Transport task

Statement of the transport problem

The transport problem (T-problem) is one of the most common special LP problems. The first rigorous statement of the T-problem belongs to F. Hitchcock, therefore in foreign literature it is often called the Hitchcock problem.

The first exact method for solving the T-problem was developed by L. V. Kantorovich and M. K. Gavurin.

The general statement of the transport problem is to determine the optimal plan for the transportation of some homogeneous cargo from m points of departure (factories, warehouses, bases, etc.) in n destinations (shops). At the same time, from each point of departure (production) it is possible to transport the product to any point of destination (consumption). As a criterion of optimality, either the minimum cost of transporting the entire cargo, or the minimum time for its delivery, is usually taken.

Choosing an Optimality Criterion

When solving a transport problem, the choice of an optimality criterion is important. As you know, the evaluation of economic efficiency approximate plan can be determined by one or another criterion, which is the basis for calculating the plan. This criterion is an economic indicator that characterizes the quality of the plan. To date, 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) The volume of transport work (criterion - distance in t / km). The minimum mileage is useful for estimating travel plans because the travel distance is easily and accurately determined 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 the transportation of homogeneous goods by cars.

2) Tariff for the carriage of goods (criterion - tariffs for carriage charges). Allows you to get a transportation scheme that is the best in terms of self-supporting indicators of the enterprise. All the surcharges, as well as existing feed-in tariffs, make it difficult to use.

3) Operating costs for the transportation of goods (criterion - the cost of operating costs). More accurately reflects the economy of transportation various types transport. Allows you to draw reasonable conclusions about the feasibility of switching from one mode of transport to another.

4) Terms of delivery of goods (criterion - the cost of time).

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

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

,

where is the operating cost,

Estimated investment efficiency ratio,

Capital investments coming 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 quantitatively-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 the transportation of a unit of cargo from the i-th point of departure to j-th item destination, through - stocks of cargo in i-th paragraph departure point, through is the demand for cargo at the jth destination, and through is the number of units of cargo transported from the i-th origin to the j-th destination. Then the mathematical formulation of the problem consists in determining the minimum value of the function

under conditions

(2)

(3)

(4)

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

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

Obviously, the total availability of cargo from suppliers is equal to , and the total need for cargo at destinations is equal to units. If the total demand for cargo at destinations is equal to the cargo stock 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 met. Such models are called open. Possible two cases:

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

Such a problem can be reduced to an ordinary transport problem as follows. In case of excess of demand over stock, i.e., a fictitious ( m+1) - th point of departure with cargo stock and tariffs are assumed to be zero:

Then it is required to minimize

under conditions

Consider now the second case.

Similarly, for , a fictitious ( n+1) – th destination with need and the corresponding tariffs are considered equal to zero:

Then the corresponding T-problem can 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 the future, we will consider a closed model of the transport problem. If the model of a specific problem is open, then, proceeding from 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 any routes. Then the costs of transportation along these routes are set so that they exceed the highest costs of possible transportation (in order to make it unprofitable to carry on inaccessible routes) - when solving the problem for a minimum. To the max, it's the other way around.

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

from the stock of the respective dispatch point;

· from the needs of the corresponding destination.

Example.

Four enterprises of this economic region Three types of raw materials are used to manufacture products. The raw material requirements of each of the enterprises are respectively equal to 120, 50, 190 and 110 units. Raw materials are concentrated in three places of their receipt, and the stocks are respectively equal to 160, 140, 170 units. Raw materials can be delivered to each of the enterprises from any point of its receipt. Freight rates are known values ​​and are given by the matrix

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

Solution. Let us denote by the number of units of raw materials transported from the i-th point of its receipt to the j-th enterprise. Then the conditions for the delivery and export of the necessary and available raw materials are ensured by fulfilling the following equalities:

(6)

With this plan transportation, the total cost of transportation will be

Thus, the mathematical formulation of this transport problem consists in finding such a non-negative solution to the system of linear equations (6), in which the objective function (7) takes the minimum value.

