Selection of optimality criterion. Objective functions. Selection of criteria As a criterion for the optimality of transport

The general formulation of the transport problem consists in determining the optimal plan for transporting some homogeneous cargo from m points of departure (suppliers) A1, A2, . . ., A m V n consumption points (consumers) B1, B2, . . . Bn so that:

Remove all cargo from suppliers;

Satisfy the demand of each consumer;

Ensure minimum total transport costs for the transportation of all goods.

Consider the transport problem as optimality criterion which uses the minimum cost of transporting the entire cargo.

Let's denote:

ai - availability of cargo in i -th point of departure https://pandia.ru/text/78/103/images/image205_0.gif" width="81" height="27 src=">;

сij - cost of transporting a unit of cargo from i th point of departure at j th point of consumption (transportation tariff);

xij - the amount of cargo transported from i th point of departure at j th destination, destination, xij ≥ 0.

The mathematical formulation of the transport problem consists of finding a non-negative solution to a system of linear equations in which the objective function takes a minimum value.

Let's write down a mathematical model of the transport problem.

It is required to determine a matrix ) that satisfies the following conditions:

https://pandia.ru/text/78/103/images/image210_0.gif" width="74" height="45">.gif" width="47" height="21">.gif" width= "63" height="20"> (5.3)

and delivers the minimum value of the objective function

L () = https://pandia.ru/text/78/103/images/image215_0.gif" width="36" height="24"> satisfy the system of linear equations (5.1), (5.2) and the non-negativity condition, This ensures delivery of the necessary cargo to each consumer, removal of existing cargo from all suppliers, and also eliminates return transportation.

Definition 1. Any non-negative solution of systems of linear equations (5.1) and (5.2), defined by the matrix ) is called feasible plan for a transport problem.

Definition 2. Plan) https://pandia.ru/text/78/103/images/image218_0.gif" width="23" height="24">, called basic or reference.

Definition 4. If in the reference plan there are a number of non-zero variable values https://pandia.ru/text/78/103/images/image219_0.gif" width="55" height="22">.gif" width="55" height ="22"> > , a fictitious (n+ 1) destination with the requirement is entered bn+1 = − https://pandia.ru/text/78/103/images/image221_0.gif" width="83 height=22" height="22">

If< https://pandia.ru/text/78/103/images/image220_0.gif" width="56 height=25" height="25">.gif" width="79" height="22 src=">

Let's consider one of the methods for constructing the first reference plan of a transport problem - the method of minimum cost or the best element of the unit cost matrix.

Definition 6. The best element of the matrix of unit costs (tariffs) will be the smallest tariff if the task is set to the minimum of the objective function, the highest tariff - if the task is set to the maximum.

Algorithm for constructing the first reference plan.

1. Among the matrix of unit costs, we find the best tariff.

2. Fill in the cell of the distribution table with the selected tariff with the maximum possible volume of cargo, taking into account row and column restrictions. In this case, either the entire cargo is removed from the supplier, or the consumer’s needs are fully satisfied. The row or column of the table is deleted from consideration and does not participate in further distribution.

3. From the remaining tariffs, we again select the best one and the process continues until all the cargo has been distributed.

If the transport problem model is open and a fictitious supplier or consumer is introduced, then distribution is first carried out for real suppliers and consumers, and lastly the unallocated cargo is sent from the fictitious supplier or to the fictitious consumer.

We further improve the first reference plan of the transport problem and obtain the optimal plan using the potential method.

Theorem 3 . The plan ) of the transport problem is optimal if there is a system (m + n) of numbers ui and vj (called potentials) that satisfies the conditions:

(5.6)

(5.7)

The potentials ui and vj are the variables of the dual problem composed of the original transport problem, and denote the evaluation of a unit of cargo at the points of origin and destination, respectively.

Let us denote: ) the estimate of a free (unoccupied) table cell.

Definition 7. The reference plan of the transport problem is optimal if all estimates of free cells of the distribution table (the problem is set to a minimum).

