Mathematical Problems in Engineering

Volume 2015, Article ID 508074, 11 pages

http://dx.doi.org/10.1155/2015/508074

## An Optimization Model for Inventory System and the Algorithm for the Optimal Inventory Costs Based on Supply-Demand Balance

^{1}College of Information Engineering, Tarim University, Alar, Xinjiang 843300, China^{2}College of Computer Science and Technology, Zhejiang University, Hangzhou, Zhejiang 310027, China

Received 13 June 2015; Revised 18 November 2015; Accepted 23 November 2015

Academic Editor: Anna M. Gil-Lafuente

Copyright © 2015 Qingsong Jiang 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

In order to investigate the inventory optimization of circulation enterprises, demand analysis was carried out firstly considering supply-demand balance. Then, it was assumed that the demand process complied with mutually independent compound Poisson process. Based on this assumption, an optimization model for inventory control of circulation enterprises was established with the goal of minimizing the average total costs in unit time of inventory system. In addition, the optimal computing algorithm for inventory costs was presented. Meanwhile, taking the agricultural enterprises in Aksu, Xinjiang, China, for example, the researchers conducted numerical simulation and sensitivity analysis. Through constantly adjusting and modifying the parameters values in model, the optimal stock and the optimal inventory costs were obtained. Therein, the numerical results showed that the uncertainty of lead time greatly influenced the optimal inventory strategy. Besides, it was demonstrated that the research results provided a valuable reference for the agricultural enterprises in terms of optimal management for inventory system.

#### 1. Introduction

Warehousing is an important part of logistics system. Inventory control of warehousing has been widely focused on by circulation enterprises and relevant scholars all the time. If the stock is high, smooth business process can be fully guaranteed to improve service level and customer satisfaction, while if the stock is low, capital backlog of enterprises and the corresponding management costs are able to be reduced (as in [1]), to optimize and control stock matters to the service quality and economic benefits of circulation enterprises. In addition, the sustainable development of circulation enterprises is crucial.

Owing to the significance of inventory optimization and control in circulation enterprises, there are many relevant scholars that begin to pay attention to this. In the last two decades, the issue has attracted much attention from many researchers. Among these researches, economic order quantity (EOQ) model based on stock-dependent demand was established (as in [2]). Besides, the production-inventory model for perishable items with definite productivity and with demand linearly depending on inventory level was considered (as in [3]). Some scholars explored the inventory issue with allowable shortages under inventory-level-dependent demand; at the same time, they also took monetary value as well as the expansion rate caused by external and internal costs into consideration (as in [4]). In addition, EOQ model for perishable items was established, where the perishable items were under the following conditions: the demand rate was related to inventory level and some stock-outs could be supplemented later (as in [5]). Cárdenas-Barrón et al. (as in [6]) studied the optimal solution of multiproduct EOQ model. Moreover, the optimal replenishment strategy for perishable items was investigated aiming at maximizing profits (as in [7]), while the inventory optimization for perishable items under stock-dependent demand was studied as well (as in [8]). Wang et al. (as in [9]) discussed the inventory control model for fresh agricultural products on Weibull distribution under the assumption that the inflation rate is higher than the natural decay rate. Paul and Rajendran (as in [10]) studied the problem of rationing mechanisms and inventory control-policy parameters for a divergent supply chain operating with lost sales and costs of review. Krishnamoorthy and Narayanan (as in [11]) considered the stability and performance analysis of a production-inventory system. Yadavalli et al. (as in [12]) studied the problem of updating service facilities for inventory system to achieve production and service synchronization. From the inventory cost and the cost of order to determine the optimal order point and quantity, Doğru et al. (as in [13]) pointed out enterprises adjust inventory quantity through a large number of buffer stocks and there is a serious bullwhip effect. Hua et al. (as in [14]) studied the carbon emissions in inventory management. The corresponding model is established by the joint of replenishment strategy, and through genetic algorithm, Zhou et al. (as in [15]) gave the solution and simulation of the model. Murray et al. (as in [16]) studied the multiproduct pricing and inventory issues. Choi and Ruszczyński (as in [17]) established a multiproduct risk-averse newsvendor, and they pointed out that the increase of risk aversion does not necessarily lead to the reduction of the order quantity. Schrijver et al. (as in [18]) studied the optimization model and algorithm of multiproduct demand inventory network design for stochastic demand and inventory decision. Based on the theory of nonlinear integer programming, Yang et al. (as in [19]) studied the integrated multiproduct optimization model. Liu et al. (as in [20]) studied the flexible service policies for a Markov inventory system with two demand classes. Zhao and Lian (as in [21]) studied the priority service rule of a queueing-inventory system with two classes of customers. Karimi-Nasab and Konstantaras (as in [22]) studied an inventory control model with stochastic review interval and special sale offer. By determining the level of customer’s anchoring effect, Liu and Shum (as in [23]) studied the joint control of pricing and inventory allocation in two periods of retailers based on constructing the customer’s disappointment aversion utility function. Mo et al. (as in [24]) researched the inventory issue for the perishable multi-items with just-in-time (JIT) inventory-level-dependent demand. Ji and Jin (as in [25]) established an inventory optimization model meeting the restrained conditions of being controllable in lead time and service level. Li (as in [26]) studied the control and optimization model for multiechelon inventory in supply chain, while the optimization method for two-echelon inventory system based on stochastic lead time was researched by Dai et al. (as in [27]). Zhao (as in [28]) presented an optimization study on multiechelon inventory in supply chain on the basis of time competition, while Wang (as in [29]) studied the optimization model for production-inventory under uncertain environments. Fu and Pan (as in [30]) mainly explored disposing the inventory management problem by using fuzzy theory under uncertainty to derive the fuzzy mathematical model for single inventory management with multiple fuzzy parameters in the case of allowing moderate shortages. Besides, supply chain inventory optimization with controllable lead time under fuzzy environment was investigated by Li and Xu (as in [31]). Wang and Guo (as in [32]) analyzed the inventory risk loss led by the EOQ and order cycles of classical inventory models under fuzzy demand to deduce the economic risk function in fuzzy situation. Kong and Jirimutu (as in [33]) researched the inventory optimization under stochastic demand based on Monte Carlo simulation. Xu et al. (as in [34]) explored the inventory control model during random replenishment interval with inventory-level-dependent demand.

