## Stochastic Inventory Management in a Shortage

Read this article. The study indicates that a stochastic inventory management system should be preceded by determining the economic feasibility of the shortage. How does using real-time statistics help determine purchasing?

### Materials and methods of the study

#### The method of determining the parameters of stochastic inventory management system in conditions of economically based shortage

The main input parameter of the inventory management system is demand. In real conditions, demand often has a random pattern. In addition, it should be considered that replenishment of stocks (supply) also occurs with certain time fluctuations. Of course, it is possible to work with averaged values of demand and time of delivery, but in some cases, neglect of the stochastic nature of consumption and replenishment of inventory may result in incorrect calculations and, consequently, increase of logistic costs.

The main parameters of the inventory management system are the maximum desired stock, the order quantity, the size of safety stock, reorder point. However, we believe that an economically based shortage should also be present among the parameters of the inventory management system. To build an effective inventory management system in stochastic conditions, the algorithm is proposed (Fig. 1). Consider each of the stages of this algorithm in more detail.

Fig. 1. Algorithm of defining the parameters of the stochastic inventory management system in conditions of economically justified shortage

1. Removal of abnormal values.

A gross error or outliers is an error of a certain measurement included in a series of measurements, which for these conditions differs dramatically from other results of this series. In logistics, outliers can be results related to the emergence of conditions that do not have to repeat in the future – stop of production equipment, car accidents, theft of warehouse or transport supplies, supplier strike, etc. If the probability of such events in the future is extremely low, observations that are registered due to the above events must be removed from a series of data as gross errors, as they "differ dramatically from other results" (according to the definition). Possible criteria for checking the values for the presence of gross errors are shown in Table 1.

Table 1 Criteria for checking the statistical values for outliers

Criterion name The number of measurements for which the criterion is applied The condition of availability of a gross error Features of use
Dixon (Variation criterion) $n \leq 10$ $\frac{x_{n}-x_{n-1}}{x_{n}-x_{1}} \text { > } Z_{q}$ The results of measurements are ranked by growth
Romanovsky $n \leq 20$ $\frac{\left|\bar{x}-x_{i}\right|}{S} \geq \beta_{q}$ -
"Three Sigma" $n \text { > } 20-50$ $\left|\bar{x}-x_{i}\right| \text { > } 3 \sigma$ -
Chauvenet $n \leq 10$ \begin{aligned} &n=3:\left|\bar{x}-x_{i}\right| \text { > } 1,6 S \\ &n=6:\left|\bar{x}-x_{i}\right| \text { > } 1,7 S \\ &n=8:\left|\bar{x}-x_{i}\right| \text { > } 0,9 S \\ &n=10:\left|\bar{x}-x_{i}\right| \text { > } 2 S \end{aligned} Determined depending on the number of measurements
Grubbs $n \text { > } 50$ $\frac{\left|\bar{x}-x_{i}\right|}{S} \geq v_{p}$ -
Irwin $\frac{x_{n+1}-x_{n}}{\sigma} \text { > } \theta_{p}$ The results of measurements are ranked by growth or in descending order. From the received series, two largest or two smallest values are chosen and the criterion is calculated

If an outlier is detected, the value must be excluded, and the numerical characteristics are recalculated. For subsequent calculations, the value of the sample mean and sample standard deviation is needed. The sample mean:

$\bar{x}=\frac{\sum_{i=1}^{N} x_{i} n_{i}}{N}=\sum_{i=1}^{n} x_{i} f_{i},$   (1)

where $x_{i}$ – the middle of the $i$-th interval; $n_{i}$ – the number of observations in the $i$-th interval; $N$ – thtal number of statistical observations; $n$ – the total number of intervals; $f_{i}$ – the frequency at which the values fall in the $i$-th interval.

Formulae for calculating dispersion and standard deviation are chosen depending on the number of observations (Table 2).

