From f6c277db46eb7092a802f64a6449bb0db94c6fae Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Wed, 16 Oct 2024 20:08:37 +0000 Subject: [PATCH] build based on a6e8a35 --- dev/.documenter-siteinfo.json | 2 +- dev/algorithms/index.html | 45 ++++++++++++++++++++++------------ dev/index.html | 2 +- dev/objects.inv | Bin 818 -> 839 bytes dev/search_index.js | 2 +- 5 files changed, 32 insertions(+), 19 deletions(-) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 6bb40a0..a0f31a4 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2024-10-16T19:26:05","documenter_version":"1.7.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.9.4","generation_timestamp":"2024-10-16T20:08:34","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/dev/algorithms/index.html b/dev/algorithms/index.html index 8cfecd5..71505ca 100644 --- a/dev/algorithms/index.html +++ b/dev/algorithms/index.html @@ -1,5 +1,5 @@ -Algorithms · OperationsResearchModels.jl
GitHub

Algorithms

Assignment Problem

OperationsResearchModels.solve — Method
solve(a)

Arguments

a::AssignmentProblem: The problem in type of AssignmentProblem

Output

AssignmentResult: The custom data type that holds problem, solution, and optimum cost.

Description

Solves an assignment problem given by an object of in type AssignmentProblem.

Example

julia> mat = [
+Algorithms · OperationsResearchModels.jl
GitHub
Algorithms
Assignment Problem
OperationsResearchModels.solve — Method
solve(a)
Arguments
a::AssignmentProblem: The problem in type of AssignmentProblem
Output
AssignmentResult: The custom data type that holds problem, solution, and optimum cost. 
Description
Solves an assignment problem given by an object of in type AssignmentProblem.
Example
julia> mat = [
                    4 8 1;
                    3 1 9;
                    1 6 7;
@@ -18,7 +18,7 @@
 
 julia> result.cost
 
-3.0
source
Transportation Problem
OperationsResearchModels.solve — Method
solve(t)
Arguments
a::TransportationProblem: The problem in type of TransportationProblem
Output
TransportationResult: The custom data type that holds problem, solution, and optimum cost. 
Description
Solves a transportation problem given by an object of in type TransportationProblem.
Example
julia> t = TransportationProblem(
+3.0
source
Transportation Problem
OperationsResearchModels.solve — Method
solve(t)
Arguments
a::TransportationProblem: The problem in type of TransportationProblem
Output
TransportationResult: The custom data type that holds problem, solution, and optimum cost. 
Description
Solves a transportation problem given by an object of in type TransportationProblem.
Example
julia> t = TransportationProblem(
                    [   1 1 1 1; 
                        2 2 2 2; 
                        3 3 3 3], 
@@ -49,7 +49,7 @@
 Cost:
 600.0
 Solution:
-[-0.0 -0.0 -0.0 100.0; 100.0 -0.0 -0.0 -0.0; -0.0 -0.0 100.0 -0.0; -0.0 100.0 -0.0 -0.0]
source
Shortest Path
OperationsResearchModels.solve — Method
solve(problem)
Description
Solves a shortest path problem given by an object of in type ShortestPathProblem.
Arguments
problem::ShortestPathProblem: The problem in type of ShortestPathProblem
Output
ShortestPathResult: The custom data type that holds path and cost.
Example
julia> conns = [
+[-0.0 -0.0 -0.0 100.0; 100.0 -0.0 -0.0 -0.0; -0.0 -0.0 100.0 -0.0; -0.0 100.0 -0.0 -0.0]
source
Shortest Path
OperationsResearchModels.solve — Method
solve(problem)
Description
Solves a shortest path problem given by an object of in type ShortestPathProblem.
Arguments
problem::ShortestPathProblem: The problem in type of ShortestPathProblem
Output
ShortestPathResult: The custom data type that holds path and cost.
Example
julia> conns = [
                    Connection(1, 2, 3),
                    Connection(1, 3, 2),
                    Connection(1, 4, 4),
@@ -70,7 +70,7 @@
  Connection(6, 7, 5, "x67")
 
 julia> result.cost
-8.0
source
Maximum Flow
OperationsResearchModels.solve — Method
solve(problem)
Arguments
problem::MaximumFlowProblem: The problem in type of MaximumFlowProblem
Output
MaximumFlowResult: The custom data type that holds path and flow.
Example
julia> conns = [
+8.0
source
Maximum Flow
OperationsResearchModels.solve — Method
solve(problem)
Arguments
problem::MaximumFlowProblem: The problem in type of MaximumFlowProblem
Output
MaximumFlowResult: The custom data type that holds path and flow.
Example
julia> conns = [
                    Connection(1, 2, 3),
                    Connection(1, 3, 2),
                    Connection(1, 4, 4),
@@ -97,7 +97,7 @@
  Connection(6, 7, 3.0, "x67")
 
 julia> result.flow
-7.0
source
Minimum Spanning Tree
OperationsResearchModels.solve — Method
solve(problem::MstProblem)
Arguments
  • problem::MstProblem: The problem in type of MstProblem
Description
Obtains the minimum spanning tree. 
Output
  • ::MstResult: A MstResult object that holds the results. 
Examples
julia> conns = Connection[
+7.0
source
Minimum Spanning Tree
OperationsResearchModels.solve — Method
solve(problem::MstProblem)
Arguments
  • problem::MstProblem: The problem in type of MstProblem
Description
Obtains the minimum spanning tree. 
Output
  • ::MstResult: A MstResult object that holds the results. 
Examples
julia> conns = Connection[
                        Connection(1, 2, 10),
                        Connection(2, 3, 10),
                        Connection(3, 4, 10),
@@ -120,7 +120,7 @@
  3-element Vector{Connection}:
   Connection(3, 4, 10, "x34")
   Connection(1, 4, 10, "x14")
-  Connection(2, 3, 10, "x23")
source
pmedian
OperationsResearchModels.PMedian.pmedian — Function
pmedian(data, ncenters)
Arguments
  • data::Matrix: Coordinates of locations 
  • ncenters::Int: Number of centers 
Description
The function calculates Euclidean distances between all possible rows of the matrix data.  ncenters locations are then selected that minimizes the total distances to the nearest rows. 
Output
  • PMedianResult: PMedianResult object. 
Example
julia> data1 = rand(10, 2);
+  Connection(2, 3, 10, "x23")
source
pmedian
OperationsResearchModels.PMedian.pmedian — Function
pmedian(data, ncenters)
Arguments
  • data::Matrix: Coordinates of locations 
  • ncenters::Int: Number of centers 
Description
The function calculates Euclidean distances between all possible rows of the matrix data.  ncenters locations are then selected that minimizes the total distances to the nearest rows. 
Output
  • PMedianResult: PMedianResult object. 
Example
julia> data1 = rand(10, 2);
 
 julia> data2 = rand(10, 2) .+ 50;
 
