DETERMINATION OF OPTIMAL RESERVE INVENTORY BETWEEN MACHINES IN SERIES, USING ORDER STATISTICS

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1 DETERMINATION OF OPTIMAL RESERVE INVENTORY BETWEEN MACHINES IN SERIES, USING ORDER STATISTICS M. Govindhan, R. Elangovan and R. Sathyamoorthi Department of Statistics, Annamalai University, Annamalai Nager , Tamil Nadu, India. ABSTRACT In Inventory control one of the interesting and important problems is the determination of optimal reserve inventory size between two machines in a serial system. The output of the machine M 1 is the raw material for machine M 2. The breakdown of the M 1 will force the machine M 2 to go to the breakdown stage. Therefore the reserve inventory in between two machines is maintained. The optimal size of the reserve inventory is determined taking it consideration a random duration of breakdown of M 1, consumption rate of M 2 the holding cost as well as shortage cost of inventory. Several authors have discussed this problem. In this paper it is assumed that machine M 1 is in the first stage and machine and the two machines are in the second stage. The optimal size of reserve inventory is derived assuming that the breakdown duration of M 1 follows first order statistics and n th order statistics Numerical illustration is also provided. Key words: Optimal Reserve Inventory, Order Statistics, Random Duration, Breakdown Duration. INTRODUCTION In inventory control theory the series system is found to be with greater viability for application in industries, which are productions oriented. The basic model is one in which it is assumed that the two machines namely M 1 and M 2 are in series and the finished product of machine M 1 happens to be a the raw material for machine M 2. If there is a breakdown of M 1, then M 2 will go to the down stage due to none availability of the semi - finished product produced by M 1. Therefore to avoid the shortage cost a reserve inventory in between M 1 and M 2 is suggested taking it consideration the inventory holding cost and also the shortage cost of the semi-finished products. Several authors have discussed this problem and the basic model is found in Hansmann F (1962), Ramachandran V and Sathiyamoorthy R (1981) have considered a modified version of this model Venkatesan et.al (2010). have discussed such a model, Now we consider the model using the concepts of order statistics. In this paper it is assumed that there is one machine M 1 in the first stage and two machines and are in the second stage. The optimal reserve inventory namely the semi-finished product between the machine M 1 in the first stage and machines and in the second stage. The break down duration of M 1 is assumed to be a random variable which follows first order Statistics and similarly n th order Statistics. The expression for optimal reserve inventory size is obtained. Numerical illustration is also provided. ASSUMPTIONS 1. M 1 is the machine in the first stage and the output of M 1 is the raw material for machine and in the second stage. 2. If there is a breakdown of M 1 the supply of raw material for and will be stopped. 3. The reserve inventory if maintained between the machine M 1 and the machines and 4. The breakdown duration of M 1 is for a random period. 5. A reserve inventory of the semi-finished product is maintained between the machines in series. 66

2 NOTATIONS: Case I : S -Level of reserve inventory µ - mean time interval between the successive breakdowns of machine M 1 - A random variable which denotes the duration of breakdown / repair time M 1 and it has p.d.f g(.) with c.d.f G(.)and exp ( ). h - Holding cost per unit of reserve inventory d 1 - Cost per unit of idle time of machine d 2 - Cost per unit of idle time of machine r 1 -Consumption rate per unit time of machine r 2 - Consumption rate per unit time of machine I (t) - Level of inventory at the time t Let a random sample of n observations be taken from the values of U. these observations can be arranged in the increasing order of magnitude as is the first order or minimum order statistics and is the highest order or maximum order statistics. Let be distributed with pdf f (.) and cdf F (.) is given by Similarly let the pdf of be given by Assuming that U exp( ) The pdf of is given by Using the concept of first order statistics for duration of breakdown of M 1 namely.the expected cost in given as. E(c) + ( )... (1) + ( ) + ( )... (2) Where A and B Now we consider, A 67

3 Differentiating (A) by using Leibnitz rule of differentiation of an integral, we get, ( ) * ( )+ ( ) Here and so, Now we have, and G ( 1-, 1 - G ( [ ] * + [ [... (3) Now we consider, B Differentiating (B) by using Leibnitz rule of differentiation of integral, we get, 0[ ] - ( ) * ( )+ ( ) [ ] 68

4 * + [ [... (4) Substitute (3) and (4) in equation (2). We get, [ + ( ) [ [ ( ) [ ( ) ( ) * ( )+ * ( )+ ( ( )) ( )... (5) The above equation (5) by taking log on both sides becomes, Therefore, the optimal reserve inventory level is given as, 69

5 Reserve Inventory Reserve Inventory Asia Pacific Journal of Research ISSN (Print) : NUMERICAL ILLUSTRATION If the value of inventory holding cost h increases the optimal inventory size namely decreases therefore a smaller reserve inventory will increase the profit. This is observed in Table 1 and Fig 1. Table 1: Optimal value of the Reserve Inventory when holding cost is fixed h S Holding cost Fig.1:Optimal value of when holding cost is fixed If the value of d 1 which denotes the shortage cost due to inadequate suppley to machine increasing the reserve inventory is suggested and this is a indicated in Table 2 and Fig 2. increases then as Table 2: Optimal value of the Reserve Inventory when shortage cost for is fixed d S Shortage cost Fig.2 : Optimal value of when shortage cost for is fixed If the value of d 2 which the shortage cost arising due to machine increase then an increase the values of is also observed and therefore if the shortage cost of machine 2 increases a higher level of reserve inventory is suggested. It is observed in Table 3 and Fig 3. Table 3: Optimal value of the Reserve Inventory when shortage cost for is fixed d S

