Inventory management: everything a manager needs to know. Inventory management models and systems Inventory management modeling
The main goal of which is to ensure an uninterrupted process of production and sales of products while minimizing the total costs of maintaining inventory.
From the point of view of current assets management industrial inventories include not only the raw materials and supplies needed for the production process, but also work in progress, finished goods and goods for resale.
Managing optimal inventory levels
An important issue is the required amount of safety stocks that an enterprise creates in case of unforeseen supply disruptions or possible seasonal surges in consumer demand. It is obvious that insurance reserves worsen the financial results of production activities (due to freezing funds in inventories), but provide the enterprise with stability and liquidity.
Inventory shortage causes a stop in production, a drop in sales volumes, and in some cases, the need to urgently purchase necessary raw materials at inflated prices. The consequence is that the enterprise does not receive any possible profit. Since inventories are liquid assets, their decline worsens the current liquidity ratio.
Excess inventory leads to increased storage costs, growth, loss of possible income due to freezing of financial resources in inventories, losses as a result of physical damage and obsolescence of inventories
Inventory management
In conditions of constant rising prices, a more objective assessment of the results obtained can be provided by the LIFO method, which smoothes out the influence of inflation when generating profits. Also, when using this method, the value of the balances and, accordingly, the amount of property tax are reduced.
If at the moment the company is pursuing a policy of minimizing prices and reducing costs, it is advisable to use the FIFO method.
Improves financial performance (), increasing the value of balances and reducing costs, increasing profits.
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 at production facilities, where it is necessary to determine the rational level of inventories 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 the theory of optimal inventory management.
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"(in foreign literature the abbreviation ROP - Reorder Point is used). 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 supply organization costs (3) decrease, since a high level of inventory allows orders to be placed less frequently.
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.
Federal State Budgetary Educational Institution of Higher Professional Education
"Volgograd State Technical University"
Department of Economics and Management
Faculty of Engineering Personnel Training
Inventory management models
Volgograd 2014
Introduction
1. Basic provisions of the theory of inventory management
2. Types of inventory management models
3. Generalized model of inventory management and its elements
4. Wilson model
Conclusion
Bibliography
Introduction
Widely used nowadays, logistics systems use inventories. Therefore, development is an important task for an enterprise; optimal inventory management strategy. Inventories can be considered raw materials, semi-finished products, components and finished products. The task of inventory management is directly related to the organization of the procurement process, that is, to the supply of the enterprise, as well as to the sale of finished products.
There is a need to have reserves if at least one of these factors exists:
· fluctuations in demand for goods;
· fluctuations in the delivery time of goods from the enterprise;
· certain conditions requiring the purchase of products in batches;
· the presence of some costs associated with shortages (lack of stock) or delays in delivery.
In most situations arising in trading activities, there is a joint appearance of these factors. Obviously, only in extremely rare cases will a company be able to operate without inventory.
1. Basic provisions of the theory of inventory management
Inventory management theory is one of the youngest branches of operations research.
Fundamentals of modern inventory management theory - problem statement, analysis of factors influencing the solution, method of taking into account uncertainty in demand. The created theory considers, for example, such problems as:
· inventory management of a homogeneous product in an isolated warehouse with a fixed delivery delay;
· inventory management in case of occasional delays in deliveries;
· multi-item inventory management, etc.
Maximum Desired Stockdetermines the inventory level that is economically feasible for a given inventory management system. This level may be exceeded. In various management systems, the maximum desired stock is used as a guideline when calculating the order quantity.
Threshold levelstock is used to determine the point in time when the next order is issued.
Current stockcorresponds to the stock level at any time of accounting. It may coincide with the maximum desired level, threshold level or safety margin.
Warranty(or insurance) stockdesigned for continuous supply of consumers in case of unforeseen circumstances.
The inventory management model consists of three blocks:
Sales forecasting block - forecasting weekly/daily product sales;
Inventory control block - optimization planning of guarantee stock, current stock, etc. taking into account the selected inventory management model for each product category;
Supply management block - optimization planning of deliveries within the company’s logistics network, taking into account planned sales, deliveries from the manufacturer, availability of stock, transport capacity, various restrictions and business rules.