Most of the above researches were conducted on the basis of continuous normal population, which made the researches convenient and operable to some extent. However, on the premise of uncertain supply and demand, there were a lot of uncertain factors for inventory optimization. In fact, most of the demand and supply in reality cannot distribute continually but present in the form of discrete random variables usually. As a result, on the assumption that the demand process of each subwarehouse submitted to the mutually independent compound Poisson process, the authors carried out the researches on some aspects, including the optimization and control of inventory system based on supply-demand balance as well as the algorithm design for the optimal inventory costs. In addition, the related researches have a certain value on theoretical research.

Although there have been quite a few researches on inventory control, it is still necessary to take many factors and variables into account due to its systematicness and complexity of inventory problem. Besides, it is difficult to quantify and define the optimal inventory because the correlation degrees between each factor are fuzzy. In view of the above facts, the optimization and control of inventory can be summarized as a complex dynamic system containing multifactors, while the quantitative model about the optimization and control of inventory is considered as a complex system with multiple variables and multiple parameters. It is very difficult to solve the problem once and for all by using a single model and a unified algorithm for the research work of the optimization and control of inventory. Most of the existing researches were focused on some specific fields of a particular region to only work out the specific issues under a certain environment. In addition, it is inevitable that the research process is influenced by the subjectivity of the researchers themselves, which suggests that the optimization and control of inventory under various situations cannot be solved. Hence, this issue will undoubtedly attract the persistent attention from the relevant experts and scholars. Actually, it still plays a very realistic role in carrying out pertinent researches on optimization and control model for inventory and algorithm with respect to some specific fields in different areas.

#### 2. Inventory System Model

From a practical point of view of research object, a necessary simplification for the research object was conducted during the research combining the actual conditions of regional circulation enterprises. For the underdeveloped regions, the two-echelon inventory system is more common. In order to improve the practical significance of research results and enhance the operability, a typical two-echelon inventory system composed of a central warehouse and several subwarehouses was emphatically studied.

#### 3. Model Assumption and Symbol Description

##### 3.1. Model Assumption

(1)The central warehouse of the two-echelon inventory system mentioned purchases products from material suppliers, while the subwarehouses order goods from the central warehouse.(2)Both the central warehouse and subwarehouses of the system carry out the ordering strategy of continuous review inventory. In other words, the inventory levels are continuously observed by the subwarehouses and central warehouse. When the inventory level reduces to order point , the warehouses will purchase with lot-size of , where the inventory level refers to the result by subtracting the stock-outs from the total of on-hand inventory and the goods of the orders in transit. In this way, the inventory level is within a range after distribution centers and retailers ordering.(3)The material suppliers can supply materials unlimitedly and the delivery time for central warehouse is a constant, while the transportation time from central warehouse to subwarehouses is a random variable. Then, the lead time of subwarehouses consists of random delay and random transportation time.(4)The product demand process of subwarehouses is a mutually independent compound Poisson process, that is to say, Poisson arrival of consumers. In addition, the demand of each consumer is a random integer.(5) are available for all order points.(6)All stock-outs in the two-echelon inventory system are waiting. Besides, the delayed order-to-delivery follows the principle of “first come first serve.”

##### 3.2. Symbol Description

The meaning of the symbols in the research is as follows: is the number of retailers. is the fixed delivery time from manufacturers to distribution centers, namely, the lead time of distribution centers. is the random transportation time of goods from distribution centers to retailers . is the random delay of retailer’s orders in distribution centers. is the lead time of retailers ; . is the order quantity of distribution centers. is the order quantity of retailers . is the order point of distribution centers. is the order point of retailers . is the storage costs of unit goods in unit time of distribution center. is the storage costs of unit goods in unit time of retailers . is the stock-out losses of unit goods in unit time of distribution center. is the stock-out losses of unit goods in unit time of retailers . is the average holding costs and shortage costs of distribution center in unit time. is the holding costs and shortage costs of retailers in unit time. is the expected gross costs of the inventory system.

