c
j
Basic Variables
Quantity
4 x_{1 }
6 x_{2 }
0 s_{1 }
0 s_{2 }
0 s_{3 }
0 6 4
s x x
2 2 1
0 3 2
0 0 1
0 1 0
1/8 1>8 1/4
1 0 0
3>20 3/20 1>10
z
j
26
4
6
1/4
0
1/2
Table A31
The Optimal Simplex Tableau for a Degenerate Problem
Irregular Types of Linear Programming Problems
A29
c j  z j
0
0

1>4
0

1>2
Notice that the optimal solution did not change from the third to the optimal simplex tableau. The graphical analysis of this problem shown in Figure A8 reveals the reason for this.
Figure A8 Graph of a degenerate solution
x_{2 }
8
6
6 x 1 + 4 x 2 = 2 4
4
A
B
x_{2 }= 3
2
5 x 1 + 1 0 x 2 = 4 0
0
2
C 4
6
8
x_{1 }
Degeneracy occurs in a simplex problem when a problem continually loops back to the same solution or tableau.
Notice that in the third tableau (Table A30) the simplex process went to point B, where all three constraint lines intersect. This is, in fact, what caused the tie for the pivot row and the degeneracy. Subsequently, the simplex process stayed at point B in the optimal tableau (Table A31). The two tableaus represent two different basic feasible solutions correspond ing to two different sets of model constraint equations.
Negative Quantity Values Occasionally a model constraint is formulated with a negative quantity value on the right side of the inequality sign—for example,
Standard form for simplex solution requires positive right handside values.
A negative righthandside value
is changed to a positive by multiplying the constraint by 1, which changes the direction of the
inequality.

6x_{1 }+ 2x_{2 }Ú 30
This is an improper condition for the simplex method, because for the method to work, all quantity values must be positive or zero.
This difficulty can be alleviated by multiplying the inequality by 1, which also changes the direction of the inequality:
6 x 1  2 x 2 … 3 0 (  1 ) (  6 x 1 + 2 x 2 Ú  3 0 )