Optimal value of the objective function. Canonical lp problem. Simplex method for solving linear programming problems

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 objective function, often encountered 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 imaged by ordinary 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 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 optimal value the objective function is not achieved where its surface has a zero gradient. Often The best decision corresponds to one of the boundaries of the design area.

Direct and functional restrictions. Direct restrictions have the form

xni ? xi? xвi at i? ,

where xнi, xвi - minimum and maximum valid values 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 HR performance area.

Any of the points X belonging to the CD is a feasible solution to the problem. Often parametric synthesis is 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 choose the best option among equals optimal options by 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. IN terms of reference for the design of a technical object, requirements for the main output parameters are indicated. 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- tolerance r-th output parameter from the TTr value specified in the technical specifications.

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

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 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 performance reserve can be obtained only for the main parameter, which is accepted as the objective function, and other output parameters will have no reserves 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 accept 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's introduce quantification degree of fulfillment of the j-th performance condition, we 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 requirements with a given tolerance, i.e. yj = TTj ± ?yj; aj=l, if it is necessary to obtain the maximum possible estimate zj.

Performance quality technical system characterized by a vector of output parameters and, therefore, a vector Z=(zm,zm,…,zm). Therefore, the target function should be formed as a certain function μ(Z) of the evaluation vector. For example, if the target function is considered to be the stock 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.

Possible various ways rationing. 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 controlled parameter ui is converted to dimensionless xi as follows:

where oi is the coefficient, numerically equal to one 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).

An objective function is a function with some variables on which the achievement of optimality directly depends. It can also act as several variables that characterize a particular object. We can say that, in essence, it shows how we have progressed towards achieving our goal.

An example of such functions is the calculation of the strength and weight of the structure, installation capacity, production volume, transportation costs, and others.

The objective function allows you to answer several questions:

Whether this or that event is beneficial or not;

Is the movement going in the right direction?

How correct was the choice made, etc.

If we do not have the opportunity to influence the parameters of the function, then we can say that we cannot do anything, except perhaps just analyze everything. But to be able to change something, there are usually mutable function parameters. the main task- this is to change the values ​​to those at which the function becomes optimal.

Objective functions cannot always be presented in the form of a formula. This could be a table, for example. Also, the condition can be in the form of several objective functions. For example, if you want to ensure maximum reliability, minimum costs and minimal material consumption.

Optimization problems must have the most important initial condition - an objective function. If we do, then we can assume that optimization does not exist. In other words, if there is no goal, then there are no ways to achieve it, much less favorable conditions.

Optimization tasks can be conditional or unconditional. The first type involves restrictions, that is, certain conditions when setting the problem. The second type is to find the maximum or at existing parameters. Often such problems involve searching for a minimum.

In the classical understanding of optimization, such parameter values ​​are selected for which the objective function satisfies the desired results. It can also be described as the process of selecting the most best option of the possible. For example, choosing the best resource allocation, design option, etc.

There is such a thing as incomplete optimization. It can form for several reasons. For example:

The number of systems that reach the maximum point is limited (a monopoly or oligopoly has already been established);

There is no monopoly, but there are no resources (lack of qualifications in any competition);

The absence of the most or rather “ignorance” of it (a man dreams of a certain beautiful woman, but it is unknown whether such a thing exists in nature), etc.

In conditions of market relations, sales and production activities For firms and enterprises, the basis for decision-making is information about the market, and the validity of this decision is checked when entering the market with the corresponding product or service. In this case, the starting point is to study consumer demand. To find solutions, a consumption target function is established. It shows the amount of goods consumed and the degree of satisfaction of consumer needs, as well as the relationship between them.

Being centralized, it performs the following functions: the function of regulating prices between new and serial products the function of targeted and constant support - the process of production of new equipment with monetary funds; the function of redistribution of funds for the development of new equipment between enterprises participating in the development to varying degrees new technology.  

As for state expenditures, they represent trust funds of funds allocated and actually used by the state to implement its functions. The main functions of targeted spending include  

Let us now move on to the description of the objective functions. PM objective function  

Target function. The objective function defines the problem that must be solved during the optimization process. For example, in this chapter we are concerned with minimizing the risk of an asset portfolio. A typical objective function for a portfolio of risky assets would be  

OBJECTIVE FUNCTION is a function that connects the goal (the variable being optimized) and the controlled variables in the optimization problem.  

