1. Chapter 4 An Introduction to Optimization

2. A schematic view of modeling/optimization process
assumptions, abstraction,data,simplifications
Real-world
problem
Mathematical
model
makes sense? change the model, assumptions?
optimization algorithm
Solution to
real-world
problem
Solution to
model
interpretation

3. Mathematical models in Optimization
The general form of an optimization model:
min or max f(x1,…,xn) (objective function)
subject to gi(x1,…,xn) ≥ (functional constraints)
x1,…,xn  S (set constraints)
x1,…,xn are called decision variables
In words,
the goal is to find x1,…,xn that
satisfy the constraints;
achieve min (max) objective function value.

4. Types of Optimization Models
Stochastic
(probabilistic
information on data)
Deterministic
(data are certain)
Discrete, Integer
(S = Zn)
Continuous
(S = Rn)
Linear
(f and g are linear)
Nonlinear
(f and g are nonlinear)

5. Linear 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?
Many different real-life situations can be modeled as linear programs (LPs).
There are efficient algorithms to solve LPs.

6. Standard form of linear program
maximize c1x1+c2x2+…+cnxn (objective function)
subject to
a11x1+a12x2+…+a1nxn  b (functional constraints)
a21x1+a22x2+…+a2nxn  b2 ….
am1x1+am2x2+…+amnxn  bm
x1, x2 , …, xn ≥ 0 (set constraints)
Note: Can also have equality or ≥ constraint
in non-standard form.

7. Standard form of linear program
In matrix form:
maximize cx (objective function)
subject to Ax  b (functional constraints)
x ≥ 0 (set constraints)
Input for IP: 1n vector c, mn matrice A, m1 vector b.
Output of IP: n1 vector x .

8. Example of Linear Program (Production Planning-Furniture Manufacturer)
Technological data:
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
Capacities for 1 week: 1500 ft pine, ft oak,
20 employees (each works 40 hrs).
Market data:
Goal: Find a production schedule for 1 week
that will maximize the profit.
profit
demand
table
$12/unit 40
chair
$5/unit
130
desk
$15/unit 30

9. Production Planning-Furniture Manufacturer: modeling the problem as linear program
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!)

10. Production Planning-Furniture Manufacturer: modeling the problem as integer program
Objective is to maximize profit:
max 12xt + 5xc + 15xd
Functional Constraints
capacity constraints:
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
Set Constraints
xt , xc , xd ≥ 0

11. Solutions to linear programs
A solution is an assignment of values to variables.
A solution can hence be thought of
as an n-dimensional vector.
A feasible solution is an assignment of values to variables such that all the constraints are satisfied.
The objective function value of a solution is obtained by evaluating the objective function at the given point.
An optimal solution (assuming maximization) is one whose objective function value is greater than or equal to that of all other feasible solutions.
Next handout will discuss solution methods in details