Inventories play both a positive and negative role in the activities of the logistics system. The positive role is that they ensure the continuity of production and sales processes, being a kind of buffer that smoothes out unforeseen fluctuations in demand, violation of the delivery time of resources, and increase the reliability of logistics management.
The negative side of creating inventories is that they immobilize significant financial resources that could be used by enterprises for other purposes, for example, investments in new technologies, market research, improving the economic performance of the enterprise. In addition, large levels of finished goods inventories prevent improving its quality, since the enterprise is, first of all, interested in selling existing products before investing in improving their quality. Based on this, problems arise in ensuring the continuity of logistics and technological processes at a minimum level of costs associated with the formation and management of various types of inventories in the logistics system.
One of the methods of effective inventory management is to determine the optimal shipments of cargo, which allows you to optimize the costs of transportation, storage of cargo, and also to avoid excess or shortage of cargo in the warehouse.
The optimal size of the delivery lot q is determined by the criterion of minimum costs for transporting products and storing inventory.
The amount of total costs is calculated using formula (3.1)
where – transportation costs for the billing period (year), c.u.
Inventory storage costs for the billing period (year), USD
The value is determined by the formula:
where n is the number of shipments delivered during the billing period,
– tariff for transportation of one lot, cu/lot.
Storage costs are determined by the formula: (3.4)
where is the average amount of stock (in tons), which is determined from the proposition that a new batch is imported after the previous one is completely used up. In this case, the average value is calculated using the following formula:
Substituting expressions and into formula (3.1), we obtain:
The total cost function C has a minimum at the point where its first derivative with respect to q is zero, i.e.
Having solved equation (3.7) for q, we obtain the optimal size of the delivery lot:
As the size of the annual volume of product consumption, we accept the data obtained as a result of forecasting using the simple average method: Q = 60.46 thousand tons/year; tariff for transportation of one batch of c.e./t; costs associated with storing the stock of c.e./t.
Let's substitute the given values and get:
q= = =269.3(t)
The total costs will be:
C= =2693.5(cu)
Solving this problem graphically consists in constructing graphs of the dependence, and, having previously performed the necessary calculations to determine, and.
Let's determine the value of , and when q changes in the range from 600 to 1000 in increments of 100. We enter the calculation results in the table. 3.1.
Table 3.1
Values , and
| Batch size | |||||
| 3627,6 | 1813,8 | 1209,2 | 906,9 | 725,5 | |
| WITH | 4127,6 | 2813,8 | 2709,2 | 2906,9 | 3225,5 |
Fig. 3.1 Dependence of costs on batch size
Analysis of the graphs in Fig. 3.1 shows that transportation costs decrease with increasing lot size, which is associated with a decrease in the number of trips. Storage costs increase in direct proportion to the lot size.
The graph of total costs has a minimum at the value of q approximately, which is the optimal value of the size of the delivery lot. The corresponding minimum total costs are 2709 cu.
Let's calculate the optimal batch size in conditions of shortage with the amount of expenses associated with the shortage = 30 cu/t.
In conditions of deficit, the value of q, calculated by formula (3.8), is adjusted by the coefficient k, which takes into account the costs associated with the deficit.
Coefficient k is calculated using formula (3.10):
The amount of expenses associated with the deficit;
accept =30 cu/t
Substituting the values, we get:
q=1.15*269.3=309.69 (t)
It follows from this that, in conditions of a possible shortage, the size of the optimal delivery lot under the given conditions has increased.
Conclusion: in this section I calculated the optimal size of the supply lot. Having solved equation (3.7) for q, I obtained the optimal delivery size.
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. Construct an inventory 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 1000 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.
