Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4391970, 9 pages

https://doi.org/10.1155/2017/4391970

## Mathematical Model of Inventory Policy under Limited Storage Space for Continuous and Periodic Review Policies with Backlog and Lost Sales

^{1}School of Manufacturing Systems and Mechanical Engineering, Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani 12121, Thailand^{2}Department of Agro-Industrial Technology, Faculty of Agro-Industry, Kasetsart University, Bangkok 10900, Thailand

Correspondence should be addressed to Jirachai Buddhakulsomsiri

Received 31 May 2017; Accepted 25 October 2017; Published 10 December 2017

Academic Editor: Thomas Hanne

Copyright © 2017 Kanokwan Singha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper involves developing new mathematical expressions to find reorder point and order quantity for inventory management policies that explicitly consider storage space capacity. Both continuous and periodic reviews, as well as backlogged and lost demand during stockout, are considered. With storage space capacity, when on-hand inventory exceeds the capacity, the over-ordering cost of storage at an external warehouse is charged on a per-unit-period basis. The objective is to minimize the total cost, consisting of ordering, shortage, holding, and over-ordering costs. Demand and lead time are stochastic and discrete in nature. Demand during varying lead time is modeled using an empirical distribution so that the findings are not subject to assumptions of demand and lead time probability distributions. Due to the complexity of the developed mathematical expressions, the problems are solved using an iterative method. The method is tested with problem instances that use real data from industry. Optimal solutions of the problem instance are determined by performing exhaustive search. The proposed method can effectively find optimal solutions for continuous review policies and near optimal solutions for periodic review policies. Fundamental insights about the inventory policies are reported from a comparison between continuous review and periodic review solutions, as well as a comparison between backlog and lost sales cases.

#### 1. Introduction

This paper focuses on the widely used inventory policy in distribution networks, where an order quantity is placed when the inventory position falls on or below the reorder point at the time of a review. Inventory can be reviewed either continuously in real time or periodically at the end of some period. Continuous reviewing mostly requires a warehouse management system that supports it, whereas periodic reviewing can be performed without such a system. Mostly, the policy is designed for a continuous review, whereas an order-up-to level policy or basestock policy is for a periodic review. However, even without the technology to support a continuous review, some industrial users prefer the policy and use it with a periodic review. This is because some users would not like to change their order quantity every order. Also, their order quantity has to be a multiple of some pack sizes or container.

There are two types of behaviors of demands during stockout: lost sales, where the demands are lost; backlog, where the demands are fulfilled when inventories become available. This behavior depends on the characteristics of products, market, and relationships between suppliers and customers. Both types of demands during stockout are considered in this paper.

The inventory system under study is motivated by and therefore modeled after real distribution networks in various industries in Thailand. In such networks, internal storage space of an item in a warehouse or distribution center (DC) is preassigned and limited. When a replenishment order from a supplier arrives, there are instances when on-hand inventory exceeds the internal storage space. When this occurs, external space can be rented with an external storage cost charged on a per-unit per-period basis. In this system, the total cost of inventory management consists of four components: fixed ordering cost, inventory holding cost at internal storage, inventory holding cost at external storage (called over-ordering cost in this paper), and shortage cost. Item demands and replenishment lead time are stochastic and discrete in nature; therefore, the demands during varying lead time (i.e., the period of shortage risk) are of the same nature. Instead of assuming probability distribution(s) of demand, lead time, and demand during lead time, they are modeled using an empirical distribution. This is to overcome a problem found in real systems that many item demands do not follow widely used probability distributions.

The objective of the paper is to develop mathematical expressions for determining optimal or near optimal and for the inventory system under study, so as to minimize the average total cost per period. The scope of the problem covers a wide range of real problems. The system considers four cost components (as mentioned above), discrete and varying demand and lead time that are not subject to assumptions of probability distribution, and four cases of review policies and demands during stockout. To the best of our knowledge, no mathematical expressions for a problem with this scope have been reported in the literature. The contributions of the paper are, therefore, the developed mathematical expressions, a method that implements them to solve a problem, and important fundamental insights gained from considering four cases of the problem.

There is a vast literature on inventory policy. Therefore, only relevant studies are included, that is, research studies that have involved storage space capacity. Those studies can be categorized into three groups: storage space capacity is considered as a hard constraint, which means over-ordering storage is not allowed; storage space capacity is a hard constraint, but with an additional cost of returning the over-ordered quantity to the supplier; and storage space is a soft constraint, where an over-ordered amount is stored at an external, rented warehouse. These studies vary considerably in other aspects of the problem, such as inventory policies used, cost components considered, nature of demands during stockout, and the solution methodology.

In the first category, recent studies include Mandal et al. [1], Zhao et al. [2], Chou et al. [3], Zhao et al. [4], Zhong and Zhou [5], Pan et al. [6], Ghosh et al. [7], Beemsterboer et al. [8], and Zhang and Rajaram [9]. Among them, studies that focused on policy are Zhao et al. [2], Zhao et al. [4], and Pan et al. [6]. These studies examined a single product, continuous review policy, where demands are stochastic, shortages are backordered, and lead time is assumed to be constant. Among the three studies, only Zhao et al. [4] extended the problem to the case of multiple products. Heuristic algorithms were proposed to solve the problem: polynomial time algorithm in Zhao et al. [2], genetic algorithm in Pan et al. [6], and a solution approach consisting of an iterative part and local search in Zhao et al. [4]. For this set of problems, since storage space is a hard constraint, this implies that the solution methodology proposed in these studies would attempt to set the inventory parameters to be within the available space, which is not the same as this paper.

