A Stochastic Model for Reserve Inventory Between Machines in Parallel

Inventory control is the process of deciding what and how much of various items are to be kept in stock. The basic objective of inventory control is to reduce investment in inventories and ensuring that production process does not suffer at the same time. In this article the optimal reserve inventory between machines in parallel is attempted. The output of first machine M1 is the input for the second and third machines M2 and M3. In between the Machines M1 and M2, M3 an inventory is maintained. One of the problems of interest in inventory control theory is the determination of the Optimal size of the buffer between operating systems, namely machines. The necessity for maintaining inventory arises in several situations in a production oriented inventory systems. The study reveals the Optimum policy for maintaining the inventory between machines. The reason for maintaining inventory between machines is, due to ageing of machines and due to some other external reasons, the time taken for receiving the finished goods between parallel system may affect the starting of the next machines. Hence here we study the optimal reserve inventory between machines in parallel. A generalized equation is derived when the number of parallel machines are ‘n’.


Introduction
One of the problem of interest in inventory control theory is the determination of the optimal size of the buffer stock between the systems.The necessity for maintaining inventory arises in several situations in a production oriented inventory system, viz., hydro-electric system, thermal power station etc.
The optimal reserve inventory between the systems, namely, the system that produces the input and the system that consumes the input obtained from the previous system is must.
If various manufacturing processes operate successively, then in the case of breakdown of one at some stage can affect the entire system.Hence stocking points of inventory are created between adjacent stages so as to achieve a certain degree of independence in operating the stages.
The recent applications in a production oriented inventory system, viz., hydro-electric system, thermal power station, etc.In the case of hydro-electric system, the input is the river-flow and the Dam is the reserve to hydro-electric station.The optimal reserve in this case is not exactly the reserve in the dam but it is only the problem of optimal discharge during the different periods of a planning horizon.The question of determination of the optimal reserve is solved.
Similarly in case of the thermal station, Coal or Lignite or Furnance Oil is used as input in the thermal generator and the consumption is at the thermal power station.So the optimal reserve inventory is the reserve between the two systems, namely, the system that produces the input and the system that consumes the input obtained from the first system.
In this paper we consider a model in which the system is conceptualized.We consider '3' machines.It is clear that the output of the first machine is the input for the second and third machine, and let us assume that, the consumption rate is constant.
The optimal size of the reserve inventory between two machines has been discussed by Hanssman (1962), which is the basic model.In that model, he assumed that, the idle time cost of second machine M 2 is very high, so a reserve in between M 1 and M 2 is suggested.If T is a random variable denoting idle time of M 2 , it may be noted that The expected total cost due to inventory holding and idle time of M 2 per unit of time is Where 1 µ denotes the number of breakdowns of M 1 per unit time.To obtain optimal 'S', we find dE(C) ds = 0, i. e., In this, the expression for optimal reserve 's' is given, has a limitation that, it should be less than unity, otherwise the solution is not feasible.Hence the new improved solution with no restriction was obtained by Sathyamoorthy.R. & Sachithanandhan, ( 2007), as follows: Then taking dE(C) ds = 0, gives the expression for optimal 's' as This solution is an improved version of (1) with no restrictions.
An extension of the above system is developed into '3' machines, in which second and third machines are parallel.

Basic Model
In this model, 3 machines M 1 , M 2 and M 3 are considered and the optimal value of the reserve inventory between M 1 and M 2 , M 3 for the system is discussed.
4. t denote the repair time duration, which is random variable.
5. U the random variable denoting the inter arrival time between successive breakdown of M 1 which is taken as exponential with parameter µ.
6. S the reserve inventories between M 1 and M 2 , M 3 .

Results
If T , the random variable denoting the idle time of M 2 , M 3 then, it may be noted that The average level of inventory is s r ∫ 0 (S − rt)g(t)dt and assuming that the rate of consumption of M 2 and M 3 is 'r' per unit of time.Where 1 µ , denotes the number of breakdowns of M 1 per unit time.The expected total cost due to inventory holding and idle time per unit time is To find optimal 'S', we find dE(C) ds = 0, i. e., Reducing further, we get From the equations 2, we get the optimal reserve inventories for S, between the machines M 1 and M 2 , M 3 .
Let us prove the result for the system of 'n' machines in parallel by the method of mathematical induction hypothesis.Let us assume that the result is true for k.
Then the equation for the reserve inventory 'S' becomes, Published by Canadian Center of Science and Education for all r, µ, h and d Where h -cost of inventory holding per unit d -idle time cost due to M 2 per unit time r -constant rate of consumption of M 2 S -reserve inventory between M 1 and M 2 t -continuous random variable denoting the repair time of M 1 with g(•) as p. d. f and G(•) as c. d. f.