Modeling and management of industrial inventories. Inventory Management. How to reduce inventory

INVENTORY MANAGEMENT MODELS

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Article topic:	INVENTORY MANAGEMENT MODELS
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CLASSIFICATION OF MATERIAL RESERVES

Stock– form of existence of a material flow. If the entire logistics chain along which the material flow moves worked like one mechanical conveyor, waiting time could be reduced to zero and inventories could be eliminated. At the same time, in real life this should not be expected.

The material flow along the path from the source to the final consumer can accumulate in the form of a reserve at any site. For this reason, stocks of raw materials, materials, finished products, etc. are distinguished. All stocks can be divided into the following groups: production, commodity. Each, in turn, is divided into groups according to the function it performs: current, insurance, seasonal.

The purpose of creating inventories– ensuring continuity production process.

Inventory– stocks of finished products at manufacturing enterprises, stocks of wholesale and retail, as well as supplies on the way. Current stocks- the main part of industrial reserves. Such reserves ensure continuity of production and trade processes between successive deliveries. Safety stocks designed to provide material and commodity resources in case of unforeseen circumstances: deviation in the frequency and size of deliveries; unexpected increase in demand. Seasonal stocks are formed due to the seasonal nature of production (agricultural products). Maximum Desired Stock- ϶ᴛᴏ stock level that is economically feasible in a given system. The threshold stock level is used to determine when to order the next batches. The current stock at any time can coincide with the maximum stock, trading level and guarantee (insurance) stock.

The logistics inventory management system is designed to ensure continuous production. The implementation of this goal is achieved by solving the following tasks: accounting for the current level of stock in warehouses; determining the size of the guarantee (insurance) stock; order size calculation; determining the time interval between orders.

Exist two main models of inventory management system: model with a fixed order quantity (Q-model); model with a fixed period between orders (P-model). Fixed order quantity management systems. When managing according to the Q-model, the next purchase order is placed at the moment when the stock of material decreases to a threshold level. This can happen at any time and depends on the volume of consumption. The use of this system involves constant monitoring of the remaining inventory. The Q model is recommended for inventory management of high-value resources because it provides the lowest average order size. At the same time, this model is characterized by high labor intensity of maintenance; therefore, for less expensive objects, a system with a fixed time between orders is used. Inventory management system with a fixed period of time between orders. When managing inventory by R-model, the next order is placed after a predetermined period. At R-model, the calculation of the remaining stock is made only after the control period of time has elapsed.

The P-model has a larger reserve since the resources must be sufficient until the next delivery at a fixed interval.

30. RULE 80–20. ABC ANALYSIS. XYZ ANALYSIS

Rule 80–20 used for structuring inventories, i.e., for identifying from the entire range of inventories those that should be optimized first: 20% of inventories account for approximately 80% of their value or sales dynamics, and 80% of inventories account for 20% of cost or sales volume. The difference between LAN analysis and the 80–20 rule is that to carry it out, the entire inventory range is divided into three groups, rather than two.

In most cases, the product range is so wide that modeling and analysis of each item is impossible. To simplify the solution of the problem, carry out ABC-analysis, on the basis of which all resources are divided into the following groups: L– high cost volume; IN– moderate cost volume; WITH– low cost volume. To classify inventories by importance (LAN analysis), the share of the cost of each item in the total cost of inventories is determined, and then they are arranged in descending order. Next in group A include approximately the top 20% of the ordered list items, whose total value represents 75–80% of the total inventory value. IN group IN include approximately 30% of items, the share of value of which is up to 70%. IN group WITH include the remaining positions, approximately 50%. The purpose of these procedures is essentially to separate the significant from the non-essential items.

This method gives the greatest effect in combination with method XYZ which allows you to classify the same range of stocks, but based on the dynamics of their consumption. The grouping of resources in XVZ analysis is carried out in increasing order of the coefficient of variation of demand for a product over a certain period of time, which is calculated for each assortment item:

Where Xi– the value of demand for the position being evaluated; X- average quarterly demand; P– number of quarters.

TO categories X include resources with a demand variation coefficient of less than 10%. Οʜᴎ are predictable, characterized by a stable amount of consumption. Category Y - ϶ᴛᴏ resources, the coefficient of variation for which changes dynamically from 10 to 25%. Category Resources Z are consumed irregularly, the accuracy of their prediction is low, and the coefficient of variation is more than 25%.

