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Advantages of Linear Programming





Linear Programming:
The term was introduced in 1950 to refer to plans or schedules for training, logistical supply and for deployment of men in the service. A linear programming is a subset of Mathematical programming, and the later field is part of operations research.

A linear programming problem differs from the general variety in that a mathematical mode or description of the problem can be stated using relationships which are called “ straight-line” or linear. The mathematical statement of the linear-programming problem includes a set of linear equation which represent the conditions of the problem.

Linear Programming is the part of mathematics deals with the study of optimization problems with required number of constraints and objective.

Advantages of Linear Programming:
Some of the real time applications are in production scheduling, production planning and repair, plant layout, equipment acquisition and replacement,logistic management and fixation. Linear programming has maintained special structure  that can be exploited to gain computational advantages.

some of the advantages of Linear Programming are:
  • Utilized to analyze numerous economic, social, military and industrial problem.
  • Linear programming is best suitable for solving complex problems.
  • Helps in simplicity and Productive management of an organization which gives better outcomes.
  • Improves quality of decision: A better quality can be obtained with the system by making use of linear programming.
  • Provides a way to unify results from disparate areas of mechanism design.
  • More flexible than any other system, a wide range of problems can be solved easily.
Limitations of Linear Programming

Linearity of relations of variables: The linearity cannot be obtained in all function since still some of the function are non linear in the  surrounding environment.
The assumptions of linear programming are also unrealistic: there is a change in relation between input,output gain, loss etc.
Limit on the possible solution.

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From Wikipedia

Linear programming

Linear programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.

More formally, linear programming is a technique for the optimization of a linearobjective function, subject to linear equality and linear inequalityconstraints. Given a polytope and a real-valued affine function defined on this polytope, a linear programming method will find a point on the polytope where this function has the smallest (or largest) value if such point exists, by searching through the polytope vertices.

Linear programs are problems that can be expressed in canonical form:

\begin{align}

& \text{maximize} && c^\top x\\ & \text{subject to} && A x \leq b \end{align} where x represents the vector of variables (to be determined), c and b are vectors of (known) coefficients and A is a (known) matrix of coefficients. The expression to be maximized or minimized is called the objective function (cTx in this case). The equations Ax' â‰¤ b are the constraints which specify a convex polytope over which the objective function is to be optimized. (In this context, two vectors are comparable when every entry in one is less-than or equal-to the corresponding entry in the other. Otherwise, they are incomparable.)

Linear programming can be applied to various fields of study. It is used most extensively in business and economics, but can also be utilized for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.

History

The problem of solving a system of linear inequalities dates back at least as far as Fourier, after whom the method of Fourier-Motzkin elimination is named. Linear programming arose as a mathematical model developed during World War II to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Postwar, many industries found its use in their daily planning.

The founders of the subject are Leonid Kantorovich, a Russian mathematician who developed linear programming problems in 1939, George B. Dantzig, who published the simplex method in 1947, and John von Neumann, who developed the theory of the duality in the same year. The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior point method for solving linear programming problems.

Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the Simplex algorithm. The theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked.

Uses

Linear programming is a considerable field of optimization for several reasons. Many practical problems in operations research can be expressed as linear programming problems. Certain special cases of linear programming, such as network flow problems and multicommodity flow problems are considered important enough to have generated much research on specialized algorithms for their solution. A number of algorithms for other types of optimization problems work by solving LP problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as duality,decomposition, and the importance of convexity and its generalizations. Likewise, linear programming is heavily used in microeconomics and company management, such as planning, production, transportation, technology and other issues. Although the modern management issues are ever-changing, most companies would like to maximize profits or minimize costs with limited resources. Therefore, many issues can be characterized as linear programming problems.

