As a criterion for the optimality of transport transportation. Optimality criteria. What problems does it solve?

From what was said in the previous paragraph, the following follows: optimality criterion for the basic solution of the transport problem: if for some basic transportation plan the algebraic sums of tariffs by cycle for all free cells are non-negative, then this plan is optimal.

This implies a method for finding the optimal solution to the transport problem, which consists in the fact that, having some basic solution, they calculate the algebraic sums of tariffs for all free cells. If the optimality criterion is met, then this solution is optimal; if there are cells with negative algebraic sums of tariffs, then they move to a new basis, recalculating according to the cycle corresponding to one of such cells. The new basic solution obtained in this way will be better than the original one - the costs of its implementation will be lower. For a new solution, the feasibility of the optimality criterion is also checked and, if necessary, a cycle recalculation is performed again for one of the cells with a negative algebraic sum of tariffs, etc.

After a finite number of steps they arrive at the desired optimal basic solution.

If the algebraic sums of tariffs for all free cells are positive, we have a unique optimal solution; if the algebraic sums of tariffs for all free cells are non-negative, but among them there are algebraic sums of tariffs equal to zero, then the optimal solution is not the only one: when recalculating through the cycle for a cell with a zero algebraic sum of tariffs, we obtain the same optimal solution, but different from the original one ( the costs for both plans will be the same).

Depending on the methods for calculating the algebraic sums of tariffs for free cells, there are two methods for finding the optimal solution to the transport problem:

Distribution method. With this method, a cycle is built for each empty cell and the algebraic sum of tariffs is directly calculated for each cycle.

Method of potentials. With this method, the potentials of bases and consumers are first found, and then the algebraic sum of tariffs is calculated for each empty cell using the potentials.

The advantages of the potential method compared to the distribution method are that there is no need to construct cycles for each of the empty cells and the calculation of algebraic sums of tariffs is simplified. Only one cycle is built - the one by which the recalculation is carried out.

Using the potential method, we can talk not about the sign of the algebraic sums of tariffs, but about comparing indirect tariffs with true ones. The requirement that the algebraic sums of tariffs be non-negative is replaced by the condition that indirect tariffs do not exceed the true ones.

It should be borne in mind that the potentials (as well as the cycles) are determined anew for each new baseline.

Above we considered a closed model of the transport problem, with the correct balance when condition (1.3) is satisfied. If (1.4) (open model) is fulfilled, the balance of the transport task can be disturbed in two directions:

1. The amount of inventory at the points of departure exceeds the amount of submitted applications (transport task with excess inventory):

a i > b j (where i=1,...,m ; j=1,...,n);

2. The amount of submitted applications exceeds the available reserves (transport problem with an excess of applications):

and i< b j (где i=1,...,m ; j=1,...,n);

Let's consider these two cases sequentially:

Transport problem with excess inventory.

Let's reduce it to the previously considered transport problem with the correct balance. To do this, in addition to the available n destinations B 1, B 2, ..., B n, we introduce one more, fictitious, destination B n +1, to which we assign a fictitious request equal to the excess inventory over requests

b n+1 = a i - b j (where i=1,...,m ; j=1,...,n) ,

and the cost of transportation from all points of departure to the fictitious destination b n +1 will be considered equal to zero. By introducing a fictitious destination B n +1 with its request b n +1, we have equalized the balance of the transport problem, and now it can be solved as a regular transport problem with the correct balance.

Transport problem with an excess of requests.

This problem can be reduced to a regular transport problem with the correct balance if we introduce a fictitious departure point A m +1 with a stock a m +1 equal to the missing stock, and the cost of transportation from the fictitious point of departure to all destinations is assumed to be zero.

When solving a transport problem, the choice of optimality criterion is important. As is known, the assessment economic efficiency rough plan may be determined by one or another criterion that forms the basis for calculating the plan. This criterion is an economic indicator characterizing the quality of the plan. Until now, there is no generally accepted single criterion that comprehensively takes into account economic factors. When solving a transport problem, the following indicators are used as an optimality criterion in various cases:

1) Volume of transport work (criterion - distance in t/km). Minimum mileage is convenient for evaluating transportation plans, since the transportation distance can be determined easily and accurately for any direction. Therefore, the criterion cannot solve transport problems involving many modes of transport. It is successfully used in solving transport problems for road transport. When developing optimal schemes for transporting homogeneous cargo by vehicles.