Algorithm of the potential method

1. Building the first reference plan transport problem using the minimum cost method.

2. Checking the degeneracy of a plan .

Potentials can only be calculated for a non-degenerate plan. If the number of occupied cells in the reference plan (the number of basic variables) is less than (m+n−1), then we enter a zero in one of the free cells of the table so that the total number of occupied cells becomes equal to (m+n−1). Zero is entered into the cell with the best tariff, which belongs to the row or column. Simultaneously crossed out when drawing up the first reference plan. In this case, a table cell fictitiously occupied by zero should not form a closed rectangular contour with other occupied table cells.

3. Calculation of the goal function value (5.4) by summing the products of tariffs (unit costs) by the volume of transported cargo for all occupied cells of the table.

4. Checking the optimality of the plan.

We determine potentials. For each occupied cell we write the equation, as a result we obtain a system of (m + n−1) equations with (m + n) variables.

Since the number of variables is greater than the number of equations, the resulting system is not defined and has an infinite number of solutions..gif" width="70" height="22">, then the remaining potentials are determined uniquely, and their values are entered in the additional row and column of the distribution tables.

For each free cell, we determine the estimates https://pandia.ru/text/78/103/images/image233.gif" width="72 height=24" height="24">(the problem is solved to the minimum of the objective function), then the optimal plan is found. If at least one estimate of a free cell does not satisfy the optimality condition, then it is necessary to improve the plan by redistributing the load.

Of all the positive estimates of free cells, we select the largest (the task is set to a minimum); of all negative ones – the largest in absolute value (the task is set to the maximum). The cell that corresponds to the highest score should be filled, i.e., the load should be sent to it. By filling the selected cell, it is necessary to change the volume of supplies recorded in a number of other occupied cells and associated with the one being filled, the so-called cycle.

A cycle or a rectangular contour in the distribution table of a transport problem is a broken line, the vertices of which are located in the occupied cells of the table, and the links are located along the rows and columns, and at each vertex of the cycle there are exactly two links, one of which is in the row, the other in the column . If a broken line forming a cycle intersects, then the intersection points are not vertices. For each free cell, a single cycle can be constructed.

The vertices of the cycle, starting from the vertex located in the selected cell for loading, are assigned alternately the signs “+” and “−”. We will call these cells plus and minus.

From the volumes of cargo in the minus cells, we select the smallest and denote it θ. We redistribute the value of θ along the contour, adding θ to the corresponding volumes of cargo in the plus cells, and subtracting θ from the volumes of cargo in the minus cells of the table. As a result, the cell that was free and selected for loading becomes occupied, and one of the occupied cells in the contour becomes free.

We check the resulting reference plan for optimality, i.e., we return to the fourth stage of the algorithm.

Notes.

1. If the minus cells of the constructed cycle contain two or more identical minimum values, then when redistributing cargo volumes, not one, but two or more cells are released. In this case, the plan becomes degenerate. To continue the solution, it is necessary to occupy one or more simultaneously vacated cells of the table with zero, and preference is given to cells with the best tariff. So many zeros are introduced so that in the newly obtained reference plan the number of occupied cells (basic variables) is exactly (m + n−1).

2. If in the optimal plan of a transport problem the estimate for some free cell is equal to zero), then the problem has many optimal plans. For a cell with a zero score, you can build a cycle and redistribute the load. As a result, the resulting plan will also be optimal and have the same value of the objective function.

3. The value of the objective function at each iteration can be calculated as follows:

(the task is set to a minimum),

(the task is set to the maximum),

where is the volume of cargo moved along the contour;

Estimation of the free cell into which the load is directed during the transition to a new reference plan;

− value of the goal function at the k-th iteration;

− value of the goal function at the previous iteration.

Example.

Three warehouses of the wholesale base have homogeneous cargo in quantities of 40, 80 and 80 units. This cargo must be transported to four stores, each of which must receive 70, 20, 60 and 60 units respectively. Delivery costs per unit of cargo (tariffs) from each warehouse ) to all stores ) are given by the matrix .

Draw up a plan for transporting homogeneous cargo with minimal transport costs (conditional numbers).

Solution.

1. Let us check the necessary and sufficient condition for the solvability of the problem:

40+80+80 = 200,

70+20+60+60 = 210.