Standard form

Standard form is the usual and most intuitive form of describing a linear programming problem. It consists of the following four parts:

  • A linear function to be maximized
e.g., Maximize: c1x1 + c2x2
  • Problem constraints of the following form
e.g.,
a1,1x1 + a1,2x2≤ b1
a2,1x1 + a2,2x2≤ b2
a3,1x1 + a3,2x2≤ b3
  • Non-negative variables
e.g.,
x1≥ 0
x2≥ 0.
  • Non-negative right hand side constants
bi≥ 0

The problem is usually expressed in From Yahoo Answers

Question:Coast-to-Coast Airlines is investigating the possibility of reducing the cost of fuel purchases by taking advantage of lower fuel costs in certain cities. since fuel purchases represent a substantial portion of operating expenses for an airline, it is important that these costs be carefully monitored. however, fuel adds weight to an airplane, and consequently, excess fuel raises the cost of getting from one city to another. in evaluating one particular flight rotation, a plane begins in Atlanta, flies from Atlanta to los angeles, from los angeles to Houston, from Houston to new Orleans, and from new Orleans to Atlanta. when the plane arrives in atlanta, the flight rotation is said to have been completed, and then it starts again. thus, the fuel on board when the flight arrived in Atlanta must be taken into consideration when the flight begins. along each leg of this route, there is a minimum and a maximum amount of fuel that may be carried. The regular fuel consumption is based on the plane carrying the minimum amount of fuel. if more than this is carried, the amount of fuel consumed is higher. specifically, for each 1,000 gallons of fuel above the minimum, 5% (or 50 gallons per 1,000 gallons of extra fuel) is lost due to excess fuel consumption. for example, if 25,000 gallons of fuel were on board when the plane takes off from atlanta, the fuel consumed on this route would be 12+0.05=12.05 thousand gallons. if 26 thousand gallons were on board, the fuel consumed would be increased by another 0.05 thousand, for a total of 12.1 thousand gallons. Formulate this as an LP problem to minimize the cost. how many gallons should be purchased in each city? what is the total cost of this? Data for problem below: Leg Atlanta-Los Angeles Los Angeles-Houston Houston-New Orleans New Orleans-Atlanta Minimum Fuel Required (1,000 Gal.) 24 15 9 11 Maximum Fuel Allowed (1,000 Gal.) 36 23 17 20 Regular Fuel Consumption (1,000 Gal.) 12 7 3 5 Fuel Price Per Gallon $1.15 $1.25 $1.10 $1.18

Answers:Atlanta to LA to Houston to NO to Atlanta for each 1,000 gallons of fuel above the minimum, 5% (or 50 gallons per 1,000 gallons of extra fuel) is lost due to excess fuel consumption. (This is clearly an approximation since the percentage loss would depend on the flight length, the weight of the aircraft, the wind, etc. and I'm thinking a more accurate formula wouldn't be linear.) Leg 1: Min 24,000 Max 36,000 Consumption 12,000 Price 1.15 Initial fuel : X1 Purchase fuel: P1 = ? Leg 2: Min 15,000 Max 23,000 Consumption 7,000 Price 1.25 X2 = ? P1=? Leg 3: Min 9,000 Max 17,000 Consumption 3,000 Price 1.10 X3 = ? P2 = ? Leg 4: Min 11,000 Max 20,000 Consumption 5,000 Price 1.18 X4 = ? P3 = ? X1, X2, X3, X4 = initial fuel when plane is ready to be fueled up P1, P2, P3, P4 = amount of fuel purchased Formulate an LP problem: This means: Identify the limits for each variable Write an equation for the quantity to be optimized (total cost) The question says to account for the fuel load in the airplane before adding fuel in Atlanta Let X1 be the initial amount in Atlanta. Then the amount it will carry is X1 + P1. The cost will be C1 = P1 x (1.15) (I apologize I used P for the amount purchased, I should have used something else since P is usually price.) The limits are 24000 < X1 + P1 < 36000 The amount left when it gets to LA will be, assuming regular consumption, X2 = X1 + P1 - 12000 The limits for P2, P3, and P4 are 15000 < X1 + P1 - 12000 + P2 < 23000 9000 < X1 + P1 + P2 + P3 - 12000 - 7000 < 17000 11000 < X1 + P1 + P2 + P3 + P4 - 12000 - 7000 - 3000 < 20000 Then to solve it you use linear programming techniques. I think it would make my answer too long if I wrote it, so it will be simpler if you repost that LP problem now that I stated it for you! Teamwork is good.

