An enterprise is a business entity that, at its own risk, carries out independent activities aimed at systematically extracting profit from the use of property, the sale of goods, the performance of work or the provision of services, and which is registered in this capacity in accordance with the procedure established by law. The enterprise independently disposes of the manufactured products, the profit received, which remains at its disposal after paying taxes and other obligatory payments.
To carry out its activities, the enterprise forms property that may belong to it on the basis of ownership or full economic ownership. The sources of the property of the enterprise are:
*Money and material contributions of the founders of the enterprise.
*Income from the sale of products, works and services.
*Securities and income from securities.
*Capital investments and subsidies from budgets of various levels.
*Gratuitous or charitable contributions.
*Donations from organizations, enterprises and citizens.
Redemption of property of state enterprises through competition, auction, tender.
*Repurchase of property state enterprise through shareholding.
*Other sources not prohibited by law.
The mechanism of the functioning of the enterprise.
The socio-economic role of the enterprise in market conditions is changing significantly. This change comes in a number of areas:
1. The main requirement that the market puts forward for an enterprise is to work in such a way that the result of the activity was not only the release of products or services, but also the receipt of profit, i.e. there should always be an excess of income over expenses, and the higher this excess, the more profitable the enterprise is.
2. The requirement to produce as many products as possible, since the social economy is permanently in deficit for a particular range of products, is being replaced by a requirement - not only to produce products, but also to sell them, to sell them. Further increase in volumes is expedient only in conditions of guaranteed sales of the latter.
3. The existing restrictions on the part of higher organizations are being replaced by the independence of enterprises in many areas of its activity, but as a price for this independence - the danger of insolvency and bankruptcy, i.e. the enterprise can be withdrawn from the economic sphere, since it could not work in the market conditions and was forced to leave it.
Consequently, the behavior of the enterprise in the market conditions of management is changing, and this change concerns many aspects of the enterprise's activities.
Any enterprise, regardless of its organizational and legal form, form of ownership, industry affiliation, manufactured products or services, is an open economic system.
1 block enterprise, whose main task is the transformation of resources.
2 block-input resources - labor, material, financial
3block resources at the output, i.e. converted resources - finished products, production waste, profit, cash
4 block - social environment with which the enterprise-state, municipal bodies, legislation interacts.
5block-natural environment-consumption natural resources, minerals, air, water.
6 block-relationship with the market, marketing, and this block interacts both at the input of resources and at the output.
7block ratio of input resources with output resources forms economic work in the enterprise, the economics of the enterprise.
As noted above, in a market environment, an enterprise solves two problems - to produce and sell products. Depending on the solution of these problems, the enterprise will flourish or become insolvent. In order to know what needs to be produced - what assortment, in what quantity, an enterprise must study the market, i.e. engage in marketing before it begins to expend resources for manufacturing specific products. Moreover, marketing
it is necessary to engage constantly - both before launching products into production, and after in the process of selling products. Therefore, in the scheme, the market is given before the start of production and its end.
To make decisions, it is necessary to collect and analyze the reliability of information, which includes:
* Characteristics of goods: whether they are end-use goods or intermediate, finished products or semi-finished products, whether a service is necessary or not, whether the price is acceptable to the consumer, like prices from competitors.
* Channels of distribution of goods: the presence of intermediaries between the manufacturer and the consumer, their number.
*Specific market conditions: are there legal restrictions that may prevent marketing activities.
*General characteristics market: a large or small number of consumers, ways of buying goods, the attitude of buyers to goods, terms and conditions of delivery, terms of sale from competitors.
*Legal restrictions: Are there any legal restrictions that may hinder marketing activities
*Levels management activities in the field of marketing: long-term goals of the company (for 10-15 years), taking into account the situation in the domestic and foreign markets and its development trends.
*Financial materials and other resources necessary to achieve these goals.
*Prospective (up to 5 years) goals of the company that arise in this case and their security necessary resources.
