Linear Programming Lecture Notes - Penn State Personal Web Server

(a) Original Problem (b) RHS Decreased (c) RHS Increased
Figure 9.3. Degeneracy in the primal problem causes alternative optimal solutions
in the dual problem and destroys the direct relationship between the resource margin
price that the dual variables represent in a non-degenerate problem.
To transform this problem to standard form, we would have to introduce surplus variables
to obtain:
min x1 + 2x2
s.t. x1 + 2x2 − s1 = 12
2x1 + 3x2 − s2 = 20
x1, x2, s1, s2 ≥ 0
In this case there is no immediately obvious initial basic feasible solution and we would have
to solve a Phase I problem. Consider the dual of the original maximization problem:
max 12w1 + 20w2
s.t. w1 + 2w2 ≤ 1
2w1 + 3w2 ≤ 1
w1, w2 ≥ 0
This is a maximization problem whose standard form is given by:
max 12w1 + 20w2
s.t. w1 + 2w2 + v1 = 1
2w1 + 3w2 + v2 = 1
w1, w2, v1, v2 ≥ 0
In this case, a reasonable initial basic feasible solution for the dual problem is to set v1 =
v2 = 1 and w1 = w2 = 0 (i.e., w1 and w2 are non-basic variables) and proceed with the
simplex algorithm from this point.
In cases like the one illustrated in Example 9.15, we can solve the dual problem directly
in the simplex tableau of the primal problem instead of forming the dual problem and solving
it as a primal problem in its own tableau. The resulting algorithm is called the dual simplex
algorithm.
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