Analysis of Two Commodity Markovian Inventory System with Lead Time
Korean J. Comput. & Appl. Math. Vol. 8(2001), No. 2, pp. 427 - 438
ANALYSIS OF TWO COMMODITY MARKOVIAN INVENTORY SYSTEM WITH LEAD TIME
N. ANBAZHAGAN AND G. ARIVARIGNAN
Abstract. A two commodity continuous review inventory system with independent Poisson processes for the demands is considered in this paper. The maximum inventory level for the i-th commodity is fixed as Si (i = 1, 2). The net inventory level at time t for the i-th commodity is denoted by Ii (t), i = 1, 2. If the total net inventory level I(t) = I1 (t) + I2 (t) −2) −2) drops to a prefixed level s [≤ (S12 or (S22 ], an order will be placed for (Si − s) units of i-th commodity(i=1,2). The probability distribution for inventory level and mean reorders and shortage rates in the steady state are computed. Numerical illustrations of the results are also provided. AMS Mathematics Subject Classification : 60J27, 90B05 Keywords and phrases : Inventory models, Joint replenishment, Stochastic lead times, Markov process.
1. Introduction With the fast expansion of activities in Business and Industrial sectors, many inventory systems are increasingly found to operate with more than single commodity. These systems unlike those dealing with single commodity, involve more complexities in the reordering procedures. In the modelling of such systems, initially models were proposed with independently established reorder points. But in situations where several products compete for common storage space or share the same transport facility or are procured from the same source, the above method overlooks potential savings associated with joint ordering and hence may not be optimal.
Received November 29, 1999. Revised August 7, 2000. The work was carried out under a Major Research Project funded by University Grants Commission, India. c 2001 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 427
428
Anbazhagan and Arivarignan
The modelling of multi-item inventory system under joint
ANALYSIS OF TWO COMMODITY MARKOVIAN INVENTORY SYSTEM WITH LEAD TIME
N. ANBAZHAGAN AND G. ARIVARIGNAN
Abstract. A two commodity continuous review inventory system with independent Poisson processes for the demands is considered in this paper. The maximum inventory level for the i-th commodity is fixed as Si (i = 1, 2). The net inventory level at time t for the i-th commodity is denoted by Ii (t), i = 1, 2. If the total net inventory level I(t) = I1 (t) + I2 (t) −2) −2) drops to a prefixed level s [≤ (S12 or (S22 ], an order will be placed for (Si − s) units of i-th commodity(i=1,2). The probability distribution for inventory level and mean reorders and shortage rates in the steady state are computed. Numerical illustrations of the results are also provided. AMS Mathematics Subject Classification : 60J27, 90B05 Keywords and phrases : Inventory models, Joint replenishment, Stochastic lead times, Markov process.
1. Introduction With the fast expansion of activities in Business and Industrial sectors, many inventory systems are increasingly found to operate with more than single commodity. These systems unlike those dealing with single commodity, involve more complexities in the reordering procedures. In the modelling of such systems, initially models were proposed with independently established reorder points. But in situations where several products compete for common storage space or share the same transport facility or are procured from the same source, the above method overlooks potential savings associated with joint ordering and hence may not be optimal.
Received November 29, 1999. Revised August 7, 2000. The work was carried out under a Major Research Project funded by University Grants Commission, India. c 2001 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 427
428
Anbazhagan and Arivarignan
The modelling of multi-item inventory system under joint
References: 1. J.L.Ballintify, On a basic class of inventory problems, Management Science 10 (1964), 287-297. 2. A.Federgruen, H.Groenvelt and H.C.Tijms, Coordinated replenishment in a multi-item inventory system with compound Poisson demands, Management Science 30 (1984), 344-357. 3. S.K.Goyal and T.Satir, Joint replenishment inventory control: Deterministic and stochastic models, European Journal of Operations Research 38 (1989), 2-13. 4. S.Kalpakam and G.Arivarignan, A coordinated multicommodity (s,S) inventory system, Mathl. Comput. Modelling 18 (1993), 69-73. 5. A.Krishnamoorthy, R.Iqbal Basha and B.Lakshmy, Analysis of two commodity problem, International Journal of Information and Management Sciences 5(1) July (1994), 127136. 6. A.Krishnamoorthy and T.V.Varghese, A two commodity inventory problem, Information and Management Sciences 5(3) Dec (1994), 55-70. 7. E.A.Silver, A control system of coordinated inventory replenishment, International Journal of Production Research 12 (1974), 647-671. G. Arivarignan received M.Sc(Statistics) from Annamalai University and his Ph.D from Indian Institute of Technology, Madras. Since 1974, he worked as Assistant Professor of Statistics in Tamilnadu Collegiate Educational service and has joined Madurai Kamraj University in 1990. His research interests are stochastic modelling and Applied Statistics. Department of Applied Mathematics and Statistics, Madurai Kamaraj University, Madurai- 625 021, India. N. Anbazhagan is carrying out his Ph.D programme in the study of Multi-Commodity Inventory Systems in the Department of Applied Mathematics and Statistics, Madurai Kamaraj University, Madurai. In 1995 he received M.Sc(Mathematics) from Cardamom Planters Association College, Bodinayakanur.