Question:can you do linear programming with a problem like this using a TI-83 graphing calculator? my teacher used the "five step method" which includes graphing and finding vertices of a geometric shape and plugging them in to find the values. but how do you do it in equations like this? a dietition must supply a d iet including at least 36 units of proten, 24 unites of carbohydrates, and 16 unites of fats. These requirements can be met with a mixutre of two foods, Frosted Snacks ($20) and Chip Starters($16.) Each unit of Frosted Snakes supplies 9 unites of proten, 3 unites of carbohydrates, and 1 unit of fat. Each unit of Chip Startes supplies 2 units of proten, 2 units of carbohydrates, and 2 units of fat. What mixture of these two foods should be used in order to minimize the cost? I can see if you have inequalities like 8x + 5y = 80 or 2x + 7y = 50, but these look like they'd have 3 variables..and how can you graph a line with an x,y, AND a z?! HELP PLEASE!!!

Answers:You can use the simplex algorithm, but it is entirely too much to explain here and unless you are a pretty bright algebra 2 student, reading it on your own probably won't do you too much good. If you know how to create the model, you can download free software called LINDO to input the linear programming model and get the solution. It's not for the TI83 though. If you understand the simplex method, I am sure you can download a SIMPLEX program for the TI 83.

Question:Any help on this problem would be greatly appreciated. My math teacher doesnt explain any of this and doesnt even use the chalk board for examples or anything like that. thanks in advance. You have 180 tomatoes and 15 onions left over from your garden. you want to use these to make jars of tomato sauce and jars of salsa to sell at a farm stand. a jar of tomato sauce requires 10 tomatoes and 1 onion and a jar of salsa requires 5 tomatoes and 1/4 onion. you make a profit of $2 on every jar of tomato sauce sold and a profit of $1.50 on every jar of salsa sold. the farm stand wants at least 3 times as many jars of tomato sauce as jars of salsa. how many jars of each should you make to maximize profit?

Answers:i knew how to do this .. but i forgot :/ maybe this will help you http://en.wikipedia.org/wiki/Linear_programming omg !! look what I found for you !!! :D http://nz.answers.yahoo.com/question/index?qid=20071010190851AAFuFLm

Question:... developed two new formulas for a new kind of soda. Formula A contains 3 grams of sugar, 1 gram of flavor, and 1 milligram of ascorbic acid. Formula B contains 1 gram of sugar, 4 grams of flavor, and 1 milligram of ascorbic acid. A bottle of the beverage must contain at least 9 grams of sugar, 8 grams of flavoring, and 5 milligrams of ascorbic acid. Formula A costs $0.02 per gram and Formula B costs $0.03 per gram. What combination of the two formulas will meet all the stated requirements at the minimum cost? I don't understand how to do this problem because it has all these gram/milligram quantities. Are each of the numbers a new variable? If someone could please help me solve this problem, I'd appreciate it. Thanks! Yes, flavor and flavoring refer to the same substance. As for the question, I didn't alter it from the original review worksheet. Thank you Carla. :)

Answers:I guess, the problem is over-determined and not consistent. Is it possible, that the real question is a little bit different? Excuse my bad English. Is flavor and flavoring the same substance? In principle we can do: Let x be the needed amount of A and y be the amount of B. Calculate the amount of sugar, flavor, acid of the mixture - these gives 3 eq. with only 2 varibales. If this system of eq. is (would be) consistent, we get x and y. Could it be, that we have 3 questions? a) Amount of sugar in the bottle is given b) flavour should be 8 g in the bottle c) vitame C is given? (I think we have very much flavor and I would not like to drink this.) Now we can calculate the price. This means, yo need no help more? I am glad with you.

From Youtube

Linear Programming :linear programming example

Linear Programming :Linear Programming www.gboyinc.net