@@ -137,7 +137,7 @@
  21
 
  julia> result.objective
- 11.531012240599605
source
pmedian with distances
OperationsResearchModels.PMedian.pmedian_with_distances — Function
pmedian_with_distances(distancematrix, ncenters)
Arguments
  • distancematrix::Matrix: n x n matrix of distances
  • ncenters::Int: Number of centers 
Description
  • ncenters locations are selected that minimizes the total distances to the nearest rows. 
Output
  • PMedianResult: PMedianResult object. 
source
CPM (Critical Path Method)
OperationsResearchModels.solve — Method
solve(problem)
Arguments
  • problem::CpmProblem: The problem in type of CpmProblem.
Output
  • ::CpmResult: The object holds the results 
Description
Calculates CPM (Critical Path Method) and reports the critical path for a given set of activities. 
Example
julia> A = CpmActivity("A", 2);
+ 11.531012240599605
source
pmedian with distances
OperationsResearchModels.PMedian.pmedian_with_distances — Function
pmedian_with_distances(distancematrix, ncenters)
Arguments
  • distancematrix::Matrix: n x n matrix of distances
  • ncenters::Int: Number of centers 
Description
  • ncenters locations are selected that minimizes the total distances to the nearest rows. 
Output
  • PMedianResult: PMedianResult object. 
source
CPM (Critical Path Method)
OperationsResearchModels.solve — Method
solve(problem)
Arguments
  • problem::CpmProblem: The problem in type of CpmProblem.
Output
  • ::CpmResult: The object holds the results 
Description
Calculates CPM (Critical Path Method) and reports the critical path for a given set of activities. 
Example
julia> A = CpmActivity("A", 2);
 julia> B = CpmActivity("B", 3);
 julia> C = CpmActivity("C", 2, [A]);
 julia> D = CpmActivity("D", 3, [B]);
@@ -162,12 +162,12 @@
  "I"
 
  julia> result.path == [B, E, G, I]
-true
source
CPM Activity
OperationsResearchModels.CPM.CpmActivity — Type
CpmActivity(name::String, time::Float64, dependencies)
Description
The object that represents an activity in CPM (Critical Path Method).
Arguments
  • name::String: The name of the activity.
  • time::Float64: The time of the activity.
  • dependencies: The dependencies of the activity in type of Vector{CpmActivity}.
Example
julia> A = CpmActivity("A", 2, []);
+true
source
CPM Activity
OperationsResearchModels.CPM.CpmActivity — Type
CpmActivity(name::String, time::Float64, dependencies)
Description
The object that represents an activity in CPM (Critical Path Method).
Arguments
  • name::String: The name of the activity.
  • time::Float64: The time of the activity.
  • dependencies: The dependencies of the activity in type of Vector{CpmActivity}.
Example
julia> A = CpmActivity("A", 2, []);
 
 julia> B = CpmActivity("B", 3, []);
 
 julia> C = CpmActivity("C", 2, [A, B]);
-
source
PERT (Project Evalutation and Review Technique)
OperationsResearchModels.solve — Method
solve(problem::PertProblem)::PertResult
Arguments
  • problem::PertProblem: The problem in type of PertProblem.
Example
julia> A = PertActivity("A", 1, 2, 3)
+
source
PERT (Project Evalutation and Review Technique)
OperationsResearchModels.solve — Method
solve(problem::PertProblem)::PertResult
Arguments
  • problem::PertProblem: The problem in type of PertProblem.
Example
julia> A = PertActivity("A", 1, 2, 3)
 PertActivity("A", 1.0, 2.0, 3.0, PertActivity[])
 
 julia> B = PertActivity("B", 3, 3, 3)
@@ -191,12 +191,12 @@
 8.0
 
 julia> result.stddev
-0.0
source
PERT Activity
OperationsResearchModels.CPM.PertActivity — Type
PertActivity(name::String, o::Float64, m::Float64, p::Float64)::PertActivity
Description
The object that represents an activity in PERT (Program Evaluation and Review Technique).
Arguments
  • name::String: The name of the activity.
  • o::Float64: The optimistic time of the activity.
  • m::Float64: The most likely time of the activity.
  • p::Float64: The pessimistic time of the activity.
  • dependencies: The dependencies of the activity in type of Vector{PertActivity}.
Example
julia> A = PertActivity("A", 1, 2, 3);
+0.0
source
PERT Activity
OperationsResearchModels.CPM.PertActivity — Type
PertActivity(name::String, o::Float64, m::Float64, p::Float64)::PertActivity
Description
The object that represents an activity in PERT (Program Evaluation and Review Technique).
Arguments
  • name::String: The name of the activity.
  • o::Float64: The optimistic time of the activity.
  • m::Float64: The most likely time of the activity.
  • p::Float64: The pessimistic time of the activity.
  • dependencies: The dependencies of the activity in type of Vector{PertActivity}.
Example
julia> A = PertActivity("A", 1, 2, 3);
 julia> B = PertActivity("B", 3, 3, 4);
-julia> C = PertActivity("C", 5, 6, 7, [A, B]);
source
Knapsack
OperationsResearchModels.solve — Method
solve(problem::KnapsackProblem)::KnapsackResult
Description
Solves the knapsack problem.
Arguments
  • problem::KnapsackProblem: The problem in type of KnapsackProblem.
Output
  • KnapsackResult: The custom data type that holds selected items, model, and objective value.
Example
julia> values = [10, 20, 30, 40, 50];
+julia> C = PertActivity("C", 5, 6, 7, [A, B]);
source
Knapsack
OperationsResearchModels.solve — Method
solve(problem::KnapsackProblem)::KnapsackResult
Description
Solves the knapsack problem.
Arguments
  • problem::KnapsackProblem: The problem in type of KnapsackProblem.
Output
  • KnapsackResult: The custom data type that holds selected items, model, and objective value.
Example
julia> values = [10, 20, 30, 40, 50];
 julia> weights = [1, 2, 3, 4, 5];
 julia> capacity = 10;
-julia> solve(KnapsackProblem(values, weights, capacity));
source
Johnson's Rule
OperationsResearchModels.Johnsons.johnsons — Function
johnsons(times::Matrix)
Given a matrix of times, returns a JohnsonResult with the permutation of the jobs.  If number of machines is 2, it uses the Johnson's algorithm for 2 machines. If number of machines is greater than 2, it uses the Johnson's algorithm by transforming the  problem into a 2-machine problem. In order to reduce the original problem to a 2-machine problem, the algorithm checks if the minimum time of the first machine is greater or equal than the maximum time of the other machines and/or if the minimum time of the  last machine is greater or equal than the maximum time of the other machines.
For example if we have 4 machines, namely, A, B, C, and D  at least one of the following conditions must be satisfied:
  • min(A) >= max(B, C)
  • min(D) >= max(B, C)
The function throws a JohnsonException if the problem cannot be reduced to a 2-machine problem.
Arguments
  • times::Matrix: a matrix of times
Returns
  • JohnsonResult: a custom data type that holds the permutation of the jobs
Example
times = Float64[
+julia> solve(KnapsackProblem(values, weights, capacity));
source
Johnson's Rule
OperationsResearchModels.Johnsons.johnsons — Function
johnsons(times::Matrix)
Given a matrix of times, returns a JohnsonResult with the permutation of the jobs.  If number of machines is 2, it uses the Johnson's algorithm for 2 machines. If number of machines is greater than 2, it uses the Johnson's algorithm by transforming the  problem into a 2-machine problem. In order to reduce the original problem to a 2-machine problem, the algorithm checks if the minimum time of the first machine is greater or equal than the maximum time of the other machines and/or if the minimum time of the  last machine is greater or equal than the maximum time of the other machines.
For example if we have 4 machines, namely, A, B, C, and D  at least one of the following conditions must be satisfied:
  • min(A) >= max(B, C)
  • min(D) >= max(B, C)
The function throws a JohnsonException if the problem cannot be reduced to a 2-machine problem.
Arguments
  • times::Matrix: a matrix of times
Returns
  • JohnsonResult: a custom data type that holds the permutation of the jobs
Example
times = Float64[
     3.1 2.8;
     4.0 7.0;
     8.0 3.0;
@@ -208,7 +208,7 @@
 
 result = johnsons(times)
 