The first expression is called the objective function (equal to the product of profit per unit of product c, - by the output of this product Xj). The remaining equations constitute linear constraints, which mean that the consumption of raw materials, semi-finished products, product quality, power, i.e., initial resources, should not exceed predetermined values ​​/ /. Coefficients a,7 are constant values ​​showing the resource consumption for / and product. The problem can be solved if the variables are non-negative and the number of unknowns is greater than the number of constraints. If the last condition is not satisfied, then the problem is inconsistent.  

As the objective function we take the production of A-76 gasoline  

The objective function has the form  

Since variable costs depend on production volume, the difference between price and variable costs must be maximized. Conditionally fixed expenses (depreciation, expenses for current repairs, wages with accruals, general shop and plant expenses) are not included in the model and are subtracted from the objective function obtained on the computer. If the operating duration of the installation for each option is taken as unknowns, then the variable costs are calculated for one day of its operation.  

Condition (4.56) characterizes the objective function, the maximum difference between the wholesale price and the cost of commercial gasoline.  

The objective function for solving this problem can be either the maximum profit for the enterprise (4.52) or the maximum volume of production of marketable products in value terms (4.53)  

The given model for calculating the cost is at the same time a model for calculating the profit of the enterprise. However, the main effect of implementing cost calculations on a computer is the ability to use the results of this calculation to optimize the enterprise's production program. In this case, the maximum profit from product sales can be taken as the objective function. When optimizing a production program, it is necessary to maximize a function of the form  

The advantages and disadvantages of a customer-centric structure are generally the same as those of a product structure, given the differences associated with different objective functions.  

Since the integral energy intensity is determined taking into account direct and indirect energy costs (through material, technical and labor resources), the reduction in the energy intensity of each of the consumed and used resources is also taken into account in the total economic savings. The energy intensity of each target effect (product, service) is calculated as the sum of energy intensity at the stages of its formation. For example, the energy intensity of a pipe consists of the energy intensity of ore mining, steel smelting, sheet rolling and the pipe manufacturing itself and is measured in kilograms of standard fuel per 1 ruble. its value. Existing forms of accounting and the proposed methodology make it possible to determine these indicators for any product, service, etc. Thus, to save energy it is necessary to reduce the consumption of production resources of all types while achieving a given target effect. These resources and the final target effect are measured in monetary terms. The costs for them depend on the scale of the technology used, the level of sophistication of the technical means in which the main target function is implemented - the target technological process, the number of scales and ramifications auxiliary functions ensuring the fulfillment of the main function, as well as the level of equipment and technology used.  

Expression (I) is usually called the original system of equations and inequalities, and expression (II) - the functional of the linear programming problem or the target function. The objective function is an optimality criterion. The first group of inequalities of the system (I) makes it possible to take into account in the calculation the limitations in the existing capacities of fuel production enterprises at the beginning of the planning period. The second group of inequalities takes into account  

To M. m. in the west. And. include the following, sections of applied mathematics, mathematical programming, game theory, queuing theory, scheduling theory, inventory management theory and the theory of wear and tear and replacement of equipment. Mathematics (or optimal) programming develops the theory and methods for solving conditional extremal problems, is the main thing. part of the formal apparatus for analyzing various management, planning and design problems. Plays a special role in optimization problems of outdoor planning. kh-va and management nronz-vom. Problems of economic planning and technology management usually come down to choosing a set of numbers (the so-called control parameters) that provide the optimum of a specific function (objective function or solution quality indicator) under restrictions of the type of equalities and inequalities determined by the operating conditions of the system. Depending on the properties of the functions that determine the quality indicator and the limitations of the problem, mathematic. programming is divided into linear and nonlinear. Problems in which the objective function is linear, and the conditions are written in the form of linear equalities and inequalities, constitute the subject of a linear program. Problems in which the quality indicator of the solution or some of the functions that determine the constraints are nonlinear, belong to the nonlinear program [) onan p go. Nonlinear programming, in turn, is divided into convex and non-convex programming. Depending on whether the initial parameters characterizing the conditions of the problem are well-defined numbers or random variables, in mathematics. programming, there are differences between management and planning methods under conditions of complete and incomplete information. Methods for setting and solving conditional extremal problems, the conditions of which contain random parameters, are the subject of stochastic programming.  