*Operational, current goals and objectives put forward by market conditions that do not contradict long-term strategic goals.
In the production of a product and the implementation of a marketing policy, it is necessary to take into account life cycle product on the market, which consists of a number of stages:
*Introduction, which requires a lot of money, so trading a product at this stage, as a rule, is unprofitable.
* Growth as a result of consumer acceptance of the product and the rapid increase in demand for it. With an increase in sales volume and, accordingly, profits, advertising costs stabilize.
*Maturity is characterized by the fact that the majority of buyers have already purchased the product, therefore, the sales growth rate, having reached a maximum, begins to fall, profits also begin to decline due to an increase in advertising and other marketing activities.
*Saturation: in this period, despite the measures taken, sales growth is no longer observed. Profit from trading continues to increase due to lower production costs.
*A recession is a period of sharp decline in sales and then profits.
The input enterprise consumes resources of a certain type, so that later as a result production process at the output, get transformed resources, resources of a different consumer value.
The ratio of resources at the input and output is the content of such a concept as the economics of the enterprise. For the economy of the enterprise, the product produced, the chosen production technology, the composition and qualification level of personnel are indifferent to a certain extent. The only thing that interests her is the ratio in the use of resources, which implies an excess of the revenue side over the expenditure side (profitable operation of the enterprise), an excess of the expenditure side over the revenue side (unprofitable operation of the enterprise) and, finally, equality of the revenue and expenditure parts (the operation of the enterprise in conditions self-sufficiency).
The enterprise consumes
Raw materials and materials Land resources Devices, tools, equipment
Component parts Equipment Household inventory
Fuel and energy Vehicles Cash
Equipment Computing facilities Transmitting devices
At the output, the enterprise releases resources in the form
Finished products in a specific nomenclature, assortment, quantity, price form
production waste
production costs
Payments different kind
How an enterprise works - profitably, unprofitably or within the framework of self-sufficiency, largely depends on the methods and methods of converting resources and can be determined by a number of both private and general performance indicators:
The total gross or clean production.
Average population workers.
The average annual cost of fixed production assets.
Current material costs for production.
Production cost
In order for the transformation of resources to be carried out, the enterprise must have a certain structure, it must take place certain processes, it must perform certain functions for the management and organization of production.
Company structure
Main production divisions
Auxiliary production divisions
Subdivisions serving side and ancillary
Management services and divisions
Processes carried out at the enterprise
Scientific research Transportation
Design, construction
Manufacturing Recovery
Repair and modernization Preservation and packaging
Control and prevention of marriage Cleaning and neutralization
Maintenance Safety Prevention
Storage Management
Enterprise management functions
Accounting, statistics Motivation
Analysis Performance control
Planning Regulation
Decision making and implementation
Functions of the organization in the enterprise
Rationing Resourcing
Coordination in time and space Routing of objects of labor
Operational planning Workplace planning
Dispatch Creation of favorable conditions
Execution of control functions Implementation finished products
Division and cooperation of labor Identification of reserves and rational
production
P the process of transforming resources in an enterprise
An important element of the production process at the enterprise is the selected technology option. It is the chosen technology that determines the composition and the required amount of resources at the exit to production system. A change in technology leads to a change in the professional and qualification structure necessary for the production of personnel, technological equipment, vehicles, tools up to changes in the types of raw materials, materials, semi-finished products, components used. An enterprise, depending on the nature of its products, the scale of investment opportunities available to it, chooses one or another technology. The selected technology is closely related to two indicators of the enterprise: the price of products and the level of its quality. At the same time, these two factors form the competitiveness of products, more often the preference is given to the second indicator. A little more about these indicators.
Under the conditions of planned management of the economy, the enterprise planned the level of prices, costs and the standard of profitability. In market conditions, these values are freely set by the enterprise.