Types of stocks
Concept of material stock
Reasons for creating inventories
The concept of material stock.Inventories are raw materials, supplies, components, finished products and other material assets awaiting entry into the process of production or personal consumption.
The main part of the inventory at the enterprise represents production items included in the material flow at various stages of its technological processing.
Reasons for creating inventories.Inventories in an enterprise are formed for two main reasons:
discrepancy between supply volumes and one-time consumption volumes;
the time gap between the moment the material is received and its consumption.
The supply of raw materials is carried out in most cases periodically, and their consumption, as a rule, occurs continuously and does not coincide in time with receipt. Therefore, to ensure uninterrupted operation, each enterprise creates certain reserves of the types of raw materials, materials, semi-finished products, fuel and other resources it needs. There are some other reasons that lead to the creation of inventories. These are seasonal price fluctuations; violation of the established delivery schedule (unpredictable decrease in the intensity of the input material flow); the possibility of fluctuations in demand (unpredictable increase in the intensity of the output flow), etc.
Types of inventories.On the path of transforming raw materials into a final product and the subsequent movement of this product to the final consumer, two main types of inventories are created: production and commodity inventories.
Industrial inventories are formed in consumer organizations and are intended to ensure the uninterrupted production process.
Inventories represent stocks of finished products at manufacturing enterprises, as well as stocks along the route of the goods from the supplier to the consumer, i.e. at wholesale and retail trade enterprises, in procurement organizations and stocks in transit. They are necessary for the uninterrupted provision of material resources to consumers.
Industrial and commodity stocks are divided into current, preparatory, insurance and seasonal:
Current inventories are necessary for the enterprise to ensure uninterrupted operation in the interval between two deliveries and to ensure the ability to produce products in batches of the optimal size. This part of the stock is formed under conditions of uniform and regular supply due to the discrepancy between supply volumes and one-time consumption, as well as delays associated with the movement of materials.
Preparatory stocks are allocated from production stocks to ensure uninterrupted operation during the period necessary to prepare materials for use and deliver them to workplaces.
Guaranteed (insurance) stocks are necessary to ensure the operation of the enterprise in case of possible interruptions in the supply process or fluctuations in production volume. With the help of these reserves, deviations of actual demand from the forecasted ones, deviations of the actual volume of output from the planned ones, and deviations of the actual execution times of various operations from the planned ones are compensated.
Seasonal inventories are due to seasonal fluctuations in production or consumption. This stock is intended to satisfy the projected (seasonal) increase in demand, as well as some unloading of the enterprise for the holiday period.
2. Types of inventory management models
Despite the fact that any inventory management model is designed to answer two main questions (when and how much), there are a significant number of models, the construction of which uses a variety of mathematical tools.
This situation is explained by the difference in initial conditions. The main basis for classifying inventory management models is the nature of demand for stored products.
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 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.
3. Generalized model of inventory management and its elements
Meeting the needs of economic, social and military facilities in various material resources includes three phases: planning, production and distribution. As a rule, by the time the supply is realized, the data underlying the request turns out to be outdated and the scope of supply no longer corresponds to the actual need. To prevent production stoppages in the event of insufficient supply, inventories are created at consumers and in the supply system. The following factors lead to the need to create reserves:
Discreteness of supplies
Random fluctuations
in demand for the interval between deliveries;
in the volume of supplies;
in the duration of the intervals between deliveries.
Expected changes in market conditions:
seasonality of demand;
seasonality of production;
inflation expectations;
expected price increases.
These factors create a tendency to increase inventories.
There are, however, a number of considerations in favor of minimizing inventories:
Fees for physical storage of inventory;
Losses in the amount of stock (evaporation, shrinkage, theft);
Qualitative changes (deterioration of consumer properties due to irreversible processes in the stored product);
Obsolescence.
