Using digital twins to create an inventory management system

. The article considers the application of the inventory management model on the example of a manufacturing enterprise, where it is planned to use a digital twin together with the inventory management system. Inventory management model is a mathematical model that allows you to determine the optimal level of stocks at the enterprise. In a manufacturing enterprise, this model can help optimize the inventory of raw materials, materials, and finished goods, which will reduce costs and reduce risks associated with the lack of necessary raw materials or required quantities of finished goods. With the help of data obtained from the enterprise, the input parameters of the model were obtained, namely the size of resource consumption, stock holding costs, delivery realization costs, and time required for delivery. The model was used to find the parameters such as optimal order size, delivery period, inventory level at which ordering should be carried out.


Introduction
Digital twin technology has already become widespread in various industries.It allows the creation of virtual models of real objects and processes that can be used for analysis, optimization and control.Digital twins can be created for various objects such as factories, warehouses, vehicles and even entire production lines.
One example of the use of digital twins is the automotive industry.Companies use digital twins to simulate the behavior of their vehicles under different operating conditions.This allows them to optimize production and improve product quality.
In addition, digital twins are used in the energy industry to simulate the operation of power plants and grids.This allows them to optimize equipment performance and reduce energy costs [1][2][3].
Digital twins also find application in medicine.For example, they can be used to model processes in the human body and determine optimal treatment methods.Digital twin technology has a wide range of applications and is an important tool for improving the efficiency and quality of production in various industries.

Methods for creating digital twins
To create digital twins, various methods can be used to create a virtual copy of a real object or system.Some of these methods include: 1.Using sensors and sensors to collect data about the real object.
Sensors and sensors are devices that are used to collect information about the real world.They are installed on a real-world object.They can, for example, measure temperature, pressure, humidity, speed of movement, etc. Sensors can be used to measure physical properties of an object such as color, texture, and shape.
Sensors and sensors can be connected to a computer or other device that processes the information received.This makes it possible to create a digital replica of the object that accurately reflects its real-time state.Using computer modeling to create a virtual copy of an object.This method can be used to create an exact copy of an object or to study its behavior under different conditions.2. Using machine learning algorithms to analyze data collected from sensors and sensors.
Machine learning algorithms are programs that can analyze large amounts of data and identify patterns.They are used to create digital twins of objects that can predict their behavior under different conditions and make decisions based on that information.
For example, a machine learning algorithm can be used to analyze room temperature, pressure, and humidity data to predict when to turn the air conditioner on or off.Machine learning algorithms can also be used to analyze data about the wear and tear of machine parts to predict when they need to be replaced [4][5][6].
The use of machine learning algorithms can create more accurate and efficient digital twins of objects, which can lead to better quality of work and more efficient production.3. Utilizing cloud technology to store and share data between different systems and devices.
Cloud technology is a way of storing and sharing information between devices and programs that uses remote servers.With cloud technologies, it becomes possible to create more complex digital models that can function in real time and interact with each other.
Cloud technologies make it easier to create digital models, as there is no need to install special software on each device.In addition, cloud services provide access to a large amount of data and computing resources, which allows for rapid information processing and decision-making.4. Utilizing virtual reality technologies.
Virtual reality (VR) technologies allow the creation of interactive digital twins.VR technologies can be used to create a virtual model of an object or system that can be explored and interacted with.This can be useful for training, designing and testing new products or systems.5. Use of mathematical models.
The use of mathematical models is one of the key elements of digital twins.They allow describing physical objects and processes and predicting their behavior.Mathematical models can be used to solve various problems such as optimization of production processes, resource management and decision making based on data analysis.
For example, when creating a digital twin of a manufacturing process, mathematical models can be used to describe the behavior of equipment, materials, and tools.These models can help determine optimal equipment settings, reduce production time, and improve product quality.Also, mathematical models are used to predict demand for goods and services.For example, if a company creates a digital twin of its store, it can use mathematical models to predict sales and optimize inventory [5][6][7][8].
In general, the use of mathematical models in digital twins can improve the efficiency and accuracy of decisions, improve product quality, and reduce production costs.

