Production, Manufacturing and Logistics
Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate

https://doi.org/10.1016/j.ejor.2006.02.024Get rights and content

Abstract

In this paper, a deterministic inventory model for deteriorating items with two warehouses is developed. A rented warehouse is used when the ordering quantity exceeds the limited capacity of the owned warehouse, and it is assumed that deterioration rates of items in the two warehouses may be different. In addition, we allow for shortages in the owned warehouse and assume that the backlogging demand rate is dependent on the duration of the stockout. We obtain the condition when to rent the warehouse and provide simple solution procedures for finding the maximum total profit per unit time. Further, we use a numerical example to illustrate the model and conclude the paper with suggestions for possible future research.

Introduction

The general assumption in classical inventory models is that the organization owns a single warehouse without capacity limitation. In practice, while a large stock is to be held, due to the limited capacity of the owned warehouse (denoted by OW), one additional warehouse is required. This additional warehouse may be a rented warehouse (denoted by RW), which is assumed to be available with abundant capacity. There exist some practical reasons such that the organizations are motivated to order more items than the capacity of OW. For example, the price discount for bulk purchase may be advantageous to the management; the demand of items may be high enough such that a considerable increase in profit is expected; and so on. In these situations, it is generally assumed that the holding cost in RW is higher than that in OW. To reduce the inventory costs, it will be economical to consume the goods of RW at the earliest. As a result, the stocks of OW will not be released until the stocks of RW are exhausted.

An early discussion on the effect of two warehouses was considered by Hartely [1]. Recently this type of inventory model has been considered by other authors. Sarma [2] developed a deterministic inventory model with infinite replenishment rate and two levels of storage. Murdeshwar and Sathe [3] extended this model to the case of finite replenishment rate. Dave [4] further discussed the cases of bulk release pattern for both finite and infinite replenishment rates. He rectified the errors in Murdeshwar and Sathe [3] and gave a complete solution for the model given by Sarma [2]. In the above literature [2], [3], [4], deterioration phenomenon was not taken into account. Assuming the deterioration in both warehouses, Sarma [5] extended his earlier model to the case of infinite replenishment rate with shortages. Pakkala and Achary [6], [7] extended the two-warehouse inventory model for deteriorating items with finite replenishment rate and shortages, taking time as discrete and continuous variable, respectively. In these models mentioned above, the demand rate was assumed to be constant. Subsequently, the ideas of time-varying demand and stock-dependent demand were considered by some authors, such as Goswami and Chaudhuri [8], [9], Bhunia and Maiti [10], [11], Benkherouf [12], Kar et al. [13] and others. In a recent paper, Yang [14] proposed an alternative model for determining the optimal replenishment cycle for the two-warehouse inventory problem under inflation, in which the inventory deteriorates at a constant rate over times and shortages were allowed. She then proved that the optimal solution not only exists but also is unique.

Furthermore, the characteristics of all above papers are that shortages are not allowed or assumed to be completely backlogged. Zhou [15] presented a multi-warehouse inventory model for non-perishable items with time-varying demand and partial backlogging. In his model, the backlogging function was assumed to be dependent on the amount of demand backlogged. In many cases customers are conditioned to a shipping delay, and may be willing to wait for a short time in order to get their first choice. Generally speaking, the length of the waiting time for the next replenishment is the main factor for deciding whether the backlogging will be accepted or not. The willingness of a customer to wait for backlogging during a shortage period declines with the length of the waiting time. Therefore, a situation is quite likely to arise in which that many savvy retailers suggest replacement items, and also provide the restocking date to allow the customer to wait during the stockout period. To reflect this phenomenon, Abad [16], [17] discussed a pricing and lot-sizing problem for a product with a variable rate of deterioration, allowing shortages and partial backlogging. The backlogging rate depends on the time to replenishment—the longer customers must wait, the greater the fraction of lost sales. However, he does not use the stockout cost (includes backorder cost and the lost sale cost) in the formulation of the objective function since these costs are not easy to estimate, and its immediate impact is that there is a lower service level to customers.

