It is to minimize the total costs of their purchase, delivery and warehousing. At the same time, delivery and storage costs demonstrate multidirectional behavior. On the one hand, an increase in the delivery lot leads to a decrease in delivery costs per unit of inventory, and, on the other hand, this leads to an increase in warehouse costs per unit of inventory. To solve this problem Wilson ( English R. H. Wilson) a calculation method was developed optimal delivery batch (English Economic Order Quantity, EOQ), also known as or Wilson's formula.
Assumptions of the EOQ model
The practical application of the EOQ model involves a number of restrictions that must be observed when calculating the optimal delivery lot:
1. The quantity of consumed stocks or purchased goods is known in advance, and their consumption is carried out evenly throughout the entire planning period.
2. The cost of organizing an order and the cost of one unit of inventory remain constant throughout the entire planning period.
3. Delivery time is fixed.
4. Rejected units are replaced instantly.
5. The minimum inventory balance is 0.
Calculation of the optimal delivery batch
The EOQ model is based on the total cost (TC) function, which reflects the costs of purchasing, delivering and holding inventory.
p– purchase price or production cost of a unit of inventory;
D– annual demand for reserves;
K– the cost of organizing the order (loading, unloading, packaging, transportation costs);
Q– volume of the delivery lot.
H– cost of storing 1 unit of inventory for a year (cost of capital, warehouse costs, insurance, etc.).
Having solved the resulting equation with respect to the variable Q, we obtain the optimal delivery quantity (EOQ).
Graphically this can be represented as follows:
In other words, the optimal delivery lot is the volume (Q) at which the value of the total cost (TC) function will be minimal.
Example. The annual demand of a building materials production company for cement is 50,000 tons at a price of 500 USD. per ton. At the same time, the cost of organizing one delivery is 350 USD, and the cost of storing 1 ton of cement for a year is 2 USD. In this case, the size of the optimal delivery lot will be 2958 tons.
In this case, the number of deliveries for the year will be 16.9 (50000/2958). The fractional part of 0.9 means that the last 17th delivery will be completed by 90%, and the remaining 10% will be transferred to the next year.
Substituting the optimal delivery batch into the total cost function, we get 25,008,874 USD.
TC = 500*50000 + 50000*350/2958 + 2*2958/2 = 25008874 c.u.
For any other delivery lot size, the total costs will be higher. For example, for 3000 tons it will be 25008833 USD, and for 2900 tons 25008934 USD.
TC = 500*50000 + 50000*350/3000 + 2*3000/2 = 25008833 c.u.
TC = 500*50000 + 50000*350/2900 + 2*2900/2 = 25008934 c.u.
Graphically, inventory consumption can be represented as follows, provided that their balance at the beginning of the year is equal to the optimal delivery lot.
Taking into account the initial assumptions of the EOQ model about uniform consumption of inventory, the optimal delivery batch will be developed to zero balance, provided that the next batch will be delivered at this moment.
Example No. 1. The store sells Q TVs daily. Overhead costs for supplying a batch of televisions to a store are estimated at S rubles. The cost of storing one TV in a store warehouse is s rub. Determine the optimal volume of a batch of televisions, the optimal average daily costs for storing and replenishing stocks of televisions in a warehouse. What will these costs be equal to for batch sizes n1 and n2 of televisions?
Download the solution.
The decision is made using the online calculator Optimal order size.
Example No. 2. Calculate the optimal order size for all components using Wilson's formula (c1=12;c2=0.3;q=1). Example No. 2(c1=5;c2=0.1;q=150).Example No. 3
(c1=1;c2=5;q=25).Example No. 4
(c1=22;c2=17;q=112).Example No. 5
(c1=150;c2=55;q=6).Example No. 6
(c1=20000;c2=150;q=3000).Example No. 7
(c1=200;c2=150;q=3000).Example No. 8
(c1=200;c2=150;q=3000).Example No. 9
(c1=20000;c2=1800;q=3000).Example No. 10
(c1=90;c2=10;q=73000).Example No. 11
(c1=90;c2=10;q=200).Example No. 12
(c1=9490.91;c2=5;q=113938.92).Example No. 13
(c1=1;c2=1;q=1).Example No. 14
(c1=3;c2=3;q=3).Example No. 15
(c1=1;c2=1;q=1).Example No. 16
(c1=1;c2=1;q=1).Example No. 17
(c1=1500;c2=20;q=30000).Example No. 18
(c1=1500;c2=20;q=3600).Example No. 19
Example No. 3. The intensity of demand is 1000 units of goods per year. Organizational costs are equal to 7 USD, storage costs - 6 USD, unit price - 6 USD. Determine the optimal batch size, number of batches per year, interval between deliveries and total costs. Create a stock chart.