2) Tariff fee for the transportation of goods (criterion - tariffs of freight charges). Allows you to obtain a transportation scheme that is the best from the point of view of the enterprise’s self-supporting indicators. All the surcharges, as well as the existing preferential tariffs, make it difficult to use.

3) Operating costs for transporting goods (criterion - cost of operating costs). More accurately reflects the cost-effectiveness of transportation various types transport. Allows you to make informed conclusions about the feasibility of switching from one type of transport to another.

4) Delivery times of goods (criterion - time consumption).

5) Levelized costs (taking into account operating costs, depending on the size of traffic and investment in rolling stock).

6) Given costs (taking into account the full operating costs of capital investments in the construction of rolling stock facilities).

where are operating costs,

Estimated investment efficiency ratio,

Capital investments per 1 ton of cargo throughout the section,

T - travel time,

C - the price of one ton of cargo.

Allows for a more complete assessment of rationalization different options transportation plans, with a fairly complete expression of the quantitative and simultaneous influence of several economic factors.

Let us consider a transport problem, the optimality criterion of which is the minimum cost of transporting the entire cargo. Let us denote by the tariffs for transporting a unit of cargo from the i-th point of departure to jth point destination, through – cargo reserves in i-th point departure, through – the demand for cargo at the jth destination, and through – the number of units of cargo transported from the i-th point of departure to the j-th destination. Then the mathematical formulation of the problem consists in determining the minimum value of the function

under conditions

Since the variables satisfy the systems linear equations(2) and (3) and the non-negativity condition (4), then the removal of existing cargo from all points of departure, delivery of the required amount of cargo to each of the destinations is ensured, and return transportation is also excluded.

Thus, the T-problem is an LP problem with m*n number of variables, and m+n number of restrictions - equalities.

Obviously, the total availability of cargo from suppliers is equal to , and the total demand for cargo at destinations is equal to units. If the total demand for cargo at destinations is equal to the supply of cargo at origins, i.e.

then the model of such a transport problem is called closed or balanced.

There are a number of practical problems in which the balance condition is not satisfied. Such models are called open. There are two possible cases:

In the first case, complete satisfaction of demand is impossible.

Such a problem can be reduced to a conventional transport problem as follows. If demand exceeds stock, i.e., a fictitious ( m+1)-th point of departure with cargo reserve and tariffs are set to zero:

Then you need to minimize

under conditions

Let us now consider the second case.

Similarly, when a fictitious ( n The +1)th destination with demand and the corresponding tariffs are considered equal to zero:

Then the corresponding T-problem will be written as follows:

Minimize

under conditions:

This reduces the problem to an ordinary transport problem, from the optimal plan of which the optimal plan of the original problem is obtained.

In what follows we will consider a closed model of the transport problem. If the model specific task is open, then, based on the above, we rewrite the table of conditions of the problem so that equality (5) is satisfied.

In some cases, you need to specify that products cannot be transported along certain routes. Then the costs of transportation along these routes are set so that they exceed the highest costs of possible transportation (so that it is unprofitable to transport along inaccessible routes) - when solving the problem at a minimum. To the maximum - the opposite.

Sometimes it is necessary to take into account that between some points of dispatch and some points of consumption contracts have been concluded for fixed volumes of supply, then it is necessary to exclude the volume of guaranteed delivery from further consideration. To do this, the volume of guaranteed supply is subtracted from the following values:

· from the stock of the corresponding dispatch point;

· according to the needs of the corresponding destination.

End of work -

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One of the areas of organizing transport logistics is optimizing not only the costs of using vehicles at the enterprise, but also optimizing the transportation itself.

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Studying what tasks are set for such activities in an organization allows you not to get confused in concepts and conduct effective business activities.

And, indeed, modern methods make it possible to predict a large number of transportations within the enterprise, whatever they may be - long-distance, international or interregional.

What it is

Optimization of road transport is the use of methods and technologies that allow the most accurate calculation of the time to manage routes and costs associated with transportation.

Such problems can be solved using calculations made by employees of the transport department, warehouses, management units for inventory control and other departments, using a computer program. Now he uses both in practice.

When defining the concept of optimization of road transportation, we can emphasize the constant, regular improvement of the transportation system (delivery, loading/unloading) of customer cargo.

Today such technologies are offered by specialists in the form software. Installation and use computer program, capable of accurately calculating routes, costs, directions and other nuances, allows you to not have a large staff in the transport department. And some organizations no longer even have one at all.