As you can see, the total demand for cargo exceeds its reserves in the warehouses of the wholesale base. Consequently, the model of the transport problem is open and has no solution in its original form. To obtain a closed model, we introduce an additional (fictitious) warehouse A4 with a cargo stock equal to A 4 = 210 – 200 = 10 units. We assume that the tariffs for transporting a unit of cargo from warehouse A4 to all stores are equal to zero.

We enter all the initial data in Table 7.

Reserves

A 1

A 2

A 3

A 4

Needs

210

2. Construction of the first reference plan using the minimum cost method.

Among the tariffs, the minimum or best is C14 = 1. We send the maximum possible load to cell A1B4, equal to min(60,40) = 40. Then x 14 = 40. All cargo has been removed from warehouse A1, but the demand of the fourth store is unsatisfied by 20 units. line A1 is out of consideration.

Among the remaining tariffs, the minimum element is C23 = 2. We send cargo min(60,80) = 60 to cell A2B3. In this case, column B3 is out of consideration, and 20 units have not been removed from warehouse A2.

Of the remaining elements, the minimum is C22 = 3. In cell A2B2 we send a load in the amount of min(20,20) = 20. In this case, row A2 and column B2 are simultaneously crossed out.

We select the minimum element C31 = 4. We send a load equal to min(70,80) = 70 to cell A3B1. In this case, column B1 is taken out of consideration, and 10 units have not been removed from warehouse A3. We send the remaining cargo from the third warehouse to tap hole A3B4, x 34 = 10. The demand of the fourth store is not satisfied by 10 units. we will send 10 units from a fictitious supplier - warehouse A4. cargo in cell A4B4, x 44 = 10.

As a result, the first reference plan is obtained, which is acceptable, since all goods have been removed from the warehouses and the needs of all stores have been satisfied.

3. Checking the degeneracy of a plan.

The number of occupied cells or basis variables in the first reference plan is six. the plan of the transport problem is degenerate, since the number of basic variables in a non-degenerate plan is equal to m + n – 1 = 4 + 4 – 1 = 7. To continue solving the problem, the reference plan must be supplemented by introducing fictitious transportation, i.e., occupy one of the free ones with zero cells.

When constructing the first reference plan, row A2 and column B2 were simultaneously crossed out, so the plan degenerated. The right of fictitious transportation is claimed by the free cells of row A2 and column B2, which have a minimum tariff and do not form a closed rectangular contour with the occupied cells. Such cells are A2B4 and A3B2. We send zero to cell A2B4.

4. Calculation of the value of the objective function.

The value of the objective function of the first reference plan is determined by summing the products of tariffs and volumes of transported cargo for all occupied cells of the table.

L(X1) = 4∙70 + 3∙20 + 2∙60 + 1∙40 + 3∙0 + 6∙10 + 0∙10 = 560 (thousand rubles).

5. Checking the optimality condition.

Let's calculate the potentials for occupied cells of the table from the condition: https://pandia.ru/text/78/103/images/image260_0.gif" width="139" height="22">Since the number of unknown potentials is greater than the number of equations (m + n > m + n – 1), then we take one of the potentials equal to zero..gif" width="115 height=154" height="154">

Assuming, we get https://pandia.ru/text/78/103/images/image265_0.gif" width="82" height="22">, ,https://pandia.ru/text/78/103/ images/image268_0.gif" width="193" height="22">

We enter the calculated potentials in Table 7. Let us calculate the estimates of free cells.

https://pandia.ru/text/78/103/images/image270_0.gif" width="167" height="22 src=">,

https://pandia.ru/text/78/103/images/image272_0.gif" width="210" height="22 src=">,

https://pandia.ru/text/78/103/images/image274_0.gif" width="183" height="22 src=">,

https://pandia.ru/text/78/103/images/image276_0.gif" width="153" height="22 src=">,

The first reference plan is not optimal, since there are positive estimates of free cells and . We choose the maximum positive estimate of a free cell - .

6. Construction of a new reference plan.

For cell A3B2, we will construct a rectangular closed circuit (0 table 7) and redistribute the load to the circuit. The vertices of the contour, starting from the vertex located in the free cell A3B2, are assigned alternately the signs “+” and “−”.

From the cargo volumes in the minus cells, select the smallest, i.e. θ = min(20,10) = 10. Add the value θ = 10 to the cargo volumes in the plus cells, subtract from the cargo volumes in the minus cells closed loop. As a result, we obtain a new reference plan shown in Table 8.

