# Probabilistic-statistical methods - Logistics and supply chain management

## Probabilistic-statistical methods

The methods of this group are based on theorems on the numerical characteristics of random variables and the transformation of random processes.

The randomness of supply and demand parameters, as well as various logistics risks are the main reasons for creating insurance stocks. In Fig. 6.4 provides a graphical interpretation of the expenditure model and replenishment, reflecting the uncertainty of demand and the random nature of the lead time for replenishment.

The process shown in Fig. 6.4, is characterized by a number of features that are important for understanding the reasons for the formation of the insurance stock.

1. In the general case, the use of the current stock is a discrete, nonincreasing random process that is not inherent

Fig. 6.4. The model of consumption and replenishment taking into account the uncertainty of demand and the duration of the order cycle:

L - the implementation of the current stock; B - delivery volumes; C - the amount of the current stock at the time of delivery; About - deficit;/(Oo) ~ is the distribution density of the stock at the beginning of the implementation;/((2) is the distribution density of the current stock at the time of delivery;/(L, m) is the distribution density of the delivery volume; & lt; p (/, ()) is the distribution density of the time value of the end of the current stock realization

stationarity and stochastic demand; the ensemble of realizations of the current stock can be with strong and weak mixing (A, see Figure 6.4).

2. The quantities of supply volumes are random variables and are subject to some distribution laws (B, see Figure 6.4); in a particular case the supply is a deterministic quantity.

3. The end date for the implementation of the current stock is also random, in some implementations the balance of the stock at the time of delivery is greater than zero, in others it is zero. In the situation of the zero balance of the current stock and in the absence of an insurance stock, a deficit (D see Figure 6.4) occurs. If there is an insurance stock, then the situation with a zero balance of the current stock can be called a "pseudo-deficit", as demand is met by the insurance part of the stock. The distribution function of the current stock (at the time of delivery) will obey the truncated normal distribution law or the distribution laws for positive random variables (C, see Figure 6.4).

4. When determining the basic parameters of the inventory management system, the model of the optimal order quantity is used, which corresponds to the Harris-Wilson formula (this formula will be discussed in Section 6.3), and a formula for calculating the time between orders.

5. In the deficit situation, when at the time ь. the total daily consumption of E ^ reaches the initial stock in the warehouse, it is assumed that unsatisfied orders accumulate up to a random moment Tk - of the time of receipt of the new order. In other words, with 5X -> gt; (I we are talking not about the real, but about the predicted process of accumulating bids in the interval АТ = Тк- Ту Random accumulated deficit values ​​are the basis for calculating the insurance stock.

Determining the size of the insurance stock under conditions of uncertainty can be achieved by the formula of the Fetter-Dalek:

where xp - the normal distribution law parameter corresponding to the probability of a product shortage in the warehouse P (x) (Table 6.5); T - the average value of the duration of the logistics cycle (the period of time between deliveries); th - The average daily inventory consumption; a, are the mean square deviations of the random variables T and th respectively.

Table 6.5. Probability of no deficit P (x) and the coefficient value xr for the normal distribution law

When carrying out insurance stock calculations using formula (6.21), it is necessary to take into account that the formula is valid if T does not change. This is the reason for some incorrectness of the calculation by formula (6.21). Let's try to prove it.

In Fig. 6.5 the line A shows the base level for the initial value T.

We will move the zero level upwards along the vertical axis (O '), thereby reducing the lead time for the order T. Given that the daily flow is characterized by strong mixing and lines, corresponding to individual implementations, diverge in the form of a "bundle" bounded by straight 0_0P, variances of random variables T line C (Figure 6.5). If you transfer the zero level to O, the cycle time increases and the line <0>

corresponds to the random values ​​ T

Thus, the incorrectness of the calculation according to the formula (6.21) is that for different (2 and, respectively, T , the same value of or is substituted random ham '/' is limited to the parallel lines B in Figure 6.5.

Equation (6.21) will be true if instead of T and op corresponding to the base level, the average value and average kv

Fig. 6.5. Graphical interpretation of the formula adjustment (6.21):

oo - the initial level of the stock; L - the base level for the lead time for the order; C, d - the changed order execution times to the smaller and larger side, respectively; About, O ', About - projection of the lead times for the vertical axis; B, G - are the direct limits that limit the variance of the stock amount for the base and changed order lead times, respectively

The deviation of the new duration of the order cycle (T *, a ^ -). If there are no data for the new conditions to determine the statistical parameters of the duration of the functional cycle, then in order to account for the changes it is necessary to introduce a parameter allowing to take into account the similarity, for example, the coefficient of variation.

