Determining the optimal lot size
Dmitry Ezepov, Purchasing Manager at Midwest © LOGISTIK&system www.logistpro.ru

One of the most difficult tasks for any purchasing manager is the selection of the optimal order size. However, there are very few real tools that facilitate its solution. Of course, there is the Wilson formula, which is presented in the theoretical literature as such a tool, but in practice its use needs to be adjusted.

The author of this article, working in several large trading firms in Minsk, has nowhere seen the Wilson formula applied in practice. Its absence in the arsenal of purchasing managers can in no way be explained by their lack of analytical skills and abilities, since modern companies pay great attention to the qualifications of their employees.

Let's try to find out why "the most common tool in inventory management" is not beyond the scope of scientific publications and textbooks. Below is the well-known Wilson formula, with which it is recommended to calculate the economic order quantity:

where Q is the volume of the purchase lot;

S - the need for materials or finished products for the reporting period;

О - fixed costs associated with the implementation of one order;

C - the cost of storing a unit of inventory for the reporting period.

The essence of this formula is to calculate what the lot sizes should be (all the same) in order to deliver a given volume of goods (that is, the total need for the reporting period) during a given period. In this case, the sum of fixed and variable costs should be minimal.

In the problem being solved, there are at least four initial conditions: 1) a given volume that needs to be delivered to the destination; 2) a given period; 3) identical batch sizes; 4) pre-approved composition of fixed and variable costs. Such a statement of the problem has little in common with the real conditions of doing business. No one knows in advance the capacity and dynamics of the market, so the sizes of ordered lots will always be different. It also makes no sense to set a period for planning purchases, since commercial companies usually exist much longer than the reporting period. The composition of costs is also subject to change due to the influence of many factors.

In other words, the conditions for applying the Wilson formula simply do not exist in reality, or at least are very rare. Do commercial companies need to solve the problem with such initial conditions? It seems not. That is why the "common tool" is implemented only on paper.

CHANGING CONDITIONS

In market conditions, sales activity is unstable, which inevitably affects the supply process. Therefore, both the frequency and the size of purchased lots never coincide with their planned indicators at the beginning of the reporting period. If we focus solely on the plan or long-term forecast (as in the Wilson formula), then one of two situations will inevitably arise: either the overflow of the warehouse or the shortage of products. The result of both will always be a decrease in net income. In the first case - due to an increase in storage costs, in the second - due to a shortage. Therefore, the formula for calculating the optimal order size should be flexible in relation to the market situation, that is, based on the most accurate short-term sales forecast.

The total cost of purchasing and holding inventory consists of the sum of these same costs for each lot purchased. Therefore, minimizing the cost of delivery and storage of each batch individually leads to minimization of the supply process as a whole. And since the calculation of the volume of each batch requires a short-term sales forecast (and not for the entire reporting period), the necessary condition for the flexibility of the formula for calculating the optimal batch size (ORP) in relation to the market situation is met. Such a condition of the problem corresponds both to the goal of a commercial company (cost minimization) and to the real conditions of doing business (variability of market conditions). The definitions of fixed and variable costs for the supply minimization approach from a lot-by-lot perspective are given in the sidebar “Cost Elements” on page 28.

OWN CALCULATION

If we assume that the loan is repaid as the cost of inventory decreases at planned intervals (days, weeks, months, etc.) (1), then using the formula for the sum of members of an arithmetic progression, you can calculate the total cost of storing one batch of inventory (fee for using credit):

where K - the cost of storing stocks;

Q is the volume of the purchase lot;

p is the purchase price of a unit of goods;

t is the time the stock is in the warehouse, which depends on the short-term forecast of the intensity of sales;

r is the interest rate per planned unit of time (day, week, etc.).

Thus, the total cost of delivery and storage of the batch of the order will be:

where Z is the total cost of delivery and storage of the batch.

There is no point in minimizing the absolute value of the cost of delivery and storage of one batch, since it would be cheaper to simply refuse to purchase, so you should move on to the relative indicator of costs per unit of stock:

where z is the cost of replenishing and storing a unit of inventory.

If purchases are made frequently, then the sales period for one batch is short, and the intensity of sales during this time will be relatively constant2. Based on this, the time the stock is in the warehouse is calculated as:

where is a short-term forecast of average sales per planned unit of time (day, week, month, etc.).

The designation is not accidental, since average sales in the past are usually used as a forecast, taking into account various adjustments (stock shortage in the past, the presence of a trend, etc.).