The goal of the model is to maximize the total discounted net income (up to profit) for a set of fields and gas pipeline systems under given technological and economic restrictions. The model allows the use of alternative criteria - minimizing the weighted sum of deviations from a given value of the objective function (target programming); calculations can be carried out for a given level of investment, for a given level of production, for a given value of the DPV.  

The success of such a business woman depends on how well the administration recognizes possible fields that can provide job satisfaction. It has been noticed that women cope well with functions that require communication with people, but if this is also an intellectual activity - teacher, journalist, tour guide, etc. - then the high efficiency of their work and their positive assessment will almost certainly coincide. In Japan, women are rarely able to obtain engineering and natural science education, especially in modern, most promising specialties, however, their inclusion in the widespread mobile target groups for solving non-standard problems turns out to be productive. The ingenuity of the female mind has been noticed for a long time and in all countries. In Japan, when they want to provide clear proof of this, they remember the competition announced by the famous company “Aji no Moto”. She offered a large cash prize for tips on how to increase sales of her condiment, which looks like salt and is sold in the likeness of salt shakers. People wrote treatises and brought in all kinds of scientific knowledge. But the winner was a housewife, whose answer fit in one line: “Make the holes in the salt shaker larger.”  

If there is only one limiting factor (for example, a scarce machine), a solution can be found using simple formulas(see link at the beginning of the article). If there are several limiting factors, the method is used linear programming.

Linear programming is the name given to a combination of tools used in management science. This method solves the problem of allocating scarce resources among competing activities in order to maximize or minimize some numerical values, such as contribution margin or expenses. In business, it can be used in areas such as production planning for maximum magnification profits, selecting components to minimize costs, selecting a portfolio of investments to maximize profitability, optimizing the transport of goods to reduce distances, assigning personnel to maximize work efficiency and scheduling work to save time.

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Linear programming involves the construction mathematical model the problem under consideration. After which the solution can be found graphically (discussed below), with using Excel(will be considered separately) or specialized computer programs.

Perhaps the construction of a mathematical model is the most difficult part of linear programming, requiring the translation of the problem under consideration into a system of variables, equations and inequalities - a process that ultimately depends on the skills, experience, abilities and intuition of the modeler.

Let's consider an example of constructing a mathematical model of linear programming

Nikolai Kuznetsov runs a small mechanical plant. Next month, he plans to produce two products (A and B), for which the specific marginal profit is estimated at 2,500 and 3,500 rubles, respectively.

Both products require machining, raw materials, and labor costs to make (Figure 1). It takes 3 hours to produce each unit of product A. machining, 16 units of raw materials and 6 units of labor. The corresponding unit requirements for Product B are 10, 4, and 6. Nicholas predicts that next month he can supply 330 hours of machining, 400 units of raw materials, and 240 units of labor. The technology of the production process is such that at least 12 units of product B must be produced in any given month.

Rice. 1. Use and provision of resources

Nikolai wants to build a model to determine the number of units of products A and B that he must produce in the next month to maximize his contribution margin.

The linear model can be built in four stages.

Step 1: Defining Variables

There is a target variable (let's call it Z) that needs to be optimized, that is, maximized or minimized (for example, profit, revenue or expenses). Nikolay seeks to maximize contribution margin, hence the target variable:

Z = total marginal profit (in rubles) received in the next month as a result of the production of products A and B.

There are a number of unknown unknown variables (let’s denote them x 1, x 2, x 3, etc.), whose values ​​must be determined to obtain the optimal value of the objective function, which, in our case, is the total marginal profit. This contribution margin depends on the quantities of products A and B produced. The values ​​of these quantities need to be calculated, and therefore they represent the desired variables in the model. So, let's denote:

x 1 = number of units of product A produced in the next month.

x 2 = number of units of product B produced in the next month.

It is very important to clearly define everything variables; Special attention Pay attention to the units of measurement and the time period to which the variables refer.

Stage. 2. Construction of the objective function

An objective function is a linear equation that must be either maximized or minimized. It contains the target variable expressed using the target variables, that is, Z expressed in terms of x 1, x 2 ... in the form of a linear equation.

