Determining the optimal size of the delivery lot. Optimal order size according to the Wilson formula Optimal size of the ordered consignment of goods

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: the volume of demand, the cost of delivering the goods, the cost of storing the stock.

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

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 delivering 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. 60.

Storage costs increase in direct proportion to the size of the order. This dependence is graphically presented in fig. 61.

Rice. 60. Dependence of transportation costs on the size of the order

Rice. 61. The dependence of the cost of storing stocks on the size of the order

Rice. 62. Dependence of the total cost of storage and transportation 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. 62). As you can see, the total cost curve has a minimum point at which the total cost will be minimal.

The task of determining the optimal order size, along with the graphical method, can also be solved analytically. For this, the Wilson formula is used.

LECTURE 11. WAREHOUSES IN LOGISTICS

The concept and types of warehouses

Warehouse functions

Brief description of warehouse operations

Loading unit

The concept and types of warehouses

Warehouses- these are buildings, structures and various devices intended for the acceptance, placement and storage of goods received on them, preparing them for consumption and release to the consumer.

Warehouses are one of the most important elements of logistics systems. An objective need for specially equipped places for keeping stocks exists at all stages of the movement of material flow, from the primary source of raw materials to the end consumer. This explains the presence of a large number of various types of warehouses:

ü Warehouses vary in size from small spaces with a total area of several hundred square meters up to giant warehouses, covering hundreds of thousands of square meters.

ü Warehouses also differ in the height of the stacking of goods. In some, the cargo is stored no higher than human height, in others, special devices are needed that can lift and accurately place the cargo in a cell at a height of 24 m or more.

ü Warehouses may have different designs: be placed in separate rooms (closed), have only a roof or a roof and one, two or three walls (half closed). Some cargoes are generally stored outdoors at specially equipped sites, in the so-called open warehouses.

ü A special mode can be created and maintained in the warehouse, for example, temperature, humidity.

ü A warehouse may be intended for the storage of goods of one enterprise (warehouse individual use), and may, on a leasing basis, be leased to individuals or legal entities (warehouse collective use or warehouse-hotel).

ü Warehouses also differ in the degree of mechanization of warehouse operations: non-mechanized, mechanized, complex-mechanized, automated and automatic.

ü An essential feature of the warehouse is the possibility of delivery and export of goods by rail or water transport. According to this feature, there are near-station or port warehouses (located on the territory of a railway station or port), railway(having a connected railway line for the supply and removal of wagons) and deep. In order to deliver cargo from a station, pier or port to a deep warehouse, it is necessary to use an automobile or other type of transport.

ü Depending on the breadth of the range of stored cargo, they distinguish specialized warehouses, warehouses with mixed or with universal range.

ü Warehouses can be divided into two groups: warehouses in the traffic area industrial and technical products and warehouses on site movement of consumer goods.

Rice. 64. Schematic diagram of the chain of warehouses on the path of the material flow from the primary source of raw materials to the final consumer

A schematic diagram of the passage of a material flow through a chain of warehouses of various enterprises is shown in fig. 64.

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. 59. Two-bunker inventory control system

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. 60.

Storage costs increase in direct proportion to the size of the order. This dependence is graphically presented in fig. 61.

Rice. 60. Dependence of transportation costs on the size of the order

Rice. 61. The dependence of the cost of storing stocks on the size of the order

Rice. 62. Dependence of the total costs of storage and transportation on the size of the order. Optimal order size S opt

The task 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.

Questions for knowledge control

1. Define the concept of "inventory".

2. List the costs associated with the need to maintain inventories.

3. What are the main reasons that force entrepreneurs to create inventories.

4. List the types of inventories known to you.

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.

This article does not claim to give a comprehensive answer to the question of the optimal sizes of production lots, its purpose is to collect in one place some aspects of one of the problems of planning complex production.

Let's start with the definition

In general, in order to really start the answer correctly, you need to define the production batch. And this attempt alone can bring to life several crusades and holy wars between adherents of one or another approach. At least in those years when I worked as a consultant in a consulting company, we broke spears over this definition for a long time, until one of the wise colleagues came up with 5 options that would more or less close the whole set of variations of production batches.

Party is:

Sales order size – external or internal (between operations)
Technological batch - simultaneously processed quantity of products
Number of products produced between changeovers
Number of products produced between shipments
The volume of the accumulator or hopper, loaded at a time before the operation

In the general case, it should be said that a production batch is the number of parts, products, products that is processed at one stage of production without interruptions, stops and switching to another type of parts, products, products. I cannot say that this is the best definition of a party that can be given, but for the purposes of this article, I think it will be enough.