Table 2 Formulas for dispersion and standard deviation

Number of observations Formula for calculating the dispersion Formula for calculating the standard deviation
More than 30 $D=\frac{\sum_{i=1}^{n} n_{i}\left(x_{i}-\bar{x}\right)^{2}}{n}$ $\sigma=\sqrt{D}$
Less than 30 $S^{2}=\frac{n}{n-1} \cdot D$ $S=\sqrt{S^{2}}$

2. Verification of the hypothesis concerning the normal distribution of demand and time of delivery.

The problem of identification of the distribution law describing statistical data is an extremely important stage of forecasting and planning.

A statistical hypothesis is any assumption concerning frequency function, probabilities (probability density function) or quantitative characteristics of numerical series.

To verify the hypothesis of the normal distribution of sales and delivery times, it is proposed to apply the Pearson chi-squared test. To do this, we must calculate the theoretical values of the probability of getting to each of the intervals. This is assisted by the Laplace transform (probability density function of the normal distribution).

The probability $p_{i}$ for the normal distribution of a random variable $X$ is calculated according to the formula:

$p_{i}=\varphi\left(x_{1}\right)-\varphi\left(x_{2}\right)=\varphi\left(\frac{x_{l i}-\bar{x}}{\sigma}\right)-\varphi\left(\frac{x_{r i}-\bar{x}}{\sigma}\right)$   (2)

where $x_{l i}$ and $x_{r i}$ – the lower and upper bounds of the intervals, respectively.

The values of the Laplace function for the positive values of the argument $x(0 \leq x \leq 5)$. For the values $x \text { > }5, \varphi(x)=0.5$ is taken. For negative $x$ values, the same table is used, given that the Laplace function is odd, i. e. $\varphi(-x)=-\varphi(x)$.

The value of the Pearson's criterion or Chi-squared criterion $\left(\chi^{2}\right)$ is calculated by the formula:

$\chi^{2}=\sum_{i=1}^{n} \frac{\left(f_{e_{i}}-f_{t_{i}}\right)^{2}}{f_{t_{i}}}$   (3)

where $n$ – the number of groups into which empirical data are divided; $f_{e i}$ – the frequency observed in the $i$-th group; $f_{t i}$ – theoretical frequency.

For the $\chi^{2}$ distribution, a table is created, where the critical value of the $\chi^{2}$ acceptance criterion for the selected significance level α and degrees of freedom is specified.

The hypothesis of the normal distribution is accepted when the actual $\chi^{2}$ value does not exceed the critical table one (Fig. 2).

Fig. 2. Illustration of deciding on a hypothesis about the normal distribution using the $\chi^{2}$ criterion graph (for degrees of freedom $k=2$)

Table values show the right border of the $\chi^{2}$ distribution graph, which means that it does not confirm the hypothesis with the significance level $\alpha$.

You can do without tabular values and use the function of the MS Excel 2010 CHISQ.INV.RT program (0.05; 2). As the first argument here is the significance level, as the second – the number of degrees of freedom. The value calculated by the MS Excel 2010 program is usually somewhat more exact than the tabular one.

To assess the correctness of the hypothesis concerning the belonging to the normal distribution, it is recommended to calculate an additional p-level or p-value (probability to obtain such or even greater value of the criterion in the equity of the null hypothesis).

3. Calculation of the optimal level of shortage (if the hypothesis for the normal distribution is accepted).

If the hypothesis of belonging of statistical data to the normal distribution is confirmed, it is possible to use the formula for the optimum level of shortage and tabular data for normal distribution.

The optimal level of shortage:

$S=\frac{c_{h}}{c_{h}+c_{d e f}}$   (4)

where $C_{h}$ – the cost of storing a unit of goods per 1 day; $C_{def}$ – losses from the shortage of a unit of goods per 1 day.

As for the shortage losses, there are certain problems with determining the value of these expenditures. There can be quite a lot of approaches. Consider a few of them:

1. if the client who wants to buy a product and is not able to do it because of the lack of goods in stock, purchases this product in another company. In this case, the shortage losses correspond to the value of lost profit from the sale of goods, which is in lack;
2. if the client who cannot buy a product because of its absence, is offered a discount with which he will be able to obtain a product if he waits for arriving of the new shipment. In this case, the shortage losses correspond to the value, which reduces the profit from the production unit through the discount;
3. if for satisfaction of the client and fast delivery of the missing goods, blitz-orders with an increased cost of delivery are done. Then the additional cost of unplanned supply is distributed between all units of the shipment, so this will be shortage losses;
4. one option can be a combination of others. For example, ordering the blitz-supply and simultaneously offering a discount, to make customer be waiting for the product and do not apply to the company-competitor. Then the shortage losses are summed up.

