Getting Started

Getting Started

using Modof, JuMP, FPBHCPLEX

Warm Up FPBHCPLEX:

FPBHCPLEX.jl has four implementations, one each for biobjective pure binary, biobjective mixed binary, multiobjective pure binary and multiobjective mixed binary program. Thus, it is recommended that before fpbhcplex is used for any actual instance, warmup_fpbhcplex is used to compile all of the four implementations.

warmup_fpbhcplex(threads=1)

Using JuMP Extension:

Providing the following multi-objective mixed integer linear program as a ModoModel:

\[\begin{aligned} \min \: & x_1 + x_2 + y_1 + y_2 \\ \max \: & x_1 + x_2 + y_1 + y_2 \\ \min \: & x_1 + 2x_2 + y_1 + 2y_2 \\ \text{s.t. } & x_1 + x_2 \leq 1 \\ & y_1 + 2y_2 \geq 1 \\ & x_1, x_2 \in \{0, 1\} \\ & y_1, y_2 \geq 0 \end{aligned}\]
model = ModoModel()
@variable(model, x[1:2], Bin)
@variable(model, y[1:2] >= 0.0)
objective!(model, 1, :Min, x[1] + x[2] + y[1] + y[2])
objective!(model, 2, :Max, x[1] + x[2] + y[1] + y[2])
objective!(model, 3, :Min, x[1] + 2x[2] + y[1] + 2y[2])
@constraint(model, x[1] + x[2] <= 1) 
@constraint(model, y[1] + 2y[2] >= 1)

Note: Currently constant terms in the objective functions are not supported

Imposing a maximum timelimit of 10.0 seconds

@time solutions = fpbhcplex(model, timelimit=10.0)

Writing nondominated frontier to a file

write_nondominated_frontier(solutions, "nondominated_frontier.txt")

Writing nondominated solutions to a file

write_nondominated_sols(solutions, "nondominated_solutions.txt")

Nondominated frontier

nondominated_frontier = wrap_sols_into_array(solutions)

Hypervolume of the nondominated frontier

For this functionality, pyModofSup.jl must be properly installed.

using pyModofSup

compute_hypervolume_of_a_discrete_frontier can be used for computing the hypervolume of a nondominated frontier, however all the objectives must be minimizations. 

nondominated_frontier[:, 2] = -1.0*nondominated_frontier[:, 2] # Converting the second objective function values into minimization 
compute_hypervolume_of_a_discrete_frontier(nondominated_frontier)

Plotting the nondominated frontier

For this functionality, pyModofSup.jl must be properly installed.

using pyModofSup

It is important to note that the plotting functions only work through IJulia. For just viewing the nondominated frontier, one can use the following instructions in an IJulia cell:

nondominated_frontier = wrap_sols_into_array(solutions)
plt_discrete_non_dom_frntr([nondominated_frontier], ["FPBHCPLEX"])

However, if the user desires to save the plot of the nondominated frontier to a file ("Plot1.png" in this case), they can use the following instructions in a julia terminal: 

nondominated_frontier = wrap_sols_into_array(solutions)
plt_discrete_non_dom_frntr([nondominated_frontier], ["FPBHCPLEX"], false, "Plot1.png")

Using LP File Format

Providing the following multiobjective mixed integer linear program as a LP file:

\[\begin{aligned} \min \: & x_1 + x_2 + y_1 + y_2 \\ \max \: & x_1 + x_2 + y_1 + y_2 \\ \min \: & x_1 + 2x_2 + y_1 + 2y_2 \\ \text{s.t. } & x_1 + x_2 \leq 1 \\ & y_1 + 2y_2 \geq 1 \\ & x_1, x_2 \in \{0, 1\} \\ & y_1, y_2 \geq 0 \end{aligned}\]

Format:

  1. The first objective function should follow the convention of LP format of single objective optimization problem

  2. The other objective functions should be added as constraints with RHS = 0, at the end of the constraint matrix in the respective order

  3. Variables and constraints should also follow the convention of LP format of single objective optimization problem

write("Test.lp", "\\ENCODING=ISO-8859-1
\\Problem name: TestLPFormat

Minimize
 obj: x1 + x2 + x3 + x4
Subject To
 c1: x1 + x2 <= 1
 c2: x3 + 2 x4 >= 1
 c3: x1 + x2 + x3 + x4  = 0
 c4: x1 + 2 x2 + x3 + 2 x4  = 0
Binaries
 x1  x2 
End\n") # Writing the LP file of the above multiobjective mixed integer program to Test.lp

Imposing a maximum timelimit of 10.0 seconds

The sense of the first objective function is automatically detected from the LP file, however the senses of the rest of the objective functions should be provided in the proper order.

@time solutions = fpbhcplex("Test.lp", [:Max, :Min], timelimit=10.0)

Using MPS File Format

Providing the following multiobjective mixed integer linear program as a MPS file:

\[\begin{aligned} \min \: & x_1 + x_2 + y_1 + y_2 \\ \max \: & x_1 + x_2 + y_1 + y_2 \\ \min \: & x_1 + 2x_2 + y_1 + 2y_2 \\ \text{s.t. } & x_1 + x_2 \leq 1 \\ & y_1 + 2y_2 \geq 1 \\ & x_1, x_2 \in \{0, 1\} \\ & y_1, y_2 \geq 0 \end{aligned}\]

Format:

  1. The first objective function should follow the convention of MPS format of single objective optimization problem

  2. The other objective functions should be added as constraints with RHS = 0, at the end of the constraint matrix in the respective order

  3. Variables and constraints should also follow the convention of MPS format of single objective optimization problem

write("Test.mps", "NAME   TestMPSFormat
ROWS
 N  OBJ
 L  CON1
 G  CON2
 E  CON3
 E  CON4
COLUMNS
    MARKER    'MARKER'                 'INTORG'
    VAR1  CON1  1
    VAR1  CON3  1
    VAR1  CON4  1
    VAR1  OBJ  1
    VAR2  CON1  1
    VAR2  CON3  1
    VAR2  CON4  2
    VAR2  OBJ  1
    MARKER    'MARKER'                 'INTEND'
    VAR3  CON2  1
    VAR3  CON3  1
    VAR3  CON4  1
    VAR3  OBJ  1
    VAR4  CON2  2
    VAR4  CON3  1
    VAR4  CON4  2
    VAR4  OBJ  1
RHS
    rhs    CON1    1
    rhs    CON2    1
    rhs    CON3    0
    rhs    CON4    0
BOUNDS
  UP BOUND VAR1 1
  UP BOUND VAR2 1
  PL BOUND VAR3
  PL BOUND VAR4
ENDATA\n") # Writing the MPS file of the above multiobjective mixed integer program to Test.mps

Imposing a maximum timelimit of 10.0 seconds

The sense of the first objective function is automatically detected from the MPS file, however the senses of the rest of the objective functions should be provided in the proper order.

@time solutions = fpbhcplex("Test.mps", [:Max, :Min], timelimit=10.0)

Using the Matrix Format - Advanced