In our example, each manufactured product A brings 2,500 rubles. marginal profit, and when producing x 1 units of product A, the marginal profit will be 2500 * x 1. Similarly, the marginal profit from producing x 2 units of product B will be 3500 * x 2. Thus, the total marginal profit received in the next month by producing x 1 units of product A and x 2 units of product B, that is, the target variable Z will be:

Z = 2500 * x 1 + 3500 * x 2

Nikolay strives to maximize this indicator. Thus, the objective function in our model is:

Maximize Z = 2500 * x 1 + 3500 * x 2

Stage. 3. Define constraints

Constraints are a system linear equations and/or inequalities that limit the values ​​of the sought variables. They mathematically reflect the availability of resources, technological factors, marketing conditions and other requirements. Constraints can be of three types: “less than or equal”, “greater than or equal”, “strictly equal”.

In our example, producing products A and B requires machining time, raw materials, and labor, and the availability of these resources is limited. The production volumes of these two products (that is, the values ​​of x 1 x 2) will therefore be limited by the fact that the amount of resources needed in production process, cannot exceed what is available. Let's consider the situation with machine processing time. The production of each unit of product A requires three hours of machining, and if x 1 units are produced, then 3 * x 1 hours of this resource will be spent. Each unit of product B requires 10 hours to produce and therefore if x 2 products are produced, then 10 * x 2 hours will be required. Thus, the total amount of machine time required to produce x 1 units of product A and x 2 units of product B is 3 * x 1 + 10 * x 2 . This total machine time cannot exceed 330 hours. Mathematically this is written as follows:

3 * x 1 + 10 * x 2 ≤ 330

Similar considerations apply to raw materials and labor, which allows us to write down two more restrictions:

16 * x 1 + 4 * x 2 ≤ 400

6 * x 1 + 6 * x 2 ≤ 240

Finally, it should be noted that there is a condition according to which at least 12 units of product B must be produced:

Stage 4. Writing non-negativity conditions

The searched variables cannot be negative numbers, which must be written in the form of inequalities x 1 ≥ 0 and x 2 ≥ 0. In our example, the second condition is redundant, since it was determined above that x 2 cannot be less than 12.

A complete linear programming model for production task Nicholas can be written as:

Maximize: Z = 2500 * x 1 + 3500 * x 2

Provided that: 3 * x 1 + 10 * x 2 ≤ 330

16 * x 1 + 4 * x 2 ≤ 400

6 * x 1 + 6 * x 2 ≤ 240

Let's consider a graphical method for solving a linear programming problem.

This method is only suitable for problems with two unknown variables. The model constructed above will be used to demonstrate the method.

The axes on the graph represent the two variables of interest (Figure 2). It doesn't matter which variable is plotted along which axis. It is important to choose a scale that will ultimately allow you to create a clear diagram. Since both variables must be non-negative, only the 1st quadrant is drawn.

Rice. 2. Linear programming graph axes

Consider, for example, the first constraint: 3 * x 1 + 10 * x 2 ≤ 330. This inequality describes the area below the line: 3 * x 1 + 10 * x 2 = 330. This line intersects the x 1 axis at x 2 = 0, that is, the equation looks like this: 3 * x 1 + 10 * 0 = 330, and its solution: x 1 = 330 / 3 = 110

Similarly, we calculate the intersection points with the x1 and x2 axes for all constraint conditions:

Range of acceptable values	Limit of acceptable values	Intersection with x-axis 1	Intersection with x-axis 2
3 * x 1 + 10 * x 2 ≤ 330	3 * x 1 + 10 * x 2 = 330	x 1 = 110; x 2 = 0	x 1 = 0; x 2 = 33
16 * x 1 + 4 * x 2 ≤ 400	16 * x 1 + 4 * x 2 = 400	x 1 = 25; x 2 = 0	x 1 = 0; x 2 = 100
6 * x 1 + 6 * x 2 ≤ 240	6 * x 1 + 6 * x 2 = 240	x 1 = 40; x 2 = 0	x 1 = 0; x 2 = 40
x 2 ≥ 12	x 2 = 12	does not cross; runs parallel to the x axis 1	x 1 = 0; x 2 = 12

Graphically, the first limitation is shown in Fig. 3.

Rice. 3. Construction of the region of feasible solutions for the first constraint

Any point within the selected triangle or on its boundaries will meet this constraint. Such points are called valid, and points outside the triangle are called invalid.

We similarly display the remaining restrictions on the graph (Fig. 4). Values ​​of x 1 and x 2 on or inside the shaded region ABCDE will satisfy all model constraints. This region is called the region of feasible solutions.

