To see why, rewrite the augmented matrix as a system of equations,

Transcription

1 Question : How do you find a basic feasible solution? The initial simplex tableau allows us to calculate the locations of the corner points as well as any other points where the lines corresponding to the equations cross or cross the axes. Let s start with the initial simplex tableau x y s s Since there are more variables than equations, there is not a unique solution to the system of equations represented by the tableau. There are five variables and three equations so we need 5 or parameters to write the solution. Any two of the variables may be parameters, but in the case of the initial simplex tableau it is convenient to choose the parameters as x and y. To see why, rewrite the augmented matrix as a system of equations, x ys 4 x y s 4. 5x6y 0 The variables s, s, and each have a coefficient of and appear nowhere else in the system. In the matrix, this corresponds to the columns consisting of all eros except for a single entry of. These variables are the basic variables for the initial simplex tableau. The other variables, x and y, are the nonbasic variables for the initial simplex tableau. To find possible corner points, we ll solve for the basic variables in term of the nonbasic variables. Solve for the basic variables variables in terms of x and y: 8

2 s 4x y s 4xy 5x6y One possible solution to this system is found by setting the nonbasic variables x and y equal to ero. In this situation, s s This situation corresponds to the corner point of the feasible region at xy, 0,0 with an objective function value of 0. At this corner point, the slack variables take on the values s 4 and s 4. Figure - The corner point corresponding to a solution of the initial simplex tableau with s = 4, s = 4 and = 0. Other corner points can be obtained by using row operations to make other variables basic instead of s, s, and. In general, should always be a basic variable so that it is easy to determine the value of the objective function. 9

3 Suppose we start from the initial simplex tableau x y s s and multiply the second row by and place the result in the second row. This gives us a in the second row, second column: x y s s If we make the rest of the second column eros, the variable y will become a basic variable. To do this, we ll use two row operations. First, multiply the second row by - and add it to the first row: : : This sum will be placed in the first row. Second, multiply the second row by 6 and add it to the third row: 6 : : This sum is placed in the third row. Putting these rows into the tableau yields 0

4 x y s s In this tableau, the variables y, s, and are basic, and x and s are nonbasic variables. Notice that by placing the in the second column in the second row, the second slack variable s (which originally was in the second row) became nonbasic. In general, when a is placed in a column to make a variable basic, the slack variable corresponding to the row it was placed in nonbasic. As with the initial simplex tableau, we ll write this matrix as a system of equations and set the nonbasic variables equal to ero. The corresponding system of equations is x s s x y s. xs If we solve each equation for a different basic variable we get s x s y x s xs. If we set the nonbasic variables x and s equal to ero, we find that s, y, and. This corresponds to the corner point at xy, 0, with an objective function value of.

5 Figure - The corner point at (0,) corresponding to s =, y =, and =. It is easier to find this corner point using the tableau without transforming it to a system of equations. The key is to realie that when nonbasic variables are set equal to ero, their coefficients in the matrix disappear. For instance, if we take the tableau we transformed with row operations and cross out the columns of the nonbasic variables we get: x y s s By ignoring these columns, we can see that s, y, and. Combining this observation with our knowledge of setting the nonbasic variables equal to ero gives us the corner point at 0, and objective function value. We cannot arbitrarily make a variable into a basic variable. For instance, we might make x a basic variable by putting a in the second row and first column and eros in the rest of the column. Starting from the original simplex tableau, we must carry out two row operations:

6 x y s s x y s s To find the corner point corresponding to this matrix, cover the nonbasic variables and read off the values of the basic variables: x y s s s x Figure - A solution to the tableau that is not a corner point of the feasible region. On the surface this might not seem all that different from earlier tableaus, but this one violates one of the assumptions for slack variables. In a standard maximiation problem, the slack variables are assumed to be non-negative. In the solution we found from this

7 tableau, the value for s is negative. This tells us that this solution does not correspond to a corner point of the feasible region. The solution we have just found, 4,0, matches a point on a line, but is not along the border of the feasible region. If we are not careful about picking the basic and nonbasic variables, we may obtain a point that is not related to the solution. If we want to make the variable x a basic variable, it is better to use row operations to transform the original matrix x y s s to x y s s By covering the columns under y and s, we observe that this new tableau corresponds to the solution x and y 0 and the objective function value 0. This does match the corner point (, 0) on the feasible region. Example Change Basic Variables The basic variables for the initial simplex tableau x y s s are s, s, and. Use row operations to change the tableau so that the basic variables are y, s, and. What corner point of the basic feasible solution does this new tableau correspond to? 4

8 Solution We want to change the basic variables from s, s, and to y, s, and. This means we need to make s nonbasic and y basic. The other two variables, s and, should remain basic variables. To make y a basic variable, we need to place two 0 s and a in the column corresponding to y. The only question is where should we put the one? The first row of the second column already has a one so all we need to do is to use row operations to place eros in the rest of the column. : : : : x y s s If you examine the new matrix, you ll notice that y is a basic variable, but s is not. Instead, s is a basic variable. If we place the one in the first row, the slack variable corresponding to that row, s, a nonbasic variable. Since we want s to remain basic and s to become nonbasic, the one should be placed in the second row of the first column. Start from the original tableau 5

9 x y s s In general, we get the one by multiplying the row by a constant: x y s s Now we can add multiples of rows to put eros in the rest of the second column. : : : : x y s s Since the second, third, and fifth columns contains two eros and a one, the basic variables are now y, s, and. The point in the feasible region corresponding to this tableau is found by covering the nonbasic variable columns x and s. This shows the corner point to be at 0, with an objective function value of. 6