An inventory model with dependent returns and disposal cost

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Introduction
Reverse Logistics deals with activities associated with handling and management of equipment, products, components, materials or even entire technical systems which are supposed to be recovered.Global warming has been one of the most important issues and the recovery of used products and materials is believed to reduce the green gas effects.Each year, auto makers sell over sixty million cars and there is a growing concern to find suitable ways on recycling car parts remanufacturing.Electronic devices are other mass produced items where one may be more interested in reusing electronic scrap recycling.Inderfurth (1997) is believed to be the first to present a simple optimal replenishment and disposal policies for a product recovery system with lead times.Fleischman and Kuik (1998) studied an optimal inventory control with stochastic item returns.Kiesmuller and van der laan (2001) introduced an inventory model with dependent product demands and returns.Fleischman et al. (2002) introduced a basic model to control inventories with stochastic item returns.There is also a good overview on quantitative models for recovery production planning and inventory control is given in Fleischmann et al. (1997).Different papers have studied inventory policies under consideration of random returns, but most of these assume that the return process is independent of the demand process (See for example Fleischmann and Kuik (2003)).In all previous works, there is no dependency between demand and returns but there are situations where it is better to consider the probabilities of dependency between demands and returns.For example, we can think about the situations in the case of rented or leased products, or if delivered items are returned to the original manufacturer only.Cohen and Pierskalla (1980) developed an inventory model for a single, reusable product where random returns depend on the demand stream and no backorders and leadtime for purchasing are considered.Also, Yuan and Cheung (1998) considered a model with dependent returns with the consideration of backorders and no purchasing lead time.De Brito and Van der laan (2009) proposed an inventory model with the uncertain time and quantity of returns.They investigated the impact of imperfect information with respect to the return process on inventory management performance.Parlikad and McFarlane (2010) proposed a model to show the value of information in product recovery decisions.Guide Jr. and Wassenhove (2009) introduced a new model for managing remanufactured products.Kiesmuller and Van der laan (2001) relaxed this assumption and allowed a positive purchasing lead time.Lieckens and Vandaele (2007) designed the reverse logistics network with stochastic lead times.The long-term behaviour of the inventory was already investigated by Van der Laan and Salomon (1998) and, in this paper we consider a finite planning horizon.The reason for this assumption is that, nowadays the life-cycle of products is getting shorter and shorter, because of fast changing trends and new developments.In this paper, we investigate an inventory model for a single, reusable product in which the random returns depend on the demand stream.We consider backorders and positive lead time for purchasing; especially we note the effect of disposal cost at the end of the planning horizon.Van der laan, Dekker andSalomon (1996, 1997) investigated the different inventory control systems with disposal.There are also situations that quality of recovered items is not suitable for remanufacturing process.Therefore, the rate of disposal in such situations would be increased.For example we can see this condition in car part factories.We also consider the remanufactured item as a new one.Corbacioglu and Van der laan (2007) studied the inventory model when the quality of remanufactured items was different with that of new ones.The paper is organized as follows.In section 2 we describe our model and its assumptions.In section 3 we determine the objective function and this followed by a numerical study to show importance of model in section 4. Finally, we summarize our results and give an outlook for further research.