For the second category where over-ordered quantity is returned to the supplier, Hariga [10] considered the following problem: an policy for a single item problem using continuous review: demands during stockout are backordered, and cycle length does not include the stockout period. The cost function was derived and solved using a simple economic order quantity (EOQ) based heuristic solution.

Studies that allowed external storage include Huang [11], Huang et al. [12], Hariga [13], Zhou et al. [14], Huang [15], Ouyang et al. [16], and Sana [17]. The inventory holding cost at an owned warehouse is normally smaller than at a rented warehouse. Huang [11] and Huang et al. [12] developed a retailer’s inventory model when demand is known and constant and considered ordering cost and holding cost while shortages are not considered. The inventory policy was to find optimal time between order and order quantity, a policy. Huang [15] later extended their previous problem by adding the perishable characteristics of the product. Hariga [13] considered multiwarehouse systems to choose where to store excessive stock. The optimal solution consisted of the optimal order quantity and leased storage space. Some studies that focused on a problem where a supplier offers trade credit to a buyer to create an incentive to place larger orders are Zhou et al. [14] and Ouyang et al. [16]. For this problem, the optimal decision for the buyer is the order quantity. The optimal decisions for the supplier are the optimal trade credit period to offer and the corresponding order quantity. Sana [17] presented an EOQ model for stochastic demand without considering lead time and with storage space capacity. Demands during stockout are lost. It can be seen that among recent studies where external rented storage is allowed, no studies have focused on determining under the same settings as this paper.

#### 2. Problem Characteristics and Notation

The mathematical model developed in this paper adheres to the following problem characteristics or assumptions.(1)The inventory policy is an policy for a single continuously stocked item.(2)Two review policies are considered: continuous review and periodic review.(3)At the time of a review, if the inventory position (IP) falls on or below , a replenishment order is placed to the supplier, and a fixed ordering cost is charged.(4)In the periodic review case, it is assumed that the IP is reviewed frequently enough so that a review period is shorter than the replenishment lead time.(5)The order arrives after a replenishment lead time that is stochastic and discrete.(6)Demands are stochastic and discrete in nature.(7)The item has a preassigned limited storage capacity. At the beginning of a cycle when a replenishment order arrives, if the on-hand inventory (OH) exceeds the storage space, over-ordered inventories will be kept at an external warehouse.(8)The over-ordered amount stored at an external warehouse is charged on a per-unit per-period basis.(9)It is assumed that the rate of external storage cost is no less than the rate of internal storage cost.(10)When demand arrives, if OH is less than the demand, shortage cost is charged on a per-unit basis.(11)Two types of shortages are considered. If there exist shortages, demands during stockout may be backlogged and may be lost.(12)An inventory cycle covers the elapsed time between replenishment orders, including the time the system has stock and the time the system is out of stock.(13)The objective is to determine the optimal or near optimal and which minimizes the total inventory management cost that consists of fixed ordering cost, inventory holding cost for internal storage space, shortage cost, and storage cost at an external warehouse (i.e., the so-called over-ordering cost in this paper).(14)Given data include historical demand data, lead time data, all cost parameters, and internal storage space capacity. External storage space is always available.

Since both continuous review and periodic reviews as well as backlog and lost sales during stockout are considered, there are a total of four cases of the problem.

##### 2.1. Notations

: replenishment order quantity, units : reorder point, units : fixed ordering cost, THB/order : inventory holding cost, THB/(unit-period) : shortage cost, THB/unit : over-ordering cost, THB/(unit-period) : random demand, units/period : average demand, units/period : probability mass function of : random lead time, periods : average lead time, periods : probability mass function of : random demand during varying lead time, units : average demand during varying lead time, units, where : probability mass function of : storage space capacity, units , : expected number of shortages, for a given* R*, under continuous and periodic reviews, respectively, units , : probability of a shortage, for a given* R*, under continuous and periodic reviews, respectively, in cases that demand during stockout is backlogged and is lost, respectively, % , , and : random level of on-hand inventory at the end of a cycle, at the time of ordering, and at the beginning of a cycle, respectively, units , , and , : the expected OH at the end of a cycle, at the time of ordering, at the beginning, and over the length of time the item is in stock in a cycle, respectively IP: random level of inventory position at the time of ordering, units , , , and : expected over-ordered amount at the beginning of an order cycle for different cases: continuous (*C*) and periodic (*P*) reviews, in combination with lost sales (*L*) and backlog (*B*), units , , , and : probability of over-ordering at the beginning of an order cycle for the four cases, % , , , and : total inventory management cost per period for the four different cases, THB.

#### 3. Methodology

##### 3.1. Probability Distribution of Demand during Varying Lead Time

The stochastic and discrete behaviors of the demand and lead time are modeled using the distribution of demand during varying lead time. Given historical demand data and lead time data, one can list all possible values of random demand, , and random lead time, , and empirically derive their probability mass functions, and. Then, the demand during a given lead time is the sum of the demands that occur during , where , and are independently and identically distributed random variables with . In addition, its probability is the product of the probability of all demands . Combining with gives the probability of demand during varying lead time: .

##### 3.2. Probability of Stockout and Probability of Over-Ordering

Given the values of , , and , in a replenishment cycle IP, OH, shortage amount, and over-ordered amount are shown in Figure 1, while the probability of stockout and the probability of over-ordering are shown in Figure 2.