Companies specializing in the production of various types of goods can organize the technological process not on a continuous basis, but on the basis of the production of batches of products. For example, a bakery might decide to produce a batch of large wholemeal loaves, followed by a batch of small buns, followed by a batch of barley scones. If a company uses batch production, then decisions must be made about the size of the batch of products produced during one production cycle and how often a batch of a certain product should be produced. The difficulties encountered are similar to those associated with determining the economic order quantity. Instead of ordering a specific quantity of a product from an external supplier, the production volume of a specific product is considered. Thus, the cost of the order, which appeared in the model outlined above, corresponds to the cost of organizing the process of producing a batch of products.
Rice. 11.5. Economic Lot Size Model
If we denote the cost of organizing each production cycle, then
where is the size of the product batch. Obviously, by analogy with the previous problem, it takes its minimum value if
The resulting optimal quantity of product per batch is called the economic batch size.
Example 11.2. A company that produces ceramic products produces several types of coffee pots. The production process is organized according to the principle of producing batches of coffee pots with a total volume of 500 pieces per week. Demand No. of the most popular model, which we will denote by X, is products per year and is evenly distributed throughout the year. Regardless of at what point in time the need arises to produce a batch of model X coffee pots, the cost of the production process is Art. According to company specialists, the cost of storing coffee pots is st. per unit.
How large should the batch of coffee pots be so that production and storage costs are minimal? How often should the production cycle be restarted and for what duration? It is assumed that there are 50 working weeks in a year.
Coffee pots per year;
Art. for one production cycle;
Art. for one coffee pot per year.
The economic batch size can be determined as follows:
Since the total cost curve is not very sensitive to small changes in values, it is likely that the value chosen as 820 will not result in a significant increase in the total cost. This statement can be easily verified.
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 cost of production 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.
With this article we open a small series of publications devoted to determining the optimal batch size of parts put into production. Obviously, this value affects economic indicators, so it is important for each manufacturer to determine it correctly. We want to talk about the history of this issue, the methods used and the latest trends.
As soon as any product is produced in quantities of more than one piece, a choice arises: either we can first completely make all the dissimilar parts of one product and only then proceed to the next one, or we make the same (or similar) parts for all products at once. The second method provides many advantages: specialization of jobs, rational use of equipment, stability of quality, increased productivity.
When producing a small quantity of goods, the number of identical parts is equal to the number of finished products. As production volume increases, production costs associated with setting up equipment, installing fixtures, and changing tools fall. But this happens up to a certain limit. Further growth leads to increased costs for storing raw materials, semi-finished products in workshops and finished products; significant funds are frozen in unfinished products.
This problem becomes noticeable even for a small artisanal workshop: “Where to place additional raw materials, where to put finished goods before they are bought and exported, where to get additional funds to buy more material?” But for a large enterprise, everything is much more serious - additional warehouses, buffer zones, and this means not only additional space, but also equipment, people, heating, organization of logistics, and accounting.
The solution is to split the total number of parts into separate batches. Production of products based on launch-release batches is called batch production.
People began to think about how many identical parts to put into production almost immediately after the transition from the manual method of manufacturing goods to the machine one. The development of high-volume and mass flow production in the early 20th century stimulated the development of theories for optimizing part lot sizes. These models have been improved over the years. At the end of the 20th and beginning of the 21st century, production began to change fundamentally, which also required new approaches to the distribution of products among production batches.
Obviously, as the batch size increases, the frequency of equipment changeovers, equipment and tool changes, and production preparation operations decreases, which means the costs of changeovers fall. At the same time, warehousing costs are increasing. The graph of total costs versus batch size has a minimum point. The nature of changes in costs is shown in the figure.
Determining the batch size that corresponds to this minimum cost is an optimization problem. Methods for calculating this point were developed at the beginning of the 20th century, and not without intrigue.
Historically, the first to propose a formula for calculating the optimal batch was the American Ford W. Harris. In 1913 he published his calculations. Frankly, derivation of the optimal batch size formula did not represent any theoretical breakthrough in mathematics. This is a fairly simple problem of finding the minimum of a function. Practical knowledge of the peculiarities of production economics was valuable. Harris worked as an engineer for an electrical engineering firm and used his experience to inform his analysis. However, he did not have a diploma - he only graduated from high school. Self-taught, he was phenomenally successful - he published 70 articles and registered 50 patents.
