{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sec4_1

# sec4_1 - 4 Basic feasible solutions To summarize the main...

This preview shows pages 1–3. Sign up to view the full content.

4 Basic feasible solutions To summarize the main idea from the last section, a basis for an m -by- n matrix A is a list of numbers chosen from { 1 , 2 , , . . . , n } such that the matrix A B with columns indexed by this list is invertible. The corresponding basic solution of a system Ax = b is the unique solution of this system satisfying x j = 0 for all indices j 6∈ B . Consider now the constraints of a linear programming problem in stan- dard equality form: Ax = b x 0 . A basic feasible solution of this system is a basic solution that is also feasi- ble. Such solutions play a special role in linear programming. To illustrate, consider the simple example (2.1): (4.1) maximize x 1 + x 2 subject to x 1 2 x 1 + 2 x 2 4 x 1 , x 2 0 . As before, we can sketch the feasible region: If we transform the constraints of this linear programming problem to stan- dard equality form, by introducing slack variables as we described in Section 2, we arrive at the following system: x 1 + x 3 = 2 x 1 + 2 x 2 + x 4 = 4 x 1 , x 2 , x 3 , x 4 0 . 25

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Feasible solutions of this system correspond one-to-one with feasible solutions of the linear programming problem (4.1), in an obvious way. If we think of this new system as Ax = b , the matrix A has ﬁve possible bases, not including permutations: [1 , 2] , [1 , 3] , [1 , 4] , [2 , 3] , [3 , 4]. The ﬁve corresponding basic solutions are x 1 x 2 x 3 x 4 = 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

sec4_1 - 4 Basic feasible solutions To summarize the main...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online