## Analytic model just in time

Exactly on time - one of the most famous and common logistics concepts. Today, the degree of its implementation allows us to talk about technologies, systems just in time & quot ;. In the literature on logistics, the concept of just in time is considered in relation to the logistics cycle, which is one of the main objects of logistics and supply chain management. In each functional area of the logistics, order cycles (logistic or functional cycles) are formed, therefore, in managing the functional areas of logistics, the actual problem is managing the order cycle precisely in time.

In Ch. 2 the concept of the logistic cycle is considered, its stages, range and expected value of the duration of the individual stages of the cycle are presented (see Table 2.3). The basic structure of all logistic cycles (links, nodes, etc.) is the same for physical distribution, logistical support of production and supply. To manage logistics cycles in functional logistics areas, you need to investigate the configuration of a single cycle in order to find out the most important relationships and control lines. The time intervals for the execution of individual operations, of which any logistic cycle consists, are random variables, this allows us to assume that the entire cycle is a random variable that obeys a certain distribution law.

Given the features of the logistics cycle, the formation of the model "just in time can be represented in the following steps.

1. Collection and statistical processing of data on the duration of individual components of the logistics cycle.

2. Calculation of statistical parameters of the logistic cycle. The average value of the logistic cycle time and the standard deviation are determined by the formulas

where 7], <7; - the average values and the mean square deviations of the time of the 1st operation of the logistics cycle, respectively; * г ~ - * is the correlation coefficient between the 1st and

*operations of the loop.*

**] - n** The sign * r < * means that the summation extends to all possible pairwise combinations of random variables. If the quantities in question are not correlated, then for all

*the formula for the mean square deviation*

**r ^ =***is simplified.*

**at** For the execution time of the order execution cycle, a correlation matrix can be compiled, in which, taking into account the sequence of operations of the cycle, for all * r>) * the correlation coefficient is zero:

3. Determination of the duration of the logistic cycle F0 with a given confidence probability P. The random nature of the execution time of individual components of the logistic cycle allows us to conclude that the concept of "just in time" should be considered taking into account the confidence limits of the duration of the cycle. For example, if the cycle time distribution function obeys the normal law, the upper confidence bound of the time of the logistic cycle

where is the index of the normal distribution corresponding to the probability * P. *

4. Determining the execution time of the logistics cycle just in time & quot ;. Consumers can put forward the requirement to deliver goods at a certain (precise) time or to set a time interval with a small deviation, which it considers permissible. If the execution of the order just in time is given by some definite value of time, i.e. the order must be executed not later than this point, then the logistic cycle time "just in time" is the upper confidence limit and can be calculated using the formula

where * Ti - * the time of the start of the logistics cycle.

If the lead time is just in time is set not only by the approximate value, but also by some deviation from it or by the time interval, it is important to estimate not only the upper limit of the order execution time by the formula (5.8), but also the lower limit:

5. Calculation of the probability of fulfilling the logistic cycle just in time & quot ;. If consumers have determined the lead time of an order "just in time" and " (that is, only an estimate of the upper confidence bound of time), then the probability of fulfilling the order accurately in time can be calculated using the formula

where Φ (...) is a tabulated function of the normal distribution law.

For an accelerated assessment of the probability of a logistics cycle, you can build a graph of the distribution function of the random value of the order time and find not only the probability of order fulfillment, but also the guaranteed time of this order with a certain probability.

In case the order execution time is set by an interval, i.e. not earlier than and no later than or a certain plus or minus value, some deviation from it, the probability of fulfilling the order will be determined as follows:

where a, p are the lower and upper bounds of the specified lead time of the order "just in time."

6. Formation of the objective function of the optimization task of performing the logistics cycle "just in time". The problem of late arrival and early arrival of an order is related to the costs of one or several participants in the supply chain. Given the random nature of the logistics cycle, we can talk about the uncertainty of such costs.

If we assume that the random variables 7) that characterize the duration of the individual operations of the cycle, and do not consider other possible limitations in the implementation of the logistics cycle (regulatory, financial, etc.), then the formally targeted function of the economic and mathematical task of performing a logistics cycle "just in time" can be represented in the form

where * Cj (t) - * the dependence of the costs of the r'th operation of the cycle on its duration

You can choose the mean values of 7] or the time estimate * Ti * for each operation with a given confidence probability

**Р.** The costs of performing logistics cycle operations * Ct (t) * are contradictory. For example, transport costs increase with decreasing delivery time, while increasing storage time leads to increased costs.

If the average values of Γ = const, then the variance * of * is a measure of the uncertainty of the logistic cycle, and the relationship (5.12) can be represented, in particular, as follows:

where C, (a) is the dependence of the execution costs of the * 1st * operation from the dispersion (uncertainty) of the execution time.

In order to reduce the risk of non-fulfillment of orders in exact contractual terms, it is important to be able to manage the order procedures, in particular, to choose the best option for performing logistic cycle operations. The range of management decisions can include:

• more precise definition of the time of the order execution;

• monitoring the duration of individual components of the logistics cycle, adjusting the lead time in the event of deviations of the actual time from the normative and subsequent agreement with the consumer;

• prompt change in the composition of the participants in the supply chain (for example, the replacement of the carrier), the route of delivery (for example, using a toll road, where the speed and the traffic flow is faster), etc.

• use of outsourcing to implement individual components of the logistics cycle. For example, customs clearance of goods with the help of a customs representative will take on average less time than self-registration. However, this intermediary renders services on a reimbursable basis, which must be taken into account when making decisions.

** ** ** Practice questions **

The customer made the order at the company "Favorite", the contract recorded the lead time - 14 days from the time of order. The enterprise keeps track of the execution time of the components of the logistics cycle, the statistical parameters of the duration of the stages of the cycle are presented in Table. 5.12. Will the company meet the contractual obligations? How likely will an order be executed on time?

Define the statistical characteristics for the general order execution cycle.

The average value (see the formula (5.5)), days:

* T = * + 2 + 3.5 + 4.5 + 1 = 12. The mean square deviation, days, is calculated by the formula (5.6) that correlation between operations of a functional cycle is absent:

We calculate the probability of fulfilling the order for 14 days. When substituting the values in the formula (5.7), we find:

According to special tables (they are in each textbook on probability theory and statistics), we determine the probability of the order in time. It is 0.7. This is a low value, since 30% of orders can be disrupted.

Suppose that the company Favorite conducted a series of activities aimed at improving the reliability of supply. Having simulated the results of the measures, they saw that the time of the operations of the logistics cycle was reduced, which led to a decrease in a ,. The changed characteristics are shown in Table. 5.13.

* Table 5.13. *

**Changed parameters of the duration of the logistics cycle**

The average quadratic deviation of the order execution time, calculated by the formula (5.6), will be, day:

Then

The statistical parameters of the duration of the logistics cycle and the probability of delivering products exactly in time after 14 days * P * = 0.855. The risk of failure of timely delivery was 0.145. Thanks to the measures, the risk was reduced by half.

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