Over the next decades, publications by other authors appeared on the topic of optimal batch size in manufacturing. Since these studies were applied, there was no tradition of citing primary sources, as is customary in fundamental science.
In 1934, a new publication appeared in the Harvard Business Review, in which the author R.H. Wilson (Wilson or Wilson) again gives a formula for the optimal batch size without reference to previous works. And by a strange coincidence, it was his name that gave the name to the formula and became entrenched in subsequent history. Some researchers believe that there was competition between various publications and business schools (Harvard and Chicago), which supported only their authors. As a result, Harris' priority was forgotten after some time. And only in 1990 in the United States was an attempt made to understand the priority and date of the first publication on this topic.
But while the Americans were figuring out who was the first to learn how to calculate the optimal size of parties, the Germans, agreeing with Harris’s primacy, claim that their compatriot Kurt Andler really developed this topic for the first time in 1929 and call the corresponding formula after him , while no mention is made of Wilson.
Andler's formula for the optimal batch size of parts in its simplest form is as follows:
where y min is the optimal batch size,
V — the required volume of products over a period of time (sales speed),
Cr — costs associated with changing batches (conditionally - for setup),
Cl— specific warehousing costs over a period of time.
Wilson's formula for the optimal batch of goods to be ordered to a warehouse (for sales or for processing) looks similar. But its components have a slightly different meaning and different designations (in the classical form):
where EOQ is the economic order quantity (EOQ)),
Q — quantity of goods per year (Quantity in annual units),
P — costs of order implementation (Placing an order cost),
C — the cost of storing a unit of goods per year (Carry costs).
By the way, Americans easily remember this formula using the mnemonic phrase: “The square root of two Q uarter P unders with C heese.” The phrase is easy to translate,
or - “the square root of two quarter pounders with cheese.” Here, for Russians and in general everyone except Americans, an explanation is required. Americans call a McDonald's cheeseburger a “quarter pound,” which traditionally weighs a quarter pound—113.4 grams.
Outside the United States, this type of hamburger has other names, and in this regard, one can recall the famous dialogue between two killers Vincent and Jules from Tarantino’s film “Pulp Fiction.” One of the bandits, played by Travolta, talks about his trip to Europe, how in Paris you can buy beer at McDonald’s and other “miracles”:
— Do you know what they call Quarter Pounder with cheese in Paris?
- Why don’t they call him Quarter Pounder?
- No, they have the metric system, and they don’t know what ... (omitting profanity) a quarter pound is. They call it the Royal Cheeseburger.
— Royal Cheeseburger??? What do they call a Big Mac then?
“Big Mac is Big Mac, but they call it Le Big Mac.”
- Le Big Mac?! Ha ha ha...
So Vincent and Jules could easily remember the formula for the optimal volume of goods and apply it in their activities.
The classical Andler-Wilson optimal batch model is based on a number of initial assumptions: production without capacity limitations, without intermediate warehouses, demand is stable, the ability to divide materials into any batch size, warehouse costs are constant, a warehouse of unlimited volume, an unlimited planning horizon, implementation goods occurs immediately after production, etc.
Each such assumption is at the same time a limitation for the application of the model in certain specific production conditions and can serve as the basis for the development and complication of the model.
However, the results of calculations using the simplest classical formula can still serve as basic values for the initial assessment - the accuracy of the assessment largely depends on how fully and accurately we take into account the costs associated with launching a new batch and storage costs.
The furniture industry has recently become increasingly individualized; work is increasingly based on orders - if not from end customers, then from a dynamically replenished warehouse, which practically acts as a customer. In this regard, the trend of the last decade has been to work according to the Losgrösse 1 principle - that is, the batch size is from one piece. We will dwell on this in more detail in the following articles.