General setting transport problem consists in determining the optimal transportation plan for some homogeneous cargo from T departure points in P destinations . 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 by tariffs for transporting a unit of cargo from i th point of departure in j th destination, through – cargo supplies in i-th point of departure, through – cargo needs in j – m destination, and through – 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

under conditions

(64)

(65)

(66)

Because the variables satisfy systems of equations (64) and (65) and the condition non-negativity(66), delivery of the required amount of cargo to each destination, removal of existing cargo from all points of departure are ensured, and return transportation is excluded.

Definition 15.

Any non-negative solution of systems of linear equations (64) and (65), defined by the matrix , called plan transport task.

Definition 16.

Plan , at which function (63) takes its minimum value is called optimal plan transport task.

Typically, the initial data of a transport task is written in the form of table 21.

Table 21

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. If the specified condition is not met, then the model of the transport problem is called open.

Theorem 13.

For the transport problem to be solvable, it is necessary and sufficient that the supplies of cargo at the points of departure are equal to the needs for cargo at the points of destination, i.e., that the equality (67).

If the stock exceeds the requirement, i.e., a fictitious ( n+1)th destination with need and the corresponding tariffs are considered equal to zero: The resulting problem is a transport problem for which equality (67) is satisfied.

Similarly, when a fictitious ( m+1)-th point of departure with cargo reserve and tariffs are assumed to be zero: 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 of a specific problem is open, then, based on the above, we will rewrite the table of conditions of the problem so that equality (67) is satisfied.

Number of variables in a transport problem with T points of departure and P destinations equal Fri, and the number of equations in systems (64) and (65) is equal to p+t . Since we assume that condition (67) is satisfied, the number of linearly independent equations is equal to p+t – 1. Consequently, the reference plan of a transport problem can have no more than P + T – 1 non-zero unknowns.

If in the reference plan the number of non-zero components is exactly P +t – 1, then the plan is non-degenerate, and if it is less, then it is degenerate.

As with any problem, the optimal plan for a transport problem is also a reference plan.

To determine the optimal plan for a transport problem, you can use the methods outlined above.

Example 19.

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

Solution. Draw up a transportation plan in which the total cost of transportation is minimal. i Let us denote by the number of units of raw materials transported from j-th point of its receipt at

(68)

-e enterprise. Then the conditions for the delivery and removal of necessary and available raw materials are ensured by fulfilling the following equalities: With this plan

transportation, the total cost of transportation will be

Thus, the mathematical formulation of this transport problem consists of finding such a non-negative solution to the system of linear equations (68), in which the objective function (69) takes on a minimum value.

Program for solving a transport problem using the potential method

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

All transport problems of optimal attachment of destinations to points of poisoning, practically implemented in optimal cargo flow patterns, are solved in terms of transportation distance based on the minimum cargo turnover. The objective function Fс of the transport problem has the form:

Fc = min xij lij, (1)

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

xij - the size of freight traffic for each correspondence between points of origin and destination, t;

As a result of research carried out by I.V. Belov, it was proven that 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 on various interacting modes of transport, i.e. when drawing up complex optimal diagrams of cargo flows on a network of different types of communication routes.

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

In accordance with the basic concept of optimization, substantiated by MIIT, in the presence of reserves of throughput and carrying capacity, as an indicator of optimality in the current planning of transportation, it is more economically feasible to use a minimum of operating costs depending on the volume of transportation, i.e. minimum cost of transportation in terms of dependent expenses. The objective function of the transport problem in this case will have the form:

Fc = min xij From factory ij, (2)

where Св ij is the cost of cargo transportation for each correspondence of the cargo flow in terms of dependent costs, k/t.

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

Fc = min xij tij, (3)

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

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

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

Fc = min xij C tare 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 a minimum of ton-kilometers and the plan for a minimum of tariff fees coincided, since freight tariffs were based on the principle of the shortest transportation distances. But this statement is not entirely correct, 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 zones, especially over long distances, change over a wide range.

It is obvious that if territorial differentiation of tariffs is possible and expedient in market economic conditions, as well as with deeper differentiation depending on the level of quality of transportation, optimal transportation plans for a minimum of ton-kilometers and a minimum of tariff fees will no longer coincide.

