1 / 21

A Better Future

A Better Future. An investigatory project on Linear Programming. George B. Dantzig. LINEAR PROGRAMMING. HISTORY. Linear Programming was developed during the second World War to plan expenditures and returns in order to reduce costs

sabin
Télécharger la présentation

A Better Future

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. A Better Future An investigatory project on Linear Programming

  2. George B. Dantzig LINEAR PROGRAMMING HISTORY • Linear Programming was developed during the second World War to plan expenditures and returns in order to reduce costs • The founders of the subject are George B.Dantzig, John von Neumann and Leonid Kantorovich. • It was kept secret until 1947, and is now being used by industries in their daily planning • The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979 • A major breakthrough came in 1984 when Narendra Karmarkar introduced the interior point method

  3. CONCEPTS DEFINITIONS A linear programming problem may be defined as the problem of maximizing or minimizing a linear function that is subjected to linear constraints, which may be equalities or inequalities. Linear Function – A function with no exponents other than one and with no products of the variables. In a rectangular co-ordinate system, the graph of a linear function is a line. Example – y=x+4, x+5=2y Optimization – In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. Objective function – The function to be maximized or minimized

  4. Constraints – In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. • They are of three types: • Equality constraints • Inequality constraints • Non-negative constraints • Optimal Solution – A solution to an optimization which minimizes or maximizes the objective function. • Feasible Solution – The space of all candidate solutions is called the feasible region, feasible set, search space, or solution space.

  5. Environment Today, the environment and its catastrophic degradation have assumed great importance, and action needs to be taken quickly Pollution has risen to hazardous levels. Pollution is a major cause of global warming, which, as we know, leads to melting of polar ice caps, rising seas, and disturbed weather patterns. Other environmental problems include the ozone hole, depletion of forests, and loss of biodiversity.

  6. Focus on Emissions Emissions from burning of fossil fuels, and in particular, from production of energy, lead to a massive amount of pollution. Last year, for instance, India’s total CO2 emission was a staggering 1342962000 metric tons. Some more facts – 1000 tonnes of CO2 are emitted every 25.8 seconds in India More than a million species face extinction from disappearing habitat, changing ecosystems, and acidifying oceans.

  7. The Connection To develop any economy requires the following : 1. Resources 2. Power 3. Control on pollution Thus, our project focuses on energy production by power plants in India (thermal, nuclear and hydro plants). We have tried to develop a model that ensures that the energy demand is met, but, at the same time, it reduces the CO2 emissions caused by the production of this energy. To The Problem

  8. Monetary Resources We have chosen monetary resources as a constraint as it is limited and is essential for the construction of new power plants, which will meet the energy demand. Data regarding costs : Cost of construction and maintenance of 1 thermal power plant : Rs. 1250 crore Cost of construction and maintenance of 1 hydro power plant : Rs. 2500 crore Cost of construction and maintenance of 1 nuclear power plant : Rs. 8000 crore Amount of money that the government is willing to spend : Rs. 5.8 trillion approx as per figures of forthcoming Five-Year Plan Home

  9. Power Since technology has developed by leaps and bounds, people’s energy demands have also spiraled. Efficient power generation is the only way to ensure that these rising demands are met. Data regarding power generation : Total power output by 1 thermal power plant : 1.56 billion KwH Total power output by 1 hydro power plant : 2.81 billion KwH Total power output by 1 nuclear power plant : 2.105 billion KwH Total energy required : 6.3 x 1018 Joules OR 6.3 exajoules Home

  10. Emission Control CO2 emissions today are at mercurial levels. India alone contributes 4.9 % of the total world CO2 emissions. Energy production, thus, must not cause a disproportionate amount of carbon emissions. Data regarding emissions : CO2 emissions from 1 thermal power plant : 1136 g / KwH CO2 emissions from 1 hydro power plant : 120 g / KwH CO2 emissions from 1 nuclear power plant : 15 g / KwH Clearly, from the above data, thermal power plants cause the maximum amount of pollution, with regard to the amount of energy produced. Our model, hence, also aims at curbing the construction of thermal power plants. Home

  11. Example only The Mathematical Model THE OPTIMIZATION EQUATION Since there is no specific limit for the amount of emissions, the parameter of the same cannot be taken as a constraint. Hence, the emission equation forms the optimization equation. The equation, as formulated from the data, is : 33.72 x + 3.1575 y + 177.2 z We have to control the emissions, so, by reducing the overall value of the above equation, we minimize the CO2 emissions while still meeting the energy requirements of our country.

  12. The Mathematical Model With the help of the data in the preceding slides, we were able to construct the following constraint equations : X = Number of hydro power plants to be constructed Y = Number of nuclear power plants to be constructed Z = Number of thermal power plants to be constructed COST CONSTRAINT EQUATION – 2512.5 x +   8115.75 y   +  1224.8 z < = 179010.8 ENERGY CONSTRAINT EQUATION – 2.81 x + 2.105 y + 1.56 z > = 1165.26 NON – NEGATIVITY CONSTRAINT – x > 0, y > 0, z >= 0 (as thermal power plants cause high amount of CO2 emissions, more of them might not actually be feasible).

  13. Various Methods Of Solution Images of simplex and graphs

  14. 2D Graph

  15. 3D Graph

  16. The Simplex Method Here are the values we entered and the outcomes we got : minimize p = 33.72x + 3.1575y + 177.2z subject to 2512.5x + 8115.75y + 1224.8z <=179010.8 2.81x + 2.105y + 1.56z >= 1165.26 No optimal solution exists for this problem. Minimize p = 33.72x + 3.1575y + 177.2z subject to 2512.5x + 8115.75y + 1224.8z <= 914878.5 2.81x + 2.105y + 1.56z>= 1165.26 Optimal Solution: p = 132362; x = 0, y = 0, z = 747 Minimize p = 33.72x + 3.1575y + 177.2z subject to 2512.5x + 8115.75y + 1224.8z <= 4492617.03 2.81x + 2.105y + 1.56z>= 1165.26 Optimal Solution: p = 1747.89; x = 0, y = 553.568, z = 0

  17. The Solution As shown in the previous slides, no feasible region exists for this Linear Programming problem. Hence, there is no possible optimal solution. This may be due to the following factors : Inadequate funds The money allocated by the Government is too less to meet the energy requirement of India in the future. We found that increasing the funds by a large amount gives us an impractical solution, with x and z equal to zero, and y assumes a large value, around 500.

  18. The Solution Excessive Energy Requirement The energy requirement may be too high, thus making it impossible to meet the energy demand within the resources that India has. Low Output By Power Plants The power output by the power plants in India is often comparatively low, thus making it necessary to build more power plants, which increases the cost. Increasing the cost calls for more funds, which again brings us back to the need for more funds.

  19. Summary Hence, as you have seen, we have tried various methods : Plottingtwo and three dimensional graphs Simplex method C++ programming (not displayed) Unfortunately, the existing constraints, which are inflexible, do not give an optimal solution. The measures that might be taken to provide a possible solution are : Increased allocation of funds by the Government Increased efficiency of power plants Saving energy to reduce the requirement

  20. The End THANK YOU By Goyal Vinayak Handa Rohan Krishna Ranjay Shukla Ayushman

More Related