-println(result.permutation)
source
Genetic Algorithm for the problems that cannot be solved with using Johnson's Rule
OperationsResearchModels.Johnsons.johnsons_ga — Function
johnsons_ga(times::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::JohnsonResult
Given a matrix of times, returns a JohnsonResult with the permutation of the jobs. The function uses a genetic algorithm to find the best permutation of the jobs.  The genetic algorithm is implemented in the RandomKeyGA module.
Arguments
  • times::Matrix: a matrix of times
  • popsize::Int: the population size. Default is 100
  • ngen::Int: the number of generations. Default is 1000
  • pcross::Float64: the crossover probability. Default is 0.8
  • pmutate::Float64: the mutation probability. Default is 0.01
  • nelites::Int: the number of elites. Default is 1
Returns
  • JohnsonResult: a custom data type that holds the permutation of the jobs
Example
times = Float64[
+println(result.permutation)
source
Genetic Algorithm for the problems that cannot be solved with using Johnson's Rule
OperationsResearchModels.Johnsons.johnsons_ga — Function
johnsons_ga(times::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::JohnsonResult
Given a matrix of times, returns a JohnsonResult with the permutation of the jobs. The function uses a genetic algorithm to find the best permutation of the jobs.  The genetic algorithm is implemented in the RandomKeyGA module.
Arguments
  • times::Matrix: a matrix of times
  • popsize::Int: the population size. Default is 100
  • ngen::Int: the number of generations. Default is 1000
  • pcross::Float64: the crossover probability. Default is 0.8
  • pmutate::Float64: the mutation probability. Default is 0.01
  • nelites::Int: the number of elites. Default is 1
Returns
  • JohnsonResult: a custom data type that holds the permutation of the jobs
Example
times = Float64[
     3.1 2.8;
     4.0 7.0;
     8.0 3.0;
@@ -220,7 +220,7 @@
 
 result = johnsons(times)
 
-println(result.permutation)
source
Makespan
Makespan
OperationsResearchModels.Johnsons.makespan — Function
makespan(times::Matrix, permutation::Vector{Int})
 
 Given a matrix of times and a permutation of the jobs, returns the makespan of the jobs.
Arguments
  • times::Matrix: a matrix of times
  • permutation::Vector{Int}: a permutation of the jobs
Returns
  • Float64: the makespan of the jobs
Example

 julia> times = Float64[
@@ -231,7 +231,7 @@
     2 5 6    
 ]
 
-julia> result = makespan(times, [1, 4, 5, 3, 2])
source
Traveling Salesman
OperationsResearchModels.TravelingSalesman.travelingsalesman — Function
travelingsalesman(distancematrix::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::TravelinSalesmenResult
Given a matrix of distances, returns a TravelinSalesmenResult with the best route and its cost.
Arguments
  • distancematrix::Matrix: a matrix of distances
  • popsize::Int: the population size. Default is 100
  • ngen::Int: the number of generations. Default is 1000
  • pcross::Float64: the crossover probability. Default is 0.8
  • pmutate::Float64: the mutation probability. Default is 0.01
  • nelites::Int: the number of elites. Default is 1
Returns
  • TravelinSalesmenResult: a custom data type that holds the best route and its cost
Example
pts = Float64[
+julia> result = makespan(times, [1, 4, 5, 3, 2])
source
Traveling Salesman
OperationsResearchModels.TravelingSalesman.travelingsalesman — Function
travelingsalesman(distancematrix::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::TravelinSalesmenResult
Given a matrix of distances, returns a TravelinSalesmenResult with the best route and its cost.
Arguments
  • distancematrix::Matrix: a matrix of distances
  • popsize::Int: the population size. Default is 100
  • ngen::Int: the number of generations. Default is 1000
  • pcross::Float64: the crossover probability. Default is 0.8
  • pmutate::Float64: the mutation probability. Default is 0.01
  • nelites::Int: the number of elites. Default is 1
Returns
  • TravelinSalesmenResult: a custom data type that holds the best route and its cost
Example
pts = Float64[
     0 0;
     0 1;
     0 2;
@@ -257,4 +257,17 @@
     end 
 end
 
-result = travelingsalesman(distmat, ngen = 1000, popsize = 100, pcross = 1.0, pmutate = 0.10)
source

+result = travelingsalesman(distmat, ngen = 1000, popsize = 100, pcross = 1.0, pmutate = 0.10)
source

Simplex

OperationsResearchModels.Simplex.createsimplexproblem — Function
createsimplexproblem(obj::Vector, amat::Matrix, rhs::Vector, dir::Vector, opttype::OptimizationType)::SimplexProblem

Description:

This function creates a SimplexProblem object from the given parameters.

Arguments:

  • obj::Vector: The objective function coefficients.
  • amat::Matrix: The LHS of the constraints.
  • rhs::Vector: The RHS of the constraints.
  • dir::Vector: The directions of the constraints. Can be a vector of LE (<=), GE (>=), or EQ (==).
  • opttype::OptimizationType: The type of the optimization. Can be Maximize or Minimize.

Returns:

A SimplexProblem object.

Example:

Suppose the linear programming problem is as follows:

Maximize: 1.0x1 + 2.0x2 + 3.0x3
+Subject to:
+1.0x1 + 2.0x2 + 3.0x3 <= 10.0
+3.0x1 + 1.0x2 + 5.0x3 <= 15.0
+x1, x2 >= 0

The following code creates a SimplexProblem object for the above problem:

using OperationsResearch.Simplex
+
+obj = Float64[1.0, 2.0, 3.0]
+amat = Float64[1.0 2.0 3.0; 3.0 1.0 5.0]
+rhs = Float64[10.0, 15.0]
+dir = [LE, LE]
+opttype = Maximize
+
+s = createsimplexproblem(obj, amat, rhs, dir, opttype)
+iters = simplexiterations(s)
source
diff --git a/dev/index.html b/dev/index.html index b3c9d1b..30d9573 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -- · OperationsResearchModels.jl
GitHub

Operations Research Models

The OperationsResearchModels package includes basic Operations Research subjects such as Transportation Problem, Assignment Problem, Minimum Spanning Tree, Shortest Path, Maximum Flow, and p-medians method for selecting location of facilities.

Package content is incrementaly updated.

Please refer the Algorithms section for the detailed documentation.

+- · OperationsResearchModels.jl
GitHub

Operations Research Models

The OperationsResearchModels package includes basic Operations Research subjects such as Transportation Problem, Assignment Problem, Minimum Spanning Tree, Shortest Path, Maximum Flow, and p-medians method for selecting location of facilities.

Package content is incrementaly updated.

Please refer the Algorithms section for the detailed documentation.