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Example No. 4. Consider all the stages of solving the problem of the optimal size of the purchased batch of goods with the following data: Q = 72, C 0 = 3 thousand rubles / m, C 1 = 400 rubles / m, C 2 = 100 rubles / m.
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Example No. 5. The annual demand for valves costing $4 per unit is 1,000 units. Storage costs are estimated at 10% of the cost of each product. The average order cost is $1.6 per order. There are 270 working days in a year. Determine the size of the economic order. Determine the optimal number of days between orders.
Solution: Download solution
Example No. 6. Grain is delivered to the warehouse in batches of 800 tons. The grain consumption from the warehouse is 200 tons per day. Overhead costs for delivering a batch of grain are 1.5 million rubles. The cost of storing 1 ton of grain for 24 hours is 80 rubles.
You need to determine:
- cycle time, average daily overhead and average daily storage costs;
- the optimal size of the ordered batch and the calculated characteristics of the warehouse in optimal mode;
To make the calculation, we use the basic formulas of the “ideal” warehouse operating model.
1) Cycle duration: T = Q/M = 800/200 = 4 days
average daily overhead costs: K/T = 1500/4 = 375 thousand rubles/day
average daily storage costs: hQ/2 = 80*800/2 = 28 thousand rubles/day
The optimal order size is calculated according to Wilson's formula:
where q 0 – optimal order size, pcs.;
C 1 = 1,500,000, cost of fulfilling one order, rub.;
Q = 200, need for inventory items for a certain period of time (year), pcs.;
C 2 = 80, cost of maintaining a unit of inventory, rub./piece. 
T
Optimal average stock level: t
days
Example No. 7. The annual demand is D units, the cost of placing an order is C 0 rubles/order, the purchase price is C b rubles/unit, the annual cost of storing one unit is a% of its price. Delivery time 6 days, 1 year = 300 working days. Find the optimal order size, costs, re-order level, number of cycles per year, distance between cycles. You can get a b% discount from suppliers if the order size is at least d units. Is it worth taking advantage of the discount? The annual cost of lack of inventory is C d rubles/unit. Compare 2 models: basic and with a deficit (orders are fulfilled).
| Item no. | D | C 0 | Cb | a | b | d | Cd |
| 21 | 400 | 50 | 40 | 20 | 3 | 80 | 10 |
We obtain the solution using a calculator. First we find the cost of storing one unit, C 2 = 40 * 20% = 8 rubles. (introduced into the main model) and at a discount, C 2 = (1-0.03)*40*20% = 7.76 rub. (for discounted model)
1. Calculation of the optimal order size.
The optimal order size is calculated using Wilson's formula:
where q 0 – optimal order size, pcs.;
C 1 = 50, cost of fulfilling one order, rub.;
Q = 400, demand for inventory items for a certain period of time (year), pcs.;
C 2 = 8, cost of maintaining a unit of inventory, rub./piece. 
Optimal average stock level: 
Optimal replenishment frequency: 
(year) or 0.18·300=53 days.
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 after which the order is repeated.
The optimal batch size of goods supplied and, accordingly, the optimal frequency of delivery depend on the following factors:
¾ volume of demand (turnover);
¾ costs of delivery of goods;
¾ costs of storing stock.
The minimum total costs for delivery and storage are chosen as an optimality criterion.
Both delivery 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 order volume is different.
