The MOSEK Python optimizer API manual Version 7.0 (Revision 141)

15.4. SENSITIVITY ANALYSIS FOR LINEAR PROBLEMS 189
The linearity intervals and shadow prices for u c i , lx j , and ux j are computed similarly to lc i .
The left and right shadow prices for c j denoted σ 1 and σ 2 respectively are computed as follows:
and
σ 1 = minimize e T j x
subject to l c + βe i ≤ Ax ≤ u c ,
c T x = z ∗ ,
l x ≤ x ≤ u x
σ 2 = maximize e T j x
subject to l c + βe i ≤ Ax ≤ u c ,
c T x = z ∗ ,
l x ≤ x ≤ u x .
Once again the above two optimization problems make it easy to interpret the shadow prices. Indeed,
if x ∗ is an arbitrary primal optimal solution, then
x ∗ j ∈ [σ 1 , σ 2 ].
The linearity interval [β 1 , β 2 ] for a c j is computed as follows:
and
β 1 = minimize β
subject to A T (s c l − s c u) + s x l − s x u = c + βe j ,
(l c ) T (s c l ) − (u c ) T (s c u) + (l x ) T (s x l ) − (u x ) T (s x u) − σ 1 β ≤ z ∗ ,
s c l , s c u, s c l , s x u ≥ 0
β 2 = maximize β
subject to A T (s c l − s c u) + s x l − s x u = c + βe j ,
(l c ) T (s c l ) − (u c ) T (s c u) + (l x ) T (s x l ) − (u x ) T (s x u) − σ 2 β ≤ z ∗ ,
s c l , s c u, s c l , s x u ≥ 0.
15.4.4 Example: Sensitivity analysis
As an example we will use the following transportation problem. Consider the problem of minimizing
the transportation cost between a number of production plants and stores. Each plant supplies a
number of goods and each store has a given demand that must be met. Supply, demand and cost of
transportation per unit are shown in Figure 15.2. If we denote the number of transported goods from
location i to location j by x ij , problem can be formulated as the linear optimization problem minimize
subject to
1x 11 + 2x 12 + 5x 23 + 2x 24 + 1x 31 + 2x 33 + 1x 34