Mathematical models of inventory management
There are several types of mathematical models for inventory management such as the minimum inventory level (EQL) model, the optimal inventory level (EOQ) model, and the dynamic programming (DP) model.
The OLQ (Ordering, Leveling and Quantity) model is an inventory management model that is used to optimize inventory under conditions of demand uncertainty.It takes into account three main factors: orders, stock levels and quantity in stock.
Orders are the quantity of goods to be ordered each time inventory reaches a certain level.Inventory levels are the levels at which goods must be held to meet demand.The quantity of goods in stock is the number of goods that are already in stock.The OLQ model uses an algorithm to determine the optimal orders, inventory levels, and quantity of goods in stock to minimize storage costs and meet the demand for goods.
The EQL (Economic Ordering Quantity with Lot Size) model is another inventory management model that also considers three main factors: orders, inventory levels, and quantity of goods in stock.However, unlike the OLQ model, the EQL model also considers order size.Order size is defined as the minimum quantity of goods required to fill an order.This minimizes the cost of transporting goods and reduces the waiting time for orders.Thus, the EQL model is more accurate and takes into account all factors related to inventory management.
Dynamic Programming (DP) model is a mathematical optimization technique that is used to solve problems with multiple states and actions.It is based on the idea of breaking the problem into smaller subproblems and using Bellman's optimality principle to find the optimal solution [7][8][9][10].
The DP model can be used to solve various problems such as task scheduling, inventory management, route optimization, etc.It finds the optimal solution by enumerating all possible paths and selecting the best one.However, the DP model has its limitations and cannot always be applied to complex problems.Therefore, other optimization methods such as genetic algorithms or neural networks are often used to solve complex problems.
The QBD model is a simplified EOQ model that takes into account the non-uniformity of consumption and allows to determine the optimal lot size and order frequency.The QBD model is based on the assumption that the demand for a product is a random process with a known probability distribution function.
To calculate the optimal lot size and order frequency in the QBD model, it is necessary to determine the parameters of the consumption process, such as the average demand, the standard deviation of demand, and the time period between orders.Special formulas can then be used to determine the optimal lot size and order interval [11][12][13].
The MRP model (Material Requirements Planning) is a material inventory management method used to optimize the inventory level of materials in a company.The MRP model is based on forecasting the demand for materials and determining the optimal amount of materials needed for production.
The MRP model includes the following steps: a) determining material requirements based on the production plan; b) calculating the optimal inventory level of materials to meet the requirements; c) planning material orders based on the optimal inventory level; d) monitoring inventory levels and adjusting orders when necessary.
One of the advantages of the MRP model is the ability to forecast the demand for materials, which reduces the cost of purchasing materials and shortens the waiting time for deliveries.In addition, the MRP-model allows you to optimize the production process and increase the productivity of the enterprise.JIT-model (Just-In-Time) is an inventory management system based on the principles of lean manufacturing.It involves delivering materials and components exactly when they are needed to manufacture products.
The JIT model provides high production efficiency by reducing delivery lead times and reducing inventory of materials and components.It also reduces the cost of warehousing and transportation of materials, which leads to increased competitiveness of the company.
ABC is an inventory management model that uses classification of goods based on their importance and frequency of use.This model helps to determine which goods should be stored in larger quantities and which should be stored in smaller quantities.
The ABC model consists of three categories, A, B, and C. Category A goods have the highest priority and should be stored in large quantities.Category B goods have medium priority and should also be stored in large quantities.Category C goods have low priority and can be stored in smaller quantities.This model helps to optimize inventory and reduce storage costs.It also helps in identifying which goods need additional support to avoid shortages or overstocking [14,15,19].
There are also models that take into account the demand characteristics of a commodity (e.g., the S&S model), as well as models that take into account the costs of producing and delivering the commodity (e.g., the EOQS and EOQSD models).The choice of a particular model depends on the demand characteristics and inventory management costs in a particular area.
The choice of a particular model depends on the specific conditions and objectives of inventory management in a particular industry.

