A MAP/PH/1 Production Inventory Model with Perishable Items and Dependent Retrials

This article studies a perishable inventory system with a production unit. The production process is governed by (s, S) policy and it is exponentially distributed. The primary arrival follows Markovian arrival process(MAP) and the service time is phase-type distributed random variable. The inventoried items are subject to decay exponentially with a linear rate. A newly arriving customer realizes that system is running out of stock or server busy either moves to infinite waiting space with a pre-assigned probability or exit system with complementary probability. Customers in the waiting space make retrials to access the free server at a linear rate. If the system is running out of stock or the server is busy upon retrial, customers go back to orbit with different pre-defined probabilities according to the level of inventory or exit the system with corresponding complementary probabilities. The system is analysed using Matrix Analytic Method(MAM) and the findings are numerically illustrated.


Introduction
Baek and Moon [1] analysed an inventory system attached to multiple server queue in which items are produced internally. The Poisson process characterizes the arrival of demand and production of items. If the system is out of stock, the unsatisfied customer wait in the system, and all new arrivals are lost. The proposed model is analysed through the regenerative process and product form solution is derived for probability vectors. Ravithammal et al. [2] dealt with a production inventory system of fixed lifetime product. The deterministic demand model considered both fixed back-orders and linear back-orders. Ioannidis et al. [3] considered a perishable inventory system with production and impatient customers. Impatient customers may balk upon arrival or renege while stock out period. The lifetime of items and patience time of customers are random variables with general distributions. Also, service time and interarrival time are assumed as exponential random variables. Chakraborty et al. [4] introduced a generalised economic manufacturing quantity model with a production machine which undergoes breakdown. Arbitrary probability distributions are assigned to time for breakdown, corrective, and preventive repair. Krishnamoorthy and Narayanan [5] studied a system to which arrivals according to a Poisson process and the production unit is controlled by (s, S) policy. The server needs a random time to serve the inventory for demanding customers and which is exponentially distributed. If the production is on, it adds single item to inventory successively and the time required to add items is exponentially distributed. A product form solution is obtained by assuming that the system loses customers during stock out period.  [6] distinguished three retrial inventory systems with a production unit. In all the three models, the demand process follows a Poisson distribution and exponential production time of items. The unsatisfied customer in each model join the orbit with predetermined probability and make retrials to achieve their demands. The inter-retrial and service time are exponentially distributed. The unsatisfied retrial customers re-join the orbit in accordance with a pre-assigned value or loss system with complementary probability. Jose and Nair [7] differentiated two (s, S) production inventory systems with retrial of customers. The production unit runs at two different rates per cycle according to the inventory level. That is, the rate is higher during the beginning of a cycle then it is lowered when the inventory level crosses a pre-defined level. The rate of loss of customers from the system is reduced through that assumption. All the underlying distributions assumed are exponential. Nair and Jose [8] proposed another work in which the variation of service rate is considered. The authors changed the service rate into two different rates; normal rate and reduced rate. The time for producing a single item follows an exponential distribution.
In this paper, the production inventory model proposed by Nair and Jose [8] is extended to perishable items and inventory dependent rejoining of customers. In real life situations, nearly all inventoried items are subject to decay and it leads us to the assumption of random lifetime of items. The lifetime of each item is distributed as exponential with a linear rate. Also, the model provides an orbit for retrial facility in which retrial customers rejoin the orbit depending on the inventory level. In existing models, the lifetime of items stored is assumed as infinite, and customers are allowed to orbit independent of the inventory level. We assume the modelling tools as Markovian Arrival Process and Phase type distribution for arrival process and service time distribution respectively. Thus, Matrix Analytic Method is well suited for analysing a practical situation considering non exponential inter-arrival and service time distribution. Also, the method reduces the problem of numerical intractability of the model. The paper is constructed follows: section 2 model the system mathematically and stability of the system is discussed in section 3. Section 4 and 5 deal with computation of stationary probability vector and performance measures respectively. In section 6, a profit function is constructed and results are numerically illustrated. 3 Consider an inventory system of perishable items governed by continuous review (s, S) policy. If the inventory diminishes to switch on level s, the production unit is switched on and it adds one by one item to the inventory. The manufacturing time of an item follows exponential distribution with rate β and the production stops when the inventory level attains its maximum capacity S. The primary arrivals follow Markovian arrival process having representation (D 0 , D 1 ) with m phases. A single server is availed in the service area and service time is phase type distributed with representation (η, T ) having n phases. If the server busy or inventory zero upon primary arrival, customers occupy the infinite waiting space called orbit with probability γ or goes out of the service area with probability (1 − γ). The unsatisfied primary customers wait in the orbit and make retrials to access the server. If the inventory is zero upon retrial, customers reoccupy the orbit with probability p 0 . If the server is busy upon retrial with inventory level j; 1 ≤ j ≤ s, then the customers reoccupy the orbit with probability p j such that p j > p j−1 (1 ≤ j ≤ s). If the inventory level is greater than switch on level s then the customers reoccupy the orbit with probability p. In all cases, retrial customers exit the system with corresponding complementary probabilities. The lifetime of an item and inter-retrial times of consecutive retrials are exponentially distributed with linear rate jω and iθ respectively. Assume that the item does not perish when the sever is providing service on the last item left in the inventory.
for arbitrary > 0, there exist N such that d r < − for all r in level i ≥ N . Now, the stability follows from Tweedi's result [9].
The stationary probability vector of the truncated system is then obtained by normalizing each ϕ i by ( i ϕ i )e. (1 − p)ψ i,1,1,j,k,l h) Overall rate of retrial, θ * 1 = θ × E orbit i) Successful rate of retrial,

Profit analysis
We define the total profit as

Numerical illustrations
In this section, we investigate the effect of positive and negative correlations on overall rate of retrial, successful rate of retrial and ratio of successful retrial. The variation is compared by making the rejoin probabilities independent of the inventory level, that is, p j = p; for all j(II). Assume that the maximum inventory level, S = 20 and the production switch on level, s = 5. If the rejoining of retrial customers to orbit is controlled by probabilities depending on the inventory level then it will reduce the congestion in the orbit and increases the fraction of successful retrials. As expected, table 2, 3, 4 and 5 shows that the ratio of successful rate of retrial θ * is greater for (I) compared to (II) having all probabilities p j = p, for all j in both MAP(+) and MAP(−). That is, rejoining of retrial customers to orbit in accordance with probabilities depending on inventory level makes retrial more successful.   with pre-determined probability. The retrial customers join back to the orbit with different probabilities in accordance with items present in the inventory. The lifetime of inventoried items and inter-retrial times of customers in the orbit is characterized by independent exponential distributions in linear rate. A profit function was constructed and the results were numerically studied. One can extend this work by considering the Batch Markovian Arrival Process(BMAP) instead of MAP. Further, another way of extension is possible if the random lifetime of the item is replaced by a common lifetime.