Model description
The proposed model of this paper considers a single stage product recovery system. .Practically, there are some limitations on the number of periods where inventory are reviewed.Also, for the sake of simplicity, we consider a fixed period of review to be equal to one with  We assume that all demands which cannot be fulfilled immediately are backordered and the rest of outstanding demands at the end of the planning horizon are lost.The parameters in Fig. 1 are described as follow: l P : Probability that customer has not returned the used product to the manufacturer.d P : Probability that returned item because of its poor quality has to be disposed.After remanufacturing, the item is as good as new.Therefore, there is a common serviceable inventory for both new and remanufactured items, since we do not need to distinguish between them.We assume that the lead times are constant.Let 1 L be the time where item is in use by the customer, 2 L be the transportation and the inspection time, 3 L and L be the remanufacturing time and the purchasing lead time, respectively.For simplifying and computational reasons we assume that the following holds, Since the returns in period t depends on demands for L previous periods, the returns in period t can be defined as the number of remanufactured items after they are used and entered to serviceable items in period t based on a periodic order-up-to strategy.At the beginning of the planning horizon, the inventory is assumed to be filled up with A items and an order is placed up to a limit of S if the inventory position is smaller than S. As we can observe, two parameters of S and A determine our strategy.To distinguish four different time phases we assume L T 3 ≥ . The details of our assumptions are depicted in Fig. 2.
Phase 1 ( ) : During the periods in this phase there are no remanufactured or new items which are delivered to the serviceable items and the on hand stock decreases over this time period and the reorder point for new orders is defined by S.
Phase 2 In the periods of this phase both remanufactured and new items enter to the serviceable items.The stock on hand can be decreased or increased during each period in this phase.In this phase, like the phase 1, the reorder point for new orders is defined by S .
Phase 3 : Because at the end of the planning horizon (last L periods), all the return items have to be disposed and on the other hand, the returns depend on the demands in L periods before, so in this phase, the reorder point must be changed to decrease the cost of disposal at the end of the planning horizon and the reorder point is defined by K : In this phase, no procurement orders take place.In other words, all returned items are disposed and no used items can enter to the serviceable inventory.In this model the set of all possible order-up-to policies is given by: In order to compare deferent parameter sets ξ ∈ ) , , ( K S A , we use the average total relevant cost as an objective function.So we define the average total cost by: They are defined as: 0 C : The starting cost for filling up the inventory before the first period.The optimal policy ξ ∈ ) , , ( is defined as the parameter set that minimizes expression (4).The starting expenditures solely depend on the number of items in the inventory at the beginning of the planning horizon and on some fixed ordering cost A C as follows, In expression (5), A C is the fixed ordering cost to order A items and p C is the procurement cost for each item.
The average relevant cost in one period is composed of the average costs for procurement, backorders and stock keeping and they are proportion to the number of items and the holding and the backorder costs which are charged at the end of each period.Therefore we have, Where the random variables are defined as follows, For the description of the inventory process it is necessary to introduce some additional random variables: Additionally, we define r P as the probability that an item is recovered, which means that it is returned to the manufacturer and then it can be remanufactured:

Determination of objective function
The objective function of the proposed model is partly formulated based on what Kiesmuller and Van der laan (2006) originally suggested.However, there are some changes on the structure of the cost items to make it compliant to our proposed method in each phase.The first step of our method is to introduce the inventory position t I which is explained in the next section.R + in situations with a high service level.Practically, there is also a strong relationship between the demands in period t and the returns in period L t + .If the demands in period t are known, we have some information about the returns at L later periods and the information could be used for the procurement decision in period L t + and this assumption holds when we define the inventory position t I , at the beginning of period t before ordering, as the stock on hand minus backorders plus the outstanding orders plus the outstanding returns.Since we normally face a high service level, we use the following as an approximation for the inventory position for

The inventory position
This means that the inventory position in period 1 + t is given by the inventory position of the previous period, the procurements and the number of demands in period t , and the number of fulfilled demands that will return to the inventory¸ periods later.Therefore, t I describes a non-homogeneous Markov-chain, because the inventory position in period t+1 depends only on the period before.If we consider the dependency of the demands and the returns, is always non-negative, since it is not possible to have more returns in period L t + .It follows from ( 9) that the inventory position never exceeds S, so that we have . As a contrast, if the dependency of demands and returns is not considered, then can also be negative, so that Z I t ∈ .For a description of the system, we need the state probabilities ) (t v j , which are the probabilities that the inventory position in period t is equal to j : At the beginning of period 1, both the inventory position and the stock on hand have the value A , which leads to: All the other state probabilities are given by the recursive formula, For the transition probabilities see Appendix A.