It is enough to train the operator-dispatcher or accountant to work in this service, and the enterprise will be fully provided with an effective solution to forwarding, intermediary and other tasks.

A transport problem is an algorithm for solving linear equations or solving them in other ways in order to find the optimal transportation plan. The variables in solving such problems are the points - from the supplier point to the consumer point (client).

The main goal of solving such problems is to reduce costs and maximize the optimization of the cargo transportation activities of the enterprise.

For example, if a transportation company can be aware of traffic jams in a timely manner, it will be easier for them to adjust the route of their vehicles in advance or along the way.

Awareness, savings, route calculation and other optimization technologies allow cargo to be delivered to the client quickly, on time and with maximum safety of the cargo.

What problems does it solve?

Many companies involved in cargo transportation are trying to find solutions to the following issues:

How to determine the optimal routes?
To what extent is it possible to identify shortest routes for cargo delivery?
What, how and where can you save money?
How to manage the management economic activity and route control if the dispatcher is not on site or the dispatcher is new to the job? And other questions.

Any task contains not only the essence of the problem that needs to be solved, but also the very method of resolving and improving it.

The tasks of a process that can optimally improve logistics in a road transport enterprise include the following suitable methods:

In some cases, there is a limitation on the number of containers, as well as the size of cargo transportation vehicles.
Minimizing reloading from one vehicle to another, from one container for transportation to another.
Organizing the convenience of loading or unloading operations in the truck body itself.
Automation of cargo labeling or its complete elimination frees the delivery process from delaying transportation times.
Consolidation of the number of units or the cargo itself as a whole for the vehicle, with mechanization of unloading on site at the customer.
Taking into account the type of vehicle for better cargo capacity.
Ensure that loads are the same size from order to order.
Increase the frequency of deliveries without increasing the cost of a unit of cargo transported.
Reducing the cost of packaging the transported goods while maintaining its integrity as much as possible during the process of moving from one point to another.
Standardization of the selection of machine types for each order, its size.
Quickly determine which deliveries to customers are maximum and which are minimum.
Minimizing the cost of transporting defective goods and return products.
Through the distribution system, reduce defects arising during transportation.
Reduce the amount of time spent on each cargo transit.
Reducing the time spent on each downtime that occurs for natural reasons - loading/unloading.
Minimizing the costs incurred in delivering cargo to the client.
Make the flow of cargo supplies uniform. This is especially true for seasonal specifics.
If we are talking about supply lots, then it is advisable to use a minimum number of machines used for these purposes.
Sorting cargo in such a way as to fit it into the vehicle as much as possible.
Reducing the time spent on receiving and receiving this or that information regarding cargo, supply requests and transportation itself. For example:
- information about downtime;
- operational information about the location of trucks;
- when the car arrived at its destination (time is indicated accurate to seconds);
- timely notification to the dispatcher about breakdowns along the way or other obstacles encountered to a safe arrival at the destination, and more.

Thus, we understand that in addition to the direct problems that are immediately obvious, there are also indirect reasons for the decrease in efficiency in the field of transport transportation.

For example, poorly organized requirements for the condition of cars and their timely repair can subsequently significantly impede the speed of delivery.

Or, if mistakes were made in the requirements for the transportation plan, then it is quite obvious that the driver may either go astray or not take into account the specifics of delivery times and other problems.

Transport optimization methods

Modern practical use Many tasks have allowed us to select today one of the most optimal methods - this is the heuristic method.

The approach consists mainly not in an emphasis on the efficiency of the route, but in maximizing the solution of the problem as a whole, which places enterprises on the basis of the client's order.

The linear programming model is mainly used. Such a targeted solution to problems of optimization of transport transportation allows them to be implemented quickly and accurately.

However, heuristics are inflexible, and therefore modern logisticians are still refining them. One thing is certain - this method clearly holds the future for the most efficient transport movement.

Let's define some of the most active methods, which transport workers use today:

Operations research.
Linear mathematical programming (LP):
- application of linear equations in calculations - a linear function of the elements of solution paths is the target (effective) function L (x);
- linear equalities/inequalities represent constraining conditions for possible solutions;
- the objective function is the largest or smallest value sought in mathematical programming, given the constraints;
- restrictions can be strict (exactly this amount and nothing else) and non-strict (no more or no less than some approximate value).
North-west corner technique (heuristics).
Mini-tariffing technique (heuristics).
Vogel's method (heuristics).
The traveling salesman method.
Application of Svir's algorithm (or as they say - “janitor-cleaner”, problems are solved by non-mathematical heuristics).