Inventory management consists of establishing the moments and volumes of orders for replenishing them and distributing the newly arrived batch to lower levels of the supply system. The set of rules by which these decisions are made is called an inventory management strategy. Finding the optimal strategy is the subject of inventory management optimization theory.
When comparing strategies, only the variable components of the cost function are taken into account, depending on the choice of strategies. Thus in many inventory management models, it is possible to ignore most of the costs of maintaining the administrative apparatus (except for the costs of processing supplies), as well as the cost of producing material assets, which is proportional to the volume of the batch, which over a sufficiently long period of time is determined by the total demand and does not depend on the organization of supply.
The mathematical formulation of the problem of finding the optimal strategy depends on the situation under study. However, the commonality of the factors taken into account allows us to speak of a unified model of inventory management. Let us give a qualitative description of it, limiting ourselves for simplicity to one warehouse, which receives a random flow of qualitatively homogeneous requirements - applications from consumers. Applications are immediately satisfied until their total volume (from the beginning of the planning period) exceeds the initial stock. All subsequent requests cannot be serviced immediately, as a result of which the consumer is idle and incurs a loss. This loss is attributed to the supply system - it pays a fine. From time to time, the stock of stored property is replenished from the warehouse of a higher association, a central base, or from industry, and certain additional costs are associated with each such replenishment. Finally, the warehouse bears the costs of storing the property located in it. It is required to choose the moment and volume of the replenishment order so that the total costs of storage, fines and supplies are minimal. Some restrictions may be imposed on the operation of the warehouse. In these cases, a conditional minimum cost is sought.
Thus, the elements of the inventory management problem are:
Supply system;
Demand for supplies;
Possibility of restocking;
Cost function;
Restrictions;
Inventory management strategy.
The supply system is understood as a set of sources of applications and warehouses, between which transport of stored property is carried out during supply operations. There are three options for building a supply system:
· Decentralized. All warehouses directly serve consumers, and shortages in one or more warehouses, at the discretion of the supply management body, can be covered by excess inventory in other warehouses.
· Linear. The production chain is considered and the distribution of buffer stocks according to the degree of product readiness is calculated.
· Echeloned. Each shortage is covered by the ending inventory of the top-level warehouse.
Storage costs:
· proportional to the average level of positive stocks for the period and time of existence of the positive stock;
· proportional to the positive balance at the end of the period;
· proportional to the maximum reserve;
· nonlinear functions of one of the above quantities.
Inventory Management.
Types of expenses
The practical implementation of the concept of material flow management is associated with the optimization of total inventories. The criterion for optimizing inventory is the total costs of fulfilling orders and storing materials.
In the system of purchasing and storing materials, expenses are divided into the following groups:
costs of order fulfillment;
direct costs determined by the purchase price;
inventory holding costs;
"costs of scarcity."
Costs for fulfilling an order related to the placement and delivery of an order. These include such cost items as the cost of developing delivery conditions and preparing them for approval; expenses for purchasing advertising catalogues; costs associated with monitoring order fulfillment and reducing the time required for their completion; transportation costs, if the cost of transportation is not included in the price of the goods received; costs for warehousing and receiving orders.
Some of them are fixed in the order and do not depend on the volume, others, for example, transport and storage costs are directly dependent on the size of the order. In general, order fulfillment costs include any type of expense, the amount of which depends on the number of orders being completed. Direct expenses are determined by the price of purchased materials and vary depending on the wholesale discount to the price, which is established when the size of the Order lot increases. Expenses for maintaining inventories are determined by the costs of storing materials and the very fact of inventory availability. This group of expenses includes such cost items as possible interest on capital invested in inventories; costs of warehouse operations and fees for the use or rental of a warehouse; current costs of maintaining warehouses belonging to the production unit; costs associated with the risk of damage and obsolescence of materials, as well as insurance and tax costs. Reducing inventories leads to a reduction in warehouse costs and operating costs for maintaining warehouse facilities. “Scarcity costs” are expenses that arise due to the limited availability of certain material resources during a certain period. This group of costs includes losses of three types:
inventory management model of losses in production associated with the suspension of the production process due to the lack of necessary materials, as well as the replacement of a material with another at a higher price;
the cost of lost sales in case of non-fulfillment of the order if the customer turns to another manufacturer (in such a situation, the cost of shortage is defined as loss of profit);
additional costs arising from waiting for an order to be completed.