Combining the results of application ABC– And XYZ- Using analysis methods, we obtain 9 groups of reserves, for each of which the company must develop its own management options. AH groups, AY And AZ require the most attention. It is worth saying that they use a model with a fixed order size, calculate the optimal order size and use just-in-time delivery technology. For CX group resources, СY And СZ simplified planning methods are used, and their management functions are, as a rule, transferred to lower levels of the production chain.

INVENTORY MANAGEMENT MODELS - concept and types. Classification and features of the category "STOCK MANAGEMENT MODELS" 2017, 2018.

1. Dependent and independent demand. Subject of inventory management theory.
2. Basic inventory management strategies.
3. Modifications to basic inventory management strategies.
4. Target functions of inventory management models.
5. Types of inventory management models.
6. The simplest models of inventory management.

6.1 Single-product static model.

6.2 Single-product static model allowing for shortages

6.3 Model with gradual replenishment of inventories.

6.4 Gradual replenishment model allowing for shortages
7. Probabilistic models of inventory management.

7.1 Fixed Order Model and Service Level.

7.2 Fixed order frequency model and service level.
8. Special inventory management models.

8.1 Model taking into account quantity discounts.

8.2 One-period model.
Appendix A. Brown Table.
Appendix B. Areas under the standard normal distribution curve.

Inventoryis a stock of any resource or items used in an organization.

From a practical point of view, the problem of inventory management is extremely serious. The losses suffered by enterprises (especially industrial ones) due to irrational inventory management are very large. It’s bad when the supply is small and insufficient. This can lead to a disruption in the rhythm of production, an increase in production costs, failure to complete work under contracts, and loss of profit. However, the situation when the stock is excessively large is also extremely undesirable. In this case, there is a “freezing” of the organization’s working capital. As a result, the money that could “work” and generate income rests in warehouses in the form of stocks of raw materials, supplies, and components.

To effectively solve problems associated with inventory management, the use of appropriate methods is required. Such methods exist, however, unfortunately, in practice (especially in Russia) they have not yet found proper distribution.

The statement of one of the foreign researchers is very revealing:

"...Too many enterprises, unfortunately, manage inventory completely unsatisfactorily; this suggests that management does not realize the full importance of production inventories. But it happens even more often that there is awareness of the problem. There is a lack of understanding of What must be done and How do this." 1

So, inventory management on a rational basis is a very urgent task. The nature of the need for the stored product is of decisive importance when building an inventory management system.

1. Dependent and independent demand. Subject of inventory management theory

The main feature that determines the methods used for planning and inventory control is the nature of the demand for these inventories. There are dependent and independent demand. Items used dependent demand, as a rule, are subassemblies and components used in the production of the final product.

Demand (i.e. use) for subassemblies and components is determined by the volume of production of finished products. A classic example here is the need for wheels for manufactured cars. If each car requires five wheels, then the number of wheels required to produce a batch of cars is a simple function of the volume of that batch. For example, 200 cars require 1000 (200∙5) wheels.

Items with independent demand- These are, most often, finished products, final products. Typically, the finished product is sold (or shipped) to the customer - it is not involved in the production of any other product. In this case, as a rule, it is impossible to accurately determine the need for a product for any period of time, since there is usually an element of randomness in demand.

Thus, with independent demand, forecasting plays a large role in inventory management, while for dependent demand, the need for inventory is determined based on the production plan.

In this section, we will consider models used to analyze situations with independent demand. To regulate inventories in the case of dependent demand, slightly different approaches are used. These are the so-called logistics concepts for managing the movement of material assets, for example, MRP, DRP, Just-in-time and others. The corresponding methods are usually considered within the disciplines of logistics and production management.

Inventory management theorycombines methods for analyzing problems of regulating stocks of a certain product with independent demand for this product.

In problems of this kind, it is necessary to find a rational amount of inventory, taking into account that losses arise both due to unsatisfied demand and due to the fact that the product is stored in a warehouse.

