TWO-COMMODITY PERISHABLE INVENTORY SYSTEM WITH BULK DEMAND FOR ONE COMMODITY

This article considers a two-commodity continuous review inventory system in which the arriving customers belong to any one of three types, such that type 1 customers demand a single item of the first commodity, type 2 customers demand bulk items of the second commodity, and type 3 customers demand one item of the first commodity and bulk items of the second commodity. The arrivals of all three types of customers are assumed to be a Markovian arrival process (MAP). It is also assumed that the number of items demanded for the second commodity is a random variable. The ordering policy is to place orders for both commodities when the inventory levels are below prefixed levels for both commodities. The lead time is assumed to have a phase type distribution, and the demands that occur during stock out period are assumed to be lost. The joint probability distribution for both commodities is obtained in the steady state case. Various measures of system performance and the total expected cost rate in the steady state are derived. The results are illustrated with numerical examples.


INTRODUCTION
One of the factors that contribute to the complexity of present day inventory systems is the multitude of items stocked, necessitating multi-commodity inventory systems.In early dealing with such systems, many models were proposed with independently established re-order points.But in situations where several products compete for limited storage space, or share the same transport facility, or are produced on (procured from) the same equipment (supplier), the above strategy overlooks the potential savings associated with joint ordering, and hence will not be optimal.Thus coordinated (also known as joint) replenishment reduces the ordering and setup costs, and allows the user to take advantage of quantity discounts, if any.Various models and references may be found in Miller [23], Agarwal [5], Silver [31], Thomstone and Silver [36], Kalpakam and Arivarignan [15], and Srinivasan and Ravichandran [35].and further references that they contain.
In continuous review inventory systems, Balintfy [6] and Silver [31] have considered a coordinated re-ordering policy, represented by the triplet , for a unit sized Poisson demand and constant lead time.In this policy, if the level of the i -th commodity at any time is below i s , an order is placed for i i s S  items; and at the same time, if any other item with available inventory is at or below its can-order level j c , an order is placed so as to bring its level back to its maximum capacity j S .Many subsequent articles have appeared with models involving the above policy.A further article of interest is that of Federgruen et al. [9], which deals with the general case of compound Poisson demands and non-zero lead times.
Kalpakam and Arivarignan [15] have introduced an ) , ( S s policy with a single re-order level s defined in terms of the total number of items in the stock.The policy avoids separate ordering of each commodity.Hence a single processing of orders for both commodities has some advantages in a situation where procurement is made from the same supplies, items are produced on the same machine, or items have to be supplied by the same transport facility.
In the case of two-commodity inventory systems, Anbazhagan and Arivarignan [1,2,3,4] have proposed various ordering policies.Yadavalli et al. [42] have analysed a model with a joint ordering policy and variable order quantities.Sivakumar et al. [33] have considered a two-commodity substitutable inventory system in which the items demanded are delivered after a random time.Sivakumar et al. [34] have considered a two-commodity perishable inventory system with a joint ordering policy.
There are some situations in which a single item is demanded for one commodity and multiple items are demanded for another commodity.For instance, a customer may buy a single razor, or a set of blades, or both.Another example is the sale of a DVD writer and a set of DVDs.It may be noted that the seller would be placing a joint order for both commodities, as these are available from the same source.Moreover, a seller may not be willing to place orders frequently, and may prefer to have one order to replenish stock in a given cycle.These situations are modelled in this work by assuming demand processes that require a single item for one commodity, multiple items for the other commodities, or both commodities, and by assuming a joint re-order for both commodities.
This paper is organised as follows.In section 2, the mathematical model and notations followed in the rest of the paper are described.The steady state solution of the joint probability distribution for both commodities, the phase of the demand process, and the phase of the lead time process are given in section 3.In section 4, various measures are derived of system performance in the steady state; and the total expected cost rate is calculated in section 5. Section 6 presents the cost analysis of the model using numerical examples.
i  e = a column vector of ones

MODEL DESCRIPTION
Consider a two-commodity perishable inventory system with the maximum capacity S i units for the i-th commodity 1,2) = (i .Assume that the demand for the first commodity is for a single item and the demand for the second commodity is for bulk items.An arriving customer may demand only the first commodity, or only the second commodity, or both.The number of items demanded for the second commodity at any demand point is a random variable Y with a probability function The three types of demand for these two commodities occur according to a Markovian arrival process ) (MAP .The life time of each commodity is exponential with parameter 1,2).= (i i