Thus, substituting formula (5) into formula (4), we obtain the objective function of minimizing the cost of delivery and storage of a unit of stock:

Equating the first derivative to zero:

find (ORP) taking into account the short-term sales forecast:

NEW WILSON FORMULA

Formally, from a mathematical point of view, formula (8) is the same Wilson formula (the numerator and denominator are divided by the same value depending on the accepted planning unit of time). And if the intensity of sales does not change, say, during the year, then, replacing the annual need for the product and r - the annual percentage rate, we will get a result that will be identical to the calculation of the EHS. However, from a functional point of view, formula (8) demonstrates a completely different approach to the problem being solved. It takes into account the operational sales forecast, which makes the calculation flexible in relation to the market situation. The remaining parameters of the ORP formula, if necessary, can be quickly adjusted, which is also an indisputable advantage over the classical formula for calculating the EOS.

The company's purchasing policy is also influenced by other, often more significant factors than the intensity of sales (current balances in the company's own warehouse, minimum lot size, delivery conditions, etc.). Therefore, despite the fact that the proposed formula eliminates the main obstacle to calculating the optimal order size, its use can only be an auxiliary tool for effective inventory management.

A highly professional purchasing manager relies on a whole system of statistical indicators in which the PPR formula plays a significant, but far from decisive role. However, the description of such a system of indicators for effective inventory management is a separate topic, which we will cover in the next issues of the journal.

1- In reality, this does not happen, so the cost of holding inventory will be higher. 2- In reality, you need to pay attention not to the frequency of the order, but to the stability of sales within the short-term sales forecast period. Just usually, the shorter the period, the less the seasonality and trend.

Workshop on the topic "Types of logistics and areas of its use"

Determination of the optimal order size

Guidelines

When determining optimal order size (delivery line) as a criterion of optimality choose the minimum total cost of delivery and storage.

where C total - the total cost of transportation and storage;

From storage - the cost of storing the stock;

With transp - transportation costs.

The creation of excess inventory increases the cost of their storage, and their irrational reduction entails an increase in the frequency of smaller shipments, which significantly increases the cost of delivering goods.

The optimal order size (delivery batch) and, accordingly, the optimal delivery frequency depend on the following factors: the volume of demand (turnover); expenses for the delivery of goods; inventory holding costs.

Let us assume that for a certain period of time T, the turnover is Q. The size of one batch ordered and delivered is S. Suppose that a new batch is imported immediately after the previous one has completely ended, then the average stock will be S / 2. The cost of storing goods for a period of time T will be:

,

where M is the cost of storing a unit of stock for period T.

The cost of transportation for the period T is determined by multiplying the number of deliveries (orders) for this period by the cost of supplying one consignment of goods.

where K is the cost of importing one consignment of goods;

Q/S - the number of deliveries for a period of time T.

After a series of transformations, the optimal size of the delivery (order) batch (S opt) is determined. The resulting formula in inventory management theory is known as Wilson's formula .

where Q is the planned volume of trade (annual volume of demand);

K - the costs of fulfilling one order (delivery), including the costs of placing one order (clerical work, administrative expenses, etc.), the costs of delivery and acceptance of a consignment of goods;

M is the cost of storing a unit of production.

Example:

Annual need for a component product - 2000 pcs.

The cost of fulfilling one order is 400 rubles.

The unit price of a component product is 200 rubles.

Determine the optimal order quantity.

The calculation is made for different values ​​of the order quantity (selection is free). According to the above formulas, we calculate the transportation, procurement and storage costs per unit of production. Summing them up, we determine the total costs. The smallest value corresponds to the optimal order quantity - 200 pcs.

order	Number of deliveries	Costs for acquisition per unit prod.	Warehouse costs / units prod.	Total supply costs / unit prod.
S	Q/S
50	40	8	0,5	8,5
100	20	4	1	5
200	10	2	2	4
400	5	1	4	5
100	2	0,4	10	10,4
2000	1	0,2	20	20,2

The result is checked according to the Wilson formula.

EXERCISE 1

The enterprise has an annual need for product A in the amount of 1500 pieces. The price of product A is 300 rubles / unit. The cost of fulfilling one order is 200 rubles. for one order. The company expects a storage cost rate and an interest rate of 20%.

Exercise:

• Make the necessary calculations and complete the table below.
• Present graphically the functions of all costs (transport, storage, general) per unit of production (on the X axis - the order volume, on the Y axis - the cost per unit of production).
• What is the optimal order quantity in this example? Do the calculation using the Wilson formula.

order	Number of deliveries	Costs for acquisition per unit prod.	Warehouse costs / units prod.	Total supply costs / units prod.