When solving a transport problem, the choice of an optimality criterion is important. As you know, the evaluation of the economic efficiency of an exemplary plan can be determined by one or another criterion, which is the basis for calculating the plan. This criterion is an economic indicator that characterizes the quality of the plan. To date, 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) The volume of transport work (criterion - distance in t / km). The minimum mileage is useful for estimating travel plans because the travel distance is easily and accurately determined 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 the transportation of homogeneous goods by cars.

2) Tariff for the carriage of goods (criterion - tariffs for carriage charges). Allows you to get a transportation scheme that is the best in terms of self-supporting indicators of the enterprise. All the surcharges, as well as existing feed-in tariffs, make it difficult to use.

3) Operating costs for the transportation of goods (criterion - the cost of operating costs). It more accurately reflects the cost-effectiveness of transportation by various modes of transport. Allows you to draw reasonable conclusions about the feasibility of switching from one mode of transport to another.

4) Terms of delivery of goods (criterion - the cost of time).

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

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

,

where is the operating cost,

Estimated investment efficiency ratio,

Capital investments coming per 1 ton of cargo throughout the section,

T - travel time,

C - the price of one ton of cargo.

Allows you to more fully evaluate the rationalization of different options for 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 through the tariffs for the transportation of a unit of cargo from the i-th point of departure to the j-th destination, through - the stocks of cargo at the i-th point of departure, through - the demand for cargo at the j-th destination, and through - the number of units of cargo transported from the i-th origin to the j-th destination. Then the mathematical formulation of the problem consists in determining the minimum value of the function

under conditions

(2)

(3)

(4)

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

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

Obviously, the total availability of cargo from suppliers is equal to , and the total need for cargo at destinations is equal to units. If the total demand for cargo at destinations is equal to the cargo stock 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 met. Such models are called open. Possible two cases:

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

Such a problem can be reduced to an ordinary transport problem as follows. In case of excess of demand over stock, i.e., a fictitious ( m+1) - th point of departure with cargo stock and tariffs are assumed to be zero:

Then it is required to minimize

under conditions

Consider now the second case.

Similarly, for , a fictitious ( n+1) – th destination with need and the corresponding tariffs are considered equal to zero:

Then the corresponding T-problem can 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 the future, we will consider a closed model of the transport problem. If the model of a specific problem is open, then, proceeding from 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 any routes. Then the costs of transportation along these routes are set so that they exceed the highest costs of possible transportation (in order to make it unprofitable to carry on inaccessible routes) - when solving the problem for a minimum. To the max, it's the other way around.

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

from the stock of the respective dispatch point;

· from the needs of the corresponding destination.

Example.

Four enterprises of this economic region use three types of raw materials for the production of products. The raw material requirements of each of the enterprises are respectively equal to 120, 50, 190 and 110 units. Raw materials are concentrated in three places of their receipt, and the stocks are respectively equal to 160, 140, 170 units. Raw materials can be delivered to each of the enterprises from any point of its receipt. Freight rates are known values ​​and are given by the matrix

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

Solution. Let us denote by the number of units of raw materials transported from the i-th point of its receipt to the j-th enterprise. Then the conditions for the delivery and export of the necessary and available raw materials are ensured by fulfilling the following equalities:

(6)

With this plan transportation, the total cost of transportation will be

Thus, the mathematical formulation of this transport problem consists in finding such a non-negative solution to the system of linear equations (6), in which the objective function (7) takes the minimum value.

Solution of the transport problem

The main steps in solving the transport problem:

1. Find an initial feasible plan.

2. Choose from non-basic variables the one that will be introduced into the basis. If all non-basic variables satisfy the optimality conditions, then finish the solution, otherwise go to the next. step.

3. Choose a variable to be derived from the basis, find a new basic solution. Return to step 2.

Any non-negative solution of systems of linear equations (2) and (3) determined by the matrix , is called the transport task plan. The reference (basic) plan of the T-problem is any of its feasible, basic solutions.

Usually, the initial data of the transport task is recorded in the form of a table.

Matrix C is called the matrix of transportation costs, the matrix X that satisfies the conditions of the T-problem (2) and (3) is called the transportation plan, and the variables are called transportation. The plan , at which the objective function is minimal, is called optimal.

