ON THE CHOICE OF THE NUMBER OF BRIGADE MEMBERS

I. A. OMELCHENKO

(Recommended by the scientific seminar of the departments of electric power stations and electrical networks and systems)

In practice, it is often necessary to determine the rational number of members of a brigade of electricians. If there are many workers in a brigade, idle time is possible because of an insufficient work front. If there are few, the brigade will not be able to complete the necessary work on time. The need arises to find the optimal number of workers in a brigade.

A solution to this problem can be found by applying queueing theory, which makes it possible to use a mathematical apparatus for evaluating phenomena and processes of a random character, occurring in any form [1, 2].

The parameters characterizing the method of queueing are the flow of requests and the time required to service one request. The flow of requests in the production of electrical-installation work is defined as the ratio of the estimated cost of an object, or group of objects, assigned to the brigade, to the output and duration of the work.

If we adopt the notation: \(\lambda\) — flow of requests, \(C\) — estimated cost of the object, \(B\) — output per worker, \(T\) — period of performance of electrical-installation work, then

\[ \lambda = \frac{C}{B \cdot T}. \tag{1} \]

The time for fulfilling one request is a random variable, since it depends on the type of work, the experience of the workers, the supply of materials, the state of construction work, etc., and obeys an exponential law [1].

Under the given conditions, one may pose the question of the number of brigade members servicing the flow of requests.

In servicing, a necessary condition is the requirement of regular fulfillment of the incoming flow of requests, so that a queue of unfulfilled requests does not form. It is also of interest what average number of requests will be fulfilled daily and what number of them will not be started on time.

It is clear that a request cannot be left unfulfilled; therefore the next request will wait for the start of service. It follows from this that the service system does not lose requests.

We assume that the flow of requests is the simplest one. The properties of the simplest flow are stationarity, ordinariness, and absence of aftereffect.

—do not contradict the properties possessed by a real flow [1]. In our case, a condition must be satisfied that ensures unlimited growth of the queue of unfulfilled requests. The number of workers must be no less than the product of the number of requests and the average time required to fulfill one; i.e.,

\[ \frac{\lambda}{\nu} \geq n, \tag{2} \]

where \(\frac{1}{\nu}\) is the average service time for one request;

\(n\) is the number of workers in the brigade.

However, this number of workers still does not guarantee good performance by the brigade. It may turn out that the fulfillment of a more labor-intensive request will lead to a lag in the work. It is also necessary to ensure that all workers are fully loaded and have time for vacations, training, and preparatory operations.

Let us assume, taking the above into account, that the brigade load should not exceed 90%.

This means that the probability that all workers are occupied must be no more than \(P=0.9\). In turn, it is necessary to achieve a situation in which the probability that the next request will not be serviced during the shift is very small, say, 0.01.

We write this as:

\[ P\{\beta > 1\}=0.01. \tag{3} \]

Then we have two contradictory conditions:

1. The waiting time before service begins for the next request must not be large.

2. The brigade load must not be greater than the specified value. In this case, the conditions written with the accepted restrictions taken into account will have the form

\[ P\{\beta > t\}=\Pi e^{-(n\nu-\lambda)t}. \tag{4} \]

Solving this formula with respect to \(n\), we obtain the rational number of brigade members.

Example. Let the estimated cost of the facility be 10 thousand rubles, the output 40 rubles per shift. The period for carrying out electrical installation work is two months.

Determine the flow

\[ \lambda=\frac{C}{BT}=\frac{10000}{40\cdot 50}=5. \tag{5} \]

The service time for one request is equal to one shift

\[ \frac{1}{\nu}=1, \tag{6} \]

whence \(\nu=1\).

Consequently, the minimum number of workers in the brigade must be

\[ n \geq \frac{5}{1}=5, \tag{7} \]

the rational number of workers in the brigade

\[ P\{\beta > 1\}=\Pi e^{-(n\nu-\lambda)t} \]

\[ 0.01=0.9e^{-(n\cdot 1-5)\cdot 1} \]

\[ n=9.5997. \tag{8} \]

Table 1

Thus, it follows that in order to satisfy our requirements it is necessary to have a brigade of at least 10 people.

To determine the distribution law of the waiting time until the start of service for the next demand for various \(n\), we shall use formula (8).

Let us compile the results of the calculations in a table.

Thus, with 6 people in the brigade, the probability that the waiting time until the start of service will not exceed one shift is equal to 0.33; that is, on average, one third of all work will not be covered during the working day. With \(n=10\), almost all work will be started on time during the shift. And with \(n=15\), after 0.1 of a shift only 0.33 demands will not have been started over the entire shift.

The calculations make it possible to conclude that, in the performance of electrical installation work, large brigades are more maneuverable and mobile. The size of a brigade may be limited only by operational considerations (the foreman’s business qualities, limited volumes of work, etc.); but in any case the number of workers must not be below the calculated value.

A large brigade, after dividing into teams, can carry out work at several sites, and, when necessary, assemble and concentrate work at facilities being handed over.

As the practice of the Omsk Installation Administration has shown, these calculations are borne out by reality.

REFERENCES

1. B. V. Gnedenko. Course in Probability Theory. Fizmatgiz, 4th ed., 1965.
2. A. Ya. Khinchin. Works on Queueing Theory. Fizmatgiz, 1963.