The re-order level for the i -th commodity is fixed at ) (1 , and the ordering quantity for the i -th commodity is items when both the inventory levels are less than or equal to their respective re-order levels.Assume that demands during stock-out period are lost, as well as unsatisfied demands.The requirement 1, >   ensures that after a replenishment the inventory levels of both commodities will always be above the respective re-order levels.Otherwise it may not be possible to place a re-order (according to the policy), leading to a perpetual shortage.That is, if represents inventory level of i -th commodity at time , t then a re-order is made when gives the rate of replenishment once an order is placed.Let 0 T be such that For the demand process, the description of MAP as given in Lucantoni [20] is an infinitesimal generator of a continuous-time Markov chain.Assume that D is irreducible and 0. 0  e D Let  be the stationary probability vector of the continuous-time Markov chain with generator .
D That is,  is the unique probability vector satisfying Let  be the initial probability vector of the underlying Markov chain governing the MAP.By choosing  appropriately, the time origin can be modelled to be: 1. an arbitrary arrival point, 2. the end of an interval during which there are at least k arrivals; or 3. the point at which the system is in a specific state, such as when the busy period ends or begins.
The important case is that of the stationary version of the MAP for .

=  
The constant , ) ( = give the arrival rate of demand for first commodity, for second commodity, and for both respectively.Note that .= For further details on MAP and phase-type distributions and their usefulness in stochastic modelling, the reader may refer to Chapter 2 in Neuts [24], Chapter 5 in Neuts [25], Ramaswami [30], Lucantoni [20,21], Lucantoni et al. [22], Latouche and Ramaswami [17], Li and Li [19], Lee and Jeon [18], Chakravarthy and Dudin [8], and references therein for a detailed introduction of the MAP and phase-type distribution.Some recent reviews can be found in Neuts [26] and Chakravarthy [7].

Let
) From the assumptions made on the demand and replenishment processes, it can be shown that {

0}
)), ( 2), ( 1), ( 2), ( 1 is a Markov process on the state space  .By placing the sets of state space in lexicographic order, the infinitesimal generator of the Markov chain governing the system, in block partitioned form, is given by and the matrices

STEADY STATE ANALYSIS
It can be seen from the structure of P that the homogeneous Markov process 0} )), ( ), ( ), ( ), ( {( on the finite state space  is irreducible.
Hence, the limiting distribution Then the vector of limiting probabilities  satisfies 1. = = e 0   and P The first equation of the above yields the following set of equations: The equations (except (2)) can be recursively solved to obtain where 2) and in the normalising condition, it follows that and From the equation ( 6), the value of ) (Q  may be obtained up to a constant multiplication.This constant can be determined by substituting the value of ) (Q  in the equation (7).

SYSTEM PERFORMANCE MEASURES
In this section, certain stationary performance measures of the system are derived.Using these measures, the total expected cost per unit time may be determined.

Mean re-order rate
A re-order for both commodities is made when the joint inventory level drops to ) , (

Mean shortage rate
and the failure rate of any one item of the second commodity is Then we have

COST ANALYSIS
The total expected cost per unit time (total expected cost rate) in the steady-state for this model is defined to be  Since the total expected cost rate is only implicitly known, the analytical properties -such as convexity of the total expected cost rate -cannot be carried out in the present form.However, the following numerical examples to demonstrate the computability of the results derived in our work, and to illustrate the existence of local optima when the total cost function is treated as a function of only two variables, are presented.

ILLUSTRATIVE NUMERICAL EXAMPLES
Since the total expected cost rate is obtained in a complex form, the convexity of the total expected cost rate cannot be studied by analytical methods.Hence, 'simple' numerical search procedures are used to find the 'local' optimal values for any two of the decision variables } , , , { by considering a small set of integer values for these variables.With a large number of numerical examples, it has been found that the total cost rate per unit time in the long run is either a convex function of both variables or an increasing function of any one variable.
The following five MAPs for arrival of demands are considered, and it may be noted that these processes can be normalized to have a specific (given) demand rate  when considered for arrival of demands.All the above MAPs are qualitatively different in that they have different variance and correlation structures.The first three processes are special cases of renewal processes, and the correlation between arrival times is 0. The demand process labelled as MNC has correlated arrivals with a correlation coefficient of -0.1254, and the demands corresponding to the process labelled MPC have a positive correlation coefficient of 0.1213.Since Erlang has the least variance among the five arrival processes considered here, the ratios of the variances of the other four arrival processes, labelled Exp, HExp, MNC and MPC above, with respect to the Erlang process, are 3.0, 15.1163, 8.1795, 8.1795 respectively.The ratios are given rather than the actual values, since the variance depends on the arrival rate, which is varied in the discussion.

MAP with negative correlation (MNC)
For the lead time distribution, the following three  PH distributions are considered.
Again these processes can be normalised to have a specific (given) rate  when considered for replenishment.