In market conditions, the price serves as a significant regulator of production and has a direct and indirect impact on the amount of generated profit, on the nature and results of competition. The price of manufactured products for the enterprise is the contractual price between it and the consumer. There are several options for contract prices (Cd):
Option 1: CD = Cost + Profit
Option 2: CD = Demand - Supply
3rd option: Competitor's price< Цена кнкурент j
The first variant of setting the contractual price is completely determined by the conditions of production and is concentrated within the enterprise. However, in market conditions, this is more an exception than a rule, since there is usually more than one manufacturer on the market. This option is applicable when the manufacturer of the goods is the leader in this market and when the results financial activities satisfy management. This option makes it possible to fairly fairly distribute profits between the producer, who does not offer a scarce product, and the consumer, for whom the price is not a decisive factor in the purchase. The second and third options are close to reality.
In the second option of setting the price, the forecasted demand and the cost of goods are taken as the basis. The task for the enterprise is set as follows: the price per unit of goods is predetermined and it is necessary to establish at what quantity of goods sold the enterprise will begin to receive real profit. The reverse problem is also possible: the approximate number of products that can be sold on the market is known, it is necessary to determine at what price this sales volume will not lead to losses.
The third pricing option takes into account the level of competition in the market and often gives good results. The manufacturer studies the price dynamics of competitors and determines what the competitor's price will be for a similar product, with the main focus on reducing costs and increasing profits.
Consideration of the three options shows that under the conditions market economy price is a factor in ensuring the competitiveness of products. Other equally important, and maybe whiter an important factor product quality.
Quality is a set of product properties designed to satisfy the need in accordance with the purpose of the product. Any product is a combination of properties, which include reliability, ergonomics, aesthetic properties, safety and environmental friendliness. Product quality is the result of quality: labor, objects of labor, means of labor, technology, management, and the working environment as a whole.
AT last years the world's leading firms are consistently implementing a new strategy for product quality management.
Quality assurance is not understood as a technical function implemented by some department, but as a systematic process that permeates the entire organizational structure firms.
The new concept of quality must meet the appropriate structure of the enterprise.
Quality improvement requires application new technology manufacturing, from design automation to automated quality control measurements.
All of this can only be done if there is a clear organized system quality management, aimed at the interests of consumers, affecting all departments and acceptable to all personnel.
Entrepreneurial activity is an initiative independent activity of citizens and their associations aimed at making a profit. Therefore, making a profit is the immediate goal of the enterprise. But an enterprise can make a profit only if it produces products or services that are sold, i.e. satisfy social needs. The subordination of these two goals - satisfaction of needs and profit - is as follows: you cannot make a profit without studying the needs and without starting to produce the product that satisfies the needs. It is necessary to produce a product that will satisfy needs and, moreover, at a price that would satisfy solvent needs. And an acceptable price is possible only if the enterprise maintains a certain level of costs, when all the costs of consumed resources are less than the revenue received. In this sense, profit is the immediate goal of the functioning of the enterprise and at the same time is the result of its activities. If an enterprise does not fit into the framework of such behavior and does not make a profit, then it is forced to leave the economic sphere, to declare itself bankrupt.
Foreign economic activity - it is economic entrepreneurial activity legal and individuals in the field of international exchange of goods, services, movement of material, financial and intellectual resources.
The choice of the type of foreign economic activity depends on the specific specialization of the enterprise, its financial situation, the mechanism state regulation, forms of ownership
Currently the main form Foreign economic activities of enterprises Russia is a foreign trade activity, which consists in the exchange of goods and services between various countries. According to the current legislation in Russia, almost any enterprise or organization has the right to independently enter the foreign market. This implies that the enterprise must itself resolve all issues related to the implementation of a foreign trade transaction with a foreign partner.
First, the question is decided what goods or services the company can offer to the foreign market. Special attention here is given to the study of the prospects for the sale of these products in the foreign market. Then a partner in the transaction is selected. When choosing a partner, it is very important to obtain information from independent sources about his conscientiousness in fulfilling obligations, about what position he occupies in this market (whether he is an intermediary or an independent producer-consumer), how strong his financial position. At the next stage of the transaction, it is supposed to establish a business contact with the selected partner - negotiations. During the negotiation process, all issues related to the future deal should be resolved and agreed upon. The result of negotiations between the parties to the transaction is a foreign trade contract.
