Analysis of two production inventory systems with buffer, retrials and different production rates

J Ind Eng Int DOI 10.1007/s40092-017-0191-0 ORIGINAL RESEARCH Analysis of two production inventory systems with buffer, retrials and different production rates K. P. Jose1 • Salini S. Nair1 Received: 20 September 2016 / Accepted: 16 February 2017  The Author(s) 2017. This article is published with open access at Springerlink.com Abstract This paper considers the comparison of two ðs; SÞ production inventory systems with retrials of unsatisfied customers. The time for producing and adding each item to the inventory is exponentially distributed with rate b. However, a production rate ab higher than b is used at the beginning of the production. The higher production rate will reduce customers’ loss when inventory level approaches zero. The demand from customers is according to a Poisson process. Service times are exponentially distributed. Upon arrival, the customers enter into a buffer of finite capacity. An arriving customer, who finds the buffer full, moves to an orbit. They can retry from there and interretrial times are exponentially distributed. The two models differ in the capacity of the buffer. The aim is to find the minimum value of total cost by varying different parameters and compare the efficiency of the models. The optimum value of a corresponding to minimum total cost is an important evaluation. Matrix analytic method is used to find an algorithmic solution to the problem. We also provide several numerical or graphical illustrations. Keywords Production inventory  Buffer  Retrial  Matrix analytic method  Cost analysis & K. P. Jose 1 Introduction A queue is formed when either there is positive service time or there are no sufficient servers for the arriving customers. Queuing systems in which an arriving customer finds the server busy and waiting positions (if any) occupied leaves the service area but repeats his demand after some random time are called retrial queues. Between trials, customer is said to be in an orbit. Retrial queues play an important role in communication and computer networks. Other applications include stacked aircraft waiting to land, ticket reservation for trains and flights and queues of retail shoppers who may leave a long waiting line hoping to return later when the line may be shorter. For detailed discussion on retrial queues, one may refer to the monograph by Falin and Templeton (1997) and the bibliography by Artalejo (2010). The analysis of inventory systems with retrials has received little attention of researchers in recent decades. Inventory is the raw materials, goods in different stages of production and finished goods, owned by a company that are ready or will be ready for sale. When customers arrive into a system and if the demanded item is available the same is provided with negligible or positive service time. If the item is out of stock, such customers need not be backlogged or lost; otherwise they move to an orbit and may retry from there. However, retrial in production inventory has received little attention of the researchers in stochastic analysis. So we considered a mathematical model in which the main contributions of this paper are summarized as follows. Salini S. Nair • PG and Research Department of Mathematics, St. Peter’s College, Kolenchery, Kerala 682311, India • Two production inventory systems with buffer are developed. Matrix analytic method is used to solve the systems. 123 J Ind Eng Int • • • • Some important performance measures of the systems are derived and a cost function is defined. The optimum value of a corresponding to the minimum expected total cost is found. The minimum value of expected total cost is found by varying different parameters of the model. The models are compared numerically and suggested best model for practical purposes. These models can be applied to manufacturing systems with stochastic environment. The rest of the paper is organized as follows. In Sect. 2, a brief review of literature is presented. In Sect. 3, we formulate the problem. In Sect. 4, we describe model I and its stability. We provide performance measures of model I in Sect. 5. We describe model II and its stability in Sect. 6 and the performance measures of model II in Sect. 7. Cost analysis is described in Sect. 8. Numerical results and graphical illustrations are presented in Sects. 9 and 10. In Sect. 11, we incorporate concluding remarks and future research. Literature review Artalejo et al. (2006) introduced retrial of unsatisfied customers in inventory systems with positive lead time. They compared numerically the efficiency of the generalized truncated model with a model based on finite truncation. There after some important works Krishnamoorthy and Jose (2007), Yadavalli et al. (2012), Jeganathan et al. (2013), etc., were reported in this direction. Recently, Padmavathi et al. (2015) analyzed a continuous review stochastic ðs; SÞ inventory system. They considered two models which differ in the way that the server goes for vacation. Here the joint probability distribution of the inventory level, the number of demands in the orbit and the server status is obtained in the steady state case. Vijaya Laxmi and Soujanya (2015) described an ðs; SÞ inventory system with service interruptions and retrial of negative customers. The arrival and service interruptions were according to a Poisson process. The lead time and inter-retrial times were exponentially distributed and solution was obtained in the steady state. Krishnamoorthy et al. (2015a) analyzed a queuing-inventory system with common life time and retrial of unsatisfied customers. The arrival of customers followed a Poisson process and all the underlying distributions were assumed to be exponential. In this, reservation and cancellation of inventory is permitted. Expected number of revisits to the maximum inventory level and sojourn times in the maximum inventory level as well as zero inventory are also computed. 123 The main area of literature related to this paper is that of inventory systems with production. Krishnamoorthy and Jose (2008) compared three production inventory systems with positive service time and retrial of customers by assuming all the underlying distributions to be exponential. They obtained that the model with buffer size equal to the inventoried items is the best profitable model for practical purposes. Benjaafar et al. (2010) analyzed a production inventory system as a Markov decision process and compared the performance of the optimal policy against several other policies and obtained that performance is poor for those models that ignore impatience of the customer. Chang and Lu (2011) studied a serial production inventory system by providing a phase-type approximation and obtained good estimates for performance measures such as fill rate and mean queue-length distributions of each station. An efficient production and service scheduling rule to a flexible production service system was proposed by Wang et (...truncated)