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ABSTRACT
The thrust on the need for effective decision making in manufacturing and service segments paved the way to develop several numerical models in the countenance of reasonable degree of certainty. Nevertheless, several kinds of decisions are to be made considering uncertainty of upcoming events. To address this, case specific stochastic models have been developed to represent, to some extent, the complexity of the real circumstances and the uncertainty existing around these circumstances. In fact it is difficult to develop a simple numerical model close to reality. As a result, dissimilar models each representing one or more parameters associated with real circumstances have been developed.
The present work contribute towards development of discrete-time stochastic model with finite state space using Markov chain that helps in arriving at optimal replacement decisions for a block of items under the influence of inflation and determining an optimal age at which the cost of replacement is minimum.
Conventional models are available to determine the replacement age for various types of items that deteriorate progressively with time and that do not deteriorate progressively with time but suddenly fail. Individual replacement models are available considering maintenance cost, resale value, cost of depreciation etc. for the items such as machine tools, equipment and machinery, automobiles, etc. that deteriorate progressively with time. While, group replacement model is applicable for the items like electric bulbs, tubes, tyres etc. that fail suddenly during usage. But, there are certain kind of items like computer systems in a network, a block of LCD or LED televisions in star hostels or restaurants, a block of pressure gauges in filling plants, a block of cutting tools in an engineering work shop etc. that may not fail completely on usage, but fail partially and can be restored to operable condition by rectifying failures. Many intermediate repairable states, that are more realistic in practice, are possible between functional and complete failure states. It is inevitable for the items to switch from one state to another state during usage over a period of time, and of course they can be restored to operable condition than allowing them directly to go to complete failure state. Block replacement decision encompasses the different intermediate repairable states.
In the present work, two intermediate states viz. minor repairable state, major repairable state have been considered between functional state and complete failure state, which has resulted in a four-state Markov Chain. A case study has been made with a reference to a block of computer systems, in which case, when the intermediate states are not considered in between the functional state and complete failure state, then it is applicable to a class of items that fail completely on usage.
First order discrete-time Markov chain process, a stochastic process, is employed to compute the probabilities of transition from a given state to any other state for future time periods. With the help of these transition probabilities, proportion of items that are in functional state, minor repair state, major repair state, and complete failure state are calculated. This helps in the estimation of maintenance costs to be incurred for block of items at various points of future time periods.
Further to make the model more realistic, the effect of inflation and time value of money on replacement decision is considered. Conventional models are available to evaluate different replacement strategies for a combination of similar computer systems of different ages considering and without considering money value. Here Net Present Value (NPV) criterion based on nominal rates of interest does not reflect the time value of money.
Real increase in the value of money or purchasing power of money depends upon many macroeconomic aggregates and variables like: Gross Domestic Product (GDP), money supply, capital formation, inflation etc. Real interest rates are computed using Fisherman’s relation that takes into account the inflation.
For this purpose the inflation (for Computer systems) over a period of time has been studied, forecasted and compared with actual values for the known periods by employing various forecasting techniques to identify the underlying model that best fits the time series data. Inflation is forecasted for the future periods by developing a regression model with trigonometric function, which yielded relatively minimal errors.
Also an attempt has been made to develop block replacement model using higher order Markov chains. In case of higher order Markov Chains, calculation and analysis of large number of transition probabilities is tedious and consumes considerable amount of time. To address this, Weighted Moving Transition Probabilities (WMTP) technique, a parsimonious model that approximates higher order Markov chains is proposed and applied. WMTP technique considers the spread of sizeable past data instead of single period data as in the case of first order Markov chain.
The cost analysis is made in evaluating the replacement strategies, - for a block of computers - without considering and with considering the influence of inflation in Indian economic environment and an age at which the block replacement is economical is determined.
Also a study is made to evaluate the behavior of the block replacement model under the influence of variable maintenance cost; and also under rapid up-trend and sluggish up-trends in inflation.
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