Companies have recognized that besides maximizing profit, customer satisfaction plays an important role for getting and keeping a successful position in a competitive market. The proper inventory level should be set based on the relationship between the investment in inventory and the service level. With a lost sale, the customer’s demand for the item is lost and presumably filled by a competitor. It can be considered as the loss of profit on the sales. Moreover, it also includes the cost of losing the customer, loss of goodwill, and of establishing a poor record of service. Therefore, if we omit the stockout cost from the total profit, then the profit will be overrated. It is true that the stockout cost is very difficult to measure. However, this does not mean that the unit does not have some specific values. In practice, the stockout cost can be easy to obtain from accounting data. In this paper, we develop a deterministic inventory model for deteriorating items with two warehouses. We assume that the inventory costs (including holding cost and deterioration cost) in RW are higher than those in OW. In addition, shortages are allowed in the owned warehouse and the backlogging rate of unsatisfied demand is assumed to be a decreasing function of the waiting time. We then prove that the optimal replenishment policy not only exists but also is unique. Moreover, a numerical example is used to illustrate the proposed model, and concluding remarks are provided.

Section snippets

Notation

To develop the mathematical model of inventory replenishment schedule with two warehouses, the notation adopted in this paper is as below:

    D

    the demand rate per unit time

    A

    the replenishment cost per order

    C

    the purchasing cost per unit

    S

    the selling price per unit, where S > C

    W

    the capacity of the owned warehouse

    Q

    the ordering quantity per cycle

    B

    the maximum inventory level per cycle

    C11

    the holding cost per unit per unit time in OW

    C12

    the holding cost per unit per unit time in RW, where C12 > C11

    C2

    the shortage

Mathematical formulation

Using above assumptions, the inventory level follows the pattern depicted in Fig. 1. To establish the total relevant profit function, we consider the following time intervals separately, [0, tw], [tw, t1], and [t1, T]. During the interval [0, tw], the inventory levels are positive at RW and OW. At RW, the inventory is depleted due to the combined effects of demand and deterioration. At OW, the inventory is only depleted by the effect of deterioration. Hence, the inventory level at RW and OW are

Inventory problem without capacity constraint in OW

When the OW is so abundant that the RW is not used, the previous model reduces to the one-warehouse inventory problem. We remove the capacity constraint of the OW, and hence the total profit per unit time in Eq. (14) becomesΠ(t1,t2)=D(S-C)-At1+t2-C11+αCα(t1+t2)Dα(eαt1-1)-Dt1-D[C2+δ(S-C+R)]δ2(t1+t2)[δt2-ln(1+δt2)].Solving the necessary conditions: Π(t1,t2)t1=0 and Π(t1,t2)t2=0 for the maximum value of Π (t1, t2), we get[C2+δ(S-C+R)]t21+δt2-(C11+αC)(eαt1-1)α=0andA+C11+αCαDα(eαt1-1)-Dt1+D[C2+δ(S

Some special cases

In this section, the two-warehouse inventory model is illustrated for some special cases. We construct them as follows:

Case 1. Without shortage

When δ  ∞ (i.e., the fraction of shortages backordered is zero), from Eq. (29), we get t2  0. The model reduce to the case where shortages are not allowed and the total profit per unit time in Eq. (14) approaches toP1(tw)P(tw,0)=D(S-C)-At1-Ct1W+Dβ(eβtw-1)-Dt1-C11αt1[W-D(t1-tw)]-DC12β2t1(eβtw-βtw-1),where t1 is a function of tw and be defined as in Eq. (8)

Numerical example

In this section, our illustration begins from a two-warehouse inventory problem under the condition W<D[C2+δ(S-C+R)]δ(C11+αC) with a precise judgment criterion, Δ. Because tw, t1, t2 and T cannot be determined in the closed forms, they have to be solved numerically by using some computer algorithm. While Δ > 0, Theorem 2 applies and we can obtain the value of tw from Eq. (20) by using Newton–Raphson Method (or any bisection method). Once the optimal tw has been determined, the optimal t1, t

Concluding remarks

In this paper, an inventory model is developed for deteriorating items with finite warehouse capacity, permitting shortage and time-proportional backlogging rate. Holding costs and deterioration costs are different in OW and RW due to different preservation environments. The inventory costs (including holding cost and deterioration cost) in RW are assumed to be higher than those in OW. To reduce the inventory costs, it will be economical for firms to store goods in OW before RW, but clear the

Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their valuable and constructive comments, which have led to a significant improvement in the manuscript. This research was partially supported by the National Science Council of the Republic of China under Grant NSC-94-2213-E-366-006.

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