A team of scientists and highly qualified specialists is engaged in the study, development and practical application of certain techniques for making optimal decisions.

This type of activity is usually carried out by large concerns and supply and cargo transportation enterprises. These are those organizations that can afford to maintain separate laboratories or research departments to develop optimization programs.

In addition to research approaches, there are also mathematical calculations of logisticians in practice. They can use not only linear equations, but matrix calculations resulting from them.

Heuristics allow you to arrange values in the matrix, starting with the left cell at the top. And the arrangement begins with the lowest price, cheaper orders in the direction of increase.

This principle is also used in the minimum tariff method, but only then the values are not assigned from the angle. This way you can find the necessary solutions - from preliminary to final optimal results.

The Vogel technique assumes the presence of some kind of auxiliary coefficient, which should be calculated both by row and column of the matrix.

The coefficient will be equal to the difference in tariffs (the two minimum ones) that are available in the row or column. The distribution of odds follows the principle “from highest to lowest”.

Traveling salesman tasks allow you to find the most profitable route with a route passing at least once through the territory of a particular city and to the final point of return to the point of the original city.

This suggests that in the conditions of tasks based on the traveling salesman principle, benefit criteria based on the cost of the route will be set. Technically, it looks like this: throughout the entire trip, the truck must pass through each city once.

Therefore, due to the large accumulation of points along the route in one order, such a problem cannot be solved using operations research methods. Here the heuristic approach will be the most beneficial.

How is the enterprise organized?

In the practical solution of transport and cargo problems, it is necessary to coordinate all the nuances. It is necessary to correctly plan routes, loading of goods, timely unloading, return of the vehicle and other issues related to the operation as a whole.

And for this they take the following steps:

Organizing a coordination department or assigning a task to one coordinating employee (most often this is a boss or manager).
Related work between departments, issuance of relevant orders and orders and other interaction between departments.
Joint planning of routes, coordination and implementation of these plans, construction of schedules between departments or divisions.
Establishing common metrics between operations so as to achieve more functional performance.
If necessary, management is involved to establish connections between divisions and departments of the enterprise.
Responsibilities between departments and employees must be clearly divided.
If any disputes arise, then those persons who will be authorized to make final decisions must be identified.
Organize a single information field that would show the status of routes and cargo transportation.

A single information space between departments can be established between the warehouse, logistics department, dispatch center and other departments related to cargo transportation.

The mathematical formulation of the transport problem consists in determining the optimal plan for transporting some homogeneous cargo from T departure points A 1 , A 2 , …, A T V P destinations IN 1 , IN 2 , …, IN P . In this case, either the minimum cost of transporting the entire cargo or the minimum time for its delivery is usually taken as an optimality criterion. Let us consider a transport problem, the optimality criterion of which is the minimum cost of transporting the entire cargo.

Let's denote:

c ij – tariffs for transporting a unit of cargo from i th point of departure in j th destination,

a i – cargo reserves in i-th point of departure,

b j – cargo needs in j– m destination,

x ij – number of units of cargo transported from i th point of departure in j th destination.

Then the mathematical formulation of the problem consists in determining the minimum value of the function: (6.1)

under conditions
(6.2)

(6.3)

Any non-negative solution of systems of linear equations (6.2) and (6.3), defined by the matrix X=(x ij ) , is called a transport problem plan.

Plan X*=(x*ij) , at which function (6.1) takes its minimum value is called the optimal plan for the transport problem.

Typically, the initial data of a transport task is written in the form of a table.

There are several methods for determining the reference plan: the northwest corner method, the least cost method, the Vogel approximation method, etc.

Northwest Angle Method

The maximum permissible transportation is entered in the most northwestern cell of the table, and either all cargo is removed from the station A 1 and all other cells of the first line are crossed out, or the needs of the first consumer IN 1 are completely satisfied, then all the cells in the first column are crossed out. After this, the northwesternmost cell becomes the cell A 1 IN 2 or IN 2 A 1 . The algorithm continues until the table is filled. Disadvantages - the cost of transportation is not taken into account, and the result is a plan that is far from optimal.