diff --git a/dev/objects.inv b/dev/objects.inv index 8fe5ed5b7efb1abd2a2d2b42cc6a7c9d4716ff0d..135bed092a93b1a3d4ac2191f352739a6a93f79a 100644 GIT binary patch delta 690 zcmV;j0!{t02FC`Fkbhlo(=Zf$&#$-=RE!4~v6sQ#${0wXh>G@oa+ABpn`4LA={6z$ zJJ(K}tnHd+Ro?P(&hh=e=5nn|VYA$TvFT;16JQ^Kw_iWNtbLB=*T0u+mRPY7_Q~TV zU!mM#ym<=_2mu&f&Er8V!gZwCwqXY-EGlwL1y}u`MW#!O#tyPMw^<)F0Koyh;b&L$ zlOF*Ve|sNWNq-77lF_dMOKyc$hW$o^yiACwQbyp%xtjT1MSAXC!-v ziDlOtE-UM^$GA!<*BgNkY>7#x#GeX#YRd~zaSfGn1?O8a91hPIKn$oE&>bGdLit!P z3a*sP*|Nl*GNPc@1PBChgy=geq#9>!8?#$oGMgbQ>hKzR;g%&t(w41|87()M*3qtv ze>-_-DUWzo*A9(nH~xS`fkYvQf)j;W6jo%P7p;@;MR!{&=J#ldW3!6tUzvi&P@UHS zwY$w|>$e*cVMf#=V(JZL%ZmE@?;K0p5dOiPyv+SDg6$)QC`~TDJky$7d?i;7@@NW9 z9lG5Lbh3Ra5cb0{$cw8YAIphO>WB~Xf7mzsG&DV(dt0f*r{F|4x*;@{H+zaloO*7} zc3F*O(Tl1_tZMqmOz)b9n>9Z*-tBq`+huCQ7ZbMA2&s(CXgTbp26Ppb?^sP0mit(K z?C8Cs62q;`U6;nP<^|1HSk&A!+(l!wH1A0c?TS3?l@sfBXOi8x4R+IAcurJp7%q_0 Yq(n}EuHTy7d8+2#AUGoQ9}@(swWUl>XaE2J delta 686 zcmV;f0#W_P2C@c_kblc=+b|Hk>nj#$jiv{>XfA1UbJMgy3!z5hyFp7EiwQ-lBIPtF z`tMy*q7>V)tQflaF*Dr9?2=rrbt!C?8!$G#Omzb6Bk=au4=`&#NAv5O1)C*SY=nLC zc*$2NcNlNpf&)STMpyHA5Q}gfX|^5M0Sb$X98O zm>m=X6l!SnLublp^^Qa;iHGBCaFg?k2H6t9eT{x9UKpM{^cQwf@(oI%*6fx`G#IoxRmzDL6VO*uu#*M&7w!kD);&+8Tw(&ybo1s$9OMc42;qZ(B z#DJOs-QiIzl&|$|!Ig47Ta?)25Gs02fIt98h`ysjvVSjYCzRdklGzMdQJ2@y7j9WX zByHIWnbC5CXAL9vsA>zPN$N&HU diff --git a/dev/search_index.js b/dev/search_index.js index 9e9cabc..3b12130 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"#Operations-Research-Models","page":"-","title":"Operations Research Models","text":"","category":"section"},{"location":"","page":"-","title":"-","text":"The OperationsResearchModels package includes basic Operations Research subjects such as Transportation Problem, Assignment Problem, Minimum Spanning Tree, Shortest Path, Maximum Flow, and p-medians method for selecting location of facilities. ","category":"page"},{"location":"","page":"-","title":"-","text":"Package content is incrementaly updated.","category":"page"},{"location":"","page":"-","title":"-","text":"Please refer the Algorithms section for the detailed documentation. ","category":"page"},{"location":"algorithms/#Algorithms","page":"Algorithms","title":"Algorithms","text":"","category":"section"},{"location":"algorithms/#Assignment-Problem","page":"Algorithms","title":"Assignment Problem","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(a::AssignmentProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{AssignmentProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(a)\n\nArguments\n\na::AssignmentProblem: The problem in type of AssignmentProblem\n\nOutput\n\nAssignmentResult: The custom data type that holds problem, solution, and optimum cost. \n\nDescription\n\nSolves an assignment problem given by an object of in type AssignmentProblem.\n\nExample\n\njulia> mat = [\n 4 8 1;\n 3 1 9;\n 1 6 7;\n ];\n\njulia> problem = AssignmentProblem(mat);\n\njulia> result = solve(problem);\n\njulia> result.solution\n\n3×3 Matrix{Float64}:\n 0.0 0.0 1.0\n 0.0 1.0 0.0\n 1.0 0.0 0.0\n\njulia> result.cost\n\n3.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Transportation-Problem","page":"Algorithms","title":"Transportation Problem","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(t::TransportationProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{TransportationProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(t)\n\nArguments\n\na::TransportationProblem: The problem in type of TransportationProblem\n\nOutput\n\nTransportationResult: The custom data type that holds problem, solution, and optimum cost. \n\nDescription\n\nSolves a transportation problem given by an object of in type TransportationProblem.\n\nExample\n\njulia> t = TransportationProblem(\n [ 1 1 1 1; \n 2 2 2 2; \n 3 3 3 3], \n [100, 100, 100, 100], # Demands \n [100, 100, 100]) # Supplies \nTransportation Problem:\nCosts: [1 1 1 1; 2 2 2 2; 3 3 3 3]\nDemand: [100, 100, 100, 100]\nSupply: [100, 100, 100]\n\njulia> isbalanced(t)\nfalse\n\njulia> result = solve(t)\nTransportation Results:\nMain problem:\nTransportation Problem:\nCosts: [1 1 1 1; 2 2 2 2; 3 3 3 3]\nDemand: [100, 100, 100, 100]\nSupply: [100, 100, 100]\n\nBalanced problem:\nTransportation Problem:\nCosts: [1 1 1 1; 2 2 2 2; 3 3 3 3; 0 0 0 0]\nDemand: [100, 100, 100, 100]\nSupply: [100, 100, 100, 100]\n\nCost:\n600.0\nSolution:\n[-0.0 -0.0 -0.0 100.0; 100.0 -0.0 -0.0 -0.0; -0.0 -0.0 100.0 -0.0; -0.0 100.0 -0.0 -0.0]\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Shortest-Path","page":"Algorithms","title":"Shortest Path","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(t::ShortestPathProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{ShortestPathProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem)\n\nDescription\n\nSolves a shortest path problem given by an object of in type ShortestPathProblem.\n\nArguments\n\nproblem::ShortestPathProblem: The problem in type of ShortestPathProblem\n\nOutput\n\nShortestPathResult: The custom data type that holds path and cost.\n\nExample\n\njulia> conns = [\n Connection(1, 2, 3),\n Connection(1, 3, 2),\n Connection(1, 4, 4),\n Connection(2, 5, 3),\n Connection(3, 5, 1),\n Connection(3, 6, 1),\n Connection(4, 6, 2),\n Connection(5, 7, 6),\n Connection(6, 7, 5),\n ];\n\njulia> solve(ShortestPathProblem(conns));\n\njulia> result.path\n3-element Vector{Connection}:\n Connection(1, 3, 2, \"x13\")\n Connection(3, 6, 1, \"x36\")\n Connection(6, 7, 5, \"x67\")\n\njulia> result.cost\n8.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Maximum-Flow","page":"Algorithms","title":"Maximum Flow","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(t::MaximumFlowProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{MaximumFlowProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem)\n\nArguments\n\nproblem::MaximumFlowProblem: The problem in type of MaximumFlowProblem\n\nOutput\n\nMaximumFlowResult: The custom data type that holds path and flow.\n\nExample\n\njulia> conns = [\n Connection(1, 2, 3),\n Connection(1, 3, 2),\n Connection(1, 4, 4),\n Connection(2, 5, 3),\n Connection(3, 5, 1),\n Connection(3, 6, 1),\n Connection(4, 6, 2),\n Connection(5, 7, 6),\n Connection(6, 7, 5),\n ];\njulia> problem = MaximumFlowProblem(conns)\njulia> result = solve(problem);\n\njulia> result.path\n9-element Vector{Connection}:\n Connection(1, 2, 3.0, \"x12\")\n Connection(1, 3, 2.0, \"x13\")\n Connection(1, 4, 2.0, \"x14\")\n Connection(2, 5, 3.0, \"x25\")\n Connection(3, 5, 1.0, \"x35\")\n Connection(3, 6, 1.0, \"x36\")\n Connection(4, 6, 2.0, \"x46\")\n Connection(5, 7, 4.0, \"x57\")\n Connection(6, 7, 3.0, \"x67\")\n\njulia> result.flow\n7.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Minimum-Spanning-Tree","page":"Algorithms","title":"Minimum Spanning Tree","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(p::MstProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{MstProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem::MstProblem)\n\nArguments\n\nproblem::MstProblem: The problem in type of MstProblem\n\nDescription\n\nObtains the minimum spanning tree. \n\nOutput\n\n::MstResult: A MstResult object that holds the results. \n\nExamples\n\njulia> conns = Connection[\n Connection(1, 2, 10),\n Connection(2, 3, 10),\n Connection(3, 4, 10),\n Connection(1, 4, 10)\n ]\n\n4-element Vector{Connection}:\n Connection(1, 2, 10, \"x12\")\n Connection(2, 3, 10, \"x23\")\n Connection(3, 4, 10, \"x34\")\n Connection(1, 4, 10, \"x14\")\n\n julia> result = solve(MstProblem(conns))\n MstResult(Connection[Connection(3, 4, 10, \"x34\"), Connection(1, 4, 10, \"x14\"), Connection(2, 3, 10, \"x23\")], 30.0)\n \n julia> result.distance\n 30.0\n \n julia> result.connections\n 3-element Vector{Connection}:\n Connection(3, 4, 10, \"x34\")\n Connection(1, 4, 10, \"x14\")\n Connection(2, 3, 10, \"x23\")\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#pmedian","page":"Algorithms","title":"pmedian","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.pmedian","category":"page"},{"location":"algorithms/#OperationsResearchModels.PMedian.pmedian","page":"Algorithms","title":"OperationsResearchModels.PMedian.pmedian","text":"pmedian(data, ncenters)\n\nArguments\n\ndata::Matrix: Coordinates of locations \nncenters::Int: Number of centers \n\nDescription\n\nThe function calculates Euclidean distances between all possible rows of the matrix data. ncenters locations are then selected that minimizes the total distances to the nearest rows. \n\nOutput\n\nPMedianResult: PMedianResult object. \n\nExample\n\njulia> data1 = rand(10, 2);\n\njulia> data2 = rand(10, 2) .