Chapter 18. Defining Inventory Size
18.1. Determining the optimal size of the current stock
Consider the procedure for determining the optimal size of the current stock of goods of one nomenclature. The nature of the current stock is reflected in its name current & quot ;. Indeed, ensuring the uninterrupted operation of a production or trading enterprise in periods between the next deliveries, this category of stock as it emerges from the warehouse, changing its value at each expense. Speaking about the size of the current stock, as a rule, they mean its maximum, average or minimum value. In the event that a new batch of consumable products arrives exactly at the end of the previous one, the minimum value of the current stock will be zero, and the average value will be half the maximum. Obviously, with this mode of supply, the maximum current stock will be equal to the size of the delivered consignment. In Fig. 82 shows how for four quarters (OX axis), as the flow and delivery, the size of the current stock (OY axis) changes from 1800 to 0 units.
The optimum size of the current stock will be considered the optimal value of its average value (3 tek.sr ), equal to half the ordered and delivered consignment. Thus, the task of finding the optimal stock size is transformed into the task of finding the optimal size of the ordered consignment of goods.
The criterion of the optimum is the minimum total costs for the period associated with the creation and content of the stock.
Fig. 82. Changing the size of the current stock
Inventory management systems use two categories of costs: unit costs and costs for the analyzed period.
Specific costs are:
o costs are specific to the creation of reserves, i.e. costs for placing and receiving one order; measured in rubles and denoted by the symbol K;
o costs of storage of stocks, i.e. the cost of storing a unit of stock per unit of time; are denoted by the symbol M and have the dimension or if the stock is measured in monetary units.
In inventory management systems, a year is usually taken as the unit of time in determining the unit cost of storage. Thus, the value of M shows how much of the cost of a unit of production is the cost of its storage during the year. For example, if the purchase price of the product is 600 rubles, and , this means that storing one product within a year costs the company 180 rubles.
Expenses for a period are:
o the cost of placing and receiving all orders made for the period (C of the order );
o The cost of storing the average stock during the period (C storage ).
Total costs for the period will be denoted by the symbol C total . The costs for the period are of the dimension , for example
In addition to unit costs and costs over a period, the inventory management system is also characterized by the following parameters:
Q  the demand for the goods for the analyzed period,
P  the purchase price of a unit of goods,
T  duration of the analyzed period,
S  the size of the ordered consignment of goods, pcs.
tek.sr  the stock is the current average, pcs.
N  the number of orders per period (the frequency of delivery),
t  the gap between deliveries,
The objective function can be represented in the following form:
Uncontrolled parameters in the objective function are obviously the unit costs for stock creation (K) and unit costs for storage of the stock (M), as well as the demand for the goods for the analyzed period (Q), the purchase price of the unit of goods (P) and the duration of the analyzed item period (T).
The remaining parameters, closely related to each other, are manageable within the framework of the problem under consideration; the manager can change them at his own discretion, getting those or other economic results.
It should be borne in mind that the optimization problem can be solved if the following conditions are satisfied:
o A new batch of goods is delivered at the time of the full flow of the current stock;
o The demand for materials for the period (demand for the goods) is known and constant (Q = const);
o The unit costs of creating inventories are known and constant (K = const), i.e. the cost of placing and receiving one order does not depend on the size of the order;
o The specific costs of stock storage are known and constant (M = const);
o the purchase price of the goods is constant and does not depend on the size of the purchased lot (P = const).
The criterion of the optimum, as already noted, is the minimum amount of total costs for the period. In this regard, we will present the objective function (C total ) as the sum of the costs for the creation and storage of stocks for the period and find the value of the order size (S onm ), costs will be minimal.
To solve the problem, we find the dependencies С Zак and С storage from S.
Dependence of costs for the period on the creation of inventories from the size of the order.
The number of orders for a period (N) is related to the demand for the goods for the corresponding period (Q) and the order size (S) by the following ratio:
The costs for the period associated with the placement and receipt of orders, calculated by the formula
or
Changing the order size (S) entails a change in the number of orders and a corresponding change in the costs for the period associated with placing and receiving orders (C order ) · C order from S, which has the form of a hyperbola, is shown in Fig. 83.
Fig. 83. Dependence of costs for the period associated with the placement and receipt of orders, on the size of the order
Changing the order size also causes a change in the average value of the current stock (3 tex ) and the corresponding change in costs for the storage period (C store ). For example, if we do not order 1800 units in our example. (see Figure 82), and for 7200 units, the number of orders for the year will decrease from four to one, and the average stock will increase from 900 to 3600 units. Accordingly, the annual storage costs will also increase 4fold.
Calculation of costs for the period for storage of the stock is performed using the formula
The substitution of the dimensions of the values included in the formula, which the reader is asked to perform independently, will make it possible to visualize the dependence more clearly and to make sure of the correctness of the formula.
Since the average value of the current stock is equal to half the order, i.e.
then we can write that
The graph of C storage from S, which is, as a rule, linear, is shown in Fig. 84.
As you can see, the change in the size of the order entails a change in the costs for the period both for the creation of the stock and for its storage. However, the nature of the dependence of each of these items of expenditure on the size of the order is different. The total costs for the period when the stock is created with the increase in the size of the order are obviously reduced, since purchases are made in larger lots and, therefore, less often. The storage costs for the period grow in direct proportion to the size of the order.
Graphically, the dependence of the total costs for the period associated with the placement and receipt of orders, as well as the storage of the stock, on the size of the order is shown in Fig. 85.
Fig. 84. Dependence of costs for the period associated with the storage of stocks, on the size of the order
Define the size of the order (5), at which the total costs are minimized:
or
Fig. 85. Dependence of the total costs for the period associated with the placement and receipt of orders, as well as the storage of the stock, on the size of the order
As you can see, in this equation there are two controllable parameters: S  independent variable and C general  dependent variable. The remaining parameters are constant coefficients. In simplified form, equation (2) takes the form, where
The total cost function has a minimum at the point at which its first derivative with respect to S is zero, and the second derivative is greater than zero. Let's find the first derivative for C total :
Let's find the value of S, which turns the derivative of the objective function to zero:
From
(3)
The check shows that the second derivative is greater than zero, hence the resulting value of S provides the minimum of the total cost of creating the stock and storing it.
Substituting the values of a and b into expression (3), we obtain a formula that allows us to calculate the optimal order size, which in the theory of stock control is known as Wilson's formula:
(4)
Consider the procedure for calculating the optimal values of the remaining control parameters.
The optimal amount of costs for the period T to create a stock (C wholesale order ):
(5)
The optimal amount of costs for the period T for storage of the stock (C wholesale ).
(6)
The minimum (the same optimal) size of the total cost for the period for creating and storing the stock C min.arch :
It follows from formulas (5) and (6) that at the point of minimum of total costs, the cost of creating a stock for a period is equal to the cost of storing the stock (for the same period). This leads to a conclusion that is of significant practical importance: if during the period the costs associated with creating the stock were equal to the costs of storing them, then the goods were purchased as optimal, i.e. correct in size by lots.
The optimal size of the average current stock value
The optimal number of orders per period (delivery frequency)
Optimal period between deliveries
The obtained period value between deliveries has an annual measurement:
i.e. The gap between orders is measured in years. In practice, the period between deliveries is more convenient to measure in months or days. The calculation formula has the form
or
We assign specific numerical values to unmanaged parameters (Table 20) in order to be able to show the order of determining the optimal size of controlled parameters by example.
Table 20
Data for calculating the optimal stock size
Indicator title Product demand for the analyzed period 
Notation Q 
Unit. dimensions

