Model of the optimal (economical) order size
One of the most popular tools for optimizing the level of stock in a warehouse is to calculate the optimal order size that replenishes the stock to the maximum level , which is understood to be the level economically feasible to meet the current consumer demand. The criterion of optimization is the minimum of total (cumulative) costs (Figure 5.15), corresponding to the optimal size of the order.
Fig. 5.15. Model of the optimal (economical) order size
In the classical model, only the costs of replenishing the stock and the costs of maintaining the stock in the warehouse are considered. The remaining components are neglected. It is assumed that the cost of replenishment decreases with the increase in the lot size due to the decrease in the number of placed orders from the supplier with the same demand for the same period of time.
On the contrary, the costs of maintaining the stock are directly proportional to the size of the order, when the volume of purchased supplies increases proportionally and the cost of their storage increases proportionally. Although this situation is more typical for the processing of stocks in a public warehouse (the forces of a logistics intermediary are outsourced). However, for leased and own warehouses, a variable part of the cost of maintaining the stock will have a similar dependence, albeit not so pronounced.
In the classical form, the formulas for calculating the optimal order size and the total costs corresponding to it have the form (notation - in Table 5.15):
(5.22)
(5.23)
The frequency of placing an order with the supplier (or the time between adjacent orders) is determined by the formula
(5.24)
Taking into account the selling price of the stock and transportation costs, the total cost formula takes the form
(5.25) Table 5.15
Components of the formula for calculating the total costs associated with procurement activities
Components of Total Costs |
Elements of components |
C • S - capital purchase costs |
C - the cost of the unit of stock S - the total need for stock for the planned period of time |
- transactional and administrative and management costs of the supply service (to replenish the stock) |
Q * - the optimal order size A - the cost of replenishing the stock, correlated to one order |
- the cost of maintaining a stock in the warehouse |
I - the cost of maintaining a unit of stock in a warehouse |
- transport costs |
E - transport costs (freight charges) per vehicle F - the freight capacity of a single vehicle |
- the time between adjacent orders |
N - the number of working days for placing/issuing an order |
In the practice of domestic and foreign companies, various modifications of the classical model of the optimal order size are applied, which additionally take into account some components that are important in each particular case. Let's consider some of the most well-known modifications.
1. Model taking into account losses from immobilization of funds invested in inventories relevant for the acquisition of capital-intensive reserves for long periods of demand, when distractions from the enterprise's turnover of significant amounts of financial resources are required:
(5.26)
where c is the price of a stock unit stored in a warehouse; i - alternative costs.
Therefore, as the i , you can use the average rate on deposits of commercial bikes or the predicted inflation rate, expressed as a percentage for the period under consideration. At the same time, losses from immobilization of funds invested in inventories ( C i), as a separate component in the calculation of total costs can be determined by the following formula:
(5.27)
It is worth noting that losses from the immobilization of financial resources refer to implicit (or mild) costs, which are neglected in the conduct of accounting and tax accounting due to the ambiguity of the mechanism for their formation.
Example
1. The magnitude of financial losses from investments in inventories, the average annual level of which in the warehouse was 2000 tons, with an average cost of 500 rubles/ton and forecasted inflation for the year of 12%:
2. The annual requirement of the enterprise in the reserve is 1200 tons, while the cost of processing one order is 1000 rubles, the cost of maintaining 1 ton of stock in the warehouse is 120 rubles per year, the number of working days per goal is 300. Then according to the classical formula of the optimal size ordering (5.22), we obtain
and the corresponding time between adjacent orders according to the formula (5.24):
We will supplement the conditions of the example with the average annual cost of the stock - 500 rubles/t and the forecasted inflation for the year - 12%. Then, using the modified formula for the optimal order size, we get:
Thus, accounting for losses from immobilization leads to a reduction in the size of the order (from 140 to 115 tons) and an increase in the frequency of its placement with the supplier (from 35 to 29 days).
2. A model taking into account losses from a deficit as a result of the temporary absence (unavailability) of stock in the warehouse (Figure 5.16).
Fig. 5.16. Stock movement with and without loss from deficit
Sometimes, instead of the term loss from deficit the term "lost profit" is used. Article 15 clause 2 of the US Civil Code treats lost profits as lost revenues, which the company could receive under normal conditions of civil turnover, had its rights not been violated. With respect to the management of supplies in supply logistics, a breach of rights can be considered non-fulfillment by the supplier of its obligations under the contract regarding delivery dates. However, a deficit can also be observed in situations of premature exhaustion of current and insurance stocks caused by a sharp increase in demand. Therefore, the term loss from deficit seems to be a broader and more relevant concept.