+ 50;\n\njulia> data3 = rand(10, 2) .+ 100;\n\njulia> data = vcat(data1, data2, data3);\n\njulia> result = pmedian(data, 3);\n\njulia> result.centers\n3-element Vector{Int64}:\n 1\n 16\n 21\n\n julia> result.objective\n 11.531012240599605\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#pmedian-with-distances","page":"Algorithms","title":"pmedian with distances","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.pmedian_with_distances","category":"page"},{"location":"algorithms/#OperationsResearchModels.PMedian.pmedian_with_distances","page":"Algorithms","title":"OperationsResearchModels.PMedian.pmedian_with_distances","text":"pmedian_with_distances(distancematrix, ncenters)\n\nArguments\n\ndistancematrix::Matrix: n x n matrix of distances\nncenters::Int: Number of centers \n\nDescription\n\nncenters locations are selected that minimizes the total distances to the nearest rows. \n\nOutput\n\nPMedianResult: PMedianResult object. \n\n\n\n\n\n","category":"function"},{"location":"algorithms/#CPM-(Critical-Path-Method)","page":"Algorithms","title":"CPM (Critical Path Method)","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(problem::CpmProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{CpmProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem)\n\nArguments\n\nproblem::CpmProblem: The problem in type of CpmProblem.\n\nOutput\n\n::CpmResult: The object holds the results \n\nDescription\n\nCalculates CPM (Critical Path Method) and reports the critical path for a given set of activities. \n\nExample\n\njulia> A = CpmActivity(\"A\", 2);\njulia> B = CpmActivity(\"B\", 3);\njulia> C = CpmActivity(\"C\", 2, [A]);\njulia> D = CpmActivity(\"D\", 3, [B]);\njulia> E = CpmActivity(\"E\", 2, [B]);\njulia> F = CpmActivity(\"F\", 3, [C, D]);\njulia> G = CpmActivity(\"G\", 7, [E]);\njulia> H = CpmActivity(\"H\", 5, [E]);\njulia> I = CpmActivity(\"I\", 6, [G, F]);\njulia> J = CpmActivity(\"J\", 2, [C, D]);\n\njulia> activities = [A, B, C, D, E, F, G, H, I, J];\n\njulia> problem = CpmProblem(activities);\n\njulia> result = solve(problem);\n\njulia> result.pathstr\n4-element Vector{String}:\n \"B\"\n \"E\"\n \"G\"\n \"I\"\n\n julia> result.path == [B, E, G, I]\ntrue\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#CPM-Activity","page":"Algorithms","title":"CPM Activity","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.CpmActivity","category":"page"},{"location":"algorithms/#OperationsResearchModels.CPM.CpmActivity","page":"Algorithms","title":"OperationsResearchModels.CPM.CpmActivity","text":"CpmActivity(name::String, time::Float64, dependencies)\n\nDescription\n\nThe object that represents an activity in CPM (Critical Path Method).\n\nArguments\n\nname::String: The name of the activity.\ntime::Float64: The time of the activity.\ndependencies: The dependencies of the activity in type of Vector{CpmActivity}.\n\nExample\n\njulia> A = CpmActivity(\"A\", 2, []);\n\njulia> B = CpmActivity(\"B\", 3, []);\n\njulia> C = CpmActivity(\"C\", 2, [A, B]);\n\n\n\n\n\n\n","category":"type"},{"location":"algorithms/#PERT-(Project-Evalutation-and-Review-Technique)","page":"Algorithms","title":"PERT (Project Evalutation and Review Technique)","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(problem::PertProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{PertProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem::PertProblem)::PertResult\n\nArguments\n\nproblem::PertProblem: The problem in type of PertProblem.\n\nExample\n\njulia> A = PertActivity(\"A\", 1, 2, 3)\nPertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[])\n\njulia> B = PertActivity(\"B\", 3, 3, 3)\nPertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])\n\njulia> C = PertActivity(\"C\", 5, 5, 5, [A, B])\nPertActivity(\"C\", 5.0, 5.0, 5.0, PertActivity[PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[]), PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])])\n\njulia> activities = [A, B, C]\n3-element Vector{PertActivity}:\n PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[])\n PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])\n PertActivity(\"C\", 5.0, 5.0, 5.0, PertActivity[PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[]), PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])])\n\njulia> problem = PertProblem(activities);\n\njulia> result = pert(activities)\nPertResult(PertActivity[PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[]), PertActivity(\"C\", 5.0, 5.0, 5.0, PertActivity[PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[]), PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])])], 8.0, 0.0)\n\njulia> result.mean\n8.0\n\njulia> result.stddev\n0.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#PERT-Activity","page":"Algorithms","title":"PERT Activity","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.PertActivity","category":"page"},{"location":"algorithms/#OperationsResearchModels.CPM.PertActivity","page":"Algorithms","title":"OperationsResearchModels.CPM.PertActivity","text":"PertActivity(name::String, o::Float64, m::Float64, p::Float64)::PertActivity\n\nDescription\n\nThe object that represents an activity in PERT (Program Evaluation and Review Technique).\n\nArguments\n\nname::String: The name of the activity.\no::Float64: The optimistic time of the activity.\nm::Float64: The most likely time of the activity.\np::Float64: The pessimistic time of the activity.\ndependencies: The dependencies of the activity in type of Vector{PertActivity}.\n\nExample\n\njulia> A = PertActivity(\"A\", 1, 2, 3);\njulia> B = PertActivity(\"B\", 3, 3, 4);\njulia> C = PertActivity(\"C\", 5, 6, 7, [A, B]);\n\n\n\n\n\n","category":"type"},{"location":"algorithms/#Knapsack","page":"Algorithms","title":"Knapsack","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(p::KnapsackProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{KnapsackProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem::KnapsackProblem)::KnapsackResult\n\nDescription\n\nSolves the knapsack problem.\n\nArguments\n\nproblem::KnapsackProblem: The problem in type of KnapsackProblem.\n\nOutput\n\nKnapsackResult: The custom data type that holds selected items, model, and objective value.\n\nExample\n\njulia> values = [10, 20, 30, 40, 50];\njulia> weights = [1, 2, 3, 4, 5];\njulia> capacity = 10;\njulia> solve(KnapsackProblem(values, weights, capacity));\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Johnson's-Rule","page":"Algorithms","title":"Johnson's Rule","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.johnsons","category":"page"},{"location":"algorithms/#OperationsResearchModels.Johnsons.johnsons","page":"Algorithms","title":"OperationsResearchModels.Johnsons.johnsons","text":"johnsons(times::Matrix)\n\nGiven a matrix of times, returns a JohnsonResult with the permutation of the jobs. If number of machines is 2, it uses the Johnson's algorithm for 2 machines. If number of machines is greater than 2, it uses the Johnson's algorithm by transforming the problem into a 2-machine problem. In order to reduce the original problem to a 2-machine problem, the algorithm checks if the minimum time of the first machine is greater or equal than the maximum time of the other machines and/or if the minimum time of the last machine is greater or equal than the maximum time of the other machines.