2. What is packing problems
Packing problems are a class of optimization problems in mathematics that
involve attempting to pack objects together into containers. The goal is to
either pack a single container as densely as possible or pack all objects
using as few containers as possible.
3. Examples of Packing Problems
Packing Commercials into station breaks
Packing files onto floppy disks (Dvds, CDs, etc.)
Packing telemetry data into fixed size pockets
Packing News articles onto a news paper
Packing a suitcase
4. Formulation of packing rectangles
using MILP
A MILP general model:
𝑧 = 𝑚𝑖𝑛 𝑖=1
𝑁
(𝑚𝑖 ∗ 𝑔 ∗ 𝑦𝑖)
Subject to
𝑤 𝑖
2
+ 𝑥𝑖 < 𝑥𝑗 −
𝑤 𝑗
2
i = 1,…,N j = 1,…,N
𝑤 𝑗
2
+ 𝑥𝑗 < 𝑥𝑖 −
𝑤 𝑖
2
i = 1,…,N j = 1,…,N
ℎ 𝑖
2
+ 𝑦𝑖 < 𝑦𝑗 −
ℎ 𝑗
2
i = 1,…,N j = 1,…,N
ℎ 𝑗
2
+ 𝑦𝑗 < 𝑦𝑖 −
ℎ 𝑖
2
i = 1,…,N j = 1,…,N
𝑎𝑖𝑗 ∗ 𝑥𝑗 ≥ 𝑏𝑖 i = 1,…,N j = 1,…,N
Coefficients:
w – width of ith rectangle
h – height of ith rectangle
A = [N,N] = {aij}, (𝑁 × 𝑁) matrix
b = [N] = N vector
g- constant variable
Variables:
x = [N] = N vector of variables
y = [N] = N vector of variables
Where 𝑥, 𝑦 = the lower left
coordinate of rectangle i
5. Why an MILP model
Integer and continuous variables are both considered
If i = j, z is a Integer Programming Problem (ILP)
If i = 0, z is a Linear Programming Problem (LP)
The constraints and the objective function are modeled using linear
equalities/inequalities
Effective solution techniques: Branch & Bound, Branch & Cut, Branch &
Price
Software available (CPLEX, XPRESS, GLPK, etc)
Huge problems can be easily solved
6. MILP model explanation
MILP Constraints
The resources used in a optimization problems always limited by a set of
constraints
MILP Variables
Variables reflect the resources to be allocated
MILP Objective Function
The objective function reflects the optimization aspect of the model
7. What is Gurobi Optimization
The Gurobi Optimizer is a commercial optimization solver for
Linear programming (LP),
Quadratic programming (QP),
Quadratical constrained programming (QCP),
Mixed integer linear programming (MILP)
Etc
8. Solving MILP using Gurobi solver
The data associated with an optimization model must be stored in a
MATLAB struct.
Fields in this struct contain the different parts of the model.
A few fields are mandatory:
the constraint matrix (A),
the objective vector (obj),
the right-hand side vector (rhs),
the constraint sense vector (sense).
A model can also include optional fields (e.g., the objective sense
model.sense).
9. Modifying Gurobi parameters
We must create a struct variable that will be used to modify the Gurobi
parameters
params.outputflag = 0;
params.resultfile = ’………’;
The Gurobi OutputFlag parameter is set to 0 in order to shut off Gurobi
output.
Also the ResultFile parameter is set to request that Gurobi produce a file as
output
10. Solving the model
The result is where the actual optimization occurs:
result = gurobi(model, params);
We pass the model and the optional list of parameter changes to the
gurobi() function.
It computes an optimal solution to the specified model and returns the
computed result.
11. Printing the solution
The gurobi() function returns a struct as its result
This struct contains a number of fields, where each field contains
information about the computed solution
The available fields depend on the result of the optimization, the type of
model that was solved,
The algorithm used to solve the model.
