Performance Analysis of Biscuit Manufacturing Plant in Steady State Using Fuzzy Availability

2422 | P a g e O c t 1 5 , 2 0 1 3 Performance Analysis of Biscuit Manufacturing Plant in Steady State Using Fuzzy Availability Pawan Kumar Chhoker Departments of Mathematics, N.C. College of Engineering, Israna Panipat, India chhoker.pawan@rediffmail.com ABSTRACT This paper deals with the performance analysis of biscuit manufacturing plant consisting of six sub-systems using fuzzy availability in the steady state. These six sub-systems are arranged in series and parallel configurations. Mathematical formulation of the problem is carried out using Markov process and the governing differential equations are solved in steady state using normalizing condition. The effect of variations of fuzzy availability for different failure, repair rates and system coverage factor for each sub-system in steady state is also studied.


MATERIAL AND METHODS
The profust reliability theory is based on the following two assumptions: 1. Probability assumption: the system failure behavior is fully characterized in the context of probability measures.
2. Fuzzy states assumption: the system success and failure are characterized by fuzzy states. At any time the system can be viewed as being in one of the two fuzzy states to some extent. This means that system failure cannot be defined in a precise way, but in a fuzzy way.
For ease of reference, in the following we provide some essential concepts and results in profust reliability theory. For more details, refer to Cai (1996).

INTRODUCTION
The reliability engineering is one of the important engineering tasks in designing and development of a technical system. Conventional reliability theory considers the assumptions of the probability theory and the binary states of a component/system as working or failed. It is well known that the binary state modeling for reliability of components as well as for the system is too simplistic and does not capture the reality for most systems which can have many levels of performance. Let us take an example of a system comprises two independent processing units which can perform their own tasks in parallel. Evidently we can take the system to be fully functioning when both processing units are functioning and take the system to be fully failed when both processing units are failed. However, when one processing unit is functioning and the other one is failed, the system operates at degraded and demonstrates a significant attribute of performance degradation. At this point the system is neither fully functioning nor fully failed, but in intermediate state. This implies that the crisp definitions for system failure and success are no longer appropriate and thus the binary state assumption should be relaxed. Zadeh (1965) introduced fuzzy set theory. This theory can handle all the possible states between a fully working state and completely failed state. Thus the binary state assumption in conventional reliability is replaced by fuzzy state assumption. This approach to the reliability is known as profust reliability and developed by Cai (1996).An important problem in the profust reliability theory is how to plan maintenance policy in the presence of the fuzzy state assumption. Cai et al. (1991) described fuzzy reliability modeling of gracefully degradable computing systems and show that the fuzziness, coverage factor and maintenance have significant effects on the fuzzy reliability behavior of gracefully degradable computing systems. Cai et al. (1990) used fuzzy reliability for street-lighting lamps replacement and make a comparison between two replacement policies, viz. the block replacement policy in a non-fuzzy environment and the periodic replacement policy without repair at lamp failures in a fuzzy environment. Cai et al. (1991A) applied fuzzy reliability to discuss the survivability index for computer communication networks (CCNs) and also evaluated fuzzy availability to discuss the effect of maintenance. Cai et al. (1993 and1995) used fuzzy states as a basis for a theory of fuzzy reliability and discuss all the forms of fuzzy reliability and typical systems including the series system, parallel system, markov model and mixture model. Verma et al. (2007) also discussed all the forms of fuzzy reliability. They evaluated fuzzy availability using two different approaches and fuzzy availability modeling of semi-markovian system. Verma et al. (2007) also found profust reliability of degraded diesel engine power plant and studied the effect of various parameters such as failure rates and system coverage factor on profust reliability. Chongshan (2009) calculated fuzzy availability of a repairable consecutive-2-out-of-3: F-system. Zuang (1995) presented a method of reliability analysis in the presence of fuzziness attached to operating time. Chowdhury and Misra (1992) presented a method to find an expression of fuzzy system reliability of a non-series parallel network taking into consideration the special requirements of fuzzy sets. Pandey and Tyagi (2007) calculated profust reliability of a power loom plant which is modeled as a two units gracefully O c t 1 5 , 2 0 1 3 degradable system. This paper proposed a method of failure rate parameter estimation using fuzzy numbers. Guan

RESULTS AND DISCUSSION
We now consider the effect of various parameters on fuzzy availability of the system in this section. Here, the fuzzy availability of the system is computed by altering the failure and repair rates of the sub-systems.  Table 1 (given in appendix-A).
The table 1 shows that the fuzzy availability of the system decreases approximately by 0.00027875 to 0.001705193 with the increase in failure rate from 0.001 to 0.004 (with increment of 0.001) for different values of system coverage factor and repair rates. Also fuzzy availability of system increases approximately by 0.000014142 to 0.001127752 with the increase in repair rates from 0.5 to 0.8 (with increment of 0.1) and increase of coverage factor with an increment of 0.1.  The table 4 (given in appendix-A) shows that the fuzzy availability of the system decreases approximately by 0.000019216 to 0.000388274 with the increase in failure rate from 0.0016 to 0.0019 (with increment of 0.0001) for different values of system coverage factor and repair rates. Also fuzzy availability of system increases approximately by 0. 000076866 to 0.002700307 with the increase in repair rates from 0.2 to 0.5 (with increment of 0.1) and increase of coverage factor with an increment of 0.1.

(v) Effect of failure rates and repair rates of Cooling Conveyor on fuzzy availability of the System: -
We have also calculated the fuzzy availability of the system after varying the failure and repair rates of cooling conveyor sub-system. Following data has been used and results are shown in Table 5 (given in appendix-A). Four levels each of failure and repair rates of this sub-system have been considered as:

CONCLUSION
Detailed study of Table 1 to Table 6 reveals that failure rates of mixing sub-system & repair rates of baking oven subsystem (also shown in figure 2 and figure 3 which are given in appendix-1) has maximum effect on fuzzy availability of the complete system. Other sub-systems are almost equally effective. Thus, we can make an inference that we should take the most care of these sub-systems in order to improve the overall fuzzy availability