Description of the mathematical model
Wilson's model is a mathematical model that is used to find the order level that satisfies the needs of an organization and at which the costs associated with it will be the lowest.Wilson's formula is as follows: Where Qw is the optimal order size.Total inventory management costs, delivery period and ordering point are defined as follows: Deciphering of all notations will be given later in the text.The following assumptions are set for Wilson's model: 1.There is an external unlimited source of goods.
The order is delivered from a supplier who has unlimited quantities of the good or from a warehouse that stores previously produced goods in sufficiently large quantities (inexhaustible source).2. Consumption intensity is a known and constant quantity, ν = const (we must know how much we consume).
It can be calculated on the basis of statistical data on the consumption of goods in the past period or determined on the basis of marketing research.Knowing the intensity of consumption allows a company to plan the production and supply of goods in such a way as to meet the needs of customers and avoid shortages or excessive stock of goods.This allows the production and delivery of goods to be planned in advance, which reduces the risks of delays and increases the company's efficiency.4. Each order is delivered as a single batch.
This means that the entire order is delivered at the same time.Delivering multiple batches of goods may incur additional costs associated with storing, packing and shipping each batch separately.It may also increase the delivery time. 5. Order fulfillment costs are independent of the size of the order; Order fulfillment costs do not depend on the quantity of materials ordered.However, if the order size is large, there may be additional costs for warehousing, transportation and handling of materials 6.The cost of storing inventory is proportional to its size.
The larger the stock, the higher the storage costs.This is due to the fact that it is necessary to use warehouse space to store stock, as well as to pay for the work of personnel who monitor the safety and storage conditions of goods.In addition, a large amount of inventory increases the risk of spoilage or damage to goods, which also requires additional costs for replacement or repair.7.No shortage of goods.
In the model, shortage of goods is unacceptable because it can lead to negative consequences for the business and customers.If the quantity of goods in stock runs out, it can cause dissatisfaction of customers who cannot get the goods, which can lead to loss of customers and decrease the profit of the company.In addition, the shortage of goods can lead to disruption of logistics chains and increase the cost of storage and delivery of goods.
In practice, it is impossible to fulfill all these conditions, so various modifications of Wilson's formula have been created, which take into account the peculiarities of the work of enterprises in real conditions [16,17,20].
A graph representing inventory cycles allows you to see how inventory levels change over time and what actions are needed to maintain the right level of inventory.For example, if inventory levels fall below a certain level, you may need to increase production or purchase additional inventory.If the inventory exceeds the optimal level, then it is possible to reduce the amount of production or sell some of the inventory.

Fig. 1. Cyclicality of inventory changes in the model
This schedule can also help to determine the optimal size of deliveries as well as their frequency.This is important to reduce inventory management costs and reduce the risks of shortages or overstocking.The representation of inventory cycles is shown in Figure 1.It can be seen that the largest figure representing the amount of product that is in stock is the order size Q.
Figure 2 shows the inventory management cost graph that illustrates the relationship between order placement costs, inventory holding costs, total costs and optimal order size.

Fig. 2. Graph of inventory management costs in Wilson's model
Graphical interpretation is an important tool for analyzing mathematical models.It allows to visualize dependencies between variables, constraints and boundary conditions, which helps to better understand the model and make decisions based on the results obtained.Graphical interpretation can also be used to verify the correctness of the model and its consistency with real data.

