O lp model o set covering set packing and set

o LP Model

o Set covering, set packing, and set partitioning
 There are m dots (cities to be covered)
 There are n patches (fire stations that cover cities)
 Each patch covers a subset of dots
 Binary matrix A
 Each row represents a dot and each column a patch
 If patch j covers dot I, then A
ij
= 1
 If patch j does not cover dot I, then A
ij
= 0
 The A matrix for the fire station example:

 Either-or constraints
o Suppose a constraint says that
 Either X
1
≤ 0 or X
1
≥ 1000
o We can reformulate constraint (1) by introducing a new binary variable Y
1
and a large finite number M
1
then constraint (1) is equivalent to the following three constraints:

 If-then constraints
o Let X
1,
X
2,
X
3,
and X
4
be four binary variables
o Suppose a constraint says that
 If X
1
= 1 then X
2
= X
3
= X
4
= 0
o Reformulation of the constraint

o Another equivalent reformulation:

 Traveling salesman problem statement
o Joe state lives in Gary, Indiana. He owns insurance agencies in Gary, For Wayne, Evansville, Tree Haute,
and South Bend. Each December, he visits each of his insurance agencies. The distance between each
agency (in miles) is shown in the table. What order of visiting his agencies will minimize the total
distance traveled?

o LP Model:
Nonlinear Programming Modeling:
 General form NLP
o