List of used literature :
V.Ya. Gorfinkel, G.P. Polyak, V.A. Schwander. "Entrepreneurship". 2000
L.V. Prykin. " Economic analysis enterprises". 2001
O.I. Volkov, V.P. Sklyarenko. "Enterprise Economics" 2001
Pskov Polytechnic Institute-branch of St. Petersburg State Technical University
FEM group 11-02
Student Alexandrov Artyom.
Pskov 2001.
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A task.
The enterprise produces science-intensive products, the production of which requires components. You can buy these components from a supplier at a price of 50 thousand rubles. for a unit. Or make them yourself. Variable expenses of the enterprise will amount to 43 thousand rubles. per unit of components. Fixed costs will amount to 64,500 thousand rubles. Determine the "supply strategy" of the enterprise with components: buy them from a supplier specializing in their production, or manufacture them at the enterprise?
Comment.
If we don’t consider that the author of the problem has little idea of what a “supply strategy” is, if we don’t consider that the author of the problem doesn’t suspect the existence of the SSS, and if we don’t consider that the problem, theoretically, cannot be solved, then let’s start in order.
Supplier price with or without VAT? If with VAT, what is the tax rate? Is this the price with delivery or Ex-Works? And the answer to these questions can be completely different. The phrase "variable costs per unit of components" amused me. The author of the problem understands that, by definition (see the dictionary), components are purchased and there can be no variable costs for them?
Now about the "main idiocy" of the task. If fixed costs (attention) amount to a certain amount, now the company is working without them? Or they already exist, they are constant, then the problem does not need to be solved at all, because the cost of production is definitely lower purchase price. And if they are not, then the enterprise does not work? But if it does not work, then why should we buy something? And if we are considering in-house production instead of purchasing components, then what is the size of the production program? Simply put, how much of the semi-fixed distributable costs should be allocated to the cost of the product? The conclusion from these arguments is sad - the problem cannot be theoretically solved.
What did the author of the problem mean?. It was assumed that it is necessary to calculate the minimum volume of output with the distribution of semi-fixed costs to the cost price in order to evaluate the production program. If we need more than this number, then more profitable production on your own, less is better purchase. But with the concept of "components" the author got into a puddle. If something else needs to be done with this product, then a completely different, unknown to us, basis for the distribution of semi-fixed costs for other products and services arises, which makes a decision impossible. If all conditionally fixed costs should be allocated to the production of only this part of the total "high technology products", then it is logical to assume that the enterprise will not produce anything other than "components". That is, there will be no "science-intensive" products. And so in a circle...
Solution.
Since a person who is familiar with the basics of economics cannot solve this problem (see above), we solve it from the point of view of the "accountant".
We assume that there is no VAT in our country, and decide ...
50,000 - 43,000 \u003d 7,000 rubles marginal difference on one product.
To cover semi-fixed costs, we need to issue
64 500 000 / 7000 ≈ 9 215 items
Note. Round up to a dot opportunity cost was not lower than the real one.
The factory produces two types of products: P 1 and P 2. Products of both types go to wholesale. For the production of these products, three initial products are used - A, B, C. The maximum possible daily reserves of these products are 6, 8 and 5 tons, respectively. The costs of raw materials A, B, C per 1 thousand products P 1 and P 2 are given in table. A study of the sales market showed that the daily demand for P 2 products never exceeds the demand for P 1 products by more than 1 thousand pieces. In addition, it has been established that the demand for products P 2 never exceeds 2 thousand pieces. per day.
Wholesale prices 1 thousand pieces. products P 1 are equal to 3 thousand rubles, 1 thousand pieces. P 2 - 2 thousand pieces.
How many products (in thousands of pieces) of each type should the factory produce in order to maximize the income from the sale of products?