The objective function
To define the objective function we have to determine the average shortage and holding costs in period t which are given for all For average holding cost we have, The probability distribution of the net stock ) ( i X P t = is given in Appendix A.4.Now, we have to consider the average procurement cost in the objective function.We have different procurement sizes in different periods.
[ ] With ( 14), ( 15) and ( 16 .At the end of the planning horizon all the items in the reuse network and the items which are remained in the serviceable items must be disposed.Therefore we have, Now with (4) we can calculate the objective function . The optimal policy can be found using the optimal solution of the following problem, Since the structure of objective function is nonlinear, a closed form solution cannot be found easily and use numerical techniques, instead.

Numerical study
In this section, we use a numerical example to demonstrate the implementation of the proposed method of this paper.We consider three cases: For the first case, we study the model when we have only one reorder point for all periods ...., , 2 , 1 T t = In the second case, we have two reorder points of . Also, demand and return sales are assumed to be constant.The other parameters are as follows, .
In Table 1 the optimal policies and the average minimal relevant costs are given for different values of d P and r P .Also the Relative difference between two systems is given, which shows the difference between the average minimal costs of each system.

Conclusion and future research
In this paper, we have developed an inventory system with periodic ordering policy which the return stream is depended on the demand stream.We consider the finite planning horizon and the backorders are allowed.We have studied the effect of recovery probability on determining the optimal policy and the average total cost.We show that with considering the cost of disposal at the end of the planning horizon, the reorder point should be changed to achieve the optimal policy.There are some limitations on the assumptions of the proposed model.For instance, we have considered only one inventory for the serviceable items.It is worth to be discussed, that a remanufactured item does not necessarily have the same quality as the new one.In this case, the recovery probability for an item is not constant but decreasing.Therefore, one can define two inventories for serviceable items, one inventory for new items and another for the remanufactured items.Moreover, it would be interesting to extent the model with the stochastic lead time structure.
itself are independent.Therefore, the sum of these terms is again Poisson distributed, which means:

The transition probabilities
Because of the non-homogeneous Markov-chain the transition probabilities are dependent on time.

= t
: The initial distribution (11) and the fact that and the state probability : In these periods from all the demands a portion will be returned to the inventory L period later, which means 0 ˆ≥ t R .Additionally, the inventory position after an order is always S , so that S O I t t

= +
.Therefore, we have These transition probabilities are independent from i and the state probabilities are given by ) : The transition probabilities of these periods are like above but instead of S we use K .
Therefore we have, , we still have procurement, but none of the delivered items to the customer will be returned to the serviceable inventory

Fig 1 :
Fig 1: The reuse network Fig. 1 demonstrates the details of the product recycling system where demand has a poison distribution over a finite planning horizon of length T [ ] ( ) T t , 0 ∈

:
Probability that returned item is remanufactured.

Fig 2 :
Fig 2: The time phase

tC:
The sum of the average relevant cost in period t .E C : The average cost at the end of the planning horizon for the disposal of the remaining items.

tO:
Number of procured items at the beginning of period t, − t X : Number of backorders at the end of period t, + t X : Stock on hand at the end of period t.The cost parameters are defined as: H C : Holding cost per item per period, B C : Backorder cost per item per period.
position at the beginning of period t before a procurement order, t R : Number of remanufactured items which are entered to the serviceable items in period t.
get the following formula for the distribution of the returns, t R ˆ for all 1

)
distribution of the net stock Different formulas are needed for different periods.

:
Because of no returns in Phase 1, the net stock only depends on the demand and is monotonously decreasing.There are also no deliveries of new items.For easily be computed, since the sum of independent Poisson distributed random variables is again Poisson distributed: 2, 3 and 4 the net stock at the end of period t is dependent on the inventory position in period L t − :

For::
In this case there are no procurements ) 0 ( 1 = O and the inventory position in period 1 is given by In this case the probabilities for the net stock can be computed as follows: In the model described in Section 2, a strong relation is given between the demands in period t and the returns in period L t + .If all the demands are fulfilled in period t, the distribution function of the random variable t I ) the average costs in period t ,

Table 1
Optimal policy and the minimal average relevant costs to compare two different systems 1) 52Therefore, the number of returns in period t is distributed with Poison distribution and parameterThe distribution of the number of items which are not returned − are dependent, while the other random variables not.We can write: