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Theory of Inventory Management with a View of Stochastic Models

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Theory of Inventory Management with a View of Stochastic Models Barbora FRANKOVÁ1, Vojtěch MEIER2 1Technical University of Ostrava, Czech Republic (e-mail: [email protected]) 2Technical University of Ostrava, Czech Republic (e-mail: [email protected]) 13. THEORY OF INVENTORY MANAGEMENT WITH A VIEW OF STOCHASTIC MODELS 13.1. Inventory All types of companies, both Czech and foreign, are struggling with the problem of the amount of inventory in stock for both raw materials and goods or products. It can be said that the supplier is unable to meet the demand immediately if he does not have enough inventory in stock (Winston 2004). Many authors are concerned with various inventory optimization models. The basic inventory management models are deterministic, but they are a decision-making model behind certainty that does not consider the variable consumption over time (Polanecký, 2018). Deterministic models are no longer used today. Many companies have a variable consumption over time and they do not know predict consumption in the future. Therefore, we will deal with stochastic models dealing with risk-taking in this article, so consumption is counted as a random variable. The aim of the article is therefore to optimize inventories using stochastic models. The main question is why company must manage its inventory? How does it manage? Inventory must manage because they take your many. If company does not manage inventory, it cannot meet the demand of its customers. The result may be a lack of money but a surplus of materials in stock. 190 MEET 2018 13.2. Stochastic models for inventory optimization If the company does not use inventory management, it may encounter several problems. The first of them is the obsolescence of the stored products. A lot of raw material on stock after a certain storage period they become unusable. The second one is the financial problems. The amount of inventories and the ways in which they are managed have a direct impact on business profitability and the need for available financial resources. This is due to the fact that inventory is tied to company capital. If those units are stored for a long time in the company, the capital that they represent cannot be used or invested somewhere else. The third issue is the cost related to the holding inventories, which includes the handling costs, the building space for inventory. These issues should be solved through efficient inventory management that matches a particular company (Baglin et al., 2001). This model, otherwise known as the Miller-Ori model is based on the assumption that the state of the current assets (inventory, cash) in the enterprise changes over time. Against the deterministic models where demand-type variables, consumption, acquisition time were known and constant (determined). In practice we usually encounter stochastic models where the values of these variables are random and only to some extent predictable (Malliaris and Brock, 1982). Which is a more realistic approach than deterministic models. Stochastic models can be seen as a regulatory tool for optimizing inventory in the company. These models work with demand forecast based on previous periods. Two quantities are used to control inventory, which is the quantity ordered and the length of the ordering interval. Stochastic models differ from deterministic models by demand, which is expressed by a random variable with probability distribution. The most common assumption is the normal distribution with mean value and standard deviation (Bartmann, 1992). There are two basic systems for stochastic models (Bartmann, 1992), namely Q system and P system. 13.2.1. Q system Model has a constant delivery quantity. To accomplish such a task, it is necessary to determine the probability distribution of random demand over a period and also to estimate its median and the standard deviation. The commonly used probability source is data on past demand for a sufficiently long past period. It is a convenient Theory of inventory management… 191 solution to categorize data of demand by intervals by days into a table divided by the frequency and then determining the required values. When inventory optimization is set, the lower limit and the value of the delivery is constant throughout. Therefore, if the stock level drops below the lower limit (level), you need to order a given amount (Mahadevan, 2009). Fig. 1. Q system Rys. 1. System Q Source: (Vaishnavi, 2018). 13.2.2. P-system Model has a fixed ordering interval for stock replenishment. This model specifies the time of the order, which repeats all the time and the upper order limit. At a given time, we will find the inventory status of stocks and add stocks up to the upper limit. Because there is fixed ordering time the uncertainty interval is longer than the previous system. For example, if there was not ordered enough quantity at one time, you need to wait until the next appointment. In order to create a sufficient supply, the uncertainty interval must be equal to the sum of the average order completion date and the average delivery cycle. The order quantity in this model is dependent on demands, safety stock and current inventory (Mahadevan,2009). 192 MEET 2018 Fig. 2. P system Rys. 2. System P Source: (Vaishnavi, 2018). 13.3. Data Data was provided to me by ABC ltd. whose main activity is the production of refractory materials. There is a single product and a single location. We choose demand for bauxite because is it is the basic raw material for production of refractory materials. It is a Czech company, so We calculated costs related with inventory in CZK. 20 15 10 Tons 5 0 1 2 3 4 5 6 7 8 9 10 11 12 Month Fig. 3. Total sum for demand for bauxite in the company Rys. 3. Całkowite zapotrzebowanie na boksyt w spółce Source: own processing. Theory of inventory management… 193 The data shows the demand for raw material (bauxite) during 2017, which you can see in Table 1, and also total sum demand per month for bauxite you can see in the Figure 3. Table 1 Demand for bauxite in tons Date 02.01. 09.01. 16.01. 23.01. 30.01. 06.02. 13.02. 20.02. 27.02. Demand 4,11 1,1 1,53 0,54 3,32 1,75 2,99 1,86 2,04 Date 06.03. 13.03. 20.03. 27.03. 03.04. 10.04. 17.04. 24.04. 01.05. Demand 1,89 1,12 2,51 1,62 2,51 5,25 2,49 0,5 2,16 Date 08.05. 15.05. 22.05. 29.05. 05.06. 12.06. 19.06. 26.06. 03.07. Demand 6 3,25 0,48 2,43 4,05 1,35 2,3 6,75 2,7 Date 10.07. 17.07. 24.07. 31.07. 07.08. 14.08. 21.08. 28.08. 04.09. Demand 3,12 1,62 2,7 7,25 2,23 3,6 2,8 0,4 2,64 Date 11.09. 18.09. 25.09. 02.10. 09.10. 16.10. 23.10. 30.10. 06.11. Demand 1 1,96 2,59 1 2 2,9 3,72 1,12 2,89 Date 13.11. 20.11. 27.11. 04.12. Demand 2,3 3,48 2,91 4,49 Source: ABC ltd. Table 2 Auxiliary calculations for probability distribution of demand Demand Centre of the Frequency Intervals 2 푠 (푠 푠 ) (tons) interval 푠 I1 0-0,5 0,25 3 0,75 15,53 I2 0,6-1 0,75 3 2,25 9,46 I3 1,1-1,5 1,25 5 6,25 8,13 I4 1,6-2 1,75 8 14 4,81 I5 2,1-2,5 2,25 8 18 0,61 I6 2,6-3 2,75 9 24,75 0,45 I7 3,1-3,5 3,25 4 13 2,1 I8 3,6-4 3,75 3 11,25 4,5 I9 4,1-4,5 4,25 2 8,5 5,95 I10 5-5,5 5,25 1 5,25 7,42 I11 5,6-6 5,75 1 5,75 10,4 I12 6,5-7 6,75 1 6,75 17,85 I13 7,1-7,5 7,25 1 7,25 22,32 Source: own processing. The first step is to find out the probability distribution of demand. Because we have a lot of data, we have to sort them into 13 intervals which we have specified the 194 MEET 2018 centre the frequency of the data at individual intervals. The other columns are helpful for determining average demand and standard deviation for determination probability distribution of demand. The most commonly used theoretical distributions are the normal, exponential, Poisson distribution, or the empirical distribution. The best way how to find out is to illustrate the data and combine it with a curve. Then we will compare the curve with the curve with a particular distribution and calculate if that's true. ∑푛 푠´ 푠 = =1 (1) 푛 2 푠 = √∑ (푠´ − 푠 ) / (2) =1 The average demand for bauxite from formula nr. 1 is therefore 2.53 tons and a standard deviation which we calculated from the formula is 1.5 tons. The standard deviation indicates how far the individual surveyed values differ from each other. If it is small, the elements of the set are mostly similar to each other and a large standard deviation signifies big differences to each other (Bartmann,1992). In the next step, it is necessary to determine the normalized variable of the normal probability distribution for the upper limit of the intervals according to the relation (3) and find out the distribution function value from the statistical tables.
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