Let's assume that the statistical parameters characterizing the daily sales volume (or the consumption of material resources) About and ap are constant and do not depend on the duration of execution of the order G; the law of distribution of daily sales (consumption) is normal. The law of distribution of the duration of the order cycle is normal with parameters: mean T and standard deviation

where ig is the coefficient of variation determined on the basis of statistical processing for the base sample.

For example, if static information is collected for the basic level of an order cycle with the parameters T = 10 days, at = 2 days and uz = 0.2, then for a cycle with T = 20 days. ar = 0.2 • 20 = 4 days.

Thus, the formula (6.21) can be written in the form

where T * - is the average value of the order cycle time that is different from the baseline.

Professor AP Dolgov proposes to calculate the standard deviation, which is used to calculate the insurance stock, but the formula

A. Dolgov believes that in the derivation of formula (6.24), the addition rule and the dispersion properties of two independent (uncorrelated) sets are used, but do not indicate for which dependency of the consumption of the reserve on time this rule is used. Obviously, we are talking about a linear dependence of the form

where (& iquest; 1, - is the non-random size of the delivery size ^ */- the intensity of the daily flow, the random variable with the parameters D a "; T - The random value of the length of the delivery cycle with the parameters T, ar

To illustrate our assumption about the linear form of the dependence of the flow in Fig. 6.6 the lines L and B, are approximated by the dependence (6.25) and are realizations of a random process with the so-called weak mixing.

Fig. 6.6. Graphical interpretation of the formula (6.24):

He is the initial level of the stock; A, B examples of the current stock implementation

To calculate the standard deviation a ", we use the method of linearizing a function of a random variable:

When the formula (6.25) is substituted into the formula (6.26), we obtain:

Thus, formula (6.24) will be valid if the flow-time dependence is a linear function.

Let's illustrate our point of view with an example. Let the demand be characterized by the parameters AND = 5 units, o0 = 2.54, the coefficient of variation for a random time value order fulfillment σ = 0,2; the parameter is D = 2.33. In Table. 6.6 we will present the results of calculations of the insurance stock according to the formulas (6.21) and (6.23), and also taking into account the formula for the standard deviation (6.24), which A. Dolgov insists.

Table 6.6. The results of calculating the safety margin for the three options

As can be seen from Table. 6.6, the size of the insurance stock, obtained by different variants of calculation, has significant discrepancies. So, the formula (6.21) was practically insensitive to the change in the size of the order, which can not but cause doubts as a result of calculations. The formula proposed by A.P. Dolgov, on the contrary, shows an unjustified increase in the insurance stock, which becomes larger than the size of the order itself. Realistic results are obtained using the formula (6.23).

In Fig. 6.7 for clarity, the graphs of the change in the size of the insurance stock are shown, depending on the volume of the order.

Fig. 6.7. The size of the insurance stock, determined by the options:

I - according to the formula (6.21); II - but with the formula (6.23); III - taking into account the formula (6.24)

Experience shows that calculation by formulas (6.21) and (6.23) is suitable for cases where a periodic strategy is used for inventory management, i.e. strategy with a constant frequency of placing an order (for more details on inventory management strategies, see paragraph 6.5) (Figure 6.8, a).

When using a periodic inventory management strategy, the order for replenishment is made at predefined times, and it is possible that there is a shortage already at the time of placing the order. During the execution of the order, the deficit will accumulate. Therefore, when calculating the safety stock, it is necessary to take into account the deviations in the demand for weight during the execution of the order (in Figure 6.8 - Hz).

When using a stock management strategy with an ordering point ENT (Figure 6.8, b), which provides the order for the replenishment of the stock when the level corresponding to the ENT is reached, the deficit at the time of the order is unlikely, and with continuous monitoring of the order level, it is excluded. Therefore, when calculating the safety stock, it is necessary to take into account deviations in demand only during the order execution time (& pound;):

where I - the average value of the order execution time (in Figure 6.8 - Г || 0СТ); x1 - is the coefficient of variation for the random value order execution time & quot ;.

It should also be noted that the Fetter-Dulleck formula and its modifications, given above, are derived for the condition of a normal distribution of the random quantities of demand and the time of the logistic cycle. Under other laws of distribution, the formulas for calculating the safety stock are modified taking into account their parameters.

Fig. 6.8. Determine the need for an insurance stock for different inventory management strategies:

a - periodic strategy; b - strategy with the order point

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