliability indices of the system effectiveness are estimated numerically using semi-Markov processes a nd regenerative point technique. Expression for the  expected profit is obtained after obtaining various measures of system effectiveness. We can conclude that cut off points for various rates/probabilities/reve nu per unit up time/costs can be obtained which he lp in deciding the upper/lower acceptable values of rates /costs so that the system is profitable.


INTRODUCTION
Literature of Reliability contains lot of studies on the reliability and cost-benefit analysis of various systems. These studies are contributed by various researchers including Rizwan et al. (2010); Manocha et al. (2011) and Kumar and Kumar (2012) where in the concepts of operating and rest periods, hot standby Programmable Logic Controller, random inspection, instructions, ash water pump systems with and without failed states, hardware based software interaction failures and different types of recovery have been taken up. These studies have considered the demand as fixed. However, there exist many practical situations where the demand of the units produced is not fixed. Such a situation may be seen in General Cable Energy System and hence there is need of studying reliability and availability analysis of a system with varying demand and hence the present paper. The present paper investigates the reliability and cost-benefit analysis of a single unit system with scheduled maintenance and variation in demand. As variation in demand affects the production and hence the system is required to be put to down state when the units produced are already in excess. The system in the down state is made operative as soon as the produced units are less in number than those demanded.

MATERIALS AND METHODS
In this study, the probabilistic analysis of the system is analyzed by making use of semi-Markov processes and regenerative point technique and have obtained various measures of system effectiveness such as Mean Time to System Failure, The Steady State Availability Science Publications JMSS when demand is not less than production, The Steady State Availability when demand is less than production, Busy period of the repairman for repair at t = 0, Expected number of visits by the repairman at t = 0, Expected Number of scheduled maintenances at t = 0, Expected down time at t = 0, Profit incurred to the system.

Notations λ
= Failure rate of the operative unit λ 1 = Rate of decrease of demand so as to become less than production λ 2 = Rate of increase of demand so as to become greater than or equal to production λ 3 = Rate of going from upstate to downstate (reason behind this is that the demand is less than production and production goes on increasing and as a result we have lot of produces in the stock. This production needs to be stopped λ 4 = Rate of change of state from down to up when there is no produce with the system and demand is there β 1 = Rate of requirement of doing scheduled maintenance β 2 = Rate of doing scheduled maintenance p 1 = Probability that during the repair time the demand is greater than or equal to production p 2 = Probability that during the repair time the demand is less than production Fr = Failed unit under repair

Symbols for the States of System
O SM = Operative unit under online scheduled maintenance Op d≥p = Operative unit when demand is not less than production Op d<p = Operative unit when demand is less than production D = Notation for down unit Fr = Failed unit under repair

DISCUSSION
It can be concluded from Fig. 2 and 3 that availability decreases with increase in the values of failure rate and increase s with increase in the values of repair rate. Figure 4 shows the behavior of profit with respect to revenue per unit up time (C 0 ) for different values of cost per visit of the repairman (C 3 ). It can be concluded from the graph that the profit increases with the increase in values of C 0 and has lower values for higher values of C 3. It is also observed from the graph that for C 3 = 100, the profit is positive or zero or negative according as C 0 > or = < 500 .So, the system is profitable only if C 0 > 500. For C 3 = 6100, the profit is positive or zero or negative according as C 0 > or = < 570. So, the system is profitable only if C 0 >570. For C 3 = 12100, the profit is positive or zero or negative according as C 0 > or = < 625. So, the system is profitable only if C 0 > 625. Fig. 5 shows the behavior of profit with respect to revenue per unit up time (C 1 ) for different values of cost per visit of the repairman (C 3 ). It can be concluded from the graph that the profit decreases with the increase in values of C 1 and has lower values for higher values of C 3 . It is also observed from the graph that for C 3 = 100, the profit is positive or zero or negative according as C 1 < or = > 1180. So, the system is profitable only if C 0 <1180. For C 3 = 6100, the profit is positive or zero or negative according as C 1 < or = > 1200. So, the system is profitable only if C 1 <1200. For C 3 = 12100, the profit is positive or zero or negative according as C 1 < or = > 1280. So, the system is profitable only if C 1 <1280.

Measures of Effectiveness 4.1.1. Mean Time to System Failure
To determine the Mean Time to System Failure (MTSF) of the system, we regard the failed states as absorbing states. By probabilistic arguments, we obtain the following recursive relations for φ i (t): Taking Laplace-Steltjes Transform (LST) of these relations and solving them for φ 0 **(s).The Mean Time to System Failure (MTSF) when the system starts from the state '0'is: where, N = µ 0 + p 01 µ 1 + p 03 µ 3 +p 01 p 15 µ 5 and D = 1-p 01 p 14 -p 03 .

Availability Analysis when Demand is not less than Production
Using the arguments of the theory of the regeneration process, the availability A i d (t) is seen to satisfy the following recursive relations: Taking Laplace transforms of the above equations and solving them for A 0 *(s).The availability of the system, in steady-state, is given by: where, N 1 = (1-p 14 p 41 ) µ 0 and D 1 = µ 0 +p 01 µ 1 +p 02 (1p 14 p 41 ) µ 2 + p 03 (1-p 14 p 41 ) µ 3 + (p 01 p 14 ) µ 4 +(p 01 p 15 ) µ 5 .

Availability Analysis when Demand is less than Production
Proceeding in the similar fashion as in 5.2, the availability of the system, in steady-state, is given by: where, N 2 = p 01 µ 1 and D 1 is already specified.

Busy Period Analysis of the Repairman
By probabilistic arguments, the total fraction of the time for which the system is under repair of the ordinary repairman, in steady-state, is given by:

Expected Number of Visits by the Repairman
The expected number of visits per unit time by the ordinary repairman is given by: where, N 5 = p 03 p 30 (1-p 14 p 41 ) and D 1 is already specified.

Expected Down Time
The total fraction of the time for which the system is in down state given by:

Cost-Benefit Analysis
DT : C 0 = Revenue per unit up time when demand is not less than production C 1 = Revenue per unit up time when demand is less than production C 2 = Cost per unit up time for engaging the repairman for repair C 3 = Cost per visit of the repairman C 4 = Cost of scheduled maintenance per unit time. (All costs are in Indian rupee) C 5 = Loss per unit time during the system remains down

CONCLUSION
From the interpretations as made above through various graphs, we can conclude that cut off points for various rates/probabilities/revenue per unit up time/costs can be obtained which help in deciding the upper/lower acceptable values of rates/costs so that the system is profitable. That is, the upper limit of the failure rate can be obtained, the lower value of the revenue per unit up time when demand is greater than or equal to production can be obtained on the basis of which the company can fix the price of the product manufactured by the company so that the system gives the positive profit. The upper/lower limits of various other rates/costs can be obtained. Obtaining such values, various suggestions can be given to the company using such systems.