TASK 2

The department store plans to sell 2,500 wall clocks a year. The cost of organizing the purchase, negotiations, delivery, acceptance of goods, etc., is 25 c.u. e. per one delivered batch. The cost of storing a unit of production - 0.4 conv. units

Determine the optimal order size.

How many times will it be necessary to import goods during the year?

TASK 3

The monthly turnover for heading A is 40 thousand rubles. The cost of storing a unit of goods for a month is 0.1 thousand rubles. Delivery costs for one consignment of goods - 0.5 thousand rubles.

Determine the optimal delivery lot size.

How many times per month will goods be imported?

What will be the total costs of the enterprise, subject to the optimal size of the delivery lot?

How will the monthly expenses of the enterprise for transportation and storage change if the delivery lot increases or decreases by 25%?

Determining the location of the distribution warehouse

Guidelines

When calculating, transport costs for the delivery of goods from the distribution warehouse to the chain stores are taken into account. The amount of transportation costs depends not only on the number of stores in the distribution network, but also on the location of the distribution warehouse in the service area.

To solve problems of this kind, various methods have been developed, the main of which are: the exhaustive search method, heuristic methods, the method for determining the center of gravity of the physical model of the distribution system.

The task of interest to us is to determine the location of the distribution warehouse. The application of the method has one limitation - the distances between the points of consumption of the material flow and the location of the distribution warehouse are measured in a straight line.

The coordinates of the center of gravity of cargo flows (X warehouse, Y warehouse), that is, the point at which a distribution warehouse can be located, are determined by the formulas:

where ri is the cargo turnover of the i-th consumer;

Xi, Yi – coordinates of the i-th consumer;

n is the number of consumers.

Example:

Using the method of determining the center of gravity, solve the problem of optimizing the placement of a distribution center serving a supermarket chain. There are coordinates of their location in the service area and cargo turnover.

Determine the location of the warehouse.

The initial data and the results of the calculation should be shown graphically.

Name	Location coordinates, km (X;Y)	cargo turnover,
Supermarket №1
Supermarket №2
Supermarket №3
Supermarket №4
Supermarket №5

TASK 4

A wholesale company that sells rolled metal serves industrial enterprises of the city, including 9 regular customers. It is necessary to determine the location of the wholesale base. The service area of ​​the wholesale company is 60 km. The coordinates of the location of consumer enterprises (X, Y) in the serviced territory and data on cargo turnover are presented in the table.

Name enterprises	Location coordinates of the enterprise, km (X; Y)	cargo turnover,
Enterprise No. 1
Enterprise №2
Enterprise No. 3
Enterprise №4
Enterprise No. 5
Enterprise No. 6
Enterprise №7
Enterprise No. 8
Enterprise No. 9

Put on the map of the service area the coordinate axes and coordinates of the points where the enterprises are located.

Determine a point where a wholesale base can be placed and plot it on the map.

TASK 5

There are 7 stores selling building materials on the territory of the district. Using the method of determining the center of gravity of cargo flows, find an approximate location for the location of a warehouse supplying stores.

The initial data for the calculation are given in the table.

Find the coordinates of the point (X warehouse, Y warehouse), where it is recommended to place a distribution warehouse.

Before starting the calculation, put on paper the coordinates of the placement of stores on the X and Y axes. Show the result on the drawing.

Shop number	Location coordinates store, km (X; Y)	cargo turnover,

Comparison of different modes of transport

TASK 6

Rank different types of transport in the context of the main factors influencing their choice.

Put down the rank from 1 to 5, considering "1" as the best value.

transport Factor	Railway		Automobile	Pipeline	Air
delivery
departures
Reliability compliance delivery
Ability transport various
Ability deliver the goods to any point territory
Price transportation

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• 2. Practical part
• Task 1
• Task 2
• Task 3
• Task 4

1. Determination of the optimal order size

volume of demand (turnover);

transportation and procurement costs;

inventory holding costs.

As an optimality criterion, choose the minimum amount of costs of transport and procurement and storage.

Transport and procurement costs decrease with an increase in the size of the order, since the purchases and transportation of goods are carried out in larger lots and, therefore, less frequently.

Storage costs increase in direct proportion to the size of the order.

To solve this problem, it is necessary to minimize the function representing the sum of the costs of transport and procurement and storage, i.e. determine the conditions under which

Common \u003d Save + Transp,

where Сtot is the total cost of transportation and storage; Store - the cost of storing the stock; Stsp - transport and procurement costs.

