1. Professor Goodchild Cee 587 April 13, 2011
Basic Optimization
Professor Goodchild
Cee 587
April 13, 2011

2. objective function
f(x)
constraints x
variable

3. Optimization Problem Elements
Objective function
The thing you want to minimize or maximize
Typically total cost
Constraints
The complexities of the problem
E.g. capacities, demands, supply
An unconstrained optimization leads to 0 or infinity

4. f(x)
objective function (nonlinear)
minimum x
Constraints can’t improve the solution

5. Route choice Objective function Subject to: where,
Minimize kckxk
Subject to:
kxk=1
xk=0 or 1 
where,
ck= cost of travel on link k
xk=1 if link k is used, 0 otherwise

6. x1 x2 x3 c3 c1 c2

7. x1 x2 x3
Objective function 1 4 6 9 2 3 y1 y2 y3
c1=4
c2=2
c3=3

8. Assigning Customers to Multiple Vehicles
Minimize ikcikyik
Subject to:
kyik=1
iaiyik≤bk
yik=0 or 1
where,
cik= cost of serving customer i with vehicle k
ai=order size from customer i
bk=capacity of vehicle k
yik=1 if customer i is assigned to vehicle k, 0 otherwise

9. Allocating Production to Factories and Customers
Minimize ikcikyik
Subject to:
kyik ≤ ai
iyik ≤ bk
where,
cik= cost per unit of serving customer i with factory k
ai=demand from customer i
bk=capacity of factory k
yik=amount produced by factory k for customer i

10. Traveling Salesman Problem
Minimize ikcikyik
Subject to:
kyik=1
iyik=1
ikyik≤│K│-1, K subset of all i,j
yik=0 or 1
where,
cik= cost of traveling from i to k
yik=1 if visit i immediately before k