Value 1800 
Specific Costs for Inventory Creation 
to 
rubles 
500 
Specific costs but stock holding 
m 

0.3 
Duration of the analyzed period in the soda dimension 
t 

0.25 (one quarter) 
The purchase price of a unit is Rapa 
P 

600 
The optimal size of the ordered batch is:
The optimal cost for the period T to create a stock
The optimal amount of costs for the period T for storage of the reserve is calculated using the same formula:
The minimum (the same optimal) size of total costs for the period of creating and storing the stock
The optimal size of the average value of the current stock
The optimal number of orders per period (delivery frequency)
The optimal period between orders (calculate this parameter in days)
The formulas and calculations above are based on the assumption that the demand for the analyzed period, as well as the order size, are calculated in kind (in pieces). The calculations will not undergo significant changes if we go over to the monetary expression of the demand and the order. Let's give an example of calculation, proceeding from the fact that the demand for a period in money terms (Q) is:
Since the size of the order, measured in monetary units (S), is
the formula for calculating costs for the storage period (formula 1) takes the form
Making the appropriate changes to the formula (1) and carrying out the subsequent transformations (see formulas 2, 3 and 4), we obtain a model for calculating the optimal order size in monetary terms:
In our example, the optimal order size in monetary terms is:
The formulas for calculating the remaining control parameters remain unchanged.
We have defined optimal supply conditions for our example. Ignoring the results will lead to overestimated costs. For example, if you import more than one product
once every ten days for 200 pcs., and once a month for 600 pcs. The total costs for the period of creation and storage of the stock will be:
that exceeds the quarterly costs (67%) that correspond to the optimal supply mode.
thematic pictures
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