Warehouse cost standards. Warehouse costs are calculated in aggregate according to a general norm, which takes into account the ratio of the fixed and variable parts of costs. The warehouse cost rate is
where N is the rate of storage costs; A is the interest rate on capital invested in inventories; B is the standard cost for storing material in a warehouse;
where G is the cost of storing materials in a warehouse for a certain period; D is the average cost of inventory.
4. Wilson model
Mathematical models of inventory management (IM) allow you to find the optimal level of inventory of a certain product, minimizing the total costs of purchasing, placing and delivering an order, storing the product, as well as losses from its shortage. The Wilson model is the simplest KM model and describes the situation of purchasing products from an external supplier, which is characterized by the following assumptions:
· consumption intensity is a priori known and constant;
· the order is delivered from the warehouse where previously produced goods are stored;
· the order delivery time is a known and constant value;
· each order is delivered in one batch;
· the costs of placing an order do not depend on the size of the order;
· the cost of holding a stock is proportional to its size;
· Lack of stock (shortage) is unacceptable.
Wilson model input parameters
) u- intensity (speed) of stock consumption, [units. Comrade / unit t];
) s - costs of storing inventory, [rub./item* unit. t];
) K - costs of ordering, including order processing and delivery, [rub.];
) t d - order delivery time, [units t].
Wilson Model Outputs
) Q - order size, [units. Comrade];
) L - total costs of inventory management per unit of time, [rub./unit t];
) t- delivery period, i.e. time between order submissions or between deliveries, [units t];
) h 0- order point, i.e. amount of stock in the warehouse at which it is necessary to place an order for delivery of the next batch, [units. Comrade].
The cycles of changes in stock levels in the Wilson model are graphically presented in Fig. 1. The maximum quantity of products that is in stock is the same as the order quantity Q.
Fig. 1 Graph of inventory change cycles in the Wilson model
Wilson model formulas
(11.1)
where Q w - optimal order size in the Wilson model;
The cost schedule for KM in the Wilson model is presented in Fig. 2
Rice. 2. Schedule of costs for KM in the Wilson model
Conclusion
Until recently, it was believed that the more reserves a company has, the better. This is true when an enterprise has problems with material and technical supplies, in conditions where it is necessary to create significant safety stocks. However, in modern conditions the problem of shortage is much less acute; enterprises can make a wide variety of investments. The owner of the enterprise is forced to create inventories, since otherwise costs will increase or profits will decrease.
consumer demand inventory management model
Bibliography
1. Blank I.A. Financial management: Training course. - 2nd ed., K.: Elga, Nika-Center, 2008.
Nerush Yu.M. Logistics: Textbook for universities. - 2nd ed. - M.: UNITY-DANA, 2007.
Financial management: theory and practice: Textbook. / Ed. Stoyanova E.S. - 2nd ed., revised. and additional - M.: Perspektiva Publishing House, 2000.
Afanasenko I.D. Workshop on supply logistics: questions and tests / Afanasenko I.D., Borisova V.V. - St. Petersburg. : Publishing house St. Petersburg State University of Economics and Economics, 2011. - pp. 131-134 (188 pp.)
http://ru. wikipedia.org -< свободная энциклопедия>
Logistclub.com.ua -< Типы моделей управления запасами>
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Inventory management models
In general, the theory of inventory management is quite well developed. Numerous order quantity optimization models have been developed. Let's consider several simple deterministic models of inventory management.
Prerequisites: 1) the rate of demand for the product is known and constant; 2) receiving the order instantly; 3) there are no quantity discounts when purchasing large quantities of goods; 4) shortages are eliminated in case of timely ordering. Determine the optimal order size, order lead time, quantity per period and recovery point (i.e., the level of inventory at which a new order must be made) so that costs over the entire planning period are minimal.