The problem of inventory management arises when considering a variety of economic objects. Inventory management problems are common in retail analysis. In this case, the stocks of some product in the store are considered. Typically, demand is considered a random variable with a given distribution. The stock is replenished by delivering goods from the wholesale base at the request of the store, and the delivery time can be fixed or be a random variable. The manager is faced with the question: when to submit an application for replenishment of stock, and what quantity of goods is required in the application? The theory of inventory management answers such questions.

Inventory management, as already mentioned, is also necessary production facilities, where it is necessary to determine the rational level of stocks of raw materials, tools, etc. Excessive inventory in this case leads to irrational use of working capital and requires significant costs for storage and care. On the other hand, shortage of raw materials, supplies or tools causes production disruptions. Therefore, establishing a rational amount of stock is a means of allowing, on the one hand, to eliminate unnecessary stocks, and on the other hand, to ensure the rhythm of production.

Inventory management consists of establishing the timing and volume of orders for their replenishment.

The set of rules by which such decisions are made is called inventory management strategy (system).

Optimal strategyis considered to be the one that ensures the minimum cost of bringing products to consumers.

Finding optimal strategies is subject of theory optimal control reserves.

2. Basic inventory management strategies

Any inventory management strategy is designed to answer two basic questions: when to order the next batch of products, and how much product to order?

There are two main strategies for inventory regulation:

1) system with a fixed order quantity;

2) a system with a fixed order frequency.

Fixed Order Quantity Systemassumes that the size of incoming batches is a constant value, and the next deliveries are carried out at different time intervals. A batch purchase order is made when the stock size is reduced to a predetermined critical level called "point of order"(V foreign literature The abbreviation ROP is used - Reorder Point). Thus, the intervals between deliveries depend on the intensity of consumption of the product.

The situation is illustrated in Figure 4.1. The figure shows:

Z(t) – the amount of product stock in the warehouse;

S – “order point”, ROP (Reorder Point);

q = const – volume of the delivered batch;

, , - duration of the procurement period.

Figure 4.1 – Movement of product inventory when using a strategy with a fixed order quantity

The adjustable parameters in such a system are: “order point” (S, ROP) and order quantity (q, ROQ - Reorder Quantity).

The time interval between submitting an application and the arrival of the batch at the warehouse is called procurement period. In the model, the duration of the procurement period can be considered constant or be a random variable with a given distribution.

The disadvantage of the first strategy is usually the need to regularly record material assets in the warehouse, so as not to miss the moment of the “order point”.

The fixed size strategy is more suitable for critical, important materials because it allows for tighter inventory control and can therefore provide a quicker response to stock-out threats.

Fixed order frequency system. In this case, products are ordered at regular intervals, and the size of the stock is adjusted by changing the batch size. The batch volume is taken to be equal to the difference between the fixed maximum level to which the stock is replenished and its actual size at the time of order.

The situation is illustrated in Figure 4.2. The figure shows:

Max – maximum (planned) level;

l – interval between orders (planned period).

Figure 4.2 – Movement of product inventory when using a strategy with a fixed order frequency

The adjustable parameters in such a system are: the maximum (planned) level (Max) and the time interval between two orders (l, also called the planned period).

The advantage of such a system is that there is no need for regular accounting of materials. Disadvantages: sometimes you have to place an order for a small amount of products, and if there is unexpectedly intense consumption, the stock may be exhausted before the next ordering time.

Figure 4.3 describes in detail and clearly how the two main inventory management strategies work.

Figure 4.3 – The order of operation of the main inventory management strategies

3. Modifications to basic inventory management strategies

Used to improve the characteristics of basic strategies.

System with fictitious stock level. It is a modification of the first of the main strategies. Used in a situation where the intensity of demand is a random variable, or the duration of the procurement period is a random variable, or both of these parameters are random variables. In this state of affairs, it is possible that upon arrival of the ordered quantity of products at the warehouse, the stock level will still be below the “order point”, i.e. I'll have to place a new order right away. But why wait for the arrival of the previous batch, if the need to quickly order the next one can be predicted?

When using this strategy, a fictitious stock level - Y(t) - is used as an indicator used to determine the moment of ordering. It represents the sum of the stock on hand in the warehouse and the quantity of products in the process of delivery. The strategy is as follows: when the fictitious stock level Y(t) reaches the “order point” S, a new order is placed.