The number of variables in the transportation problem with m departure points and n destinations equals m*n, and the number of equations in systems (2) and (3) is m+n. Since we assume that condition (5) is satisfied, the number of linearly independent equations is equal to m+n-1. Therefore, the basic plan of the transport task can have no more than m+n-1 non-zero unknowns.

If in the reference design the number of non-zero components is exactly equal to m+n-1, then the design is non-degenerate, and if less, then degenerate.

As for any linear programming problem, the optimal plan of the transport problem is also a base plan.

Construction of an admissible (reference) plan in the transportation problem

By analogy with other problems of linear programming, the solution of the transport problem begins with the construction of an admissible basic plan. There are several methods for constructing initial base plans for the T-problem. Of these, the most common northwest corner method And minimum element method.

The simplest way to find it is based on the so-called northwest corner method. The essence of the method is the sequential distribution of all stocks available at the first, second, etc. points of production, according to the first, second, etc. points of consumption. Each distribution step is reduced to an attempt to completely deplete stocks at the next point of production or to an attempt to fully satisfy the needs at the next point of consumption. At every step q current unallocated reserves are indicated and i (q ), and current unmet needs - b j (q ) . The construction of an acceptable initial plan, according to the northwest corner method, starts from the upper left corner of the transport table, while we assume a i (0) = a i , b j (0) = b j . For the next cell located in the row i and column j , we consider the values ​​of unallocated stock in i -th point of production and unmet need j -th point of consumption, of which the minimum is selected and assigned as the volume of transportation between these points: x i, j =min(a i (q) , b j (q) ) . After that, the values ​​of unallocated stock and unmet demand in the respective points are reduced by this amount:

a i (q+1) = a i (q) - x i , j , b j (q+1) = b j (q) - x i , j

Obviously, at each step, at least one of the equalities is satisfied: and i (q+1) = 0 or b j (q+1) = 0 . If the first is true, then this means that the entire stock of the i-th production point is exhausted and it is necessary to proceed to the distribution of the stock at the production point i+1 , i.e. move to the next cell down the column. If b j (q+1) = 0, This means that the need for j -th point, after which the transition to the cell located to the right of the line follows. The newly selected cell becomes current, and all the listed operations are repeated for it.

Based on the condition of the balance of supplies and needs, it is not difficult to prove that in a finite number of steps we will obtain an admissible plan. By virtue of the same condition, the number of steps of the algorithm cannot be more than m+n-1 , so will always remain free (zero) mn-(m+n-1) cells. Therefore, the resulting plan is basic. It is possible that at some intermediate step, the current unallocated stock turns out to be equal to the current unmet need (a i (q) = b j (q)) . In this case, the transition to the next cell occurs in a diagonal direction (the current points of production and consumption change simultaneously), which means the “loss” of one nonzero component in the plan, or, in other words, the degeneracy of the constructed plan.

A feature of an acceptable plan constructed by the northwest corner method is that the objective function on it takes a value, as a rule, far from the optimal one. This is because it does not take into account the values c i , j . In this regard, in practice, to obtain the original plan, another method is used - minimum element method, in which, when distributing traffic volumes, the cells with the lowest prices are occupied first.

An example of finding a baseline

F=14 x 11 + 28 x 12 + 21 x 13 + 28 x 14 + 10 x 21 + 17 x 22 + 15 x 23 + 24 x 24 + 14 x 31 + 30 x 32 + 25 x 33 + 21 x 34

The original plan was obtained using the northwest corner method. The problem is balanced (closed).

Table 1

The cost of transportation under this plan is: 1681:

F=14 *27 + 28* 0 + 21*0 + 28*0 + 10 *6 + 17 *13 + 15*1 + 24 *0 + 14 *0 + 30 *0 +25*26 + 21 *17 = 1681

Estimated work No. 4: TRANSPORT PROBLEM

The general formulation of the transport problem is to determine the optimal plan for the transportation of some homogeneous cargo from points of departure (production) to points of destination (consumption). 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 us denote through the tariffs for the transportation of a unit of cargo from the -th point of departure to the -th point of destination, through - the stocks of cargo at the -th point of departure, through - the demand for cargo at the -th destination, and through - the number of units of cargo transported from the -th point departure to the th destination. Usually, the initial data of the transport task is recorded in the form of a table.

production	Consumption points				production
consumer

Let's make a mathematical model of the problem.