Suppose that for a certain period of time the turnover is Q. The size of one ordered batch S. Let's say that a new batch is imported after the previous one has completely ended. Then the average value of the stock will be S / 2. Let's introduce the tariff (M) for the storage of goods. It is measured by the proportion of the cost of storage for the period T in the value of the average inventory for the same period.

The cost of storing goods for period T can be calculated using the following formula:

Save = M (S / 2).

The amount of transport and procurement costs for the period T will be determined by the formula:

Store = K (Q/S)

where K - transportation and procurement costs associated with the placement and delivery of one order; Q/S - number of orders for a period of time. Substituting the data into the main function, we get:

So6sch \u003d M (S / 2) + K (Q / S).

The minimum Ctot is at the point where its first derivative with respect to S is equal to zero, and the second derivative is greater than zero.

Let's find the first derivative:

Once the choice of a replenishment system has been made, it is necessary to quantify the size of the ordered batch, as well as the time interval through which the order is repeated.

The optimal batch size of the delivered goods and, accordingly, the optimal frequency of importation depend on the following factors:

volume of demand (turnover);

shipping costs;

inventory holding costs.

As an optimality criterion, a minimum of total costs for delivery and storage is chosen.

Rice. 1. Dual bin inventory control system

The graph of this dependence, which has the form of a hyperbola, is shown in Fig.1.

Both shipping costs and storage costs depend on the size of the order, however, the nature of the dependence of each of these cost items on the volume of the order is different. The cost of delivery of goods with an increase in the size of the order obviously decreases, since shipments are carried out in larger consignments and, therefore, less frequently.

The graph of this dependence, which has the form of a hyperbola, is shown in Fig. 2.

Storage costs increase in direct proportion to the size of the order. This dependence is graphically presented in fig. 3.

Rice. 2. Dependence of transportation costs on the size of the order

Rice. 3. The dependence of the cost of storing stocks on the size of the order

Adding both graphs, we get a curve that reflects the nature of the dependence of the total costs of transportation and storage on the size of the ordered lot (Fig. 4). As you can see, the total cost curve has a minimum point at which the total cost will be minimal. The abscissa of this point Sopt gives the value of the optimal order size.

Rice. 4. Dependence of the total cost of storage and transportation on the size of the order. Optimal order size S opt

Thus, the problem of determining the optimal order size, along with the graphical method, can also be solved analytically. To do this, you need to find the equation of the total curve, differentiate it and equate the second derivative to zero.

As a result, we obtain a formula known in the theory of inventory management as the Wilson formula, which allows us to calculate the optimal order size:

where Sopt - the optimal size of the ordered lot;

O - turnover value;

St - the costs associated with delivery;

Сх - costs associated with storage.

The resulting formula, which allows you to calculate the optimal order size, is known in the theory of inventory management as the Wilson formula.

The task of determining the optimal order size can be solved graphically and analytically. Consider the analytical method.

"To do this, it is necessary to minimize the function representing the sum of transportation and procurement costs and storage costs from the size of the order, i.e. to determine the conditions under which:

With total = From storage + transp. Min

where, C total. - the total cost of transportation and storage of the stock;

From storage - the cost of holding stock;

With transp. - transportation and procurement costs.

Suppose that for a certain period of time, the turnover is Q. The size of one batch ordered and delivered is S. Let's say that a new batch is imported after the previous one has completely ended. Then the average value of the stock will be S / 2.

Let's introduce the size of the tariff M for stock storage. M is measured by the share that storage costs for period T make up in the cost of the average stock for the same period. For example, if M = 0.1, then this means that the cost of holding the stock for the period amounted to 10% of the cost of the average stock for the same period. We can also say that the cost of storing a unit of goods during the period amounted to 10 5 of its value.

Now you can calculate how much it will cost to store goods for period T:

From storage = M x S/2

The amount of transportation and procurement costs for period T is determined by multiplying the number of orders for this period by the amount of costs associated with the placement and delivery of one order.

With transp. = K x Q/S

where

K - transportation and procurement costs associated with the placement and delivery of one order; Q/S - the number of deliveries for a period of time.

Having performed a number of transformations, we will find the optimal size of a one-time delivered lot (S opt.), at which the total cost of storage and delivery will be minimal.

With total = M x S/2 + K x Q/S

Next, we find the value of S, which turns the derivative of the objective function to zero, from which a formula is derived that allows you to calculate the optimal order size, known in inventory management theory as the Wilson formula.