Economically optimal lot size per operation

For each individual stage of production, it is possible to reliably determine the economically optimal batch size, for which the Wilson formula is used.

where EOQ is the economic order quantity (EOQ)),
Q - quantity of goods per year (Quantity in annual units),
P - costs for the implementation of the order (Placing an order cost),
C - the cost of storing a unit of goods per year (Carry costs)

or its analogue Andler's formula

where min is the optimal batch size,
V - the required volume of production for a period of time (sales rate),
Cr- costs associated with the change of batches (conditionally - for adjustment),
Cl- unit costs for warehousing in a period of time.

The general view of the graph is as follows:

Actually, here you need to look for the minimum of the “Total Costs” curve, and the value of X that corresponds to it will be the “economically optimal lot size”.

Naturally, all this looks simple only on the graph, in order to calculate the exact value, you need to have a good understanding of the setup costs (green curve) and the amount of storage costs (lilac curve).

Setup costs may include:

equipment downtime cost
operator downtime cost
installer costs
tool costs
tooling costs
additional costs of materials and energy carriers for the shutdown / start-up time
etc.

Warehouse costs include:

cost of stored objects
warehouse space cost
warehouse staff costs
lighting and heating costs
costs for warehouse equipment (stackers / loaders)
etc.

All in all, there are quite a few things to consider.

The total cost curve does not have a break at the current minimum, which means that if you get, for example, an economically optimal lot size of 1327 pieces, then most likely you can start production in batches of 1300 to 1400 pieces without any significant deviations in the cost, and certainly if the optimal batch size is 4.6 pieces, then you can run batches of 4 pieces and 5 pieces each.

Problem: different technologies - different batches

The problem with real production is that setup and storage costs are not the same throughout the production cycle, and this introduces disagreement about what should be the size of a lot that goes through several stages of production, and not just one.

For example, it is profitable to bring raw materials by trucks, because. the cost of the vehicle is “smeared” over the entire volume of raw materials, no matter how much it is, heat treatment must be performed for such a number of parts that can be put into the furnace as much as possible, and shipment must be done only in the quantity ordered by a particular customer, otherwise everything is superfluous, what you send him will just get him for free.

Storage of small and bulky objects also costs different amounts of money, and if some raw materials also need to be kept warm or in other “special climatic conditions”, then the cost of storing such raw materials will be higher than for other types of raw materials.

2000 pieces per lot
200 pieces per lot
540 pieces per lot
34 pieces per lot

And it's good if the units of measurement are the same in each case. And then it might turn out like this:

2000 kg per batch
200 pieces per lot
540 pairs per lot
34 sets per lot

In this case, the problem of the optimal batch size is only exacerbated.

Extreme solutions to the problem

In order not to get confused, I want to have one batch size for all occasions. After all, if at one stage of production a batch consists of ten pieces, and at another of thirteen, it is necessary to organize some kind of intermediate warehouse in order to accumulate the missing pieces of semi-finished products.

What are the extreme options?

use the maximum of the estimated lot sizes
use the minimum of the estimated lot sizes

Let's take the piece example described above (2000, 200, 530 and 34 pieces) and see how to implement both options on it.

Max Lot Size

The maximum batch size of all four options is 2000 pieces. Having agreed to use it, we come to production planning, in which only batches of 2000 pieces are used:

2000 pieces per lot
2000 pieces per lot
2000 pieces per lot
2000 pieces per lot

What does this result in?

At the first stage, we get the optimal batch size - no more, no less. And those who work in this area, and even more so those who manage it, should be absolutely satisfied with this decision.

In the second stage, the batch size is 10 times the optimal one. What does this mean? We spend 10 times less time to change this stage of production, but at the same time we fill the intermediate warehouse between stages 2 and 3 with ten times the amount of inventory that our managers could arrange.

At the third stage, the batch size is almost 4 times larger than the optimal one, and this can also lead to a large amount of inventory.

But that's where the reserves are definitely VERY MUCH - this is after the fourth stage. There, you can work on 34 pieces, which means that the batch size is almost 60 times larger than the optimal one.

What is good and what is bad such a decision.

The good result is that the equipment will be fully loaded, changeover downtime will be kept to a minimum, and if we can synchronize equipment changeovers and pass one batch through all stages in order, then we will only need three intermediate warehouses for 2000 pieces of semi-finished products (between the first and second stages, between the second and third stages, between the third and fourth stages) and then the whole process will work as a conveyor. If any of the stages stops, then the limit of the intermediate warehouse size of 2000 pieces will quickly force all production to stop and overproduction will not occur: the subsequent stages will exhaust their stocks of semi-finished products and stop, because. the emergency stage will not allow them to be replenished, and the previous stages will fill the intermediate warehouses and also stop, because emergency stage will not allow them to be released).