The construction of a mathematical model should begin with the identification of variables (desired values). After that, the objective function and constraints are determined through the corresponding variables.
In this example, we have the following:
Variables.
Since it is necessary to determine the volume of production of each type of product, the variables are:
X 1 - daily output of the product P 1 in thousand pieces;
X 2 - daily output of the product P 2 in thousand pieces.
objective function. Since the cost of 1 thousand products P 1 is equal to 3 thousand rubles, the daily income from its sale will be 3X 1 thousand roubles. Similarly, income from the sale of X 2 thousand pieces. P 2 will be 2X 2 thousand rubles. per day. Assuming the independence of sales volumes of each of the products, the total income is equal to the sum of two terms - income from the sale of products P 1 and income from the sale of products P 2.
Denoting income (in thousand rubles) through f(X), we can give the following mathematical formulation of the objective function: determine the (admissible) values of X 1 and X 2 that maximize the value of the total income:
f(X) \u003d 3X 1 + 2X 2, X \u003d (X 1, X 2)
Restrictions. When solving the problem under consideration, restrictions on the consumption of initial products A, B and C and the demand for manufactured products should be taken into account, which can be written as follows: Consumption of the initial product for the production of both types of products
≤
The maximum possible supply of a given input product
This leads to three restrictions:
X 1+ 2X 2 ≤ 6 (for A),
2X 1+ X 2 ≤ 8 (for B),
X 1+ 0.8X 2 ≤ 5 (for C).
Restrictions on the magnitude of demand for products have the form:
X 2 - X 1 ≤ 1 (the ratio of the demand for products P 1 and P 2),
X 2 ≤ 2 (maximum demand for products P 2).
The conditions for the non-negativity of variables are also introduced, i.e., restrictions on their sign:
X 1 ≥ 0 (output P 1)
X 2 ≥ 0 (output P 2).
These restrictions lie in the fact that the volume of production cannot take negative values.
Therefore, the mathematical model is written as follows.
Determine the daily production volumes (X 1 and X 2) of products P 1 and P 2 in thousand pieces, at which
max f(X) = 3 X 1 + 2 X 2 (objective function)
with restrictions:
X 1 + 2X 2 ≤ 6
2X 1 + X 2 ≤ 8
X 1 + 0.8X 2 ≤
5 - X 1 + X 2 ≤ 1
X 2 ≤ 2
X 1 ≥0, X 2 ≥ 0
resource task
For the manufacture of products of type A and B, three types of raw materials are used, the stocks of each of which are P1, P2, P3. For the production of one product of type A, it is required to spend a1 kg of raw materials of the first type, a2 kg of raw materials of the second type, and a3 kg of raw materials of the third type. For one product of type B, b1, b2, b3 kg of each type of raw material is consumed, respectively. Profit from the sale of a unit of product A is α monetary units, and products B - β monetary units. Draw up a plan for the production of products A and B, providing the maximum profit from their sale. Solve the problem using the simplex method. Give a geometric interpretation of the problem.Mathematical model of the problem
2x1 + x2 ≤438
3x1 + 6x2 ≤747
4x1 + 7x2 ≤812
F(X) = 7x1 + 5x2 => max
Resource usage problem (production planning problem)
For the manufacture of two types of products P 1 and P 2 use three types of resources S 1 , S 2 , S 3 . Stocks of resources, the number of units of resources spent on the manufacture of a unit of output, are given in Table. one.Table 1
Resource type | The number of units of production spent on the manufacture of a unit of production | Resource reserves |
|
| R 1 | R 2 | ||
| S1 | 2 | 3 | 20 |
| S2 | 3 | – | 18 |
| S3 | 1 | 4 | 10 |
Profit received from a unit of production Р 1 and Р 2 – respectively 2 and 3 c.u. It is necessary to draw up a plan for the production of products in which the profit from its sale will be maximized.