Supply volume
insurance T
period between
1st and 2nd deliveries
Let us accept the following notation:
p – price of the product;
Q – size of one order batch;
T – length of the planning period;
d – intensity of demand per unit of time (D – for the entire planning period);
h – storage costs of a unit of goods per unit of time (H – for the entire period);
s – organizational costs for one batch of goods.
Total order costs for the entire period S = (D/Q) ∙ s;
Total storage costs – (Q/2) ∙ H;
The total cost of goods for the entire period is p ∙ D;
Total costs for the entire planning period
To find the optimal order size Q* (the minimum point for the function C(Q)), we find the first derivative of the function C and equate it to zero:
The resulting formula is called the Wilson (Harris) formula or the optimal order size formula.
Optimal number of orders per period: N = D/Q*.
Optimal time between orders (or cycle time between orders): t* = Q*/d or t* = T/N.
Example 1. Annual demand is estimated at 500 units. goods. Average order costs – 10,000 rubles. Annual storage costs – 20,000 rubles. Product price – 50,000 rubles. Determine the optimal batch size, the number of batches ordered, and the total annual costs, provided that the order is completed instantly and the store is open seven days a week.
Initial data: D = 500; s = 10000; p = 50000; H = 20000.
Optimal order size:
Total costs:
The optimal number of ordered batches: N = D/Q* = 500/22.4 = 22.3.
In this case, the number of days between orders: t* = T/N = 365/22.3 = 16.4 days.
A model of the optimal order size for order receipt over time. The basic model assumed that an order for a new batch of goods is fulfilled instantly. A new batch of goods arrives within one day or night. But more often there are situations when it takes some time to complete an order. In this case, the order must be placed in advance, having in stock the stock of goods necessary to meet the demand during the order execution time. The level of inventory at which a new order must be made is called the replenishment point. It is calculated using the following formula:
where R is the inventory replenishment point (order point);
d – intensity of demand per unit of time;
G – order fulfillment time.
Example 2. Let's consider the initial data of example 1, but on the condition that the order lead time is 5 days, and the number of working days in the year is 250. Annual demand is estimated at 500 units. goods. Average order costs – 10,000 rubles. Annual storage costs – 20,000 rubles. Product price – 50,000 rubles.
The optimal batch size, the number of batches ordered and the total annual costs are calculated in the same way as in the first example.
Demand for a product per unit of time d = annual demand divided by the number of working days in a year = 500/250 = 2.
Recovery point (order point):
R = demand per unit time multiplied by lead time = 2∙5 = 10.
Model of the optimal order size, provided that a shortage of goods is allowed (model of accelerated use of inventory). In some cases, if the intensity of consumption of material resources is such that a certain shortage of materials may arise, the inventory management model allows for a shortage of goods and, accordingly, lost profit associated with the shortage. If this deficit is comparable to the costs of maintaining inventories, for example, if storage costs are quite high, or warehouse premises do not allow storing a large batch of goods, or the goods are perishable. In this case, the optimal order size is calculated using the formula:
where P is lost profit for the entire planning period associated with a shortage of a unit of goods.
T 1 is the time during which the stock is available in the warehouse.
q opt t 2 - time that the stock is absent.
q n - initial stock size (usually less
q is lower than the optimal batch size)
Production supply model (delayed supply model). In industrial production, very often the production of goods to replenish inventories occurs simultaneously with the satisfaction of demand for the goods.
q The time period is divided into 2 parts.
q max in period t 1: receipt of mat. resources and their consumption (u>d,
(u-d)t u-dt u-receipt, production)
In this case, the following parameters are added to the initial data of the main model: u – line productivity (product units per year), s – production costs.
Then the rate of replenishment will be equal to u – d. During the execution time of the order at production intensity u, Q units of goods were produced. Then u ∙ t = Q, or t = Q/u. Maximum stock level reached at the end of this time:
The average inventory level is half the maximum:
Total storage costs:
Total order costs: .