6 Reserve Inventory Reserve Inventory Asia Pacific Journal of Research ISSN (Print) : Shortage cost Fig.3 : Optimal value of when shortage cost for is fixed If the values of which is the average inter arrival time between the successive failure of machine M 1 increases then a smaller reserve inventory is suggested.tis is obsevrved in Table 4 and Fig 4. Table 4: Optimal value of the Reserve Inventory when Successive breakdown of the machine is fixed S Inter Arrival Time CONCLUSION Fig 4 : Optimal value of when Successive breakdown of the machine is fixed The following conclusions are drawn on the basis of the numerical illustration that has been worked out for this model. 1. As the value of inventory holding cost h increases the optimal inventory size namely decreases therefore a smaller reserve inventory will increase the prophet. 2. If the value of d 1 which denotes the shortage cost due to inadequate supply to machine has an increase then an increase in the reserve inventory is suggested. 3. If the value of d 2 which the shortage cost arising due to machine increases an increase the value of is also observed. 4. If the value of which is the average inter arrival time between the successive failure is of machine M 1 increases a smaller reserve inventory is suggested. Case II : The model discussed above is extended to the case where the duration of breakdown of M 1. Which is a random variable denoted as as the distribution of n th order statistics. The optimal inventory level is derived taking the random variable follows exponential distribution with parameter and the case is extended under the assumption that follows the n th order statistics. 71

7 The pdf of be given by Under the above assumption the expression for expected cost is given as follows. E(c) + ( )... (6) Here and so, and G ( 1-, 1 - G ( [ ] + ( ) [ ] C + ( )... (7) Where C [ ] and D [ ] Now we consider, C [ ] Differentiating (A) by using Leibnitz rule of differentiation of an integral, we get, ( ) [ ] [ ] [ ] Now we have, * + 72

8 ... (8) Now we consider, D [ ] Differentiating (A) by using Leibnitz rule of differentiation of an integral, we get, ( ) [ ] [ ] [ ] Now we have, * +... (9) Substituting (8) and (9) values in equation (7). We get, ] * + 73

9 Reserve Inventory Asia Pacific Journal of Research ISSN (Print) : [ 1- * + ] Taking log on both sides becomes, * + Therefore, the optimal reserve inventory level is given as, [ ] NUMERICAL ILLUSTRATION If the value of inventory holding cost h increases the optimal inventory size namely decreases therefore a smaller reserve inventory will increase the profit. This is observed in Table 5 and Fig 5. Table 5 : Optimal value of the Reserve Inventory when holding cost is fixed h Holding cost Fig.5:Optimal value of when holding cost is fixed If the value of d 1 which denotes the shortage cost due to inadequate suppley to machine increasing the reserve inventory is suggested and this is a indicated in Table 6 and Fig 6. increases then as Table 6: Optimal value of the Reserve Inventory when shortage cost for is fixed d

10 Reserve Inventory Reserve Inventory Asia Pacific Journal of Research ISSN (Print) : Shortage cost Fig 6 : Optimal value of when shortage cost for is fixed If the value of d 2 which the shortage cost arising due to machine increase then an increase in the values of is also observed and therefore if the shortage cost of machine 2 increases a higher level of reserve inventory is suggested. It is observed in Table 7 and Fig 7. Table 7: Optimal value of the Reserve Inventory when shortage cost for is fixed d shortage cost Fig 7: Optimal value of when shortage cost for is fixed If the values of which is the average inter arrival time between the successive failure of machine M 1 increases then a smaller reserve inventory is suggested.tis is obsevrved in Table 8 and Fig 8. Table 8 : Optimal value of the Reserve Inventory when Successive breakdown of the machine is fixed

11 Reserve Inventory Asia Pacific Journal of Research ISSN (Print) : Inter Arrival Time Fig 8: Optimal value of when Successive breakdown of the machine is fixed CONCLUSION The following conclusions are drawn on the basis of the numerical illustration that has been worked out for this model. 1. As the value of inventory holding cost h increases the optimal inventory size namely decreases therefore a smaller reserve inventory will increase the prophet. 2. If the value of d 1 which denotes the shortage cost due to inadequate supply to machine has an increase then an increase in the reserve inventory is suggested. 3. If the value of d 2 which the shortage cost arising due to machine increases an increase the value of is also observed. 4. If the value of which is the average inter arrival time between the successive failure is of machine M 1 increases a smaller reserve inventory is suggested. REFERENCE 1) Hansmann F (1962). Operation Research in Production and Inventory Control. John Wiley and Sons, Inc. New York. 2) Ramachandran V and Sathiyamoorthy R (1981). Optimal Reserve for Two Machines. IEE Trans on Reliability, Vol. R- 30, No.4, P ) Srinivasan S, Sulaimen A and Sathiyamoorthy R (2007). Optimal Reserve Inventory Between Two Machines under SCBZ Property of Interarrival times Between Break downs, International Journal of Physical Sciences Ultra Science, Vol 19(2) M, PP ) T. Venkatesan, C Muthu and R Sathiyamoorty (2010). Determination of Optimal Reserve inventory between two Machines in Series, Ultra Scientist, Vol 22(3) M, pp