For the time of shortage, relations with customers can be organized in two alternative ways:
- incoming orders are not satisfied (left part of the figure);
- the satisfaction of incoming orders is postponed until the next delivery (right side of the figure).
In the nervous case, temporary deficit situations do not have serious consequences for the company. In the second case, it is assumed that the losses from the deficit exceed the costs of maintaining the stock, and therefore an increase in the order size is required to cover the deferred customer demand. However, such a decision will inevitably lead to an increase in the level of stored stocks, and the expected increase in the company's profitability may not take place. Therefore, it is necessary to find the size of the order, which would balance the costs associated with the content of stocks, and the losses from their deficit:
where H is the loss from a stock unit deficit.
Losses from stock shortages can be classified as follows:
1. Rigid losses related to the actions necessary to eliminate the effects of the deficit:
- missed revenue, if the company keeps records of unsatisfied customer orders;
- fines, penalties imposed on the company by customers in accordance with the terms of the contract;
(5.28)
- the cost of finding additional sources of supply;
- the cost of placing unscheduled orders;
- losses from higher selling prices of inventories to unscheduled orders;
- additional transportation costs;
- payment for overtime work of personnel on unscheduled orders, etc.
Rough losses can be taken into account when planning costs related to inventory, because they are easy to identify and fix.
2. Soft losses, whose dependence on the consequences of the deficit is ambiguous, and therefore they are neglected or use expert estimates in planned calculations:
- Loss of customer loyalty;
- loss of the customer (market segment);
- loss of company image, etc.
Example
The monthly demand of the enterprise in the reserve is 1200 tons, while the cost of processing one order is 1000 rubles, the cost of maintaining 1 ton of stock in the warehouse is 120 rubles/month, the number of working days in the month is 30. Then, formulas (5.22 and 5.24), we obtain:
Let's add to the conditions of the example the losses from the deficit of 1 ton of the stock - 150 rubles, then by the formulas (5.28 and 5.24) we get (Figure 5.17):
Thus, accounting for losses from stock shortages leads to an increase in the order size (from 140 to 190 tons) and a decrease in the frequency of its placement with the supplier (from 3.5 to 4.7 days). Such a decision assumes that the additional costs for maintaining an increased stock will not exceed the partially excluded losses from the stock deficit.
Fig. 5.17. Stock movement with and without loss from deficit
3. A model with a gradual replenishment, taking into account the time it takes to accept the stock when it arrives from the supplier. When organizing deliveries in large consignments (for example, by rail), the time required for unloading the stock, their quantitative and qualitative acceptance, and the registration of the receipts can be longer than the interval between the times at which the stock remains in the warehouse. Therefore, the stock accumulation occurs gradually, and not instantaneously, as in the classical model of inventory management.
In addition, in most cases, simultaneously with the acceptance of stocks, they can also be shipped to the company's customers (or leave for production), the rate of which is usually less, but not enough to neglect it in planned calculations. Otherwise, the batch taken for some time will be partially spent and the estimated level of the maximum reserve will not be reached, which may lead to an increase in the probability of a deficit (Figure 5.18).
Fig. 5.18. Stock movement with account for incremental replenishment
In this situation it is necessary to increase the size of the order, comparing the rate of receipt and shipment of stocks:
(5.29)
where s is the average daily rate of consumption (shipment) of the stock; d is the average daily rate of supply from the supplier.
Example
Let's add the conditions of the previous example to the values of the average daily rate of consumption of the stock - 40 tonnes and the average daily rate of its acceptance - 180 tonnes. Then, using formula (5.29), we get (see also Figure 5.19)
As you can see from the figure, the actual stock level turned out to be less than the calculated level (108 instead of 140 tons), since during the acceptance of the received lot, which took approximately 0.8 days (140 tons/180 tons), the stock was shipped to the company's customers in the size of 32 tons (40 tons • 0.8). Thus, in order to take into account the ratio of the average daily consumption rate and the receipt of stocks, it is necessary to increase the order size (from 140 to 160 tons) and reduce the frequency of its placement with the supplier (from 3.5 to 4 days).
