Under the situation which customer orders are cancelled unless all products in the order are delivered all at once, this paper concentrates on the purchase dependent demands and explores the systematic approach to implant the purchase dependence into the multi-product inventory model. First, by acknowledging that it is a challenging task to formulate a suitable inventory model for the purchase dependence, we derive the optimal solution condition using an EOQ model and extend the optimal solution condition to periodic review models. Then, through the comparison simulation of four inventory policies regarding several degrees of purchase dependence, we demonstrate that the inventory models which consider the purchase dependence generate less total cost than the inventory models which ignore the purchase dependence. In general, the inventory models which consider the purchase dependence reduce the loss of sales by maintaining more inventories, which results in reducing the total cost. Consequently, the simulation result supports the effectiveness of this paper’s approach. In addition, this paper uses the individual order period and joint order period obtained from the EOQ model for the multi-product inventory model. Through the in-depth analysis of comparing the two models, we observe that the model of using the joint order period produces less total cost when the degree of purchase dependence is high, but the model of using the individual order period produces less total cost when the degree of purchase dependence is low.

For the competitive business environment under purchase dependence, this paper proposes a new approximate calculation of order fill rate which is a probability of satisfying a customer order immediately using the existing inventory. Purchase dependence is different to demand dependence. Purchase dependence treats the purchase behavior of customers, while demand dependence considers demand correlation between items, between regions, or over time. Purchase dependence can be observed in such areas as marketing, manufacturing systems, and distribution systems. Traditional computational methods have a difficulty of the curse of dimensionality for the large cases, when deriving the stationary joint distribution which is utilized to calculate the order fill rate. In order to escape the curse of dimensionality and protect the solution from diverging for the large cases, we develop a greedy iterative search algorithm based on the Gauss-Seidel method. We show that the greedy iterative search algorithm is a dependable algorithm to derive the stationary joint distribution of on-hand inventories in the retailer system by conducting a comparison analysis of a greedy iterative search algorithm with the simulation. In addition, we present some managerial insights such as : (1) The upper bound of order fill rate can be calculated by the one-item pure system, while the lower bound can be provided by the pure system that consists of all items; (2) As the degree of purchase dependence declines while other conditions remain same, it is observed that the difference between the lower and upper bounds reduces, the order fill rate increases, and the order fill rate gets closer to the upper bound.

This paper introduces the existence of purchase dependence that was identified during the analysis of inventory operations practice at a sales agency of dealing with spare parts for ship engines and generators. Purchase dependence is an important factor in designing an inventory replenishment policy. However, it has remained mostly unaddressed. Purchase dependence is different from demand dependence. Purchase dependence deals with the purchase behavior of customers, whereas demand dependence deals with the relationship between item-demands. In order to deal with purchase dependence in inventory operations practice, this paper proposes (Q, r) models with the consideration of purchase dependence. Through a computer simulation experiment, this paper compares performance of the proposed (Q, r) models to that of a (Q, r) model ignoring purchase dependence. The simulation experiment is conducted for two cases : a case of using a lost sale cost and a case of using a service level. For a case of using a lost sale cost, this paper calculates an order quantity, Q and a reorder point, r using the iterative procedure. However, for a case of using a service level, it is not an easy task to find Q and r. The complexity stems from the interactions among inventory replenishment policies for items. Thus, this paper considers the genetic algorithm (GA) as an optimization method. The simulation results demonstrates that the proposed (Q, r) models incur less inventory operations cost (satisfies better service levels) than a (Q, r) model ignoring purchase dependence. As a result, the simulation results supports that it is important to consider purchase dependence in the inventory operations practice.

One of the most usual indicators to measure the performance of any inventory policy is the mean physical stock. In general, when estimating the mean physical stock in periodic review inventory systems, approximate approaches are often utilized by practitioners and researchers. The mean physical stock is generally calculated by a simple approximation. Still these simple methods are frequently used to analyze various single stockpoint and multi-echelon inventory systems. However, such a simple approximation can be very inaccurate. This is particularly true for low service levels. Even though exact methods to calculate the mean physical stock have been derived, they are available for specific cases only and computationally not very efficient, and therefore less useful in practice. In literature, approximate approaches, such as the simple, the linear, and Simpson approximations, were derived for the periodic review inventory systems that allow backorders. This paper modifies the approximate approaches for the lost sales case and evaluates the modified approximate approaches. Through computational experiments, average (and maximum) percentage deviations of mean physical stock between the exact method and the modified approximations are compared in the periodic review inventory system with lost sales. The same comparison between the modified and the original approximations are also conducted, in order to examine the performance of modified approximations. The results show that all modified approximations perform well for high service levels, but also that the performance may deteriorate fast with decreasing service level. The modified Simpson approximation is clearly better. In addition, the comparison between the modified and the original approximations in the periodic review inventory system with lost sales shows that the modified approximation outperforms the original approximation.