(1)

under restrictions

Plan , in which the function (1) takes its minimum value, is called optimal plan transport task.

The condition for the solvability of the transport problem

Theorem: For the solvability of the transport problem, it is necessary and sufficient that the stocks of cargo at the points of departure are equal to the demand for cargo at the destinations, i.e., that the equality

The model of such a transport problem is called closed, or closed, or balanced, otherwise the model is called open.

In the case, a dummy is entered - th destination with need ; similarly, when a fictitious th point of departure with a cargo reserve is entered and the corresponding tariffs are considered equal to zero: . In this way, the problem is reduced to the usual transport problem. In the future, we will consider a closed model of the transport problem.

The number of variables in the transport problem with points of departure and destination is , and the number of equations in the system (2)-(4) is . Since we assume the fulfillment of condition (5), the number of linearly independent equations is . Therefore, the reference design can have no more than zero unknowns. If in the base plan the number of non-zero components is exactly equal to , then the plan is called non-degenerate, and if less, then degenerate.

Building the initial baseline

There are several methods for determining the baseline: northwest corner (diagonal method), method lowest cost (minimum element), method double preference and method Vogel approximations.

Let's briefly consider each of them.

1.Northwest corner method. In finding the baseline, at each step, the first of the remaining origins and the first of the remaining destinations are considered. Filling in the cells of the condition table starts from the upper left cell for the unknown ("northwest corner") and ends with the cell for the unknown, i.e. like a diagonal table.

2. Least cost method. The essence of the method lies in the fact that the smallest one is chosen from the entire cost table and in the cell that corresponds to it, the smallest of the numbers and is placed, then either the row corresponding to the supplier, whose stocks are completely used up, or the column corresponding to the consumer, whose needs are excluded from consideration fully satisfied, or both row and column if the supplier's inventory is used up and the consumer's needs are satisfied. From the rest of the table of costs, the lowest cost is again selected, and the process of placing stocks is continued until all stocks have been allocated and requirements have been satisfied.

3. Double Preference Method. The essence of the method is as follows. In each column, mark the cell with the lowest cost with a "√" sign. Then the same is done in each line. As a result, some cells are marked "√√". They contain the minimum cost, both by column and by row. The maximum possible volumes of traffic are placed in these cells, each time excluding the corresponding columns or rows from consideration. Then the transportation is distributed among the cells marked with the sign "√". In the remainder of the table, shipments are distributed at the lowest cost.

4. Vogel approximation method. When determining the base plan by this method, at each iteration, in all columns and all rows, the difference between the two minimum tariffs recorded in them is found. These differences are entered in the line and column specially designated for this in the table of the conditions of the problem. Among these differences, choose the maximum. In the row (or column) that this difference corresponds to, the minimum tariff is determined. The cell in which it is written is filled in at this iteration.

Definition of the optimality criterion

With the help of the considered methods for constructing the initial reference plan, one can obtain a degenerate or non-degenerate reference plan. The constructed plan of the transport problem as a linear programming problem could be brought to an optimal one using the simplex method. However, due to the bulkiness of simplex tables containing tp unknown, and a large amount of computational work to obtain the optimal plan use more simple methods. The most commonly used method of potentials (modified distributive method).

Potential method.

The method of potentials allows you to determine, starting from some basic transportation plan, to build a solution to the transportation problem in a finite number of steps (iterations).

The general principle of determining the optimal plan for a transport problem by this method is similar to the principle of solving a linear programming problem using the simplex method, namely: first, a reference plan for the transport problem is found, and then it is successively improved until an optimal plan is obtained.

Let's make a dual problem

1., - any

3.

Let there be a plan

Theorem(optimality criterion): In order for the feasible transportation plan in the transportation problem to be optimal, it is necessary and sufficient that there are such numbers , , that

If. (7)

numbers and are called the potentials of the points of origin and destination, respectively.