Consider an example of calculating the optimal size of the ordered batch. We take the following values ​​as initial data. The cost of a unit of goods is 40 rubles. (0.04 thousand rubles).

Monthly warehouse turnover for this item: Q = 500 units/month. or Q = 20 thousand rubles. /month The share of costs for storing goods is 10% of its value, i.e. M = 0.1.

Transport and procurement costs associated with the placement and delivery of one order: K = 0.25 thousand rubles.

Then the optimal size of the imported lot will be:

Obviously, it is advisable to import goods twice a month:

20 thousand rubles / 10 thousand rubles = 2 times.

In this case, transport and procurement costs and storage costs:

With total \u003d 0.1 H 10/2 + 0.25 H 20/10 \u003d 1 thousand rubles.

Ignoring the results obtained will lead to inflated costs.

An error in determining the volume of the ordered batch by 20% in our case will increase the monthly costs of the enterprise for transportation and storage by 2%. This is commensurate with the deposit rate.

In other words, this mistake is tantamount to the unacceptable behavior of a financier who kept money without movement for a month and did not allow them to "work" on a deposit."

The reorder point is determined by the formula:

Tz \u003d Rz x Tc + Zr

where, Pz is the average consumption of goods per unit of order duration;

Tc - the duration of the order cycle (the time interval between placing an order and receiving it);

Зр - the size of the reserve (guarantee) stock.

Consider an example of calculating the reorder point.

The company buys cotton fabric from a supplier. The annual volume of fabric demand is 8,200 m. We assume that the annual demand is equal to the volume of purchases. At the enterprise, the fabric is consumed evenly, and a reserve supply of fabric equal to 150 m is required. (Assume that there are 50 weeks in a year).

The average consumption of fabric per unit of order duration will be:

Rz = 8200 m. / 50 weeks = 164 m.

The reorder point will be equal to:

Tz \u003d 164 m. X 1 week. + 150 m. = 314 m.

This means that when the level of stock of fabric in the warehouse reaches 314 m, the next order should be made to the supplier.

It is worth noting that many enterprises have accessible and very important information that can be used in the control of inventories. Grouping of material costs should be carried out for all types of inventory in order to identify the most significant among them.

As a result of ranking by the cost of certain types of raw materials and materials, a specific group can be distinguished among them, the control over the state of which is of paramount importance for managing the working capital of an enterprise. For the most significant and expensive types of raw materials, it is advisable to determine the most rational order size and set the value of the reserve (insurance) stock.

It is necessary to compare the savings that can be obtained by the enterprise due to the optimal order size, with the additional transportation costs that arise when implementing this proposal.

For example, the daily supply of raw materials and materials may require the maintenance of a significant fleet of trucks. Transportation and operating costs can exceed the savings that can be obtained by optimizing the size of stocks.

transportation size order commodity

At the same time, it is possible to create a consignment warehouse of used raw materials near the enterprise.

In the management of stocks of products in a warehouse, the same techniques can be used as in the management of goods and materials, in particular the ABC method.

With the help of the methods presented above, as well as on the basis of an analysis of consumer requests and production capabilities, the most rational schedule for the receipt of finished products at the warehouse and the size of the safety stock can be determined.

The costs of storage, accounting and other costs associated with ensuring the rhythm of the supply of manufactured products must be weighed against the benefits that come from the uninterrupted supply of traditional buyers and the fulfillment of periodic urgent orders.

2. Practical part

Task 1

Plot an ABC analysis curve for the following set:

Let's sort all the objects in the table according to the share of the object in the total contribution, while calculating the share of the object on an accrual basis. Let's divide all the materials into groups as follows: objects belong to group A until the share of the cumulative total reaches 80%; in group B - 95%, the rest of the objects will be assigned to group C.

ABC analysis

primary list	ordered list
object number	Object Contribution		item number	Object Contribution	The share of the object in the total contribution,%	Share of cumulative total, %

Task 2

The annual demand is D units, the cost of placing an order, rubles/order, the purchase price, C rubles/unit, the annual cost of storing one unit is a % of its purchase price. Delivery time 6 days, 300 working days a year. Find the optimal order level, costs, reorder level, number of cycles per year, distance between cycles. Compare two models: the main one and the one with a deficit (bids are fulfilled).

1) The basic model of inventory management.

Optimal order level:

Thus, during each order cycle, it is necessary to place an order for 86 units of products.