Thus, the total costs are calculated using the formula:
We obtain the optimal supply size by equating the derivative to zero:
Hence the optimal order size:
Therefore, the maximum inventory level is:
Delivery model with quantity discounts. Many companies often offer quantity discounts to wholesale buyers for large quantities of goods. In other words, if the batch size of a product Q is less than a certain value Q 1, then the product is sold at a price p 1. When the batch size is Q ≥ Q 1, the product is sold at a price p 2 (p 2 > p 1), etc.
In this case, the total costs are determined as follows:
The function reaches a minimum at point Q*:
IfQ*≥Q 1, therefore, Q* is the optimal batch size if Q*< Q 1 , то сравниваем С(Q*) и С(Q 1). Если С(Q 1) окажется меньше С(Q*) то оптимальным размером партии товара будет Q 1 .
Example 3. Let's consider the initial data of example 1, provided that if the batch size is at least 30 units, the price is reduced to 40,000. Annual demand is estimated at 500 units. goods. Average order costs – 10,000 rubles. Annual storage costs – 20,000 rubles. The original price of the product is 50,000 rubles. Determine the optimal batch size, the number of batches ordered, and the total annual costs, provided that the order is completed instantly and the store is open seven days a week.
The optimal order size, as already discussed in example 1, is equal to:
Since Q<30, то:
250000 +223214+224000=697214.
200000 +166666,7+300000=666666,7.
Therefore, the optimal batch size is 30 units. goods.
The optimal number of ordered batches of goods is N = D/Q* = 500/30 = 16.7.
In this case, the number of days between orders t* = T/N = 365/16.7 = 21.9 days.
Inventory management is an important area of management activity in many enterprises in various industries, both in the production of goods and in the provision of services.
In a market economy, the issues of rational and effective organization of management and control processes over the movement of material and financial flows in an enterprise become especially relevant in order to increase the efficiency of the logistics of the enterprise itself and the marketing of its finished products. This is necessary to optimize inventory levels and their efficient use, reduce their levels, and also minimize working capital invested in these inventories.
The lack of production reserves at an enterprise leads to a disruption in the rhythm of its production, a decrease in labor productivity, overexpenditure of material resources due to forced irrational replacements and an increase in the cost of products. The lack of sales reserves does not allow for the uninterrupted process of shipment of finished products; accordingly, this reduces the volume of its sales, reduces the amount of profit received and the loss of potential clientele of consumers of the products manufactured by the enterprise. At the same time, the presence of unused inventories slows down the turnover of working capital, diverts material resources from circulation and reduces the rate of reproduction and leads to high costs for maintaining the inventories themselves.
The functioning of an enterprise with a relatively high level of reserves will be completely ineffective. In this case, the enterprise has, for certain groups of inventory items, inventories that are greater than their actually necessary values - excess inventories (“staying”). In this regard, it must additionally invest significant working capital in them, which consequently leads to a lack of free financial resources - a decrease in the solvency of the enterprise, the inability to timely acquire the material resources and equipment necessary for production, pay off taxes and wages with the budget and extra-budgetary funds. staff, etc.
In addition, a high level of excess inventory leads to an increase in the company’s costs for maintaining the inventory itself: the need to have large warehouse areas, the need to have increased staff (storekeepers, loaders, accountants) to process and account for materials in the warehouse, these are additional utility bills and taxes on property. All this leads to increased costs for: depreciation deductions due to the creation of additional warehouse premises for storing excess stocks, wage costs for increased accounting and warehouse personnel (storekeepers, loaders processing these stocks), increased utility bills - for lighting, heating of additional warehouses premises, etc. Additional costs increase the cost of finished products produced by an industrial enterprise and reduce its competitiveness in the goods market.
Literature
- Richard Thomas. Quantitative methods for analyzing economic activity/Trans. from English – M.: Publishing House “Delo and Service”, 1999. – 432 p.
- Taha Hamdi. Introduction to Operations Research. M.: Mir, 1985. Description
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