### Use of continuous Markov chains to determine the functional relation between the availability of equipment and the stock level

Objectives
The determination of the stock level is one of the most important problems in the programming and planning of the production. The most frequently used method is to consider to types of costs: the money immobilization cost and the administrative cost of launching a provision order. The first item increases with the spare parts stock level and the second remain almost constant independently of the spare parts quantity. The total cost, in general, has a minimum. This methodology has several deficiencies. Among them the following can be mentioned:

• The difficulties to compute the real administrative cost of a provision order since most of them are indirect costs.

• The lack of consideration of the stock breakdown cost that produces a decrease of the final product production.

When unit cost of spare parts is reduced this approach gives a reasonable value of the optimum stock level but in the opposite case, when they are high, the most significant cost is the loss of production. In some special circumstances such as in the petroleum industry using submersible pumps, considering only the administrative costs does not give useful results.

The goal of the present project is to develop a method to compute the availability of the production equipment based on a fundamental spare part stock level.

Introduction
The process of failure has a stochastic nature. The most used distribution is the exponential distribution of failure intervals that gives origin to the Poisson and Weibull process. The exponential distribution approach represents a good approximation of real world process and has the advantage that mathematical closed results can be obtained. But the Poisson process is memory less so hysteresis and material fatigue phenomena cannot be modeled.

Also questions such as: “Are the results obtained by exponential models valid if the stochastic process has other distribution?” must be analyzed and answered to obtain useful results.

To solve the problem three different approaches were considered:

• Intuitive method

• Theoretical method for exponential distributions

• Computational method for both

• Numerical solutions of the theoretic results and

• Montecarlo simulation.

The three approaches obtained similar results.

Implementation
Continuous exponential Markov chains were used for modeling the phenomenon. Continuous Markov chains are models that have a finite number of states and a function that computes the next state based on the actual state. The time that the system remains in one state is a random exponential stochastic variable whose mean value depend on the current state. The state is defined by the number of running machines and the stock of the spare parts.

Having in account these hypotheses the result of the model is a set of total linear differential equations with constant coefficients. These equations are known as the Kolmogorov equations. The matrix of this linear system has range n – 1 where n is the order of the system of differential equation.

The solution to the system was implementing using Java programming language and Apache mathematical package for matrix algebra for both the transient and the stationary case.

The simulation system was also implemented in Java to accept different types of distributions (Exponential, Uniform, Deterministic and Weibull).

Conclusions
The simulations running for different values of the mean time between failures, the mean time of repair and the mean time of provision give results that permits to conclude that:

• The mean availability in function of the stock level depends weakly on the distribution type but only on the mean value.

• The qualitative analysis gives a reasonable approach of the mean availability in function of the stock level

• The theoretical and simulation results are almost identical

• The deterministic distribution gives a better availability that the stochastics distributions for each stock level. In other words the uncertainty has a price.

Future development
Results so far were obtained considering a set of identical equipment that use one spear part. A logical extension should be to retain the hypothesis of identical equipment but extend to the case of several spear parts.