Fig. 5.19. Stock movement with and without incremental refill
4. The model taking into account the losses from the deficit with a gradual replenishment, combining the conditions for applying the two previous modifications of the classical model of the optimal order size:
(5.30)
Example
We use the conditions of the two previous examples to calculate the optimal order size using formula (5.30) (see also Figure 5.20):
Thus, the joint accounting of losses from the deficit, as well as the ratio of the average daily consumption rate and the receipt of stocks leads to the need to increase the size of the order (from 140 to 210 tons) and reduce the frequency of its placement with the supplier (from 3.5 to 5.2 day). It can also be seen from the figure that the incoming batch of 210 tons exceeds the daily throughput of the company's warehouse but the acceptance of the stock, the total time of which will be 1.2 days (210/180 grams).
Fig. 5.20. Stock movement with joint account of losses from deficits and incremental replenishment
5. The model with allowances for wholesale discounts. The classic model of the optimal order size does not take into account the costs of purchasing and stockpiling, in which they are considered constant and independent of the lot size. The total cost curve in this case is shifted upward by the product of the unit price ( C ) and the total demand ( S ) without changing the optimal delivery schedule size (Figure 5.21).
Fig. 5.21. Model of the optimal order size taking into account the cost of purchased inventory
Often in practice, suppliers of material resources motivate buyers to increase the volume of purchases, providing wholesale discounts. In such a situation, the costs incurred by customers per unit of stock (unit costs) are reduced, while the amount of financial resources that they withdraw from the turnover is growing at a time. Then the optimization calculations for the classical or modified formula for the optimal order size are meaningless, since the total cost curve will have breaks at the start of the action of lower prices ( C 1, C 2 , C 3) (Figure 5.22). The figure shows that the optimal order size is Q 2, because it corresponds to the minimum aggregate costs.
Fig. 5.22. Total costs associated with the purchase of stocks in the presence of bulk discounts
If there are wholesale discounts, buyers will have to go through the options for the organization of supplies but the criterion of the minimum total costs associated with the purchase of stocks.
Example
The Department of Material and Technical Supply of the combined heat and power plant (CHPP) purchases from the coal producer of the SSPK brand (weakly baking, fraction 50-200 mm) for the production of electricity and heat.
The total annual requirement is 1800 tonnes. Deliveries are made by rail cars with a payload capacity of 60 tonnes at a freight rate of 18,000 rubles per car. The cost of storing coal in the warehouse is 1800 rubles. per tonne per year. To calculate the alternative costs, a deposit rate of 13% is used. The cost of placing an order is calculated on the basis of the total annual costs for the maintenance of OVTS, reduced in proportion to the quantitative share of coal of this brand in the total nomenclature of purchased stocks and the average number of orders placed - 10,000 rubles. for one order.
The characteristics of the possible options for the organization of coal supplies, cited in the calendar year, are summarized in Table. 5.16, and the graphical representation of the components of total costs - in Fig. 5.23.
Table 5.16
Characteristics of possible strategies for supplying coal to the SSPK for the needs of the CHP
Number of orders per year |
30 |
15 |
10 |
6 |
5 |
3 |
2 |
1 |
Order size, t |
60 |
120 |
180 |
300 |
360 |
600 |
900 |
1800 |
Price, rub/t |
1500 |
1500 |
1500 |
1300 |
1300 |
1300 |
1100 |
1100 |
Cost, thousand rubles/goal |
2700 |
2700 |
2700 |
2340 |
2340 |
2340 |
1980 |
1980 |
TZR, thousand rubles/year |
540 |
540 |
540 |
540 |
525 |
525 |
525 |
460 |
Cost of replenishment, thousand rubles/year |
300 |
150 |
100 |
60 |
50 |
30 |
20 |
10 |
Maintenance costs, thousand rubles/year |
54 |
108 |
162 |
270 |
324 |
540 |
810 |
1620 |
Cost of capital immobilization, thousand rubles/year |
76.0 |
81.9 |
87.8 |
95.6 |
98.7 |
119.0 |
132.6 |
188.5 |
Total costs, thousand rubles/year |
3670 |
3580 |
3590 |
3306 |
3338 |
3554 |
3468 |
4258 |
Fig. 5.23. Components of the total costs of various supply strategies for CHP Coal
Thus, according to the criterion of minimum total costs, six orders per year (order size - 300 tons) is the optimal strategy for supplying CHP with coal.
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