Least cost method

The method to some extent takes into account transportation costs and is constructed as follows: the matrix is considered and the cell with the lowest cost is found, which is filled with the maximum allowable transportation. In most cases, this method gives a plan close to optimal.

Vogel approximation method

At each iteration, using all columns and all rows, find the difference between the two minimum tariffs written in them. These differences are recorded in a specially designated row and column in the table of problem conditions. Among the indicated differences, the minimum is chosen. In the row or column to which this difference corresponds, the minimum tariff is determined.

A widely used method for solving transport problems is the method of potentials.

This method allows you to evaluate the initial reference solution and find the optimal solution using a sequential improvement method.

Theorem 1. If the reference plan X=(x ij ) is optimal, there is a system of ( t+p) numbers called potentials U i , V j, such that:

U i + V j =C ij ,for x ij >0 (basic variables);

U i + V j =C ij ,for x ij =0 (free variables).

Thus, to check the optimality of the initial optimal plan, the following conditions must be met:

for each occupied cell, the sum of the potentials is equal to the cost of transporting a unit of cargo standing in this cell:

U i + V j =C ij

for each free cell, the sum of potentials is less than or equal to the cost of transporting a unit of cargo standing in this cell:

U i + V j £ WITH ij

Theorem 2. Any closed transport problem has a solution that can be achieved in a finite number of steps of the potential method.

Construction of a cycle and determination of the amount of load redistribution.

A cycle in a transportation table is a broken line with vertices in cells and edges located along the rows or columns of the matrix, satisfying two requirements:

the broken line must be connected, i.e. from any of its vertices you can get to another vertex by moving along the edges;

At each vertex of the cycle exactly two edges converge - one in a row, the other in a column.

Free recalculation cycle cell is such a cycle, one of the vertices of which is in a free cell, and the rest are in basic cells.

Let's give examples of some cycles.

Theorem 3. In the transportation table, for each free cell there is one recalculation cycle.

Algorithm of the potential method

Let's assign each station A i potential And i, and each station IN j potential v j. To do this, for each filled cell X ij ≠0 let's make an equation And i + v j =c ij . Let's add And 1 =1 (any other value is possible) and find all other potentials.

Let's check the optimality of the found reference plan. To do this, we calculate the sum of potentials for free cells. If this amount less cost transportation With ij, standing in this cell, then the optimal solution is found. If it is more, then there is a violation in this cell equal to the difference between this amount and the cost of transportation. We will find all such violations (we will write them down in the same cells at the bottom right). Let's choose the largest of these violations and build a cycle for recalculating a free cell that will start from the marked cell with the greatest violation.

3. The recalculation cycle begins in a free cell, where we put a plus sign, and the remaining cells are occupied. The signs in these cells alternate. Let us choose the smallest of the transportations in the cells with a minus sign. Then we add this transportation to the transportation with a plus sign and subtract it from the transportation with a minus sign. We obtain a new optimal solution. Let's check it for optimality.

4. For new potentials, we check the optimality condition. If the optimality conditions are met, then an optimal solution is obtained; if not, then we continue to search for the optimal solution using the potential method.

Example 7.1. Four enterprises in this economic region use three types of raw materials to produce products. The raw material requirements of each enterprise are respectively 120, 50, 190 and 110 units. Raw materials are concentrated in three places where they are received, and reserves are respectively equal to 160, 140, 170 units. Raw materials can be imported to each of the enterprises from any point of receipt. Transportation tariffs are known quantities and are specified by the matrix
.

Draw up a transportation plan in which the total cost of transportation is minimal.

Solution:

closed type task.

Let's draw up the first plan of the transport problem using the northwestern angle method. Let's start filling out the table cells from the top left cell.

S 1 =120·7+40·8+10·5+130·9+60·3+110·6=3120

Let's draw up the first plan using the minimum cost method. We will fill the cells with minimum tariffs.

S 2 =160 1+120 4+20 8+50 2+30 3+90 6=1530

The cost of this transportation plan is almost two times less. Let's start solving the problem with this plan. Let's check it for optimality. Let us introduce the potentials α i – respectively, departures, β j – respectively, destinations. Based on the occupied cells, we compose a system of equations α i + β j =c ij:

For free cells of the table, we check the optimality criterion

Let's make up the differences

The plan is not optimal because there is a positive assessment
Let's build a recalculation cycle from it. This is a broken line of links that are located strictly vertically or horizontally, and the vertices are in occupied cells. In the bad cell we will put a (+) sign. At the remaining vertices the signs alternate. From the negative vertices we select the smallest number and shift it along the cycle. We moved on to a new reference plan.