Exponential (Exp)
Example 1: In this example, the effect of the demand rate ,  the lead time rate  , the five types of demand processes, and the three types of lead time processes on the optimal values ) , (

S S TC
is illustrated.The following fixed values for the parameters and costs are assumed: S that minimise the total expected cost rate for each of the five MAPs for arrivals of demands considered against each of the three PHs for lead times.The associated total expected cost rate values are also given in the table.The lower entry in each cell gives the optimal expected cost rate, and the upper entries correspond to * 1 S and .* 2

S
The following observations from Table 1 may be noticed: 1.As  increases, the optimal total cost rate decreases for all five demand processes and for all three lead time processes.Similarly, as  increases the optimal total cost rate decreases.2. The optimal total expected cost rate has a higher value for a demand process having an hyper-exponential distribution, and a lower value for the Erlang demand process.3. The Erlang distributed lead time has a low optimal total cost rate, except for an HExp distributed demand process; and HExp distributed lead time has high optimal total cost rate, except for HExp distributed demand process.For HExp distributed demand process this observation reverses -i.e.HExp distributed lead time has a low optimal total cost rate, and Erl distributed lead time has a high optimal total cost rate.
The optimum values * 1 s and * 2 s that minimise the expected total cost for each of the five MAPs for arrivals of demands considered against each of the three PHs for lead times is given in Table 2.The associated total expected cost rate values are also given.The lower entry in each cell gives the optimal expected cost rate, and the upper entries correspond to As  increases, the optimal total cost rate increases, except for an HExp distributed demand process.For an HExp distributed demand process, the optimal total cost rate decreases as the demand rate  increases.

2.
When  increases, the optimal total cost rate increases for all combinations of five arrival processes and three demand processes.

3.
The optimal cost rate is high in cases where the demand process is an HExp, and it is low when the demand process is Erlang.

4.
The optimal total cost rate is low when the lead time is Erl, except for the HExp distributed demand process.For HExp distributed lead time, the optimal total cost rate is high, except for HExp distributed demand process.For HExp distributed demand process this observation reverses -i.e. the HExp distributed lead time is associated with a low optimal total cost rate, and Erl is associated with a high optimal total cost rate.
The graphs of the total expected cost rate as a function of 1 The graphs of the total expected cost rate as a function of 1 1 h c  For all the demand processes, the Erlang lead time process has a low total expected cost rate, and a hyper exponential lead time process has a high total expected cost rate.


The difference between the total expected cost rates for any two lead time processes is high, except for the HExp demand process.For the HExp demand process, the difference between the total expected cost rates for any two lead time processes is low.This research has extended the existing work on two-commodity continuous review inventory systems by introducing perishability for both commodities, Markov arrival processes for demand time points, and a phase type distribution for lead time.It is also assumed that one of the commodities may accept bulk demands.Steady state solutions are also provided for the joint distribution of inventory levels.For a suitable cost structure, the total expected cost rate in the steady state has been determined.To demonstrate the computability of results derived here, ample numerical illustrations are given.Numerical analysis of the effect of the parameters and costs on the total expected cost rate is given.

Figure 1 :
Figure 1: Space of inventory levels the fundamental rate, gives the expected number of demands per unit time in the stationary version of the MAP .The quantities ,


in the equation(5) we get the values of .


is the steady state probability vector for the inventory level of first commodity i and the second commodity j , it follows that shortage rate of  i th type demand in the steady state 1Let the mean failure rate of commodity-i in the steady state be denoted by 1when any one of the stocked items ceases to function or perishes.Since the rate of failure of a single item is j  for commodity , j the rate at which any one of i items for the th j  commodity fail is given by 1failure of any one item of the first commodity is given by 1 inventory carrying cost of i -th commodity per unit item per unit time 1,2) ordering cost per order.

:
failure cost of i -th commodity per unit item per unit time 1,2) Shortage cost due to type i demand per unit time 1,2

Example 2 :
In this example, the effect of the arrival rate ,  the lead time rate  , and the type of arrival and lead time processes on the optimal values ) is illustrated.Assume the following fixed values for the parameters and cost:

f c and 2 fch c and 2 hc
for the three lead time processes and the five demand processes are shown in Figures7 to 11.In all the figures the plots for the lead time distributions Exp, Erl, and HExp are coloured blue, black, and red respectively.The following may be observed:In all five arrival processes, as 1 increase, the total expected cost rate increases.But the increasing rate for 2 h c is high compared with that of .

Figure 10 :Figure 11 :
Figure 10: MNC demand process Note that the phase type distribution is defined as the time until absorption in a finite state irreducible Markov chain with one absorbing state.The mean of the phase type distribution .Let  denote the reciprocal of this mean.That is, 2 m Table 1 gives the optimum values, * 2

s
The key observations are summarised below.