The formulated theorem makes it possible to construct an algorithm for finding a solution to the transport problem. It consists of the following. Let one of the methods discussed above find a reference plan. For this plan, in which there are basic cells, it is possible to determine the potentials and so that condition (6) is satisfied. Since system (2)-(4) contains equations and unknowns, one of them can be set arbitrarily (for example, equated to zero). After that, the remaining potentials are determined from equations (6) and the values ​​are calculated for each of the free cells. If it turns out that , then the plan is optimal. If at least in one free cell, then the plan is not optimal and can be improved by transferring along the cycle corresponding to this free cell.

cycle in the table of conditions of the transport problem, a broken line is called, the vertices of which are located in the occupied cells of the table, and the links along the rows and columns, and at each vertex of the cycle there are exactly two links, one of which is in the row, and the other in the column. If the polyline forming the cycle intersects, then the self-intersection points are not vertices.

The plan improvement process continues until the if conditions (7) are met.

An example of solving a transport problem.

Task. Four bases A 1 , A 2 , A 3 , A 4 received a homogeneous cargo in the following quantity: a 1 ton - to the base A 1 , and 2 tons - to the base A 2 , and 3 tons - to the base A 3 , and 4 tons - to the base A 4. The received cargo needs to be transported to five points: b 1 tons - to base B 1, b 2 tons - to base B 2, b 3 tons - to base B 3, b 4 tons - to base B 4, b 5 tons - to base B5. Distances between destinations are shown in the distance matrix.

departure points	destinations
needs

The cost of transportation is proportional to the amount of cargo and the distance over which this cargo is transported. Plan transportation so that their total cost is minimal.

Solution. Let's check the balance of the transport problem, for this it is necessary that

, .

1. Solve the problem using the diagonal method or the northwest corner method.

The process of obtaining a plan can be arranged in the form of a table:

departure points

It is necessary to distinguish between the optimization criterion and the indicators of the optimality of freight transportation plans. The optimization criterion should reflect the essence of the national economic approach to its choice, taking into account the strategy economic policy states in the field of transport. The choice of optimization indicators that reflect various aspects of the global economic optimization criterion is a difficult task.

All transport tasks of optimal attachment of destinations to poisoning points, practically implemented in optimal schemes of cargo flows, are solved in terms of the transportation distance based on the minimum cargo turnover. The objective function Fc of the transport problem has the following form:

Fс = min хij lij, (1)

where m, n - the number of points of departure and destination, respectively;

хij - the amount of cargo traffic transportation for each correspondence between the points of departure and destination, t;

lij - transportation distance for each correspondence of the cargo flow, km.

As a result of studies carried out by I. V. Belov, it was proved that the optimization of cargo transportation plans for a minimum of ton-kilometers does not reflect the main characteristics of the national economic optimality criterion and, therefore, does not allow obtaining a truly optimal plan.

The shortest distance as an indicator of optimality is obviously unsuitable for optimizing freight transportation plans for various interacting modes of transport, i.e. when compiling complex optimal schemes of cargo flows on the network different types ways of communication.

When optimizing freight transportation plans, the shortest cost direction is also not always the most profitable. The bottom line is that the amount of costs in the directions of transportation is influenced not only by the distance (range), but also by a number of other operational, technical and socio-economic factors. Comprehensive indicators that the best way all the most important characteristics of the national economic criterion of optimization can be reflected in the development of freight transportation plans, are cost indicators. Their use in solving transport optimization problems fully complies with modern requirements for improving the quality of planning and regulation of transportation.

In accordance with the basic concept of optimization, justified by MIIT, in the presence of reserves of throughput and carrying capacity, it is more economically expedient to use the minimum of operating costs dependent on the volume of traffic, i.e. minimum cost of transportation in terms of dependent costs. The objective function of the transport task in this case will look like:

Fс = min хij С factory ij, (2)

where C head ij is the cost of transportation of goods for each correspondence of the cargo flow in terms of dependent costs, c / t.