Solution. Compose an economic mathematical model tasks. Denote by x 1, x 2 - the number of units of production P 1 and P 2, respectively. Then the total profit F will be 2x CU 1. from the sale of products Р 1 and 3х 2 f.u. from the sale of products P 2, that is
F = 2x1 + 3x2. (one)
Since the amount of resources required for the production of products is limited, we will compose a system of resource restrictions. Production will require (2x 1 + 3x 2) units of resource S 1 ,
3x 1 units of resource S 2 and (x 1 + 4x 2) units of resource S 3 . Since the consumption of resources S 1 , S 2 , S 3 should not exceed their reserves, 20, 18, 10 units, respectively, the relationship between resource consumption and their reserves will be expressed by a system of inequality constraints:
(2)
So, the economic and mathematical model of the problem: to find such a production plan that satisfies the system of constraints (2), in which the objective function (1) takes the maximum value.
The resource utilization problem can be generalized to the case of producing n types of products using m types of resources.
Denote by x (j = 1, 2,…,n) the number of units of production P j planned for production; b 1 (i = 1, 2,…,m) – stocks of resources S i , a ij – number of units of resource S i spent on manufacturing a unit of product P j ; c j - profit from the sale of a unit of production P j . Then the economic-mathematical model of the problem in the general formulation will take the form:
F=c 1 x 1 +c 2 x 2 +…+c n x n →(max) (3)
(4)
Find such a plan 
output that satisfies system (4), at which function (3) takes the maximum value.
Comment. This task It is also called the problem of determining the optimal range of products.
Production target to the minimum
The team accepted an order for the manufacture of 57 pieces. products P1, 68 pcs. products P2 and 80 pcs. products P3. Products are produced on machines A and B. For the manufacture of a unit of production P1 on machine A, 15 minutes are required, units of production P2 - 50 minutes, units of production P3 - 27 minutes, on machine B - respectively 11, 15 and 13 minutes.Build a mathematical model of the problem, on the basis of which you can find how many products and what kind should be produced on machines A and B so that the order is completed in the minimum time.
Solution. Mathematical model of the problem.
x 1 - manufactured products P1 on machine A, pcs.
x 2 - manufactured products P1 on machine B, pcs.
x 3 - manufactured products P2 on machine A, pcs.
x 4 - manufactured products P2 on machine B, pcs.
x 5 - manufactured products P3 on machine A, pcs.
x 6 - manufactured products P3 on machine B, pcs.
Quantity restrictions:
x 1 + x 2 ≥ 57
x 3 + x 4 ≥ 68
x 5 + x 6 ≥ 80
Target function:
15x1 + 11x2 + 50x3 + 15x4 + 27x5 + 13x6 = min
The problem of optimal production
Example #1. The enterprise plans to produce two types of products I and II, the production of which consumes three types of raw materials A, B, and FROM. The need a ij for each unit of the j-th type of product of the i-th type of raw material, the stock b i of the corresponding type of raw material and the profit c j from the sale of a unit of the j-th type of product are given by the table:- For the production of two types of products I and II with a plan x 1 and x 2 units, make up the target profit function Z and the corresponding system of restrictions on stocks of raw materials, assuming that it is required to produce a total of at least n units of both products.
- Under the conditions of task 1, draw up an optimal plan (x 1 , x 2) for the production of products that ensures maximum profit Z max . Determine the balance of each type of raw material. (Problem to be solved by the simplex method)
- Construct a polygon of feasible solutions according to the obtained system of restrictions and find the optimal production plan geometrically. Determine the corresponding profit Z max .
Example #2. A farmer can grow 4 crops on an area of 80 ha. He has already invested agreements for the sale of certain products (sales volume) and may purchase 250c of mineral fertilizers.
The area of tilled crops (sunflower, sugar beet, potatoes, corn) should be 20 ha.
Labor and fertilizer costs, profit per 1 ha are shown in Table 2.
Determine how much space should be devoted to each crop in order to maximize profit.
Develop an economic-mathematical model and solve the problem.