The costs of delivering goods when the order size increases obviously decrease, since transportation is carried out in larger quantities and, therefore, less frequently. The graph of this dependence, which has the shape of a hyperbola, is presented in Fig. 12.1
Rice. 12.1 Dependence of transportation costs on order size
Storage costs increase in direct proportion to the size of the order. This dependence is graphically presented in Fig. 22.2
Rice. 12.2 Dependence of inventory storage costs on order size
By adding both graphs, we obtain a curve reflecting the nature of the dependence of the total costs of transportation and storage on the size of the ordered batch (Fig. 22.3).
Rice. 12.3 Dependence of total costs for storage and transportation on order size (Optimal order size Q*)
The total cost curve has a minimum point at which total costs will be minimal. The abscissa of this point Q* gives the value of the optimal order size.
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.
The costs (R) of maintaining inventories in a certain period are made up of the following elements:
1) the total cost of submitting orders (cost of documentation forms, costs of developing delivery conditions, catalogs, order control, etc.);
2) the price of the ordered component;
3) the cost of storing inventory.
Mathematically, costs can be represented as follows:
R = A*S/Q+ S*C+ I*Q/2, (12.1)
where C is the unit price of the ordered component product.
Q – order size;
A – cost (expenses) of submitting one order, rub.;
S – need for inventory items for a certain period, pcs.;
I – costs (cost) for maintaining a unit of inventory, rub./piece.
The amount of costs must be minimized: RÞmin.
Differentiation by Q gives a formula for calculating the optimal order size (Wilson’s formula, Wilson’s surname is sometimes found):
where Q* – optimal order size, pcs.;
According to cost accounting data, it is known that the cost of submitting one order is 200 rubles, the annual need for a component product is 1550 pcs., the unit price of a component product is 560 rubles, the applicable order size is 50 pcs., the cost of maintaining a component product in a warehouse is 20 % of its price. Determine the optimal order size Q* for a component product and the total costs R.
Solution. Using formula 12.2, we determine the optimal order size based on the available initial data:
To avoid shortages of components, you can round up the optimal order size. Thus, the optimal order size for a component product is 75 pcs.
R = A*S/Q+ S*C+ I*Q/2=200*1550/50+1550*560+0.2*560*50/2=877000 rub.
Batch size- this is the amount of sequentially produced goods without interruptions or switching in the technological process .
What is the importance of determining the optimal batch size?
The optimal batch size leads to a reduction in warehouse losses, interest on property, and reconfiguration costs. Consequently, dividing the volume of goods produced per year into shares leads to a significant reduction in costs.
The best lot size for the manufacturer is counteracted by the best lot size for distribution. With this option, reconfiguration costs become costs for order registration.
What is the peculiarity of mass production?
Serial production is optimal for groups of goods with similar manufacturing processes. After some time, there is a need to reconfigure to produce a different product. The above figure shows that products A, B, C are produced sequentially on the same production line.
A break in the technological process to put a new product into production leads to downtime and the appearance of costs not related to the batch size - constant serial costs. These are the costs of reconfiguring and adjusting production facilities.
As batch size increases, fixed serial costs also increase. In terms of per unit of production, these costs are reduced as the batch size increases, produced without interruptions or reconfiguration of the technological process - digressive cost behavior.
Serial production requires precise coordination of production volume, series and sequence of production of goods. Requirements for various goods must be fulfilled by the enterprise without delay.
What are the options for meeting the annual demand for a product?
A businessman has several options for satisfying the need for a product throughout the year:
1) A single batch equal to the volume of annual demand:
- an increase in proportional serial costs, namely warehouse costs and interest on property;
- single costs for reconfiguration;
- low level of fixed serial costs;
- the likelihood that needs for other types of goods will not be satisfied.
2) A certain number of batches that satisfy the annual need:
- reduction of warehouse and property costs;
- increase in reconfiguration costs.
So, the main task is to find the most effective batch size, at which a unit of goods produced will bring minimal constant and proportional serial costs.
What are the main costs for mass production?
When mass producing goods at an enterprise, costs arise that require more complete consideration:
A) Warehouse costs:
- warehouse expenses - wages, costs of maintaining the functionality of warehouse space;
- calculation interest is an expense that correlates with the volume of property stored in a warehouse.