Rice. 4. Region of feasible solutions for the model as a whole

Now, in the region of feasible solutions, it is necessary to determine the values ​​x 1 and x 2 that maximize Z. To do this, in the objective function equation:

Z = 2500 * x 1 + 3500 * x 2

divide (or multiply) the coefficients before x 1 and x 2 by the same number, so that the resulting values ​​fall within the range reflected on the graph; in our case, this range is from 0 to 120; so the odds can be divided by 100 (or 50):

Z = 25x 1 + 35x 2

then assign Z a value equal to the product of the coefficients before x 1 and x 2 (25 * 35 = 875):

875 = 25x 1 + 35x 2

and finally, find the points of intersection of the line with the x 1 and x 2 axes:

Let's plot this target equation on a graph similar to the constraints (Fig. 5):

Rice. 5. Applying the objective function (black dotted line) to the region of feasible solutions

The Z value is constant throughout the objective function line. To find the values ​​x 1 and x 2 that maximize Z, you need to parallelly move the line of the objective function to a point within the boundaries of the region of feasible solutions, which is located at the maximum distance from the original line of the objective function up and to the right, that is, to point C (Fig. 6).

Rice. 6. The line of the objective function has reached a maximum within the region of feasible solutions (at point C)

We can conclude that the optimal solution will be located at one of the extreme points of the decision area. Which one will depend on the slope of the objective function and on what problem we are solving: maximization or minimization. Thus, it is not necessary to plot the objective function - all that is necessary is to determine the values ​​of x 1 and x 2 at each extreme point by reading from a diagram or by solving the appropriate pair of equations. The found values ​​of x 1 and x 2 are then substituted into the objective function to calculate the corresponding Z value. The optimal solution is the one at which the maximum value of Z is obtained when solving the maximization problem, and the minimum value is obtained when solving the minimization problem.

Let us determine, for example, the values ​​of x 1 and x 2 at point C. Note that point C is located at the intersection of the lines: 3x 1 + 10x 2 = 330 and 6x 1 + 6x 2 = 240. Solving this system of equations gives: x 1 = 10, x 2 = 30. The calculation results for all vertices of the region of feasible solutions are given in the table:

Dot	Value x 1	Value x 2	Z = 2500x 1 + 3500x 2
A	22	12	97 000
IN	20	20	120 000
WITH	10	30	130 000
D	0	33	115 500
E	0	12	42 000

Thus, Nikolai Kuznets must plan for the next month the production of 10 products A and 30 products B, which will allow him to receive a marginal profit of 130 thousand rubles.

Briefly the essence graphic method solutions to linear programming problems can be stated as follows:

1. Draw two axes on the graph, representing the two parameters of the solution; draw only the 1st quadrant.
2. Determine the coordinates of the points of intersection of all boundary conditions with the axes, substituting alternately the values ​​x 1 = 0 and x 2 = 0 into the equations of the boundary conditions.
3. Plot the model's constraint lines on the graph.
4. Determine the region on the graph (called the feasible decision region) that meets all of the constraints. If there is no such region, then the model has no solution.
5. Determine the values ​​of the target variables at the extreme points of the decision area, and in each case calculate the corresponding value of the target variable Z.
6. For maximization problems, the solution is the point at which Z is maximum; for minimization problems, the solution is the point at which Z is minimum.

) in order to solve some optimization problem. The term is used in mathematical programming, operations research, linear programming, statistical decision theory and other areas of mathematics primarily of an applied nature, although the goal of optimization may also be the solution itself mathematical problem. In addition to the objective function in the optimization problem, restrictions can be specified for variables in the form of a system of equalities or inequalities. In general, the arguments of the objective function can be specified on arbitrary sets.

Examples

Smooth functions and systems of equations

$\backslash left\backslash (\; \backslash begin(matrix)\; F\_1(x\_1,\; x\_2,\; \backslash ldots,\; x\_M)\; =\; 0\; \backslash \backslash \; F\_2(x\_1,\; x\_2,\; \backslash ldots,\; x\_M)\; =\; 0\; \backslash \backslash \; \backslash ldots\; \backslash \backslash \; F\_N(x\_1,\; x\_2,\; \backslash \; ldots,\; x\_M)\; =\; 0\; \backslash end(matrix)\; \backslash right.$

can be formulated as a problem of minimizing the objective function

$S\; =\; \backslash sum\_(j=1)^N\; F\_j^2(x\_1,\; x\_2,\; \backslash ldots,\; x\_M)\; \backslash qquad\; (1)$

If the functions are smooth, then the minimization problem can be solved using gradient methods.

For any smooth objective function can be equated to $0$ partial derivatives with respect to all variables. The optimum of the objective function will be one of the solutions to such a system of equations. In case of function $(1)$ this will be a system of least squares (LSM) equations. Every decision original system is a solution to the least squares system. If the original system is inconsistent, then the least squares system, which always has a solution, allows us to obtain an approximate solution of the original system. The number of equations in the least squares system coincides with the number of unknowns, which sometimes facilitates the solution of joint initial systems.

Linear programming

To others famous example The objective function is a linear function that arises in linear programming problems. Unlike the quadratic objective function, optimization linear function is possible only if there are restrictions in the form of a system of linear equalities or inequalities.

Combinatorial optimization

A typical example of a combinatorial objective function is the objective function of the traveling salesman problem. This function is equal to the length of the Hamiltonian cycle on the graph. It is defined on a set of permutations $n-1$ vertices of the graph and is determined by the matrix of edge lengths of the graph. The exact solution to such problems often comes down to enumerating options.

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Notes

see also

Literature

• Burak Ya. I., Ogirko I. V. Optimal heating of a cylindrical shell with temperature-dependent material characteristics // Mat. methods and physical-mechanical fields. - 1977. - Issue. 5. - P.26-30

An excerpt characterizing the objective function

My poor husband endures labor and hunger in Jewish taverns; but the news I have makes me even more excited.
You probably heard about the heroic feat of Raevsky, who hugged his two sons and said: “I will die with them, but we will not waver!” And indeed, although the enemy was twice as strong as us, we did not waver. We spend our time as best we can; but in war, as in war. Princess Alina and Sophie sit with me all day long, and we, unfortunate widows of living husbands, have wonderful conversations over lint; only you, my friend, are missing... etc.
Mostly Princess Marya did not understand the full significance of this war because the old prince never talked about it, did not acknowledge it and laughed at Desalles at dinner when he talked about this war. The prince's tone was so calm and confident that Princess Marya, without reasoning, believed him.
Throughout the month of July, the old prince was extremely active and even animated. He also laid out a new garden and new building, building for courtyards. One thing that bothered Princess Marya was that he slept little and, having changed his habit of sleeping in the study, changed his sleeping place every day. Either he ordered his camp bed to be set up in the gallery, then he remained on the sofa or in the Voltaire chair in the living room and dozed without undressing, while not m lle Bourienne, but the boy Petrusha read to him; then he spent the night in the dining room.
On August 1, a second letter was received from Prince Andrei. In the first letter, received shortly after his departure, Prince Andrei humbly asked his father for forgiveness for what he had allowed himself to say to him, and asked him to return his favor to him. The old prince responded to this letter with an affectionate letter and after this letter he alienated the Frenchwoman from himself. The second letter from Prince Andrei, written from near Vitebsk, after the French occupied it, consisted of brief description the entire campaign with the plan outlined in the letter, and with considerations for the further course of the campaign. In this letter, Prince Andrei presented his father with the inconvenience of his position close to the theater of war, on the very line of movement of the troops, and advised him to go to Moscow.
At dinner that day, in response to the words of Desalles, who said that, as heard, the French had already entered Vitebsk, the old prince remembered Prince Andrei’s letter.
“I received it from Prince Andrei today,” he said to Princess Marya, “didn’t you read it?”
“No, mon pere, [father],” the princess answered fearfully. She could not read a letter that she had never even heard of.
“He writes about this war,” said the prince with that familiar, contemptuous smile with which he always spoke about the real war.
“It must be very interesting,” said Desalles. - The prince is able to know...
- Oh, very interesting! - said Mlle Bourienne.
“Go and bring it to me,” the old prince turned to Mlle Bourienne. – You know, on a small table under a paperweight.
M lle Bourienne jumped up joyfully.
“Oh no,” he shouted, frowning. - Come on, Mikhail Ivanovich.
Mikhail Ivanovich got up and went into the office. But as soon as he left, the old prince, looking around uneasily, threw down his napkin and went off on his own.
“They don’t know how to do anything, they’ll confuse everything.”
While he walked, Princess Marya, Desalles, m lle Bourienne and even Nikolushka silently looked at each other. The old prince returned with a hasty step, accompanied by Mikhail Ivanovich, with a letter and a plan, which he, not allowing anyone to read during dinner, placed next to him.