Results from the use of the model
For modeling purposes, data on the consumption of pine boards used for the production of products at the enterprise were obtained.The data on the number of boards used for the production of various products, as well as on the production volumes of these products were analyzed.Based on this data, a mathematical model was created that takes into account all the factors that affect the consumption of pine board.The model has taken into account such parameters as number of working days in a year, number of shifts of work, number of employees in the company, etc.
The input data of the model are: 1.The size of resource consumption (v) is the amount of consumption of a given resource over a given period of time (usually a year, but in this problem consumption per hour has been taken).The size of consumption can depend on various factors such as equipment capacity, productivity, duration of use, operating conditions, quality of materials, qualifications of personnel, storage and transportation conditions, consumption rate, and availability of stock.2. Inventory holding costs (s) are the costs associated with maintaining inventory in a warehouse, which include charges for renting space, payroll of warehouse employees, depreciation of equipment, etc.These costs can be significant, especially if the stock is large or stored for a long time.3. Delivery costs (K) are the costs associated with getting the goods from the supplier to the buyer.They include transportation, cargo insurance, customs duties, and other costs.Costs can be high, especially when shipping long distances or using expensive modes of transportation.4. The time required for delivery (td) depends on many factors.Here are some of them: a) the distance between the supplier and the recipient.the farther the destination is, the longer it will take to deliver the goods; b) transportation conditions.Different modes of transportation may have different speeds, which also affects delivery time; c) the presence of delays at customs, border posts or other places along the shipment's route; d) the quantity and volume of the shipment.The larger the shipment, the longer it will take to be delivered; e) the quality of packaging and labeling of the shipment.good packaging and labeling can speed up the delivery process and avoid damage to the shipment.
The outputs of the model are the following parameters: 1.The delivery lot size (Qw)is the number of units of a commodity that is ordered from a supplier at one time.It is determined based on the demand for the product, order value, delivery conditions and other factors.The size of the delivery lot can be constant or change depending on changes in demand.2. Inventory management costs (L) are the costs associated with maintaining the required level of inventory in the warehouse.They include the cost of renting space, salaries of warehouse employees, payment of utilities, insurance, etc. 3. Delivery period (τ) is the period of time that elapses between the time a product is ordered from a supplier and the time it is delivered.4. Ordering point (h0) is the inventory level at which an order must be placed with the supplier.
After analyzing the data about the enterprise and making calculations, the input parameters for the model were obtained, which are presented in table 1.
The result of using the model was the data shown in table 2. According to the results of modeling, we can see that the optimal order size is 4.74 m 2 or approximately 5 m 2 , and the total cost of inventory management will be 189.74rub./h.The size of the stock at which it is necessary to make replenishment is 60 m 2 .
The resulting order size will minimize the cost of buying and storing boards, as well as increase the productivity of the enterprise through more efficient use of available resources.

Conclusion
Inventory management is an important aspect for businesses of all sizes as it helps to optimize the processes of purchasing, production, sales and cash flow management.It helps to reduce costs, increase profits and improve customer service [18,21].
There are many factors to be considered in inventory management such as product demand, level of competition, raw material prices, seasonality, production capacity, etc.It is also important to analyze inventory data and forecast inventory changes in order to make decisions about purchasing, producing and selling products according to market needs.
In this paper, an example of using a mathematical model of inventory management is presented to implement a digital twin to be used in the inventory management system of a company.
Mathematical models are important tools for inventory management.They can determine optimal inventory levels, calculate order intervals, and forecast demand for goods.They can also be used to optimize production processes and improve customer service.
Wilson's model can be used to create a digital twin of an inventory management system.This model can be used to determine the optimal inventory level and order spacing for a particular inventory management system, helps optimize the cost of storing and purchasing goods, and allows for improved system efficiency.
The digital twin of an inventory management system can include information about the demand for goods, stock on hand, costs of storing and producing goods, and other factors that affect system performance.Using Wilson's model, the digital twin can automatically optimize inventory management to reduce decision-making time and improve the accuracy of goods demand forecasting.

Table 1 .
Input data

Table 2 .
Output data