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# section2 - 1 1.1 Solving LPs The Simplex Algorithm of...

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1 Solving LPs: The Simplex Algorithm of George Dantzig 1.1 Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm , by implementing it on a very simple example. Consider the LP max 5 x 1 + 4 x 2 + 3 x 3 (1.1) s.t. 2 x 1 + 3 x 2 + x 3 5 4 x 1 + x 2 + 2 x 3 11 3 x 1 + 4 x 2 + 2 x 3 8 0 x 1 , x 2 , x 3 In devising our solution procedure we take a standard mathematical approach; reduce the problem to one that we already know how to solve. Since the structure of this problem is essentially linear, we will try to reduce it to a problem of solving a system of linear equations, or perhaps a series of such systems. By encoding the problem as a system of linear equations we bring into play our knowledge and experience with such systems in the new context of linear programming. In order to encode the LP (1.1) as a system of linear equations we must first transform linear inequalities into linear equation. This is done by introducing a new non-negative variable, called a slack variable , for each inequality: x 4 = 5 - [2 x 1 + 3 x 2 + x 3 ] 0 , x 5 = 11 - [4 x 1 + x 2 + 2 x 3 ] 0 , x 6 = 8 - [3 x 1 + 4 x 2 + 2 x 3 ] 0 . To handle the objective, we introduce a new variable z : z = 5 x 1 + 4 x 2 + 3 x 3 . Then all of the information associated with the LP (1.1) can be coded as follows: (1.2) 2 x 1 + 3 x 2 + x 3 + x 4 = 5 4 x 1 + x 2 + 2 x 3 + x 5 = 11 3 x 2 + 4 x 2 + 2 x 3 + x 6 = 8 - z + 5 x 1 + 4 x 2 + 3 x 3 = 0 0 x 1 , x 2 , x 3 , x 4 , x 5 , x 6 . The new variables x 4 , x 5 , and x 6 are called slack variables since they take up the slack in the linear inequalities. This system can also be written using block structured matrix notation as 0 A I - 1 c T 0 z x = b 0 , 1

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where A = 2 3 1 4 1 2 3 4 2 , I = 1 0 0 0 1 0 0 0 1 , b = 5 11 8 , and c = 5 4 3 . The augmented matrix associated with the system (1.2) is (1.3) 0 A I - 1 c 0 b 0 and is referred to as the initial simplex tableau for the LP (1.1). Again consider the system x 4 = 5 - 2 x 1 - 3 x 2 - x 3 (1.4) x 5 = 11 - 4 x 1 - x 2 - 2 x 3 x 6 = 8 - 3 x 1 - 4 x 2 - 2 x 3 z = 5 x 1 + 4 x 2 + 3 x 3 . This system defines the variables x 4 , x 5 , x 6 and z as linear combinations of the variables x 1 , x 2 , and x 3 . We call this system a dictionary for the LP (1.1). More specifically, it is the initial dictionary for the the LP (1.1). This initial dictionary defines the objective value z and the slack variables as a linear combination of the initial decision variables. The variables that are “defined” in this way are called the basic variables , while the remaining variables are called nonbasic . The set of all basic variables is called the basis . A particular solution to this system is easily obtained by setting the non-basic variables equal to zero. In this case, we get x 4 = 5 x 5 = 11 x 6 = 8 z = 0 . Note that the solution (1.5) x 1 x 2 x 3 x 4 x 5 x 6 = 0 0 0 5 11 8 is feasible for the extended system (1.2) since all components are non-negative. For this reason, we call the dictionary (1.4) a feasible dictionary for the LP (1.1), and we say that this LP has
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section2 - 1 1.1 Solving LPs The Simplex Algorithm of...

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