\n\nFor example if we have 4 machines, namely, A, B, C, and D at least one of the following conditions must be satisfied:\n\nmin(A) >= max(B, C)\nmin(D) >= max(B, C)\n\nThe function throws a JohnsonException if the problem cannot be reduced to a 2-machine problem.\n\nArguments\n\ntimes::Matrix: a matrix of times\n\nReturns\n\nJohnsonResult: a custom data type that holds the permutation of the jobs\n\nExample\n\ntimes = Float64[\n 3.1 2.8;\n 4.0 7.0;\n 8.0 3.0;\n 5.0 8.0;\n 6.0 4.0;\n 8.0 5.0;\n 7.0 4.0\n]\n\nresult = johnsons(times)\n\nprintln(result.permutation)\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#Genetic-Algorithm-for-the-problems-that-cannot-be-solved-with-using-Johnson's-Rule","page":"Algorithms","title":"Genetic Algorithm for the problems that cannot be solved with using Johnson's Rule","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.johnsons_ga","category":"page"},{"location":"algorithms/#OperationsResearchModels.Johnsons.johnsons_ga","page":"Algorithms","title":"OperationsResearchModels.Johnsons.johnsons_ga","text":"johnsons_ga(times::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::JohnsonResult\n\nGiven a matrix of times, returns a JohnsonResult with the permutation of the jobs. The function uses a genetic algorithm to find the best permutation of the jobs. The genetic algorithm is implemented in the RandomKeyGA module.\n\nArguments\n\ntimes::Matrix: a matrix of times\npopsize::Int: the population size. Default is 100\nngen::Int: the number of generations. Default is 1000\npcross::Float64: the crossover probability. Default is 0.8\npmutate::Float64: the mutation probability. Default is 0.01\nnelites::Int: the number of elites. Default is 1\n\nReturns\n\nJohnsonResult: a custom data type that holds the permutation of the jobs\n\nExample\n\ntimes = Float64[\n 3.1 2.8;\n 4.0 7.0;\n 8.0 3.0;\n 5.0 8.0;\n 6.0 4.0;\n 8.0 5.0;\n 7.0 4.0\n]\n\nresult = johnsons(times)\n\nprintln(result.permutation)\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#Makespan","page":"Algorithms","title":"Makespan","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.makespan","category":"page"},{"location":"algorithms/#OperationsResearchModels.Johnsons.makespan","page":"Algorithms","title":"OperationsResearchModels.Johnsons.makespan","text":"makespan(times::Matrix, permutation::Vector{Int})\n\nGiven a matrix of times and a permutation of the jobs, returns the makespan of the jobs.\n\nArguments\n\ntimes::Matrix: a matrix of times\npermutation::Vector{Int}: a permutation of the jobs\n\nReturns\n\nFloat64: the makespan of the jobs\n\nExample\n\n\njulia> times = Float64[\n 3 3 5;\n 8 4 8;\n 7 2 10;\n 5 1 7;\n 2 5 6 \n]\n\njulia> result = makespan(times, [1, 4, 5, 3, 2])\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#Traveling-Salesman","page":"Algorithms","title":"Traveling Salesman","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.travelingsalesman","category":"page"},{"location":"algorithms/#OperationsResearchModels.TravelingSalesman.travelingsalesman","page":"Algorithms","title":"OperationsResearchModels.TravelingSalesman.travelingsalesman","text":"travelingsalesman(distancematrix::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::TravelinSalesmenResult\n\nGiven a matrix of distances, returns a TravelinSalesmenResult with the best route and its cost.\n\nArguments\n\ndistancematrix::Matrix: a matrix of distances\npopsize::Int: the population size. Default is 100\nngen::Int: the number of generations. Default is 1000\npcross::Float64: the crossover probability. Default is 0.8\npmutate::Float64: the mutation probability. Default is 0.01\nnelites::Int: the number of elites. Default is 1\n\nReturns\n\nTravelinSalesmenResult: a custom data type that holds the best route and its cost\n\nExample\n\npts = Float64[\n 0 0;\n 0 1;\n 0 2;\n 1 2;\n 2 2;\n 3 2;\n 4 2; \n 5 2;\n 5 1;\n 5 0;\n 4 0;\n 3 0;\n 2 0;\n 1 0;\n]\n\nn = size(pts, 1\ndistmat = zeros(n, n)\n\nfor i in 1:n\n for j in 1:n\n distmat[i, j] = sqrt(sum((pts[i, :] .- pts[j, :]).^2))\n end \nend\n\nresult = travelingsalesman(distmat, ngen = 1000, popsize = 100, pcross = 1.0, pmutate = 0.10)\n\n\n\n\n\n","category":"function"}] +[{"location":"#Operations-Research-Models","page":"-","title":"Operations Research Models","text":"","category":"section"},{"location":"","page":"-","title":"-","text":"The OperationsResearchModels package includes basic Operations Research subjects such as Transportation Problem, Assignment Problem, Minimum Spanning Tree, Shortest Path, Maximum Flow, and p-medians method for selecting location of facilities. ","category":"page"},{"location":"","page":"-","title":"-","text":"Package content is incrementaly updated.","category":"page"},{"location":"","page":"-","title":"-","text":"Please refer the Algorithms section for the detailed documentation. ","category":"page"},{"location":"algorithms/#Algorithms","page":"Algorithms","title":"Algorithms","text":"","category":"section"},{"location":"algorithms/#Assignment-Problem","page":"Algorithms","title":"Assignment Problem","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(a::AssignmentProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{AssignmentProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(a)\n\nArguments\n\na::AssignmentProblem: The problem in type of AssignmentProblem\n\nOutput\n\nAssignmentResult: The custom data type that holds problem, solution, and optimum cost. \n\nDescription\n\nSolves an assignment problem given by an object of in type AssignmentProblem.\n\nExample\n\njulia> mat = [\n 4 8 1;\n 3 1 9;\n 1 6 7;\n ];\n\njulia> problem = AssignmentProblem(mat);\n\njulia> result = solve(problem);\n\njulia> result.solution\n\n3×3 Matrix{Float64}:\n 0.0 0.0 1.0\n 0.0 1.0 0.0\n 1.0 0.0 0.0\n\njulia> result.cost\n\n3.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Transportation-Problem","page":"Algorithms","title":"Transportation Problem","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(t::TransportationProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{TransportationProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(t)\n\nArguments\n\na::TransportationProblem: The problem in type of TransportationProblem\n\nOutput\n\nTransportationResult: The custom data type that holds problem, solution, and optimum cost. \n\nDescription\n\nSolves a transportation problem given by an object of in type TransportationProblem.\n\nExample\n\njulia> t = TransportationProblem(\n [ 1 1 1 1; \n 2 2 2 2; \n 3 3 3 3], \n [100, 100, 100, 100], # Demands \n [100, 100, 100]) # Supplies \nTransportation Problem:\nCosts: [1 1 1 1; 2 2 2 2; 3 3 3 3]\nDemand: [100, 100, 100, 100]\nSupply: [100, 100, 100]\n\njulia> isbalanced(t)\nfalse\n\njulia> result = solve(t)\nTransportation Results:\nMain problem:\nTransportation Problem:\nCosts: [1 1 1 1; 2 2 2 2; 3 3 3 3]\nDemand: [100, 100, 100, 100]\nSupply: [100, 100, 100]\n\nBalanced problem:\nTransportation Problem:\nCosts: [1 1 1 1; 2 2 2 2; 3 3 3 3; 0 0 0 0]\nDemand: [100, 100, 100, 100]\nSupply: [100, 100, 100, 100]\n\nCost:\n600.0\nSolution:\n[-0.0 -0.0 -0.0 100.0; 100.0 -0.0 -0.0 -0.0; -0.0 -0.0 100.0 -0.0; -0.0 100.0 -0.0 -0.0]\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Shortest-Path","page":"Algorithms","title":"Shortest Path","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(t::ShortestPathProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{ShortestPathProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem)\n\nDescription\n\nSolves a shortest path problem given by an object of in type ShortestPathProblem.