Both positions can be reduced through a planned reduction in the volume of goods in stock. The lower limit in this case is the safety stock.
Reducing warehouse costs and calculation interest causes resistance from the increasing costs of reconfiguring the technological process and the likelihood of not saturating the need for a certain type of goods. The way out of this situation is to find the optimal batch size.
B) Reconfiguration costs:
- depend on the duration of the reconfiguration process;
- do not depend on the size of the batch;
- in terms of unit of goods decrease with increasing batch size;
- consist of: 1) downtime costs; 2) costs for the necessary technical means and equipment; 3) wages; 4) support costs.
Steps to finding the optimal batch size
To find the most appropriate batch size option you need to:
1. Find the number of games:
where n is the number of batches, M is the annual volume of goods sold, m is the most acceptable batch size, produced without interruption or reconfiguration of the technological process.
2. Calculate the constant serial costs of all series:
where K F is the total fixed costs for reconfiguring all batches, K f is serial costs for one batch.
where K L is the amount of total warehouse costs, K l is the rate of warehouse costs and calculation interest in terms of per unit of goods for the period.
4. Determine total costs (K):
5. Minimizing total costs leads us to the function:
6. The most acceptable batch size (m) is found by reducing the equation to differential form:
7. Setting the condition
8. Solving the equation for m
Consider this with an example. Projected sales next year will be 400,000 units of product T. The amount of fixed serial costs reaches 6,000 DM. Warehouse costs are equal to 20 DM per unit of goods per year. Let's calculate the most acceptable batch size option.
So, cost minimization will be achieved with a batch size of 15,491 pieces. goods.
Are there any assumptions in the formula for calculating the optimal batch size?
Assumptions in the formula for calculating the most acceptable lot size:
- infinity of speed of the production process;
- constant speed of implementation;
- warehouse losses were not taken into account;
- constancy of fixed serial costs;
- directly proportional change in other production costs;
- restrictions on warehouse space were not taken into account.
Is calculating the optimal batch size feasible today?
You should not refuse to calculate the optimal batch size under the pretext of excessive expenditure of labor resources. Of course, there is no need to determine the optimal lot size for each type of product, but for goods A and B these calculations are necessary.
To begin with, the optimal batch size is calculated for A-products, which make up 5 percent of the volume of all products, but give about 75 percent in terms of profitability. Improved planning and regulation of the production of A-products will lead to a significant reduction in costs.
Implementing batch size optimization in combination with ABC analysis will significantly reduce production costs. This effect will be more significant when efficiency increases and warehouse costs decrease.
The widespread and active use of personal computers facilitates the task of finding the optimal batch size.
The EOQ model is based on the total cost (TC) function, which reflects the costs of purchasing, delivering and holding inventory.
p– purchase price or production cost of a unit of inventory;
D– annual demand for reserves;
K– the cost of organizing the order (loading, unloading, packaging, transportation costs);
Q– volume of the delivery lot.
H– cost of storing 1 unit of inventory for a year (cost of capital, warehouse costs, insurance, etc.).
Having solved the resulting equation with respect to the variable Q, we obtain the optimal delivery quantity (EOQ).
Graphically this can be represented as follows:
In other words, the optimal delivery lot is the volume (Q) at which the value of the total cost (TC) function will be minimal.
Example. The annual demand of a building materials production company for cement is 50,000 tons at a price of 500 USD. per ton. At the same time, the cost of organizing one delivery is 350 USD, and the cost of storing 1 ton of cement for a year is 2 USD. In this case, the size of the optimal delivery lot will be 2958 tons.
In this case, the number of deliveries for the year will be 16.9 (50000/2958). The fractional part of 0.9 means that the last 17th delivery will be completed by 90%, and the remaining 10% will be transferred to the next year.
Substituting the optimal delivery batch into the total cost function, we get 25,008,874 USD.
TC = 500*50000 + 50000*350/2958 + 2*2958/2 = 25008874 c.u.
For any other delivery lot size, the total costs will be higher. For example, for 3000 tons it will be 25008833 USD, and for 2900 tons 25008934 USD.