S 3 =70 1+90 2+120 4+20 8+50 2+120 3=1350

The cost of transportation is lower, i.e. the plan has been improved. Let's check now new plan for optimality. By occupied cells:

For free cells:

The plan is not optimal because there is a positive assessment
We build a recalculation cycle and move on to a new plan.

S 4 =50 1+110 2+120 4+20 5+30 2+140 3=1330

We check the new plan for optimality.

By occupied cells:

For free cells:

The optimality criterion is satisfied, i.e. the last plan is optimal.

Answer:

Design parameters. This term refers to independent variable parameters, which completely and unambiguously define the design problem being solved. Design parameters are unknown quantities whose values are calculated during the optimization process. Any basic or derived quantities that serve to quantitatively describe the system can serve as design parameters. So, these can be unknown values of length, mass, time, temperature. The number of design parameters characterizes the degree of complexity of a given design problem. Usually the number of design parameters is denoted by n, and the design parameters themselves by x with the corresponding indices. Thus, n design parameters of this problem will be denoted by

X1, X2, X3,...Xp.

It should be noted that design parameters may be referred to as internal controllable parameters in some sources.

Target function. This is an expression whose value the engineer strives to make maximum or minimum. The objective function allows you to quantitatively compare two alternative solutions. From a mathematical point of view, the objective function describes some (n+1)-dimensional surface. Its value is determined by the design parameters

M = M (x1,x2,…,xn).

Examples of objective functions often found in engineering practice are cost, weight, strength, dimensions, efficiency. If there is only one design parameter, then the objective function can be represented by a curve on the plane (Fig. 1). If there are two design parameters, then the objective function will be depicted as a surface in three-dimensional space (Fig. 2). With three or more design parameters, the surfaces specified by the objective function are called hypersurfaces and cannot be depicted by conventional means. The topological properties of the surface of the objective function play a big role in the optimization process, since the choice of the most efficient algorithm depends on them.

Figure 1. One-dimensional objective function.

Figure 2. Two-dimensional objective function.

The objective function in some cases can take the most unexpected forms. For example, it cannot always be expressed in a closed mathematical form; in other cases, it can be a piecewise linear function. Specifying an objective function may sometimes require a table of technical data (for example, a table of the state of water vapor) or may require an experiment. In some cases, design parameters take only integer values. An example would be the number of teeth in a gear train or the number of bolts in a flange. Sometimes design parameters have only two meanings - yes or no. Qualitative parameters, such as the satisfaction experienced by the buyer who purchased the product, reliability, aesthetics, are difficult to take into account in the optimization process, since they are almost impossible to characterize quantitatively. However, in whatever form the objective function is presented, it must be an unambiguous function of the design parameters.

A number of optimization problems require the introduction of more than one objective function. Sometimes one of them may be incompatible with the other. An example is aircraft design, where maximum strength, minimum weight and minimum cost are simultaneously required. In such cases, the designer must introduce a system of priorities and assign a certain dimensionless factor to each objective function. As a result, a “compromise function” appears, which allows the use of one composite objective function during the optimization process.

Search for minimum and maximum. Some optimization algorithms are designed to find the maximum, others - to find the minimum. However, regardless of the type of extremum problem being solved, you can use the same algorithm, since the minimization problem can easily be turned into a maximum search problem by reversing the sign of the objective function. This technique is illustrated in Fig. 3.

Figure 3. When the sign of the objective function changes to the opposite in a minimum problem, it turns it into a maximum problem.

Design space. This is the name of the area defined by all n design parameters. The design space is not as large as it may seem, since it is usually limited by a number of conditions related to the physical nature of the problem. The restrictions may be so strong that the problem will not have a single satisfactory solution. Constraints are divided into two groups: constraints - equality and constraints - inequality.

Equality constraints are dependencies between design parameters that must be taken into account when finding a solution. They reflect the laws of nature, economics, law, prevailing tastes and availability necessary materials. The number of constraints - equalities can be any. They look like

C1 (X1, X2, X3, . . ., Xn) = 0,

C2 (X1, X2, X3, . . ., X n) = 0,

..……………………………..

Cj(X1, X2, X 3,..., Xn) = 0.