\n\nArguments\n\nproblem::ShortestPathProblem: The problem in type of ShortestPathProblem\n\nOutput\n\nShortestPathResult: The custom data type that holds path and cost.\n\nExample\n\njulia> conns = [\n Connection(1, 2, 3),\n Connection(1, 3, 2),\n Connection(1, 4, 4),\n Connection(2, 5, 3),\n Connection(3, 5, 1),\n Connection(3, 6, 1),\n Connection(4, 6, 2),\n Connection(5, 7, 6),\n Connection(6, 7, 5),\n ];\n\njulia> solve(ShortestPathProblem(conns));\n\njulia> result.path\n3-element Vector{Connection}:\n Connection(1, 3, 2, \"x13\")\n Connection(3, 6, 1, \"x36\")\n Connection(6, 7, 5, \"x67\")\n\njulia> result.cost\n8.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Maximum-Flow","page":"Algorithms","title":"Maximum Flow","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(t::MaximumFlowProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{MaximumFlowProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem)\n\nArguments\n\nproblem::MaximumFlowProblem: The problem in type of MaximumFlowProblem\n\nOutput\n\nMaximumFlowResult: The custom data type that holds path and flow.\n\nExample\n\njulia> conns = [\n Connection(1, 2, 3),\n Connection(1, 3, 2),\n Connection(1, 4, 4),\n Connection(2, 5, 3),\n Connection(3, 5, 1),\n Connection(3, 6, 1),\n Connection(4, 6, 2),\n Connection(5, 7, 6),\n Connection(6, 7, 5),\n ];\njulia> problem = MaximumFlowProblem(conns)\njulia> result = solve(problem);\n\njulia> result.path\n9-element Vector{Connection}:\n Connection(1, 2, 3.0, \"x12\")\n Connection(1, 3, 2.0, \"x13\")\n Connection(1, 4, 2.0, \"x14\")\n Connection(2, 5, 3.0, \"x25\")\n Connection(3, 5, 1.0, \"x35\")\n Connection(3, 6, 1.0, \"x36\")\n Connection(4, 6, 2.0, \"x46\")\n Connection(5, 7, 4.0, \"x57\")\n Connection(6, 7, 3.0, \"x67\")\n\njulia> result.flow\n7.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Minimum-Spanning-Tree","page":"Algorithms","title":"Minimum Spanning Tree","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(p::MstProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{MstProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem::MstProblem)\n\nArguments\n\nproblem::MstProblem: The problem in type of MstProblem\n\nDescription\n\nObtains the minimum spanning tree. \n\nOutput\n\n::MstResult: A MstResult object that holds the results. \n\nExamples\n\njulia> conns = Connection[\n Connection(1, 2, 10),\n Connection(2, 3, 10),\n Connection(3, 4, 10),\n Connection(1, 4, 10)\n ]\n\n4-element Vector{Connection}:\n Connection(1, 2, 10, \"x12\")\n Connection(2, 3, 10, \"x23\")\n Connection(3, 4, 10, \"x34\")\n Connection(1, 4, 10, \"x14\")\n\n julia> result = solve(MstProblem(conns))\n MstResult(Connection[Connection(3, 4, 10, \"x34\"), Connection(1, 4, 10, \"x14\"), Connection(2, 3, 10, \"x23\")], 30.0)\n \n julia> result.distance\n 30.0\n \n julia> result.connections\n 3-element Vector{Connection}:\n Connection(3, 4, 10, \"x34\")\n Connection(1, 4, 10, \"x14\")\n Connection(2, 3, 10, \"x23\")\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#pmedian","page":"Algorithms","title":"pmedian","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.pmedian","category":"page"},{"location":"algorithms/#OperationsResearchModels.PMedian.pmedian","page":"Algorithms","title":"OperationsResearchModels.PMedian.pmedian","text":"pmedian(data, ncenters)\n\nArguments\n\ndata::Matrix: Coordinates of locations \nncenters::Int: Number of centers \n\nDescription\n\nThe function calculates Euclidean distances between all possible rows of the matrix data. ncenters locations are then selected that minimizes the total distances to the nearest rows. \n\nOutput\n\nPMedianResult: PMedianResult object. \n\nExample\n\njulia> data1 = rand(10, 2);\n\njulia> data2 = rand(10, 2) .+ 50;\n\njulia> data3 = rand(10, 2) .+ 100;\n\njulia> data = vcat(data1, data2, data3);\n\njulia> result = pmedian(data, 3);\n\njulia> result.centers\n3-element Vector{Int64}:\n 1\n 16\n 21\n\n julia> result.objective\n 11.531012240599605\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#pmedian-with-distances","page":"Algorithms","title":"pmedian with distances","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.pmedian_with_distances","category":"page"},{"location":"algorithms/#OperationsResearchModels.PMedian.pmedian_with_distances","page":"Algorithms","title":"OperationsResearchModels.PMedian.pmedian_with_distances","text":"pmedian_with_distances(distancematrix, ncenters)\n\nArguments\n\ndistancematrix::Matrix: n x n matrix of distances\nncenters::Int: Number of centers \n\nDescription\n\nncenters locations are selected that minimizes the total distances to the nearest rows. \n\nOutput\n\nPMedianResult: PMedianResult object. \n\n\n\n\n\n","category":"function"},{"location":"algorithms/#CPM-(Critical-Path-Method)","page":"Algorithms","title":"CPM (Critical Path Method)","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(problem::CpmProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{CpmProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem)\n\nArguments\n\nproblem::CpmProblem: The problem in type of CpmProblem.\n\nOutput\n\n::CpmResult: The object holds the results \n\nDescription\n\nCalculates CPM (Critical Path Method) and reports the critical path for a given set of activities. \n\nExample\n\njulia> A = CpmActivity(\"A\", 2);\njulia> B = CpmActivity(\"B\", 3);\njulia> C = CpmActivity(\"C\", 2, [A]);\njulia> D = CpmActivity(\"D\", 3, [B]);\njulia> E = CpmActivity(\"E\", 2, [B]);\njulia> F = CpmActivity(\"F\", 3, [C, D]);\njulia> G = CpmActivity(\"G\", 7, [E]);\njulia> H = CpmActivity(\"H\", 5, [E]);\njulia> I = CpmActivity(\"I\", 6, [G, F]);\njulia> J = CpmActivity(\"J\", 2, [C, D]);\n\njulia> activities = [A, B, C, D, E, F, G, H, I, J];\n\njulia> problem = CpmProblem(activities);\n\njulia> result = solve(problem);\n\njulia> result.pathstr\n4-element Vector{String}:\n \"B\"\n \"E\"\n \"G\"\n \"I\"\n\n julia> result.path == [B, E, G, I]\ntrue\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#CPM-Activity","page":"Algorithms","title":"CPM Activity","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.CpmActivity","category":"page"},{"location":"algorithms/#OperationsResearchModels.CPM.CpmActivity","page":"Algorithms","title":"OperationsResearchModels.CPM.CpmActivity","text":"CpmActivity(name::String, time::Float64, dependencies)\n\nDescription\n\nThe object that represents an activity in CPM (Critical Path Method).\n\nArguments\n\nname::String: The name of the activity.\ntime::Float64: The time of the activity.\ndependencies: The dependencies of the activity in type of Vector{CpmActivity}.\n\nExample\n\njulia> A = CpmActivity(\"A\", 2, []);\n\njulia> B = CpmActivity(\"B\", 3, []);\n\njulia> C = CpmActivity(\"C\", 2, [A, B]);\n\n\n\n\n\n\n","category":"type"},{"location":"algorithms/#PERT-(Project-Evalutation-and-Review-Technique)","page":"Algorithms","title":"PERT (Project Evalutation and Review Technique)","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(problem::PertProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{PertProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem::PertProblem)::PertResult\n\nArguments\n\nproblem::PertProblem: The problem in type of PertProblem.