The situation is illustrated in Figure 4.4. The figure shows:

Y(t) – dotted line, fictitious stock level;

Z(t) – solid line, actual stock level in the warehouse;

– duration of the procurement period.

Figure 4.4 – Movement of product inventory when using a strategy with a fictitious inventory level

System with fixed periodicity and two fixed levels. It is a modification of the second of the main strategies. Here, in addition to the upper maximum stock level, a minimum one is also set. If the stock size decreases to the minimum level before the next order, then an extraordinary order is made. The rest of the time, this system functions as a system with a fixed order frequency. The movement of product inventory when using a strategy with a fixed frequency and two fixed levels is illustrated in Figure 4.5.

Figure 4.5 – Movement of product inventory when using a strategy with a fixed frequency and two fixed levels

The advantage of the strategy is that it eliminates the possibility of material shortages. The need to regularly monitor inventory levels may be cited as a disadvantage.

4. Objective functions of inventory management models

The criterion for the optimality of the strategy is the minimum of total costs associated with the formation and storage of inventories, and losses arising in the event of interruptions in supply to consumers. In this case, only those costs that depend on the size of the supply lots and the amount of stock are taken into account.

As a rule, the minimum amount of the following types of costs is taken as the objective function in inventory management models.

1. Costs associated with supply interruptions (losses from shortages). Let us introduce the notation. The letter a denotes the amount of losses from a shortage of a unit of production.

2. Costs associated with storing inventory. Let us denote b - the cost of storing a unit of production per unit of time.

3. Costs associated with organizing supplies; let c be the cost per batch. In the simplest case:

c(q) = c 0 + c 1 q ,

(4.1)

where q is the quantity of products ordered,

c 0 - costs that do not depend on the volume of the order and are associated with the very fact of its production;

c 1 - purchase price per unit of production.

The presence of a non-zero value c0 in costs c(q) leads to a limitation on the number of orders and, in fact, to the need to have a warehouse.

Let's try to analyze the dependence of the amount of costs of each type on the level of inventory in the warehouse. From Figure 4.6 it can be seen that as the stock level increases, the costs of the first type decrease, which is natural, since this reduces the risk of stock depletion. Storage costs (2) increase (linearly or nonlinearly), and the costs of organizing supplies (3) decrease, since high level inventory allows you to place orders less often.

Note that the total cost curve (dashed line) has a clear minimum point. This allows us to conclude that there must be a level of stock Z * at which the total costs reach the minimum value V min.

Figure 4.6 – Dependence of costs on average stock levels

Since the inventory changes over time, requests for its replenishment are also submitted periodically; when studying inventory storage systems, the average cost of operating the system per unit time is usually minimized. Such costs can be represented as follows:

This situation is explained by the difference in initial conditions. The main basis for classifying inventory management models is the nature of the demand for stored products (recall that from the point of view of a more general gradation, we are now considering only cases with independent demand).

So, depending on the nature of demand, inventory management models can be

· deterministic;

· probabilistic.

In turn, deterministic demand can be static, when the intensity of consumption does not change over time, or dynamic, when reliable demand can change over time.

Probabilistic demand can be stationary, when the probability density function of demand does not change over time, and non-stationary, where the probability density function changes depending on time. The above classification is explained in Figure 4.7.

Figure 4.7 – Types of inventory management models depending on the nature of demand

The simplest case is the case of deterministic static demand for products. However, this type of consumption is quite rare in practice. The most complex models are non-stationary type models.

In addition to the nature of demand for products, when building inventory management models, many other factors must be taken into account, for example:

· deadlines for order fulfillment. The duration of the procurement period can be constant or be a random variable;

· inventory replenishment process. Can be instantaneous or distributed over time;

· presence of restrictions on working capital, warehouse space, etc.

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Generalized model of inventory management.

Any inventory management model must ultimately answer two questions:

1. How many products should I order?

2. When to order?

The answer to the first question is expressed through order size, which determines the optimal quantity of resources that must be supplied whenever an order is placed. Depending on the situation under consideration, the size of the order may vary over time.

