An Inventory Model for Returns

distributed. Moreover there is an inventory cost for holding each unit of product and a shortfall cost for each unit of backlog. In [37, 49], the production process of the machines is then a Markov Modulated Poisson Process (MMPP). Each machine has two states, either “normal operation” or “under repair”. Since there are m machines, there are2^mstates for the system of machines. The states of the machines and the inventory level can then be modeled as an irreducible continuous time Markov chain. For different values of the hedging pointn, the average running cost can be written in terms of the steady-state probability distribution of the Markov chain. Therefore the optimal hedging point can be obtained by varying the value ofn. In [49], numerical algorithm based on circulant preconditioning discussed in Chap. 2 has been designed to obtain the steady-state probability distribution efficiently. An extension to the case of batch arrival of demands can be found in [44].

3.3 An Inventory Model for Returns

In this section, a single-item inventory system is presented. The demands and returns of the product are assumed to follow two independent Poisson processes with mean rates and respectively. The maximum inventory capacity of the system isQ. When the inventory level is Q, any arrived return will be disposed of. A returned product is checked/repaired before being put into the serviceable inventory. Here it is assumed that only a stationary proportion, let us saya100% of the returned product is repairable and a non-repairable return will be disposed of.

The checking/repairing time of a returned product is assumed to be negligible. The notations for later discussions are listed below:

(a) ¹, the mean inter-arrival time of demands (b) ¹, the mean inter-arrival time of returns

(c) a, the probability that a returned product is repairable (d) Q, maximum inventory capacity

(e) I, unit inventory cost

(f) R, cost per replenishment order

An (r; Q) inventory control policy is employed as inventory control. Here, the lead time of a replenishment is assumed to be negligible. For simplicity of discussion, here we assume that r D 0. In a traditional (0; Q) inventory control policy, a replenishment size of Q is placed whenever the inventory level is 0. Here, we assume that there is no loss of demand in our model. A replenishment order of size.QC1/is placed when the inventory level is0and there is an arrived demand.

This will then clear the arrived demand and bring the inventory level up toQ, see Fig.3.3(Taken from [72]). In fact, State ‘1’ does not exist in the Markov chain, see Fig.3.4(Taken from [72]) for instance.

The states of the Markov chain are ordered according to the inventory levels in ascending order and give the following Markov chain.

Returns

Fig. 3.3 The single-item inventory model λ Fig. 3.4 The Markov chain

The.QC1/.QC1/system generator matrix is given as follows:

A2 D

The steady-state probability distribution p of the system satisfies

A2pD0 and 1^TpD1: (3.6)

By direct verification the following proposition can be obtained.

Proposition 3.3. The steady-state probability distribution p is given by

pi DK.1^iC1/; iD0; 1; : : : ; Q (3.7)

By using the result of the steady-state probability in Proposition3.3, the following corollary is obtained.

3.3 An Inventory Model for Returns 85

Corollary 3.4. The expected inventory level is XQ

the average rejection rate of returns is

pQDK.1^QC1/ and the mean replenishment rate is

p0¹

¹C.a/¹ D K.1/ .1C/ : Proposition 3.5. If < 1andQis large then

K.1CQ/¹

and the approximated average running cost (inventory and replenishment cost) is C.Q/ QI

2 C .1/R

.1C/.1CQ/: The optimal replenishment size is

QC1

One can observe that the optimal replenishment sizeQeitherorRincreases or I decreases. We end this section with the following remarks:

1. The model can be extended to the multi-item case when there is no limit in the inventory capacity. The trick is to use independent queueing networks to model individual products. Suppose there are s different products and their demand rates, return rates, unit inventory costs, cost per replenishment order and the probability of getting a repairable return are given byi; i; Ii; Ri andai

respectively. Then the optimal replenishment size of each productiwill be given by (3.8)

2. To include the inventory capacity in the system, one can have approximations for the steady-state probability distributions for the inventory levels of the returns

and the serviceable product by assuming that capacity for storing returns is large.

Then the inventory levels of the returns forms an M/M/1 queue and the output process of an M/M/1queue in steady-state is again a Poisson process with the same mean rate, see the lemma below.

Lemma 3.6. The output process of an M/M/1queue in steady-state is again a Poisson process with the same mean as the input rate.

Proof. We first note that ifX andY are two independent exponential random variables with means¹ and ¹ respectively. Then the probability density function for the random variableZ DXCY is given by

f .z/D

e^z

e^z:

Let the arrival rate of the M/M/1 queue beand the service rate of the server be. There are two cases to be considered: the server is idle (the steady-state probability is.1=/(see Chap. 2) and the server is not idle (the steady-state probability is=.)

For the former case, the departure time followsf .z/(a waiting time for an arrival plus a service time). For the latter case, the departure time followse^z. Thus the probability density functiong.z/for the departure time is given by

.1

/f .z/C

.e^z/D

e^z

e^{z 2}

e^zC ²

e^zCe^z: Thus

g.z/De^z

is the exponential distribution. This implies that the departure process is a Poisson process. From Proposition 1.37 it is evident that the departure process is a Poisson process with meanif and only if the inter-departure time follows the exponential distribution with mean¹.

3. One can also take into account the lead time of a replenishment and the check-ing/repairing time of a return. In this case, it becomes a tandem queueing network and the analytic solution for the system steady-state probability distribution is not available in general. A numerical method based on a preconditioned conjugate gradient method has been applied to solve this type of tandem queueing system, see for instance [36, 37, 41, 43, 46, 49].