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Advantages of Linear Programming
 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.
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From Wikipedia
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 realvalued 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 (c^{T}x 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 lessthan or equalto 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 FourierMotzkin 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 subproblems. 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 everchanging, 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: c_{1}x_{1} + c_{2}x_{2}
 Problem constraints of the following form
 e.g.,
 a_{1,1}x_{1} + a_{1,2}x_{2}≤ b_{1}
 a_{2,1}x_{1} + a_{2,2}x_{2}≤ b_{2}
 a_{3,1}x_{1} + a_{3,2}x_{2}≤ b_{3}
 Nonnegative variables
 e.g.,
 x_{1}≥ 0
 x_{2}≥ 0.
 Nonnegative right hand side constants
 b_{i}≥ 0
The problem is usually expressed in From Yahoo Answers
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.
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.
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
Answers:I guess, the problem is overdetermined 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.
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