In accordance with the transitional concept of optimization in the absence of reserves of throughput and carrying capacity, the cost indicators of current transportation planning are also unacceptable. The optimization problem in this case should be solved not for a minimum of current costs, but for a maximum of results in the level of meeting the needs of production in transportation. These goals are best met by the optimization indicator - the minimum time for the delivery of goods, i.e.

Fс = min хij tij, (3)

where tjj is the time of delivery of goods for each correspondence of the cargo flow, h.

This optimality indicator, being simple, best meets the conditions for optimizing the transportation of perishable goods, since it simultaneously provides a minimum of national economic costs (including loss of goods) during transportation.

In the context of the transition of transport to market relations, the optimization of transportation plans based on the minimum tariff fees, when the objective function has the form

Fс = min хij С tar ij, (4)

where C tar ij is the profitable tariff rate for the transportation of goods for each correspondence of the cargo flow, k / t.

Previously, it was believed that the plan for the minimum ton-kilometers and the plan for the minimum tariff payments coincide, since freight rates are based on the principle of the shortest transportation distances. But this statement is not entirely true, since the tariff fee is charged each time not for a specific shortest transportation distance, but for the average distance of a given tariff zone. Tariff belts, especially at long distances, change in a wide range.

Obviously, with the possible and expedient territorial differentiation of tariffs in market conditions, as well as with their deeper differentiation depending on the level of transportation quality, the optimal transportation plans for the minimum ton-kilometers and the minimum tariff fees will no longer coincide.

One more important circumstance should be borne in mind. Optimization of transport links at the minimum of tariffs means minimization of transport revenues, which may adversely affect its profits and profitability, i.e. on the self-supporting interests of transport. Some experts argue that the optimization of transportation plans for this indicator is generally unacceptable, since it deliberately puts transport in an unequal economic situation compared to other sectors of the economy. There is a serious objection to this argument. Transport revenues are at the same time the tariff transportation costs of the national economy, which we must constantly strive to save by eliminating all kinds of irrational transportation and the unproductive costs associated with them. Thus, in the context of the development of market relations, the optimization of transportation plans for a minimum of tariff fees should have a wider scope. But at the same time, it must move from the field of transport as such to the field of logistics as an optimization of supply plans.

The above costs as an indicator of optimality can be used in solving transport problems on the network of communication routes of different interacting modes of transport in the conditions of both current and long-term planning and regulation of work, as well as on one type of transport for long-term conditions of planning and regulation of work with the development of throughput . The objective function of the optimal plan here can be expressed in two ways: without taking into account the cost of the freight mass in transit, if there are no significant differences in the time of delivery of goods by interacting modes of transport:

Fс = min хij (сij + En kij), (5)

taking into account the cost of the cargo mass in transit, when the interacting modes of transport differ significantly in the time of delivery of goods:

Fс = min хij (сij + Ен (кij + mij), (6)

where kij - specific investment in rolling stock and permanent devices for each correspondence of the cargo flow, to / t;

mij is the unit cost of the freight mass en route for each correspondence of the freight traffic, c/t.

When choosing cost indicators for the purposes of optimizing the transportation of goods, it is necessary to ensure the greatest completeness in these indicators of all their constituent elements of costs and losses that change depending on changes in the conditions of the transportation process for specific transport and economic links between the points of departure and destination of goods. Back in the late 60s and 70s, it was pointed out that, in necessary cases, especially when transporting with the participation of different modes of transport, it is necessary to additionally take into account the losses associated with the non-preservation of the cargo. This meant those cases where differences in the amount of losses by mode of transport or options for attaching consumers to suppliers on a given mode of transport significantly affect the choice of a truly optimal transportation plan.

Similar judgments were expressed by experts in relation to the problem of optimizing the country's fuel and energy balance and determining the role of coal in it. It was argued that the correct solution of the optimization problem is possible if the formation of economic information on fuel is carried out on the basis of comparable and comparable indicators for all stages social production according to identical methodology and on the basis of the same methodological prerequisites. In this case, it is especially important to accurately take into account the costs caused by the loss of fuel during transportation.