The bad result is that you will most likely need a lot of storage space to organize three intermediate warehouses: most often production is organized like this. that until all 2000 semi-finished products appear in the previous warehouse, the next stage of production does not start, which means that you need to have the appropriate space for these semi-finished products (in some cases, you can work “from wheels”, i.e. start production at the next stage yet before the entire batch of 2000 semi-finished products is completed, but this is not possible for every technology). Worst of all, the situation will be with the warehouse of finished products, because. there we will get a catastrophic supply of excess production.

Minimum Lot Size

The minimum lot size of all four options is 34 pieces. Having agreed to use it, we come to production planning, in which only batches of 34 pieces are used:

34 pieces per lot
34 pieces per lot
34 pieces per lot
34 pieces per lot

What does this result in?

At the first stage, changeover will be performed 60 times more often than required for the optimal variant. This is a lot. If each changeover takes a significant amount of time, this can have a disastrous effect on the performance of the entire process - it simply will not have time to release everything that you want to get from it.

Further readjustment will also be performed non-optimally - 6 times more often than is required for the optimal variant. Worse, if, for example, at the start of each batch, expensive equipment or materials are used that are consumed once for the entire batch, these costs will increase significantly and will place an exorbitant burden on the cost of production.

The same will happen with the third stage, and only at the fourth stage everything will be as it should be.

In general, the entire manufacturing process will be slower, held back by the longest changeover phase.

The advantage of this option is that you minimize the need for storage space - they only need as much as is required to store 3 types of semi-finished products of 34 pieces, a little more - for 34 units of raw materials and 34 units of finished products. A microscopic figure, compared to the previous stage.

Cons - increased tooling losses during changeovers and reduced due to large losses of time for changeover the productivity of the entire process as a whole.

Let's leave everything as it is

Now, having dealt with what happens in extreme cases, we can figure out how production will work if we leave the batch sizes such that they are equal to the economically optimal batch size of each stage separately:

2000 pieces per lot
200 pieces per lot
540 pieces per lot
34 pieces per lot

So how will it work?

To start such a production, we need 2000 units of raw materials before the first stage. Then we will be able to set up and put the optimal batch into production and everything will be fine.

After that, 2000 semi-finished products will go to an intermediate warehouse. Of these, only 200 pieces will be selected for the first run in order to start the second stage of production in an optimal way. Everything is good here too.

After the second stage, 200 pieces will go into stock and will wait for the next batch, since at least 540 pieces are needed to launch the third stage. And if the second stage will produce semi-finished products of the same type, then two more batches of 200 pieces will need to be produced. In this case, stocks between the second and third stages will reach 600 pieces and it will be possible to start the third stage of production.

The third stage of production will release 540 semi-finished products to the last intermediate warehouse and they will be consumed from there in small batches of 34 pieces. In this case, we will ensure a minimum stock in the finished goods warehouse, but still not get rid of the stock in the semi-finished products warehouse between 3 and 4 stages of production.

What can be seen in this situation?

The size of the intermediate warehouse is proportional to the larger quantity of the economically optimal batches of these two stages.

Those. the warehouse of semi-finished products between the first and second stages of production must contain at least 2000 products. The warehouse for semi-finished products between the second and third stages of production should contain 540, and not 200 products at all. And the warehouse of semi-finished products between the third and fourth stages of production must also accommodate 540 products. The warehouse of finished products should contain batches of 34 finished products, and this, apparently, will be enough in our case.

Interestingly, this is the first change worth making to the planning system.

Since the size of our warehouses is larger than the optimal one (2000, 540, 540 and 34), it makes no logical sense to launch batches of 200 pieces instead of 540 at the second stage - we still pay for the warehouse as “for 540” and accumulate parts there to launch at the next stage at (minimum) 540 pieces, so it is worth changing the size of the economically optimal lot of the second stage from 200 to 540, despite the fact that we received the figure of 200 by calculation using the above formula.

In reality, the adoption of such a decision looks like this: the foreman of the site where the second stage of production is being carried out looks at the statistics of the stocks of semi-finished products in both warehouses and says something like this: “Why are we soaring and doing changeovers all the time, nobody needs this! »

Thus, we smoothly move on to option 2:

2000 pieces per lot
540 pieces per lot
540 pieces per lot
34 pieces per lot

And this is not arbitrariness, this is just the common sense of the master or planner, because in this case, working in batches of 200 pieces is really not needed for anything other than to meet the calculated economically justified batch size. And if this is not a game situation, but a life situation, then no one gives a damn about the calculated figures - it is obvious that in this case the peculiarities of the entire process were not taken into account in the calculation.

