This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 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 nonnegative variable, called a slack variable , for each inequality: x 4 = 5 [2 x 1 + 3 x 2 + x 3 ] , x 5 = 11 [4 x 1 + x 2 + 2 x 3 ] , x 6 = 8 [3 x 1 + 4 x 2 + 2 x 3 ] . 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 = 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 A I 1 c T z x = b , 1 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) A I 1 c b 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 nonbasic variables equal to zero. In this case, we get x 4 = 5 x 5 = 11 x 6 = 8 z = 0 ....
View
Full
Document
This note was uploaded on 07/05/2010 for the course MTH 407 taught by Professor Burke during the Summer '10 term at University of Washington.
 Summer '10
 BURKE

Click to edit the document details