Now, we look for other basic feasible solutions which gives better objective values than the current

1

• Now, we look for other basic feasible solutions
  which gives better objective values than the
  current solution. Such solutions can be
  examined by setting 7 4 3 variables at 0
  (called nonbasic variables) and solve the
  equations for the remaining 4 variables (called
  basic variables). Here z may be regarded as a
  basic variable and it remains basic at any time
  during the simplex iterations.

2

• Initial feasible solution
• To find a better solution, find a nonbasic
  variable having positive coefficient in z row
  (say x1) and increase the value of the chosen
  nonbasic variable while other nonbasic variables
  remain at 0.
• We need to obtain a solution that satisfies the
  equations. Since x1 increases and other nonbasic
  variables remain at 0, the value of basic
  variables must change so that the new solution
  satisfies the equations and nonnegativity. How
  much can we increase x1?

3

• (continued)
• x1 ? (5/2) most binding (ratio test), get new
  solution
• x1 (5/2), x2, x3 0, x4 0, x5 1, x6
  (1/2), z 25/2
• This is a new b.f.s since x4 now can be treated
  as a nonbasic variable (has value 0) and x1 is
  basic.
• (We need a little bit of caution here in saying
  that the new solution is a basic feasible
  solution since we must be able to obtain it by
  setting x2, x3 and x4 at 0 and obtain a unique
  solution after solving the remaining system of
  equations)

4

• Change the dictionary so that the new solution
  can be directly read off
• x1 0 ?(5/2), x4 5 ? 0
• So change the role of x1 and x4 . x4
  becomes independent (nonbasic) variable and x1
  becomes dependent (basic) variable.
• Why could we find a basic feasible solution
  easily?
• 1) all independent(nonbasic) variables appear at
  the right of equality (have value 0)
• 2) each dependent (basic) variable appears in
  only one equation
• 3) each equation has exactly one basic variable
  appearing
• ( z variable may be interpreted as a basic
  variable, but usually it is treated separately
  since it always remains basic and it is
  irrelevant to the description of the feasible
  solutions)
• So change the dictionary so that it satisfies
  the above properties.

5

6

7

Equivalent to performing row operations
8

• Note that the previous solution
• x1 x2 x3 0, x4 5, x5 11, x6 8, z
  0
• and the new solution
• x1 (5/2), x2, x3 0, x4 0, x5 1, x6
  (1/2), z 25/2
• satisfies the updated system of equations. Only
  difference is that the new solution can be read
  off directly from the dictionary.
• We update the dictionary to read a new basic
  solution directly, but the set of solutions is
  not changed.

9

• Next iteration
• Select x3 as the nonbasic bariable to increase
  the value (called entering nonbasic variable).
  
• x6 becomes 0 (changes status to nonbasic
  variable from basic variable)
• Perform substitutions (elementary row operations)

10

• New solution is
• It is optimal since any feasible solution must
  have nonnegative values and
• implies that z ? 13 for any nonnegative
  feasible solution
• Hence if the coefficients of the nonbasic
  variables in z- row are all non-positive, current
  solution is optimal (note that it is a sufficient
  condition for optimality but not a necessary
  condition)

11

• Moving directions in Rn in the example

x1 (5/2), x2, x3 0, x4 0, x5 1, x6
(1/2), z 25/2
Then we obtained x1 x0 t d, where d (1, 0,
0, -2, -4, -3) and t 5/2 Note that the d vector
can be found from the dictionary.( the column for
x1) We make t as large as possible while x0 t d
? 0.
12
Geometric meaning of an iteration

• Notation

x10
x20
x3
x30
x1
x2
13

• Our example assume x2 does not exist. It makes
  the polyhedron 2 dimensional since we have 5
  variables and 3 equations (except nonnegativity
  and obj row)

x30
x60
A
We move from A, which is an extreme point defined
by 3 eq. and x1x3 0 to B defined by the 3 eq.
and x3 x4 0.
x10
x40
B
14
Terminology

• Assume that we have max cx, Ax b, x ? 0,
  where A is m ? (n m) and full row rank.
• A solution x is called a basic solution (???)
  if it can be obtained by setting n of the
  variables equal to 0 and then solving for the
  remaining m variables, where the columns of the A
  matrix corresponding to the m variables are
  linearly independent. (Hence provides a unique
  solution.)
• In the text, basic solution is defined as the
  solution which can be obtained by setting the
  right-hand side variables (independent var.) at
  zero in the dictionary. This is the same
  definition as the one given above. But the text
  does not make clear distinction between basic
  solution and basic feasible solution.

15

• For a basic solution x, the n variables which
  are set to 0 are called nonbasic variables
  (?????) (independent var.) and the remaining m
  variables are called basic variables (????)
  (dependent var.)
• The z-row may be considered as part of system of
  equations. In that case, z var. is regarded as
  basic variable. It always remains basic during
  the simplex iterations.
• On the other hand, z-row may be regarded as a
  separate equation which is used to read off
  objective function values and other equations and
  nonnegativity describes the solution set. Both
  viewpoint are useful.
• A solution x is called a basic feasible solution
  (?????) if it is a basic solution and satisfies x
  ? 0. (feasible solution to the augmented LP)

16

• The set of basic variables are called basis (??)
  of the basic solution. (note that the set of
  basic variables spans the subspace generated by
  the columns of A matrix.)
• In a simplex iteration, the nonbasic variable
  which becomes basic in that iteration is called
  entering (nonbasic) variable (????) and the
  basic variable which becomes nonbasic is called
  leaving (basic) variable (????)
• Minimum ratio test (??????) test to determine
  the leaving basic variable
• Pivoting computational process of constructing
  the new dictionary (elementary row operations)

17
Remarks

• For standard LP, the basic feasible solution to
  the augmented form corresponds to the extreme
  point of the feasible set of points.
• (If the given LP is not in standard form, we
  should be careful in saying the equivalence,
  especially when free variables exist.)
• Simplex method searches the extreme points in its
  iterations.
• Note that we used (though without proof) the
  equivalence of the extreme points and the basic
  feasible solution

18

• Maximum number of b.f.s. in augmented form is
• In the simplex method, one nonbasic variable
  becomes basic and one basic variable becomes
  nonbasic in each iteration (except the z
  variable, it always remains basic.)
• In real implementations, we do not update entire
  dictionary ( or tableau). We maintain
  information about the current basis. Then entire
  tableau can be constructed from that information
  and the simplex iteration can be performed
  (called revised simplex method).

19
Obtaining all optimal solutions
If all coefficients in the z- row are lt 0, it
gives a sufficient condition for the uniqueness
of the current optimal solution.
20

• Another example

Any feasible solution with x3 0 is an optimal
solution. The set of feasible solutions with x3
0 is given by
21
Tableau format

22

• Tableau format only maintains coefficients in the
  equations.
• It is convenient to carry out a simplex
  iteration in the tableau.