The answer to the second question depends on the type of inventory management system. If the system provides periodic control inventory status at regular intervals (weekly or monthly), the moment a new order arrives usually coincides with the beginning of each time interval. If the system provides continuous monitoring stock status, order point usually determined stock level, at which it is necessary to place a new order.

Thus, the solution to the generalized inventory management problem is determined as follows:

1. In case of periodic monitoring of stock status It is necessary to ensure the supply of new quantities of resources in the volume of the order size at regular intervals.

2. In case of continuous monitoring of stock status need to place a new order in size stock volume when its level reaches order points.

The size and order point are usually determined from the conditions for minimizing the total costs of the inventory management system, which can be expressed as a function of these two variables.

The total costs of the inventory management system are expressed as a function of their main components:

Acquisition costs becomes an important factor when the unit price depends on the order size, which is usually expressed as wholesale discounts in cases where the unit price decreases with increasing order size.

Ordering costs represent fixed costs related to its placement. When satisfying demand for a given period of time by placing smaller orders (more frequently), costs increase compared to when demand is satisfied by placing larger orders (and therefore less frequently).

Inventory holding costs, which are the costs of holding inventory in a warehouse (conversion costs, depreciation costs, operating costs) typically increase as inventory levels increase.

Deficit Losses represent expenses due to the lack of stock of necessary products.

The following figure illustrates the dependence of the four cost components of the generalized inventory management model on inventory levels.

The optimal inventory level corresponds to the minimum total costs.

An inventory management model does not have to include all four types of costs, since some of them may be insignificant, and sometimes accounting for all types of costs overly complicates the total cost function. In practice, a cost component may be ignored as long as it does not constitute a significant portion of the total cost.

Inventory management problems constitute one of the most numerous classes of economic problems in operations research, the solution of which is of great national economic importance. Correct and timely determination of the optimal inventory management strategy, as well as the standard inventory level, allows you to free up significant working capital, frozen in the form of stocks, which ultimately increases the efficiency of the resources used.

Let's consider the main characteristics of inventory management models.

Demand. The demand for the stocked product can be deterministic(in the simplest case, constant in time) or random. Randomness of demand is described by either a random moment of demand or a random object of demand at deterministic or random points in time.

Warehouse replenishment. Replenishment of the warehouse can be carried out either periodically at certain intervals, or as stocks are exhausted, i.e. reducing them to a certain level.

Order quantity. With periodic replenishment and occasional stock outs, the order quantity may depend on the condition observed at the time the order is placed. An order is usually placed for the same amount when the stock reaches a given level - the so-called order points.

Delivery time. Idealized inventory management models assume that ordered replenishment is delivered to the warehouse instantly. In other models

a delay in deliveries for a fixed or random period of time is considered.

Delivery cost. As a rule, it is assumed that the cost of each delivery consists of two components - one-time costs that do not depend on the volume of the ordered batch, and costs that depend (most often linearly) on the volume of the batch.

Storage costs. In most inventory management models, the warehouse volume is considered to be practically unlimited, and the volume of stored inventory serves as the controlling variable. It is assumed that a certain fee is charged for storing each unit of inventory per unit of time.

Deficiency penalty. Any warehouse is created in order to prevent shortages of a certain type of product in the serviced system. Lack of stock at the right time leads to losses associated with equipment downtime, irregular production, etc. We will further call these losses penalty for deficit.

Stock nomenclature. In the simplest cases, it is assumed that the warehouse stores a stock of similar products or a homogeneous product. In more complex cases it is considered multi-menclature stock.

Structure of the warehouse system. Mathematical models of a single warehouse have been most fully developed. However, in practice there are more complex structures: hierarchical warehouse systems with different replenishment periods and order delivery times, with the ability to exchange inventory between warehouses of the same hierarchy level, etc.

The criterion for the effectiveness of the adopted inventory management strategy is cost function (costs), representing the total costs of storage and delivery of the stored product (including losses from spoilage of the product during storage and its obsolescence, loss of profit from the death of capital, etc.) and the cost of fines.

Inventory management consists of finding a strategy for replenishment and consumption of inventories in which the cost function takes on a minimum value.

The simplest inventory management models are discussed below.

Let the functions A(J), B(t) And R(t) express respectively the replenishment of inventories, their consumption and the demand for the stored product over a period of time)

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Generalized model of inventory management.