The annual total variable cost of orders is determined according to the formula:

The sales volume for 6 days of delivery will be:

Reorder level 16 units.

That is, new stock is supplied when the stock level is 16 units. Number of cycles per year

Distance between cycles

2) Consider a model with a deficit (orders are fulfilled).

Planned deficit

Optimal order level:

In this situation, it is necessary to submit orders of 116 units.

Maximum deficit:

The total variable cost per year is defined as follows:

Compared to the basic model, the savings are

1396.42-1073.26 = 323.16 rubles per year.

Thus, if we use the deficit planning model, then we can achieve savings in the total variable cost of inventory, equal to 323.16 rubles per year.

Task 3

The table shows the coordinates of eight consumers, the monthly turnover of each of them is indicated. Find the coordinates of the supply center.

consumer number	X coordinate	Y coordinate	Cargo turnover

We will solve the problem of choosing the location of the supply center for a distribution system that includes one supply center. The main factor influencing the choice of the location of the supply center is the size of the cargo turnover of each of the eight consumers. Costs can be minimized by placing a supply center in the vicinity of the center of gravity of cargo flows.

Coordinates of the center of gravity of cargo flows (X center, Y center), i.e. - the points where the distribution warehouse can be located are determined by the formulas:

where G i g freight turnover of the i-th consumer;

Xi, Yj are the coordinates of the i-th consumer.

The point of the territory, which provides a minimum of transport work for delivery, in the general case does not coincide with the found center of gravity, but, as a rule, is located somewhere nearby. To select an acceptable location for the supply center will allow the subsequent analysis of possible locations in the vicinity of the found center of gravity. At the same time, it is necessary to assess the transport accessibility of the area, the size and configuration of the possible site, as well as the plans of local authorities regarding the intended territory.

Let's make a drawing for the task.

Find the coordinates of the center of gravity of cargo flows.

X center = 21.7

Y center = 17

Let's put a point with such coordinates on the drawing.

Task 4

Select a supplier if the dynamics of prices for the supplied goods is known. The data are given in the table.

Dynamics of prices for supplied goods

Price growth rate for the i-th type of goods from the j-th supplier

where C ij2 is the price of the i-th product from the j-th supplier in the second quarter;

C ij1 - the price of the i-th product from the j-th supplier in the first quarter.

Under the conditions of this problem, for the first supplier for goods A, B and C, respectively

For the second supplier for goods A, B and C, respectively

The share of the i-th product in the total supply of the j-th supplier

where S ij is the amount for which the goods of the i-th type are supplied by the j-th supplier;

G ij - the volume of supply of goods of the i-th type by the j-th supplier;

УS ij - the amount for which all goods are supplied by the j-th supplier.

The share of goods of type A in the total supply of the first supplier

The share of goods of type B in the total supply of the first supplier

The share of goods of type C in the total supply of the first supplier

The share of goods of type A in the total supply of the second supplier

The share of goods of type B in the total supply of the second supplier

The share of goods of type C in the total supply of the second supplier

The weighted average price growth rate of the j-th supplier

Then the weighted average price growth rate of the first supplier.

The weighted average price growth rate of the second supplier.

The growth rate of the price reflects the increase in the negative characteristics of the supplier, so preference should obviously be given to one of them, whose rating is lower. In this example, supplier #1 should be preferred.

List of used literature

1. Anikin, B.A. Logistics: Textbook [Text] / B.A. Anikin. M.: INFRA - M, 2008.

2. Gadzhinsky, A.M. Logistics: Textbook [Text] / A.M. Gadzhinsky - M.: "Dashkov and Co", 2008. - 484 p.

3. Nerush, Yu.M. Logistics: Textbook [Text] / Yu.M. Nerush. - M.: Prospekt, TK Velby, 2008. - 520 p.

4. Workshop on logistics / Ed. B.A. Anikina. - M.: INFRA - M, 2007. - 280 p.

5. Chudakov, A.D. Logistics [Text]: Textbook / A.D. Chudakov. - M.: RDL Publishing House, 2003. - 480 p.

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test, added 01/18/2010


Inventory management as a basic logistics function of commodity supply. Comparative characteristics of basic inventory management systems. ABC-analysis of materials purchased by the enterprise for production and determination of the optimal order size.

Consider the operation of a warehouse that stores inventories spent on supplying consumers. The work of a real warehouse is accompanied by many deviations from the ideal regime: a batch of one volume is ordered, but a batch with a different volume arrives; according to the plan, the party should arrive in two weeks, and it arrived in 10 days; at the unloading rate of one day, the unloading of the batch lasted three days, etc. It is practically impossible to take into account all these deviations, therefore, the following assumptions are usually made when modeling the work of a warehouse.

