Get Complete Project Material File(s) Now! »

**A FINITE SOURCE MULTI-SERVER INVENTORY SYSTEM WITH SERVICE FACILITY**

**INTRODUCTION**

One implicit assumption made by many previous stochastic inventory models is that the item whose inventory is kept is made available to the customer immediately it is demanded. This is not generally true, however, as many items are delivered only after some work has been done on them. This is a particularly growing trend as many organisations are strategically shifting their production approach from a make-to-stock system to an assemble-to-order system. Such systems have longer lead time but maintain smaller inventory levels than the make-to-stock system. The implication of such increase in lead time on the level of service available to customers is an area that is now being actively researched by many authors.

Berman et al (1993) considered an inventory management system at a service facility which uses one item of inventory for each service provided. They assumed that both demand and service rates are deterministic and constant and queues can form only during stock outs. They determined optimal order quantity that minimizes the total cost rate. Berman and Kim (1999) analysed a problem in a stochastic environment where customers arrive at a service facility according to a Poisson process. The service times are exponentially distributed with mean inter-arrival time which is assumed to be larger than the mean service time. Under both the discounted and the average cost cases, the optimal policy of both the finite and infinite time horizon problem is a threshold ordering policy. A logically related model was studied by He et al. (1998), who analyzed a Markovian inventory – production system, in which demands are processed by a single machine in a batch of size one. Berman and Sapna (2000) studied an inventory control problem at a service facility which requires one item of the inventory. They assumed Poisson arrivals, arbitrarily distributed service times and zero lead times. They assumed that their the system has finite waiting room. Under a specified cost structure, the optimal ordering quantity that minimizes the long-run expected cost per unit time was derived. Schwarz et al. (2006) considered an inventory system with Poisson demand and exponentially distributed service time with deterministic and randomized ordering policies.

In all the above models the authors assumed that the service facility had a single server. But in many real life situations the service facility may provide more than one server so that more customers are handled at a time. Moreover if a customer’s request cannot be processed for want of stock or free server he/she may prefer to leave the system and make an attempt at later time. The concept of having unserviced customers in an orbit and allowing them to retry for the service have been considered in queueing systems. A complete description of situations where queues with retrial customers arise can be found in Falin and Templeton (1997). A classified bibliography is given in Artalejo (1999).

For more details on multi-server retrial queues see Anisimov and Artalejo (2001), Artalejo et al. (2001) and Chakravarthy and Dudin (2002). Multi server inventory system with service facility was considered by Arivarignan et al (2008). They assumed a continuous review (K, >) perishable inventory system in which the customers arrive according to a Markovian arrival process. The service time, the lead time for the reorders and the life time of the items were assumed to be exponential. The customer who arrive during the stock-out period or all the items in the inventory are in service or all the servers are busy entered into the orbit of infinite size and these customers compete for their service after an exponentially distributed time interval. Using matrix geometric method, they derived the steady state probabilities and under a suitable cost structure, they calculated the long run total expected cost rate.

In this chapter, the focus is on the case in which the population of demanding customers under study is finite so that each individual customer generates his own flow of primary demand. The inventory system with finite source was received only a little attention.

This concept was introduced by Sivakumar (2009). But the analysis of finite source retrial queue in continuous time have been considered by many authors, the interested reader see Falin and Templeton (1997), Artalejo (1998) and Falin and Artalejo (1998) Almasi et al., (2005) and Artalejo and Lopez-Herero (2007) and references therein. The chapter utilises the quasi-random distribution for the arrival process. A good reading on quasirandom distribution is Sharafali et al (2009).

The rest of the chapter is organized as follows. In the next section, the mathematical model and the notation used were described. The steady state analysis of the model is presented in section 3. In section 4, the various system performance measures in the steady state were derived. In the final section, the total expected cost rate in the steady state were calculated.

**MODEL DESCRIPTION**

Consider a service facility which can stock a maximum of > units and ? (≥ 1) identical servers. It is assumed that the arrival process of customers is quasi random with parameter á. The number of sources that generate the customers is assumed to be 8. The customers demand a single item and the item is delivered to the customer after performing some service on the item. The service time is assumed to have exponential distribution. If a customer finds any one of the server is idle and at least one item is not in service, then he/she immediately accedes to the service. The customer who finds either all the servers are busy or all the items are in service enters the orbit of unsatisfied customers. These orbiting customers send requests at random time points for possible selection of their demands. The time intervals describing the repeated attempts are assumed to be independent and exponentially distributed with rate fà AU + , when there are customers in orbit. The service times are independent exponential random variables with rate (. As and when the on-hand inventory level drops to a prefixed level K(≥ ?), an order for L(= > − K > K) units is placed. The lead time distribution is exponential with parameter N(> 0). The streams of arrival of customers, intervals separating successive repeated attempts, service times and lead times are assumed to be mutually independent.

**ACKNOWLEDGEMENTS **

**TABLE OF CONTENTS **

**LIST OF TABLES **

**LIST OF FIGURES **

**CHAPTER 1 **

1.1. SUPPLY CHAIN CONCEPTS

1.2. SUPPLY CHAIN SYSTEMS AND MODELLING

1.3. LITERATURE REVIEW

1.4. STOCHASTIC PROCESSES

1.5. POPULAR MANAGEMENT PHILOSOPHIES

1.6. RESEARCH FOCUS AND CONTRIBUTION

1.7. CHAPTER OVERVIEW

**CHAPTER 2**

2.1. INTRODUCTION

2.2. MODEL DESCRIPTION

2.3. ANALYSIS

2.4. SYSTEM PERFORMANCE MEASURES

2.5. COST ANALYSIS

2.6. NUMERICAL ILLUSTRATIONS

CONCLUSION

**CHAPTER 3 **

3.1. INTRODUCTION

3.2. MODEL DESCRIPTION

3.3. ANALYSIS

3.4. SYSTEM PERFORMANCE MEASURES

3.5. TOTAL EXPECTED COST

3.6. CONCLUSION

**CHAPTER 4 **

4.1. INTRODUCTION

4.2. MODEL DESCRIPTION

4.3. STEADY STATE ANALYSIS

4.4. SYSTEM PERFORMANCE MEASURES

4.5. COST ANALYSIS

4.6. ILLUSTRATIVE NUMERICAL EXAMPLES

4.7. CONCLUSION

**CHAPTER 5 **

5.1. PART A: BUFFERING WITH ZERO SHORTAGE COST

5.2. PART B: BUFFERING WITH POSITIVE SHORTAGE COST

**CHAPTER 6**

6.1. CONCLUDING OVERVIEW

6.2. SOME POSSIBLE APPLICATIONS OF DERIVED MODELS

6.3. POSSIBLE AREAS FOR FUTURE RESEARCH

BIBLIOGRAPHY

APPENDIX

*Philosophiae Doctor*