One more important circumstance should be kept in mind. Optimizing transport connections at a minimum of tariffs means minimizing transport revenues, which can negatively affect its profits and profitability, i.e. on self-supporting interests of transport. Some experts argue that optimizing transportation plans according to this indicator is generally unacceptable, since it obviously puts transport in an unequal economic position compared to other sectors of the national economy. A serious objection to this argument arises. Transport revenues are at the same time tariff transport costs of the national economy, which we must constantly strive to save by eliminating various types of irrational transportation and associated unproductive expenses. Thus, in the context of the development of market relations, optimization of transportation plans at a minimum of tariffs should have a wider scope of application. But at the same time, it must move from the field of transport as such to the field of logistics as the optimization of supply plans.

The given costs as an indicator of optimality can be used when solving transport problems on a 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 capacity . The objective function of the optimal plan here can be expressed in two ways: without taking into account the cost of the cargo mass en route, if there are no significant differences in the time of delivery of goods by interacting modes of transport:

Fc = min xij (сij + En kij), (5)

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

Fc = min xij (сij + En (кij + mij), (6)

where kij - specific capital investments in rolling stock and permanent devices for each correspondence of freight traffic, k/t;

mij is the specific cost of the cargo mass en route for each correspondence of the cargo flow, k/t.

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

Similar judgments were made 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 to 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 of social production using an identical methodology and on the basis of the same methodological premises. In this case, it is especially important to accurately take into account the costs associated with fuel losses 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 type of transport are distorted. In order to remove distortions caused by the incomparability of cost indicators when optimizing the country's fuel and energy balance, these indicators must take into account the losses of the corresponding cargo.

Some works by 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 transported national economic products themselves and 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 the assortment and quality of their composition. It is meant that optimization of transportation of interchangeable products of different range and quality with commensurate consideration of their consumer properties (tire mileage, fuel calorie content, proportion of nutrients in fertilizers, iron in ore, etc.) will give an optimal plan that differs significantly from the optimal one 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.). The works of MIIT established that when developing operational current and long-term optimal plans for transportation by rail, the cost indicators of optimality must necessarily take into account the losses of many cargoes, especially perishable, bulk and bulk goods. When solving complex transport problems of transportation optimization for any period and planning involving two or more interacting modes of transport, losses must be included in the cost indicators of optimality for all groups of cargo in accordance with the classification. Differences, if any, in the consumer properties and quality of interchangeable cargo should be reflected through their corresponding prices in the cost of the cargo mass en route. The functionalities of the optimal plan can be expressed in general form: without taking into account the cost of the cargo mass in transit

Fс = min хij (сij + Enкij + у е ij), (7)

taking into account the cost of cargo weight en route

Fc = min xij (сij + En (кij + mij + y pe ij), (8)

where е ij is the specific value of current cargo losses in value terms for each correspondence of 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 cargo flow schemes based on 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 transportation costs. Therefore, it seems possible to optimize current and operational plans for the transportation of perishable goods based on the minimum of current losses while ensuring that the specified delivery times are met. 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:

Fc = 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 cargo flow schemes is fraught with great difficulties. The fact is that the preliminary calculation of site-by-site 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 individual components of cost indicators of optimality do not always provide the necessary reliability of the results.

Excess transportation capacity increases transportation costs and production costs. The optimality criterion is proposed to take minimal losses, on the one hand, from underutilization of rolling stock, and on the other, losses of consignees from late 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 transport capacity of the transport must be no less than the size of the cargo flow.

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

Consequently, 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 compared to the cargo flow.

When solving a transport problem, the choice of optimality criterion is important. As is known, an assessment of the economic efficiency of an approximate plan can 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 by various modes of 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 a more complete assessment of 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 transporting a unit of cargo from the i-th point of departure to the j-th point of destination, through – the cargo reserves at the i-th point of departure, through – the requirements for cargo at the j-th point of 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 of linear equations (2) and (3) and the non-negativity condition (4), the removal of existing cargo from all points of departure, delivery of the required amount of cargo to each destination are ensured, and return transportation is excluded.

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

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 of a specific problem is open, then, based on the above, we will 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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