• 1. The rate of consumption of stocks from the warehouse is a constant value, which we denote M(units of commodity stocks per unit of time); in accordance with this, the graph of changes in the value of reserves in terms of spending is a straight line segment.
• 2. Batch volume replenishment Q is a constant, so the inventory control system is a system with a fixed order quantity.
• 3. The unloading time of the arriving batch of replenishment is short, we will assume it to be zero.
• 4. The time from the decision to replenish to the arrival of the ordered batch is a constant value Δ t, so we can assume that the ordered batch arrives instantly: if you need it to arrive exactly at a certain moment, then it should be ordered at a time on At previously.
• 5. There is no systematic accumulation or overrun of stocks in the warehouse. If through T denote the time between two consecutive deliveries, then the following equality is obligatory: Q = MT. From what has been said, it follows that the work of the warehouse occurs in the same cycles of duration T, and during the cycle, the stock value changes from the maximum level S to the minimum level s.
• 6. Finally, we will consider it obligatory to fulfill the requirement that the absence of stocks in the warehouse be unacceptable, i.e. the inequality s > 0. From the point of view of reducing the costs of the warehouse for storage, it follows that s= 0 and, therefore, S = Q.

The final graph of the ideal operation of the warehouse in the form of a dependence of the value of stocks at from time t will have the form shown in Fig. 12.3.

Earlier it was noted that the efficiency of the warehouse is estimated by its costs for replenishment of stocks and their storage. Costs that do not depend on the size of the lot are called overhead. This includes postal and telegraph expenses, travel expenses, some part of transportation expenses, etc. Overhead expenses will be denoted by TO. The costs of storing stocks will be considered proportional to the value of the stored stocks and the time of their storage. The cost of storing one unit of inventory for one unit of time is called the amount of unit storage costs; we will denote them by h.

Rice. 12.3.

With a changing amount of stored stocks, the cost of storage for some time T obtained by multiplying the value h and T on the average value of stocks during this time T. Thus, the cost of the warehouse for the time T at the size of the replenishment batch Q in the case of an ideal warehouse operation mode, shown in Fig. 12.3 are equal

After dividing this function by a constant value T subject to equality Q = MT we obtain an expression for the value of the cost of replenishment and storage of stocks per unit of time:

This will be the objective function, the minimization of which will allow you to specify the optimal mode of operation of the warehouse.

Find the volume of the ordered batch Q, which minimizes the function of average warehouse costs per unit of time, i.e. function . On practice Q often take discrete values, in particular due to the use of vehicles of a certain carrying capacity; in this case, the optimal value Q find by enumeration of valid values Q. We will assume that the restrictions on the accepted values Q no, then the problem for the minimum of a function (it is easy to show that it is convex, Fig. 12.4 can be solved by methods of differential calculus:

where we find the minimum point:

This formula is called Wilson's formula(named after the English scientist-economyist, who received it in the 20s of the last century).

The optimal lot size, calculated by the Wilson formula, has a characteristic property: lot size Q is optimal if and only if the cycle time storage cost T equal to overhead TO.

Rice. 12.4.

Indeed, if, then storage costs

per cycle are equal

If the storage costs per cycle are equal to the overhead costs, i.e.

We illustrate the characteristic property of the optimal lot size graphically.

On fig. 12.4 it can be seen that the minimum value of the function is achieved at that value Q, at which the values ​​of the other two functions that make it up are equal.

Using the Wilson formula (12.18), in the assumptions made earlier about the ideal warehouse operation, a number of calculated characteristics of the warehouse operation in the optimal mode can be obtained.

Optimal average stock level:

Optimal restocking frequency:

Optimal average cost of inventory holding per unit of time:

(12.21)

Example

Let's consider a typical task. 1500 tons of cement is delivered to the warehouse on a barge. Consumers take 50 tons of cement from the warehouse per day. Overhead costs for the delivery of a batch of cement are 2 thousand rubles. The cost of storing 1 ton of cement during the day is 0.1 rubles. It is required to determine: 1) the duration of the cycle, the average daily overhead and the average daily storage costs; 2) the same values ​​for lot sizes of 500 tons and 3000 tons; 3) the optimal size of the ordered batch and the calculated characteristics of the warehouse in the optimal mode.