Inequality constraints are a special type of constraint expressed by inequalities. In the general case, there can be as many of them as you like, and they all have the form

z1 ?r1(X1, X2, X3, . . ., Xn) ?Z1

z2 ?r2(X1, X2, X3, . . ., Xn) ?Z2

………………………………………

zk ?rk(X1, X2, X3, . . ., Xn) ?Zk

It should be noted that very often, due to restrictions, the optimal value of the objective function is not achieved where its surface has a zero gradient. Often the best solution corresponds to one of the boundaries of the design space.

Direct and functional restrictions. Direct restrictions have the form

xнi? xi? xвi at i? ,

where xнi, xвi are the minimum and maximum permissible values of the i-th controlled parameter; n is the dimension of the space of controlled parameters. For example, for many objects, the parameters of elements cannot be negative: xнi ? 0 (geometric dimensions, electrical resistance, mass, etc.).

Functional restrictions, as a rule, represent conditions for the performance of output parameters that are not included in the target function. Functional restrictions may be:

1) type of equalities
w(X) = 0; (2.1)
2) type of inequalities

tz (X) › 0, (2.2)

where w(X) and q(X) are vector functions.

Direct and functional restrictions form the permissible search area:

ХД = (Х | w(Х) = 0, ц (Х)›0, xi › xнi ,

xi ‹ xвi for i ? ).

If restrictions (2.1) and (2.2) coincide with the performance conditions, then the permissible area is also called the XP performance area.

Any of the points X belonging to the CD is a feasible solution to the problem. Parametric synthesis is often posed as the problem of determining any of the feasible solutions. However, it is much more important to solve the optimization problem - to find the optimal solution among the feasible ones.

Local optimum. This is the name of the point in the design space at which the objective function has highest value compared to its values at all other points in its immediate vicinity. Figure 4 shows a one-dimensional objective function that has two local optima. Often the design space contains many local optima and care must be taken not to mistake the first one for the optimal solution to the problem.

Figure 4. An arbitrary objective function can have several local optima.

The global optimum is the optimal solution for the entire design space. It is better than all other solutions corresponding to local optima, and it is what the designer is looking for. It is possible that there are several equal global optima located in different parts design space. This allows you to select best option from equal optimal options according to the objective function. IN in this case the designer can choose an option intuitively or based on a comparison of the resulting options.

Selection of criteria. The main problem in setting extremal problems is the formulation of the objective function. The difficulty in choosing an objective function lies in the fact that any technical object initially has a vector nature of optimality criteria (multi-criteria). Moreover, an improvement in one of the output parameters, as a rule, leads to a deterioration in the other, since all output parameters are functions of the same controlled parameters and cannot change independently of each other. Such output parameters are called conflict parameters.

There must be one target function (uniqueness principle). Reducing a multi-criteria problem to a single-criteria problem is called convolution of a vector criterion. The task of finding its extremum is reduced to a problem of mathematical programming. Depending on how the output parameters are selected and combined in the scalar quality function, partial, additive, multiplicative, minimax, statistical criteria and other criteria are distinguished. The technical specifications for the design of a technical object indicate the requirements for the main output parameters. These requirements are expressed in the form of specific numerical data, the range of their variation, operating conditions and acceptable minimum or maximum values. The required relationships between output parameters and technical requirements (TR) are called performance conditions and are written in the form:

yi< TTi , i О ; yi >TTj, j O;

yr = TTr ± ?yr; r O .

where yi, yj, yr - set of output parameters;

TTi, TTj, TTr - required quantitative values of the corresponding output parameters according to the technical specifications;

Yr is the permissible deviation of the r-th output parameter from the TTr value specified in the technical specifications.

Operating conditions are of decisive importance in the development of technical devices, since the design task is to select a design solution in which all operating conditions are best met over the entire range of changes in external parameters and when all requirements of the technical specifications are met.

Particular criteria can be used in cases where among the output parameters one main parameter yi(X) can be identified, which most fully reflects the effectiveness of the designed object. This parameter is taken as the objective function. Examples of such parameters are: for an energy facility - power, for a technological machine - productivity, for a vehicle - load capacity. For many technical objects, this parameter is cost. The operating conditions of all other output parameters of the object are referred to as functional restrictions. Optimization based on such a formulation is called optimization according to a particular criterion.

The advantage of this approach is its simplicity; a significant drawback is that a large margin of efficiency can be obtained only for the main parameter, which is accepted as the objective function, and other output parameters will have no margins at all.