Take the output according to the option (Table 2)
Solution.
x 1 - area under buckwheat, ha; x 2 - area under barley, ha; x 3 – area under millet, ha; x 4 - area for potatoes, ha
Objective function: 140x 1 + 110x 2 + 120x 3 + 380x 4 → max
Fertilizer cost limits
Sales volume restrictions
Area restrictions
x 1 + x 2 + x 3 + x 4 ≤ 80
Row crop area restrictions
x4 ≤ 20
In total, we have the following ZLP
3x1 + 3x2 + 2x3 + 5x4 ≤ 250
10x1 + 30x2 + 25x3 + 180x4 ≤ 200
x 1 + x 2 + x 3 + x 4 ≤ 80
x4 ≤ 20
x 1 , x 2 , x 3 , x 4 ≤ 0
Objective function: 140x 1 + 110x 2 + 120x 3 + 380x 4 → max
Analyze the situation and choose the most beneficial solution for the enterprise:
A. Make the parts yourself.
B. Acquire parts on the side and use your own released equipment for the production of other products that can bring a profit in the amount of 18,000 rubles.
Problem 24
The company has 1,000 obsolete computer parts purchased earlier for 200,000 rubles. What is more profitable: to process parts with a cost of 40,000 rubles. and sell them for 64,000 rubles. or sell them for 17,000 rubles. without any processing?
Problem 25
The company produces technically complex products, the production of which requires a significant number of components. The company can buy these parts at a price of 100 rubles. / PCS. or make them yourself. Determine what is more profitable for the enterprise (the threshold of profitability has been crossed).
The cost of production of parts at the enterprise itself:
straight variable costs- 85 rubles. / PCS.;
Initial data for calculations:
fixed costs - 60,000 rubles; selling price of a unit of production - 20 rubles; variable costs per unit of production - 12 rubles; current sales volume - 8200 units; the acceptable range of production volumes is 4,500–12,500 units.
Problem 46
Based on the table. 1 and 2 initial data to calculate the retail price of the old and new goods.
Table 1 Expert assessments quality parameters of old and new goods
Table 2 Conditions for calculating the retail price
The maximum allowable operating time on devices I, II, III, IV is 84, 42, 21 and 42 hours, respectively.
Solution
Let's place the table with the initial data in cells A1:G9 of the Excel worksheet and perform the necessary preliminary calculations (see Fig. 4.3).
Rice. 4.3 - Initial data of the optimization problem
Find a solution to the problem under the following conditions:
The final form of the problem statement is shown in fig. 4.4:
Fig.4.4 - Formulating the task in terms of an Excel worksheet
The final result is shown in Figure 4.5:
Fig.4.5 - Optimization result
Analysis of the solution shows that all the requirements of the optimization problem are met without exception. At the same time, it is clear that in order to maximize profits, it is not advisable to produce product C.
The calculation results are presented in the report on the results (Fig. 4.6):
Fig.4.6 - Report on the results
The Solve Finder utility can also be used to solve more challenging tasks optimization.
Task 2. Optimization of the transportation plan (transport task).
The firm has 4 factories and 5 distribution centers for its products. Factories are located in the city. Slutsk, Borisov, Molodechno and Bobruisk with production capabilities respectively 200, 150, 225 and 175 units daily.
Distribution centers are located in Vitebsk, Minsk, Orsha, Mogilev and Gomel with a daily demand of 100, 200, 50, 250 and 150 units respectively.
Storage at the factory of a unit of production that is not delivered to the distribution center costs 0.75 c.u. per day, and the penalty for late delivery of a product unit ordered by the consumer in the distribution center, but not located there, is 2.5 c.u. in a day.
The cost of transporting a unit of production from factories to distribution points is shown in table 4.1
Table 4.1. Transportation plan
It is necessary to plan transportation in such a way as to minimize the total transportation costs.