Fuel losses are included in the cost of transportation only through oil and gas pipelines, as well as power lines. Losses of coal during transportation are not fully taken into account and, as a rule, are not reflected in economic calculations. This leads to the fact that ideas about the degree of efficiency of a particular mode of transport are distorted. In order to remove the distortions caused by the incomparability of cost indicators when optimizing the country's fuel and energy balance, these indicators should take into account the losses of the corresponding cargoes.

Some works of economists noted the need to take into account, when optimizing transport and economic relations, not only the quality of transportation, but also the quality of the most transported national economic products, their consumer properties. In this case, we are talking about reflecting in the cost indicator of optimality not only the losses of transported goods, but also differences in their assortment and quality composition. This means that the optimization of transportation of interchangeable products of different assortment and quality, with a commensurate consideration of its consumer properties (mileage car tires, calorific value of fuel, share nutrients in fertilizers, iron in ore, etc.) will give an optimal plan that differs significantly from the optimal plan drawn up without taking into account these differences.

The economic and mathematical model of the optimization problem, taking into account the consumer properties of interchangeable products, was implemented in specific solutions, in particular, in the work of NIIMS (authors E. P. Nesterov, V. A. Skvortsova, etc.). In the works of MIIT, it was established that in the development of operational current and prospective optimal plans for transportation by rail transport, the cost indicators of optimality must necessarily take into account the loss of many goods, especially perishable, bulk and bulk. When solving complex transport problems of optimizing transportation for any period and planning with the participation of two or more interacting modes of transport, losses must be included in the cost indicators of optimality for all groups of goods in accordance with the classification. Differences, if any, in the consumer properties and quality of interchangeable goods should be reflected through their corresponding prices in the cost of the cargo mass in transit. The functionals of the optimal plan can be expressed in general terms: without taking into account the cost of the cargo mass in transit

Fс = min хij (сij + Enkij + y pe ij), (7)

taking into account the cost of the cargo mass in transit

Fс = min хij (сij + Ен (кij + mij + y pe ij), (8)

where y pe ij is the specific value of the current losses of goods in terms of value for each correspondence of the cargo flow, k / t.

Optimization of cargo transportation, taking into account their losses, can practically be carried out only after the transition to the development of simple or complex optimal schemes of cargo flows in terms of cost indicators of optimality - current and reduced costs. A very important task in this case is the advance preparation of reliable regulatory economic information for calculating losses during the transportation of goods.

When transporting perishable goods, their losses, as a rule, are much, and often several times, higher than the actual costs of transportation. Therefore, it seems possible to optimize the current and operational plans for the transportation of perishable goods based on the minimum current losses with the obligatory fulfillment of the specified delivery times. It can be argued that the optimal plan for minimizing losses coincides with the optimal plan for minimizing the delivery time of perishable goods. The objective function of this optimal plan is:

Fс = min xij y pe ij. (9)

However, it should be borne in mind that the practical use of cost optimality indicators for solving transport problems and drawing up optimal schemes for cargo flows is fraught with great difficulties. The fact is that the preliminary calculation of individual cost indicators is very complicated. These indicators are unstable over time due to constant changes in conditions and factors affecting the amount of costs. The initial data for calculating the individual components of the cost indicators of optimality do not always provide the necessary reliability of the results.

An excess of carrying capacity increases the cost of transportation and the cost of production. The criterion of optimality is proposed to accept the minimum losses, on the one hand - from the underuse of the rolling stock, on the other - the loss of consignees from untimely delivery.

Any cargo flow is characterized by a four-index number: the point of production, the point of consumption of the cargo, the class of the cargo and the time of delivery of the cargo to the consumer. In order to deliver all manufactured products from the place of production to the place of consumption, the carrying capacity of transport must be no less than the value of the freight traffic.

It is known that the carrying capacity of the rolling stock is a probabilistic value, which is influenced by many factors: road and climatic conditions, type and age composition of the rolling stock, driver qualifications, compliance of the production and technical base with the capacity of the fleet, etc. Therefore, at certain moments, the magnitude of the freight traffic can exceed the carrying capacity of the rolling stock and part of the cargo will not be delivered to the place of consumption in a timely manner.

Therefore, the main condition for the timely transportation of goods to the place of their consumption is the excess of the carrying capacity of the rolling stock in comparison with the freight traffic.