# Determination of the optimal size of the delivery lot.

Stocks play both a positive and a negative role in the operation of the logistics system. The positive role lies in the fact that they ensure the continuity of the processes of production and marketing of products, being a kind of buffer that smooths out unforeseen fluctuations in demand, violation of the timing of the supply of resources, and increases the reliability of logistics management.

The negative side of creating stocks is that they immobilize significant financial resources that could be used by enterprises for other purposes, for example, investment in new technologies, market research, and improvement of the economic performance of the enterprise. Based on this, the problem arises of ensuring the continuity of logistics and technological processes at a minimum level of costs associated with the formation and management of various types of stocks in the logistics system.

One of the methods for effective inventory management is the determination of optimal consignments of cargo, which allows you to optimize the cost of transportation, storage of cargo, as well as 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 stocks.

The amount of total costs is calculated by the formula (3.1):

C u003d C tr + C xp (3.1)

where С tr – transportation costs for the billing period (year), c.u.;

C xp – the cost of storing stock for the billing period (year), c.u.

The value of C tr is determined by the formula:

С tr =n c tr (3.2)

where n is the number of batches delivered during the billing period,

n= (3.3)

c tr – the tariff for the transportation of one batch, c.u. / batch.

Storage costs are determined by the formula (3.4):

C xp = q cf * c xp (3.4)

where q cf is the average value of reserves (in tons), which is determined on the assumption 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:

q cf =q/2 (3.5)

Substituting the expression C tr and C xp into formula (3.1), we obtain:

C= with tr + c xp (3.6)

The total cost function C has a minimum at the point where its first derivative with respect to q is equal to zero, i.e. = – with tr + =0 (3.7)

Solving equation (3.7) with respect to q, we obtain the optimal size of the delivery lot:

q * = (3.8)

As the size of the annual consumption of products, we accept the data obtained as a result of forecasting using the moving average method: Q=105.919 thousand tons/year; tariff for the transportation of one batch С tr =50 c.u./t; the costs associated with the storage of the stock С хр =8 c.u./t.

Substituting the given values, we get:

q * = ≈1151 (t)

In this case, the total costs are:

C= 60+ ∙10=5522+5775=11277 (c.u.)

The solution of this problem in a graphical way is to plot C tr (q), C xp (q) and C (q), having previously performed the necessary calculations to determine C tr, C xp and C.

Let’s determine the values of С tr, С хр and С. when changing q in the range from 600 to 1000 in increments of 100. We will enter the results of the table in Table 3.1.

Table 3.1.

Values С tr, С хр, С total

 Costs, c.u. Lot size, q(t) C tr C xp C total

According to table 3.1. graphs of the dependence of costs (transport, storage and total) on the size of the lot were built (Fig. 3.1.). Fig.3.1. Dependence of costs on lot size

Analysis of the graphs in fig. 3.1 shows that the cost of transportation decreases with increasing lot size, which is associated with a decrease in the number of flights. Storage costs increase in direct proportion to lot size.

The schedule of total costs has a minimum at a value of q approximately equal to 820 tons, which is the optimal value of the size of the delivery lot. The corresponding minimum total costs are \$8212.

Let’s calculate the optimal batch size in the conditions of a deficit with the amount of costs associated with the deficit C def =30 c.u./t.

In conditions of deficit, q * calculated by formula (3.8) is adjusted by the coefficient k, which takes into account the costs associated with the deficit.

q * = (3.9)

The coefficient k is calculated by the formula (3.10):

k= (3.10)

C def – the amount of expenses associated with the deficit;

Accept C def =30 c.u./t.

Substituting the values, we get:

k= =1.18

q=1.18∙1151≈1359 t.

It follows from this that in conditions of a possible shortage, the size of the optimal value of the batch for given data must be increased by 15%.