Integer Programming. Integer Programming. Programming = Planning in this context Origins go back to military logistics in WWII (1940s). In a survey of Fortune 500 firms, 85% of those responding said that they had used linear or integer programming. Why is it so popular?
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maximizec1x1+c2x2+…+cnxn (objective function)
a11x1+a12x2+…+a1nxn b1 (functional constraints)
x1, x2 , …, xn Z+(set constraints)
Note: Can also have equality or ≥ constraint
in non-standard form.
maximizecx (objective function)
subject toAxb (functional constraints)
x (set constraints)
Input for IP: 1n vector c, mn matrice A, m1 vector b.
Output of IP: n1 integer vector x .
mixed integer programs (MIP),
that is, some variables are integer,
the others are continuous.
Production of 1 table requires 5 ft pine, 2 ft oak, 3 hrs labor
1 chair requires 1 ft pine, 3 ft oak, 2 hrs labor
1 desk requires 9 ft pine, 4 ft oak, 5 hrs labor
20 employees (each works 40 hrs).
that will maximize the profit.
The goal can be achieved
by making appropriate decisions.
First define decision variables:
Let xt be the number of tables to be produced;
xc be the number of chairs to be produced;
xd be the number of desks to be produced.
(Always define decision variables properly!)
max 12xt + 5xc + 15xd
pine: 5xt + 1xc + 9xd 1500
oak: 2xt + 3xc + 4xd 1000
labor: 3xt + 2xc + 5xd 800
market demand constraints:
tables: xt ≥ 40
chairs: xc ≥ 130
desks: xd ≥ 30
xt , xc , xd Z+
as an n-dimensional vector.
IP modeling techniques