#### 4. Hypothesis Testing and Demonstration of Poisson Distribution

In order to test whether the order demand is subject to Poisson distribution, we give an empirical research by taking the agricultural warehousing company in Akesu area of Xinjiang as an example. Based on balance theory between supply and demand, the fertilizers supply quantity of the agricultural warehousing company within a week is determined by the demand quantity of farmers, while the supply quantity of the agricultural warehousing company decides its order quantity. Hence, the demand-based order quantity of the agricultural warehousing company within a week depends on the demand quantity of the farmers in this area in this period. In order to simplify the problem, the order quantity of each time with little fluctuation is taken as a constant. By doing so, the order quantity can be determined by controlling the order times within a single period.

In general, agricultural production presents seasonality and periodicity. This epically can be obviously found in agriculture plantation of Aksu area in Xinjiang province. As the demand quantity of fertilizers for the farmers in Aksu area is shown to be stable within some period or a given time in a season, to some extent, the order times of the agricultural warehousing company for an arbitrary time interval within the demand period merely rely on the span of time interval, instead of the end point of time interval. In addition, the order events of the agricultural warehousing company happen independently in the nonoverlapping intervals. Moreover, the probability for the occurrence of two or more than two order times can be nearly neglected when the time interval is enough small. As above mentioned, it is indicated that the order times of the agricultural warehousing company conform to Poisson stream, indicating stability, no aftereffect stream, and ordinary (as in [35]). On this basis, it is supposed that the order demand of the agricultural warehousing company is subject to Poisson distribution.

In order to verify our supposing, we conducted on-site interviews to the large and middle size agricultural warehousing company and the farmers in various corps. for surveying the demands of various fertilizers. With in-depth investigation, we have acquired large volume of information and first-hand data: through the close communication and contact with the managers in each corp., the total arable areas in Aksu and actual demand quantity of various fertilizers of each corp. were acquired. Afterwards, we verified the goodness of fit for Pearson of the sample data, to judge whether the sample data are subject to Poisson distribution or not. It is noteworthy that the actual demand quantity of fertilizers is the order demand quantity of the agricultural warehousing company in fact. The method is described as follows.

By randomly selecting a set of sample data with each sample size more than 50, the order quantity of each time is assumed to be a constant when the order quantity of each time shows little fluctuation. In this case, the order demand can be directly presented by controlling order times. The analysis of chosen data indicates that third- or fourth-order times of the agricultural warehousing company are shown to be reasonable.

According to statistical quantity, is the number of purchase groups, where .

The empirical and theoretical frequencies of stochastic events are and , respectively, and they indicate significant difference, with a given confidential level ; the distribution of does not agree with that of ; if there is no significant difference between and , the random variable distribution is subject to the theoretical scheme. To validate whether or not the empirical scheme of the fertilizers demand in a time interval in some areas is subject to Poisson distribution, a significant difference between the theoretical and empirical frequencies needs to be checked. In the case of , refers to the value of when the degree of freedom is 1. On this basis, is calculated through software programming this is obviously obtained that . As can be seen, the selected sample data have passed the verification of Poisson distribution. This is to say, these data are subject to the Poisson distribution; the detailed processes and methods can be seen in the researches (as in [35, 36]). By frequent sampling on the sample data, various sets of sample data are obtained by repeating abovementioned methods and processes. All chosen sample data have been validated in Poisson distribution. This further confirmed the feasibility of our assumption.

Apart from verifying the goodness of fit for Pearson , the repeated samplings have been conducted on the sample data. Moreover, the density function was performed on the simulation of different samples. The simulating results indicate that the density function approximates to normal density with increasing quantity of sample size; considering that the order quantity or order number of a warehouse is stochastic and independent, the data obtained are discrete data, as the order number is relatively large. This is because the corps. in Aksu area mainly make a living by planting; the demand quantity of various fertilizers is relatively large: the quantities are usually more than 1,000 tons on average. The average value of real sample data investigated is 1,093 tons. The total average value estimated is the average value of the sample data. Based on these two points, the parameter lambda (total average value) of Poisson distribution is large according to probability knowledge; as its limit distribution is normal, it is supposed that the demand data are subject to Poisson distribution which is assumed to be reasonable according to the abovementioned analysis results.

As shown in Figure 1, represents the density function image of all data investigated, where refers to 16 corps. in Aksu. The density functions are obtained by the simulation when three sample sizes are 30, 60, and 300, respectively. The analysis and investigation into the simulation of density function images reveal that, in the average value range of the samples , there is a good similarity between Poisson and normal distributions. The large the sample size, the more favorable the approximation effect.