Warehouse operation parameters:

1. Cycle duration ( T):

Average daily overhead:

Average daily storage costs:

2. We will carry out similar calculations for m:

3. Find the optimal size of the ordered lot according to the Wilson formula (12.18):

The optimal average stock level is calculated by the formula (12.19):

The optimal frequency of replenishment of stocks is calculated by the formula (12.20):

Formula (12.21) is used to calculate the optimal average cost of holding inventory per unit of time.

Definition: The optimal order lot is the amount of product that must be ordered to optimally satisfy the current level of demand.

The size of the optimal order lot depends on a large number of factors:

• Demand for goods (demand for goods among buyers);
• order period;
• Remaining stock;
• Insurance stock;
• Frequency of deliveries;
• Minimum lot of the order;
• Multiplicity of deliveries;
• Service level, %;
• Expiration date (when ordering, you need to take into account the risk of delay of the goods)

In general, the optimal order lot is the difference between the optimal stock for the delivery period (how much you need to store the goods to meet demand) and the balance of the goods (what will be the balance of the goods on the delivery date).

The main factor influencing the volume of the order is the demand for the goods.

Model of the optimal order lot on the example in the Forecast NOW!

For example, a product was sold in quantities of 50 pieces. per week, but due to the increase in prices, the demand for it decreased to 40 pcs. in Week. Accordingly, the optimal inventory and the optimal order lot can be reduced based on these changes.

Forecast NOW! allows you to take into account changes in demand and many other factors affecting the order. In this case, all formulas are calculated automatically, you only need to check and change the necessary parameters.

Let's take a step-by-step look at how you can take into account the factors that affect the model of the optimal order lot in the Forecast NOW! :

Step 1. We go to the "Parameters" tab and check the parameters we need for the goods for the order or change individual parameter indicators.

The Options tab has 6 sections:

• Main settings,
• Delivery features,
• Delivery schedule,
• forecasting,
• seasonality,
• Trend.

Step 2 We add the necessary goods, the parameters of which we want to check or change.

The green arrow in the figure below indicates the addition of the product. Further, the parameter - expiration date is marked with a red arrow. This parameter, as well as others, can be changed if necessary. For example, for the test item "Breakfast Cookies", we will set the expiration date to 7 days (red arrow). If this parameter value needs to be entered for all products added to the table, then you must click on the "Apply to all" button (blue arrow).

With a set expiration date, the program will not order more goods than the optimal demand for this period (in the example for "Biscuits for breakfast" - 7 days)

Step 3 Go to the next tab - "Peculiarities of deliveries". In the same way, we look through the parameters and note what needs to be taken into account in calculating the batch size of the optimal order.

Here you can, for example, set the supplier's restrictions on the multiplicity (if the goods can only be ordered in batches of a certain size) and the minimum order lot.

For seasonal goods, it is necessary to set the parameters in the "Seasonality" tab in the calculation of the optimal batch of the order.

Seasonality is best calculated for a group of products with similar seasonality:

If the demand for goods changes predictably, but is not related to seasonality, then you need to mark the parameters in the "Forecast" and "Trend" tabs.

Let's check how changing the parameters affects the size of the optimal order. To begin with, we will not take into account any additional parameters, go to the "Order" tab and create an order.

Select the desired products and click "Place an order".

There are three products in the order: Marmalade "Little Princess", Zephyr and Waffles. The program calculated that at the moment it is necessary to order only chocolate wafers in the amount of 29 units. Now let's go to the "Parameters" tab and see what these items are taken into account in the calculation and what needs to be taken into account.

In the main parameters, we put down the expiration date of the products (red arrow) and add this parameter to the calculated ones by checking the box above the desired column and clicking on the "Apply to all" button.

Go to the next tab "Features of deliveries". Let's pay attention to such parameters as the minimum stock, which is necessary in order to limit the system, and even in the absence of demand for a product, maintain a stock and multiplicity for it.

Now let's see how the optimal order size for these products changes based on the new parameters. To do this, go to the "Order" tab and again create an order.

The volume of the order has changed. Order options have changed. Before the introduction of new parameters, it was required to order only Wafers in the amount of 29 units, now the order includes Wafers - 28 units (The order has been rounded up). and Zephyr in the amount of 35 pack.

Automatic calculation of the optimal order, taking into account all the necessary parameters, ensures that there is no excess of goods in the warehouse, and demand will always be maintained at the required level. By adjusting the different conditions of supply, demand and storage of goods, you can automatically adjust the size of the optimal order lot.