It is important to note that since this model is balanced, i.e. the total volume of manufactured products is equal to the total volume of needs for it, then in this model it is not necessary to take into account the costs associated with both warehousing and underdeliveries of products. Otherwise, the model must be entered:
In case of overproduction, a fictitious distribution point; the cost of transporting a unit of production to this fictitious point is assumed to be equal to the cost of warehousing, and the volumes of transportation to this point are equal to the volumes of warehousing of excess products in factories;
In case of shortage, a fictitious factory; the cost of transportation of a unit of production from a fictitious factory is assumed to be equal to the cost of fines for underdelivery of products, and the volumes of transportation from this factory are equal to the volumes of underdeliveries of products to distribution points.
To solve this problem, we construct a mathematical model. The volumes of traffic are unknown here. Let xij be the volume of transportation from the i-th factory to j-th center distribution. The goal function is the total transportation costs, i.e.
,
where cij is the cost of transporting a unit of product from the i-th factory to the j-th distribution center. In addition, the unknowns must satisfy the following constraints:
Non-negativity of traffic volume;
Because model is balanced, then all products must be taken out of the factories and the need of all distribution centers must be fully satisfied.
Thus, we have the following model:
Minimize:
,
With restrictions:
IО, jО,
where a i is the volume of production per i-th factory, b j – demand in j-th center distribution.
Statement of the problem in terms of an Excel worksheet for using the Solver utility.
1. Place the initial data, as shown in Fig.4.7, 4.8.
2. Select cells B8:F11 for the unknowns (traffic volumes).
3. Enter in cells H8:H11 the volumes of production in factories.
4. Enter in cells B13:F13 the need for products at distribution points.
5. In cell B16, enter the goal function = SUMPRODUCT(B3:F6;B8:F11).
6. In cells G8:G11, enter formulas that calculate production volumes at factories, in cells B12:F12 - the volumes of products delivered to distribution points.
Fig.4.7 - Initial data
Rice. 4.8 - Raw data in formula mode
In the window of the "Search for a solution" utility, specify the target cell, cells to be changed and restrictions (see Fig. 4.9).
Rice. 4.9 - Parameters of the "Search for a solution" window
The optimal plan that provides the minimum cost for transporting products from producers to consumers, found using the Search for a Solution utility, is shown in Fig. 4.10.
Fig.4.10 - Results of the search for a solution
As described above, the Find Solution utility can generate a report on the results.
Tasks for independent work
Exercise 1. Solve a linear programming problem using the add-in "Search for a solution" TP MS Excel.
For the production of two types of products A and B, three types of technological equipment are used. For the production of a unit of product A, equipment of the first type is used a 1 hour, equipment of the second type - a 2 hours, equipment of the third type - a 3 hours. For the production of a unit of product B, equipment of the first type is used in 1 hour, equipment of the second type - in 2 hours, equipment of the third type - in 3 hours.
For the manufacture of all products, the administration of the enterprise can provide equipment of the first type for no more than t 1 hours, equipment of the second type for no more than t 2 hours, equipment of the third type for no more than t 3 hours.
Profit from the sale of a unit of finished product A is α rubles, and products B - β rubles.
Draw up a plan for the production of products A and B, providing the maximum profit from their sale.
Task options are shown in Table 4.2.
Table 4.2. Task options
| Option | a 1 | a 2 | a 3 | in 1 | in 2 | at 3 | t1 | t2 | t3 | α | β |
Task 2. Solve the problem of optimizing the transportation plan (transport problem).
There are n production points and m product distribution points. The cost of transporting a unit of production from the i-th production point to the j-th distribution center c ij is given in the table, where the line is the point of production, and the column is the point of distribution. In addition, in this table, the i-th line indicates the volume of production in i-th paragraph production, and the j-th column indicates the demand in the j-th distribution center. It is necessary to draw up a transportation plan for the delivery of the required products to distribution points, minimizing the total transportation costs.
Option 1.
Option 2.
Option 3.
Option 4.
Option 5.
Option 6.
Option 7.
Option 8.
Option 9.
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