The purpose of this study is to investigate two stochastic periodic review inventory models affected by the lead time and lost sales rate. In our models, we ...
Journal of the Chinese Institute of Industrial Engineers, Vol. 22, No. 5, pp. 355-368 (2005)
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PERIODIC REVIEW INVENTORY MODELS WITH CONTROLLABLE LEAD TIME AND LOST SALES RATE REDUCTION Liang-Yuh Ouyang* Graduate Institute of Management Sciences Tamkang University 151, Ying-Chuan Road, Tamsui, Taipei, Taiwan, 251 Bor-Ren Chuang Department of International Trade Ta Hwa Institute of Technology Yu-Jen Lin Graduate Institute of Management Sciences Tamkang University Holistic Education Center ST. John’s & ST. Mary’s Institute of Technology
ABSTRACT The purpose of this study is to investigate two stochastic periodic review inventory models affected by the lead time and lost sales rate. In our models, we consider that the lead time can be shortened at an extra crashing cost, which depends on the length of lead time. Moreover, we assume that the lost sales rate can also be reduced by capital investment. The objective of our study is to simultaneously optimize the review period, the lost sales rate and the lead time. We first assume that the protection interval (i.e., review period plus lead time) demand follows a normal distribution. Then, the assumption is relaxed to consider a distribution-free case where only the mean and standard deviation of protected interval demand are known. For each case, an algorithm is developed to find the optimal investment strategy. Finally, two numerical examples are given to illustrate the results. Keywords: inventory, periodic review, lost sales rate, crashing cost, minimax distribution free procedure
1. INTRODUCTION *
In traditional economic order quantity (EOQ) literature dealing with inventory problems, either using deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable. Therefore, lead time is not subject to control (see, e.g., [3, 4, 6]). However, this may not be realistic. As pointed out by Tersine [15], lead time usually consists of the following components: order preparation, order transit, supplier lead time, delivery time, and setup time. In some practical cases, lead time can be shortened at an added crashing cost; in other words, it is controllable. Through the Japanese experience of using Just-In-Time (JIT) production, the advantages and benefits associated with efforts to control the lead time can be clearly perceived. As a result, by shortening the lead time, we can lower the safety *
Corresponding author:liangyuh@mail.tku.edu.tw
stock, reduce the stockout loss and improve the service level to the customer so as to increase the competitive edge in business. On the other hand, inventory problems assumed that demand during the stockout period is either completely backordered or completely lost in the past. However, in many market situations, we can often observe that some of customers are willing to wait for their demand when the inventory system is out of stock, while other may refuse the backorders if stockout occurs. Therefore, for inventory models in which shortages are allowed, it is more reasonable to assume that some of the excess demand is backordered and the rest is lost. Recently, some studies discussed the partial lost sales rate as a fixed constant and further explored its optimal solutions. Montgomery et al. [4] is among the first who analyzed a fraction of demand is backordered and the remaining fraction is lost. The framework proposed by Montgomery et al. [4] has encouraged many researchers to present various types of inventory
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Journal of the Chinese Institute of Industrial Engineers, Vol. 22, No. 5 (2005)
models with partial lost sales. Ouyang et al. [10] proposed a continuous review inventory model with lead time reductions by allowing shortages with backorders. Later, Ouyang and Wu [11] extended Ouyang et al’s [10] model. They considered an inventory model with a mixture of backorders and lost sales in which a service level constraint was used instead of shortages in the objective function. Moon and Choi [5] and Hargia and Ben-Daya [2] generalized that of Ouyang et al. [10] by allowing the reorder points as one of the decision variables. In a recent paper, Ouyang et al. [8] further extended Moon and Choi’s [5] model and permitted the setup cost as a decision variable. We note that the above papers are all focused on the benefits from lead time reductions and lost sales (or backorder) rate is a prescribed constant, which is not easily controlled. However, in real markets, many factors may affect customers’ willingness of accepting backorders during the stockout period. It is obvious that, for some well-famed products or fashionable goods such as certain brand gumshoes, hi-fi equipment, and clothes, customers may prefer to wait for backorders. Cost and operation of inventory depend greatly on what happens to demand when the system is out of stock. Especially, for products, which have high direct profitability and/or high sales value, the cost of lost demand will be high. A manager’s intention should always be to explore the possibility of improving the current system so as to minimize (maximize) the total cost (profit). Therefore, in addition to the traditional consideration of keeping safety stock, any possible way that could prevent the loss caused by stockout may be tried, even if there is risk involved. It is noticed that the above papers [2-6, 8, 10] are all focused on the continuous review inventory model to derive the benefits from reducing lead time, and the lost sales rate treated as a fixed constant. However, for the periodic review inventory model, lead time and lost sales rate as decision variables has rarely been discussed. The applications of the periodic review inventory model can often be found in managing inventory cases such as smaller retail stores, drugstores and grocery stores (see, for example Taylor III [14, p.779]). For the reason, contrary to the continuous review inventory model, we seek to investigate a periodic review inventory model, and consider the review period, lead time and lost sales rate are treated as decision variables. In this paper, we attempt to examine a stochastic periodic review inventory model with controllable lead time and lost sales rate reduction. The objective is to find the optimal review period, lost sales rate and lead time such that the total expected annual cost is minimum. We start with a protection interval demand that follows a normal distribution, and try to determine the optimal ordering policy.
Then, we relax this assumption by only assuming that the first and second moments of the probability distribution of the protection interval demand to be known and finite, and then solve this inventory model by using the minimax distribution free approach. Furthermore, two numerical examples are provided to illustrate the proposed models.
2. NOTATIONS AND ASSUMPTIONS The mathematical models in this paper are developed on the basis of the following notations and assumptions.
2.1 Notations D A h R π
π0 T L X
= = = = = =
average demand per year fixed ordering cost per order inventory holding cost per item per year target level shortage cost per unit short marginal profit (i.e., cost of lost demand) per unit = length of a review period = length of lead time = the protection interval, T + L , demand which has a probability density function (p.d.f.) f X with finite mean D (T + L )
and standard deviation σ T + L , where σ denotes the standard deviation of the demand per unit time α = fraction of the shortage that will be lost, 0
