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![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](https://image.slidesharecdn.com/15746ae9-718c-4eae-9735-cdaa9e4424d4-160308221733/85/Packing-Problems-Using-Gurobi-4-320.jpg)
Packing Problems Using Gurobi
- 1.
- 2. What is packingproblems 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 PackingProblems 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 packingrectangles 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 MILPmodel 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 GurobiOptimization 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 usingGurobi 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.
- 12.