TC = 500*50000 + 50000*350/3000 + 2*3000/2 = 25008833 c.u.
TC = 500*50000 + 50000*350/2900 + 2*2900/2 = 25008934 c.u.
Graphically, inventory consumption can be represented as follows, provided that their balance at the beginning of the year is equal to the optimal delivery lot.
Taking into account the initial assumptions of the EOQ model about uniform consumption of inventory, the optimal delivery batch will be developed to zero balance, provided that the next batch will be delivered at this moment.
67. Operating lever and determining the strength of its influence;
Operating leverage manifests itself in cases where an enterprise has fixed costs, regardless of production (sales) volume.
The production leverage effect arises due to the heterogeneous cost structure of the enterprise. Changes in variable costs are directly proportional to changes in production volume and sales revenue, and fixed costs over a fairly long period of time almost do not respond to changes in production volume. A sharp change in the amount of fixed costs occurs due to a radical restructuring of the organizational structure of the enterprise during periods of mass replacement of fixed assets and quality
"technological leaps".
The strength of the production lever depends on the share of fixed costs in the total costs of the enterprise.
The effect of production leverage is one of the most important indicators of financial risk, since it shows by what percentage the balance sheet profit, as well as the economic profitability of assets, will change when the sales volume or revenue from the sale of products (works, services) changes by one percent.
In practical calculations, to determine the strength of the impact of operating leverage on a specific enterprise, the result from product sales after reimbursement of variable costs (VC), which is often called marginal income, is used.
Operating leverage is always calculated for a certain sales volume. As sales revenue changes, so does its impact. Operating leverage allows you to assess the degree of influence of changes in sales volumes on the size of the organization's future profit. Operating leverage calculations show by what percentage profit will change if sales volume changes by 1%.
Operating leverage effect comes down to the fact that any change in sales revenue (due to a change in volume) leads to an even stronger change in profit. The action of this effect is associated with the disproportionate influence of fixed and variable costs on the result of the financial and economic activities of the enterprise when production volume changes.
Operating leverage force shows the degree of business risk, that is, the risk of loss of profit associated with fluctuations in sales volume. The greater the effect of operating leverage (the greater the share of fixed costs), the greater the business risk.
Thus, modern cost management involves quite diverse approaches to accounting and analysis of costs, profits, and business risk. You have to master these interesting tools to ensure the survival and development of your business.
Production risk is associated with the concept of operational, or production, leverage, and financial risk is associated with the concept of financial leverage.
There are three main measures of operating leverage:
a) the share of fixed production costs in the total amount of costs, or, which is equivalent, the ratio of fixed and variable costs,
b) the ratio of the rate of change in profit before interest and taxes to the rate of change in sales volume in natural units;
c) the ratio of net profit to fixed production costs
Any major improvement in the material and technical base towards an increase in the share of non-current assets is accompanied by an increase in the level of operational leverage and production risk.
Method of controlling the level of fixed expenses- method for calculating the critical sales volume. Its meaning is to calculate at what production volumes in natural units the marginal profit (i.e., the difference between sales revenue and non-financial variable expenses or direct variable expenses) will be equal to the amount of conditionally fixed expenses. This method allows you to find the minimum volume of production that is necessary to cover conditionally fixed costs, i.e. expenses that do not depend on production volumes.
Among the indicators for assessing the level of financial leverage, two are most famous: the ratio of debt to equity capital and the ratio of the rate of change in net profit to the rate of change in earnings before interest and taxes.
As part of the overall financial strategy of a business entity debt management involves a preliminary analysis of their attraction and use, adjustment of the attraction policy or development of a new policy. The analysis involves studying the volumes, dynamics, forms of attraction, types of loans, terms of attraction, lending conditions, composition of creditors, efficiency of use and repayment of borrowed funds. Borrowing Policy includes the determination of: a) the reasons and prerequisites for such attraction; b) the targeted nature of the use of borrowed funds; c) limits (maximum volumes) of attraction; d) conditions (including terms and prices of attraction); e) general composition, structure; f) forms of attraction; g) creditors, etc.
68. Features of planning depreciation charges using the linear method;