To demonstrate this approach with another example, let's assume that the production consists of 10 stages instead of 4, and the optimal batches for each stage were calculated as follows:

4000 pieces
70 pieces
320 pieces
15 pieces
645 pieces
90 pieces
425 pieces
64 pieces
130 pieces
70 pieces

Obviously, stocks between stages should contain no less than:

4000 products between the first and second stages
320 products between the second and third stages
320 products between the third and fourth stages
645 products between the fourth and fifth stages
645 products between the fifth and sixth stages
425 products between the sixth and seventh stages
425 products between the seventh and eighth stages
130 products between the eighth and ninth stages
130 products between the ninth and tenth stages

After reflecting a little on the optimal batch sizes, we can come to the conclusion that we can just as well set the batch sizes as follows:

4000 items
320 products
320 products
645 items
645 items
425 items
425 items
130 items
130 items
70 items

Now it becomes clear that a buffer of 645 pieces is needed between the third and fourth stages, and then it turns out that the same buffer is actually needed between the second and third stages of production. As a result, the optimal sizes of production batches by stages will be the following sequence:

4000 items
645 items
645 items
645 items
645 items
425 items
425 items
130 items
130 items
70 items

Those. at steady state, any set of batches in the production stages tends to such a set when the next stage has a batch size equal to or less than the batch size of the previous stage.

Let's call it the paradox of "home canning": first we harvest all the harvest we can and put it in jars, then on holidays we take out a jar of cucumbers from the stock, open it, and for several days we hastily eat an open jar of cucumbers so that they not spoiled - at each stage of the "consumption" of the cucumber crop, the batch size is smaller and smaller until it reaches the size of the batches that the consumer takes the products with.

If we initially had the batch sizes would be as follows:

34 pieces
540 pieces
200 pieces
2000 pieces

then it is quite reasonable to expect that after some time the set of batch sizes would come to the variant

2000 pieces
2000 pieces
2000 pieces
2000 pieces

since there is no need to reconfigure the equipment of the third stage of production 10 times in order to run one batch of 2000 identical products in the fourth stage.

Warning about conditions that remained "behind the text"

All these layouts are given for one type of product without taking into account other types of products - we just mean that the changeover is made for the manufacture of a “different” type of product.

The paradox of “home-made canning” can only be seen in its purest form in a production facility where there is enough production and storage space to store all these growing stocks. Otherwise, they will be limited by the physical scale of production, but the essence of the paradox will be the same: the sizes of the batches at the previous stages will increase until the limit of the space occupied by the stock is reached, or until this same batch size reaches the size of the batches of subsequent ones. stages.

An important conclusion about the maximum optimal lot size

The batch size at each stage of production will be no less than the size of the batches of the last stage of production or the last stage of transportation of products to the customer.

Those. if you're shipping dental bike pumps to a customer in 40-foot containers, there's no point in producing them in batches of 10 rather than 50 or 1,000—you end up needing a full container of pumps anyway.

Calculation of the minimum allowable lot size

In the logic of lean manufacturing, one of the goals of production planning is to reduce the lot size until the ideal state is reached, which is described by the concept of “one piece flow” - One Piece Flow.

If the calculation of the economically optimal lot size is done within the framework of the generally accepted management logic, when certain inventory sizes are a boon, not evil, then in lean manufacturing, when any inventory is considered harmful to one degree or another, the question of the optimal lot size is posed a little differently: how small can there be batches of production, provided that the required level of production productivity is maintained?

Here is the calculation.

Suppose we need to produce a certain number n of products or semi-finished products in time T. The average cycle time is CT. In this case, the time that we can spend on changeovers will be equal to

Tcho = (T - n x CT)

If one changeover takes about the time of ChT, then we can afford a certain number of changeovers in this period of time:

Ncho = (T - n x CT) / ChT

And then the average number of products in the batch will be equal to:

Batch = n / Ncho = n x ChT / (T - n x CT)

For the maximum changeovers performed over a certain period of time, this will be the minimum of products per batch, at which production still has time to fulfill its plan.

Here is an example.

Shift duration = 8 hours or 480 minutes

Cycle time = 1 minute / item

Planned release of 400 items

Changeover time 5 minutes

Batch = 450 x 5 / (480 - 400 x 1) = 450 x 5 / 80 = 29 items (round up)

For reliability, it is worth introducing an equipment availability factor to take into account the time for maintenance and repair.

Then the formula will look like this:

Batch = n x ChT / (T x k - n x CT)

in this case, if we add an availability factor of 90% to our example, then the batch size will be equal to:

Batch = 450 x 5 / (480 x 0.9 - 400 x 1) = 450 x 5 / (432 - 400) = 450 x 5 / 32 = 71 items.

Here are a few implications of this formula:

The larger the planned output, the less changeovers can be made and the larger lot sizes need to be applied.
The lower the availability factor, the fewer changeovers and the larger the batch sizes.
The longer the changeover time, the fewer changeovers and the larger the lot sizes
The shorter the changeover time, the more changeovers can be made and the smaller lot sizes can be used.

In this formula, two simplifications are made, taking into account the following assumptions.

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, which 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 stock, which depends on the short-term forecast of sales intensity;

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 ESP. 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.