4. Calculation of the amount of safety stock (considering the optimal level of shortage).

If the demand (sales, need) and time of delivery (the time of order fulfillment) obey the normal distribution law, the calculation of the safety stock requires a table value Z (the number of entailing no deficit standard deviations).

The number of entailing no deficit standard deviations and service level are connected by the NORMSINV function of the MS Excel 2010 program. In order to find the number of entailing no deficit standard deviations at a certain service level, it is necessary to specify the =NORMSINV function (service level).

The probability of shortage (in this technique – the optimum level of shortage) is defined as

$\mathrm{d}=1-L$   (5)

where $L$ – service level

Considering the $Z$ values and parameters of the distribution of consumption volume and delivery time, the safety stock is calculated according to the formula:

$Z_{s}=z \cdot \sqrt{\bar{t} \cdot \sigma_{s}^{2}+\bar{S}^{2} \sigma_{t}^{2}}$   (6)

where $Z_{s}$ – the value of safety stock, un.; $z$ – the number of entailing no deficit standard deviations; $\bar{t}_{n}$ – average execution time of the order, days; $s_{s}$ – standard deviation of demand, un./day; $s_{t}$ – average consumption, un./day; st - standard deviation of delivery time, days.

5. Calculation of the amount of optimal order quantity considering the possibility of shortage.

The model of inventory management with the possibility of shortage provides the calculation of the optimal order quantity by the formula

$Q_{\text {opt } t}=\sqrt{\frac{2 D C_{s}}{C_{h}}} \cdot \sqrt{\frac{C_{h}+C_{d e f}}{C_{d e f}}}$   (7)

where $D$ – demand for goods during a certain period of time, un.; $C_{s}$ – the cost of order delivery, which does not depend on the size of the order, UAH/un.; $C_{h}$ – the cost of storing a unit of goods during a certain period of time, UAH/un.; $C_{def}$ – the cost of shortage of a unit of goods during a certain period of time, UAH/un.

6. Calculation of the interval between orders.

To calculate the optimal interval between orders, first the number of deliveries per year should be determined:

$K=D / Q_{o p t^{-}}$   (8)

The interval between orders is calculated by dividing the number of days in the period by the number of deliveries for that period:

$I=T / K$   (9)

7. Calculation of the optimal order quantity in stochastic conditions.

The order quantity can be calculated as follows:

$Q_{i}^{\prime}=S \cdot\left(t_{i}+\bar{t}\right)+Z \cdot \sqrt{\left(t_{i}+\bar{t}\right) \cdot \sigma_{s}^{2}+\bar{S}^{2} \sigma_{t}^{2}}-Z_{T i}-Z_{t i}$   (10)

where $Q_{i}^{\prime}$ – the quantity of the order $i$, un.; $t_{i}$ – time interval between orders, days; $Z_{ti}$ – the level of the current inventory when issuing an order $i$, un; $Z_{ti}$ – the quantity of the stocks on the way not received at the moment of issuing an order $i$, un.

If the inventory level at the time of the order is zero and there are no stocks on the way, the order quantity can be obtained by the formula:

$Q_{i}^{\prime}=S \cdot\left(t_{i}+\bar{t}\right)+Z \cdot \sqrt{\left(t_{i}+\bar{t}\right) \cdot \sigma_{s}^{2}+\bar{S}^{2} \sigma_{t}^{2}}$   (11)

8. Calculation of the threshold level of the inventory management system.

The threshold level of inventory determines the level at which it is necessary to make the next order for replenishment of stocks. In stochastic conditions of consumption and receiving orders, the threshold level is determined by the formula:

$Q_{R O P}=\overline{S t}+Z \cdot \sqrt{\bar{t} \sigma_{s}^{2}+\bar{S}^{2} \sigma_{t}^{2}}=\overline{S t}+Z_{S}$   (12)