The weighted additive criterion is used when the performance conditions make it possible to distinguish two groups of output parameters. The first group includes output parameters, the values of which should be increased during the optimization process y+i(X) (performance, noise immunity, probability of failure-free operation, etc.), the second group includes output parameters, the values of which should be decreased y-i (X) ( fuel consumption, duration of the transient process, overshoot, displacement, etc.). Combining several output parameters, which generally have different physical dimensions, into one scalar objective function requires preliminary normalization of these parameters. Methods for normalizing parameters will be discussed below. For now, we will assume that all y(X) are dimensionless and among them there are no ones that correspond to performance conditions of the type of equality. Then, for the case of minimizing the objective function, the convolution of the vector criterion will have the form

where aj>0 is a weighting coefficient that determines the degree of importance of the j-th output parameter (usually aj is selected by the designer and remains constant during the optimization process).

The objective function in the form (2.1), expressing the additive criterion, can also be written in the case when all or the main performance conditions have the form of equalities. Then the objective function

determines the root-mean-square approximation of yj(X) to the given technical requirements TTj.

The multiplicative criterion can be used in cases where there are no equality-type performance conditions and the output parameters cannot take zero values. Then the multiplicative objective function to be minimized has the form

One of the most significant drawbacks of both additive and multiplicative criteria is the failure to take into account the technical requirements for output parameters in the formulation of the problem.

The function form criterion is used when the task is set of the best match of a given (reference) characteristic yCT(X, y) with the corresponding output characteristic y(X, y) of the designed object, where y is some variable, for example, frequency, time, selected phase variable. These tasks include: designing an automatic control system that provides the required type of transient process for the controlled parameter; determining the parameters of the transistor model that give maximum agreement with its theoretical current-voltage characteristics with experimental ones; search for parameters of beam sections, the values of which lead to the best coincidence of the given stress diagram with the calculated one, etc.

The use of a particular optimization criterion in these cases comes down to replacing continuous characteristics with a finite set of nodal points and choosing one of the following objective functions to be minimized:

where p is the number of nodal points uj on the axis of the variable u; aj - weighting coefficients, the values of which are greater, the smaller the deviation y(X, φj) - yTT(X, φj) must be obtained at the j-th point.

Maximin (minimax) criteria allow one to achieve one of the goals of optimal design - the best satisfaction of performance conditions.

Let us introduce a quantitative assessment of the degree of fulfillment of the j-th performance condition, denote it by zj and call it the performance reserve of the parameter yj. The calculation of the margin for the jth output parameter can be performed in various ways, for example,

where aj is the weighting coefficient; yjnom - nominal value of the j-th output parameter; dj is a value characterizing the spread of the jth output parameter.

Here it is assumed that all relations are reduced to the form yi< TТj. Если yi >TTj, then -yj< -TТj . Следует принимать аj >1 (recommended values 5 ? aj ? 20), if it is desirable to achieve the j-th technical requirement with a given tolerance, i.e. yj = TTj ± ?yj; aj=l, if it is necessary to obtain the maximum possible estimate zj.

The quality of functioning of a technical system is characterized by a vector of output parameters and, therefore, by a vector Z=(zm,zm,…,zm). Therefore, the target function should be formed as some function μ(Z) of the evaluation vector. For example, if the target function considers the reserve of only that output parameter that at a given point X is the worst from the standpoint of meeting the requirements of the technical specifications, then

where m is the number of working capacity reserves.

It is natural now to pose the problem of choosing a search strategy X that would maximize the minimum of the reserves, i.e.

where HD is the searchable area.

The optimization criterion with objective function (2.6) is called the maximin criterion.

Statistical criteria. Optimization using statistical criteria is aimed at obtaining the maximum probability P of performance. This probability is taken as the objective function. Then we have the problem

Normalization of controlled and output parameters. The space of controlled parameters is metric. Therefore, when choosing the directions and values of search steps, it is necessary to introduce one or another norm, identified with the distance between two points. The latter assumes that all controlled parameters have the same dimension or are dimensionless.

Various rationing methods are possible. As an example, consider the method of logarithmic normalization, the advantage of which is the transition from absolute increments of parameters to relative ones. In that case i the controlled parameter ui is converted into dimensionless xi as follows:

where oi is a coefficient numerically equal to unity of the parameter ui.

Normalization of output parameters can be performed using weighting coefficients, as in the additive criterion, or by moving from уj to performance reserves zj according to (2.5).