\n\nExample\n\njulia> A = PertActivity(\"A\", 1, 2, 3)\nPertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[])\n\njulia> B = PertActivity(\"B\", 3, 3, 3)\nPertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])\n\njulia> C = PertActivity(\"C\", 5, 5, 5, [A, B])\nPertActivity(\"C\", 5.0, 5.0, 5.0, PertActivity[PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[]), PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])])\n\njulia> activities = [A, B, C]\n3-element Vector{PertActivity}:\n PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[])\n PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])\n PertActivity(\"C\", 5.0, 5.0, 5.0, PertActivity[PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[]), PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])])\n\njulia> problem = PertProblem(activities);\n\njulia> result = pert(activities)\nPertResult(PertActivity[PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[]), PertActivity(\"C\", 5.0, 5.0, 5.0, PertActivity[PertActivity(\"A\", 1.0, 2.0, 3.0, PertActivity[]), PertActivity(\"B\", 3.0, 3.0, 3.0, PertActivity[])])], 8.0, 0.0)\n\njulia> result.mean\n8.0\n\njulia> result.stddev\n0.0\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#PERT-Activity","page":"Algorithms","title":"PERT Activity","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.PertActivity","category":"page"},{"location":"algorithms/#OperationsResearchModels.CPM.PertActivity","page":"Algorithms","title":"OperationsResearchModels.CPM.PertActivity","text":"PertActivity(name::String, o::Float64, m::Float64, p::Float64)::PertActivity\n\nDescription\n\nThe object that represents an activity in PERT (Program Evaluation and Review Technique).\n\nArguments\n\nname::String: The name of the activity.\no::Float64: The optimistic time of the activity.\nm::Float64: The most likely time of the activity.\np::Float64: The pessimistic time of the activity.\ndependencies: The dependencies of the activity in type of Vector{PertActivity}.\n\nExample\n\njulia> A = PertActivity(\"A\", 1, 2, 3);\njulia> B = PertActivity(\"B\", 3, 3, 4);\njulia> C = PertActivity(\"C\", 5, 6, 7, [A, B]);\n\n\n\n\n\n","category":"type"},{"location":"algorithms/#Knapsack","page":"Algorithms","title":"Knapsack","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.solve(p::KnapsackProblem)","category":"page"},{"location":"algorithms/#OperationsResearchModels.solve-Tuple{KnapsackProblem}","page":"Algorithms","title":"OperationsResearchModels.solve","text":"solve(problem::KnapsackProblem)::KnapsackResult\n\nDescription\n\nSolves the knapsack problem.\n\nArguments\n\nproblem::KnapsackProblem: The problem in type of KnapsackProblem.\n\nOutput\n\nKnapsackResult: The custom data type that holds selected items, model, and objective value.\n\nExample\n\njulia> values = [10, 20, 30, 40, 50];\njulia> weights = [1, 2, 3, 4, 5];\njulia> capacity = 10;\njulia> solve(KnapsackProblem(values, weights, capacity));\n\n\n\n\n\n","category":"method"},{"location":"algorithms/#Johnson's-Rule","page":"Algorithms","title":"Johnson's Rule","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.johnsons","category":"page"},{"location":"algorithms/#OperationsResearchModels.Johnsons.johnsons","page":"Algorithms","title":"OperationsResearchModels.Johnsons.johnsons","text":"johnsons(times::Matrix)\n\nGiven a matrix of times, returns a JohnsonResult with the permutation of the jobs. If number of machines is 2, it uses the Johnson's algorithm for 2 machines. If number of machines is greater than 2, it uses the Johnson's algorithm by transforming the problem into a 2-machine problem. In order to reduce the original problem to a 2-machine problem, the algorithm checks if the minimum time of the first machine is greater or equal than the maximum time of the other machines and/or if the minimum time of the last machine is greater or equal than the maximum time of the other machines.\n\nFor example if we have 4 machines, namely, A, B, C, and D at least one of the following conditions must be satisfied:\n\nmin(A) >= max(B, C)\nmin(D) >= max(B, C)\n\nThe function throws a JohnsonException if the problem cannot be reduced to a 2-machine problem.\n\nArguments\n\ntimes::Matrix: a matrix of times\n\nReturns\n\nJohnsonResult: a custom data type that holds the permutation of the jobs\n\nExample\n\ntimes = Float64[\n 3.1 2.8;\n 4.0 7.0;\n 8.0 3.0;\n 5.0 8.0;\n 6.0 4.0;\n 8.0 5.0;\n 7.0 4.0\n]\n\nresult = johnsons(times)\n\nprintln(result.permutation)\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#Genetic-Algorithm-for-the-problems-that-cannot-be-solved-with-using-Johnson's-Rule","page":"Algorithms","title":"Genetic Algorithm for the problems that cannot be solved with using Johnson's Rule","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.johnsons_ga","category":"page"},{"location":"algorithms/#OperationsResearchModels.Johnsons.johnsons_ga","page":"Algorithms","title":"OperationsResearchModels.Johnsons.johnsons_ga","text":"johnsons_ga(times::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::JohnsonResult\n\nGiven a matrix of times, returns a JohnsonResult with the permutation of the jobs. The function uses a genetic algorithm to find the best permutation of the jobs. The genetic algorithm is implemented in the RandomKeyGA module.\n\nArguments\n\ntimes::Matrix: a matrix of times\npopsize::Int: the population size. Default is 100\nngen::Int: the number of generations. Default is 1000\npcross::Float64: the crossover probability. Default is 0.8\npmutate::Float64: the mutation probability. Default is 0.01\nnelites::Int: the number of elites. Default is 1\n\nReturns\n\nJohnsonResult: a custom data type that holds the permutation of the jobs\n\nExample\n\ntimes = Float64[\n 3.1 2.8;\n 4.0 7.0;\n 8.0 3.0;\n 5.0 8.0;\n 6.0 4.0;\n 8.0 5.0;\n 7.0 4.0\n]\n\nresult = johnsons(times)\n\nprintln(result.permutation)\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#Makespan","page":"Algorithms","title":"Makespan","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.makespan","category":"page"},{"location":"algorithms/#OperationsResearchModels.Johnsons.makespan","page":"Algorithms","title":"OperationsResearchModels.Johnsons.makespan","text":"makespan(times::Matrix, permutation::Vector{Int})\n\nGiven a matrix of times and a permutation of the jobs, returns the makespan of the jobs.\n\nArguments\n\ntimes::Matrix: a matrix of times\npermutation::Vector{Int}: a permutation of the jobs\n\nReturns\n\nFloat64: the makespan of the jobs\n\nExample\n\n\njulia> times = Float64[\n 3 3 5;\n 8 4 8;\n 7 2 10;\n 5 1 7;\n 2 5 6 \n]\n\njulia> result = makespan(times, [1, 4, 5, 3, 2])\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#Traveling-Salesman","page":"Algorithms","title":"Traveling Salesman","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.travelingsalesman","category":"page"},{"location":"algorithms/#OperationsResearchModels.TravelingSalesman.travelingsalesman","page":"Algorithms","title":"OperationsResearchModels.TravelingSalesman.travelingsalesman","text":"travelingsalesman(distancematrix::Matrix; popsize = 100, ngen = 1000, pcross = 0.8, pmutate = 0.01, nelites = 1)::TravelinSalesmenResult\n\nGiven a matrix of distances, returns a TravelinSalesmenResult with the best route and its cost.\n\nArguments\n\ndistancematrix::Matrix: a matrix of distances\npopsize::Int: the population size. Default is 100\nngen::Int: the number of generations. Default is 1000\npcross::Float64: the crossover probability. Default is 0.8\npmutate::Float64: the mutation probability. Default is 0.01\nnelites::Int: the number of elites. Default is 1\n\nReturns\n\nTravelinSalesmenResult: a custom data type that holds the best route and its cost\n\nExample\n\npts = Float64[\n 0 0;\n 0 1;\n 0 2;\n 1 2;\n 2 2;\n 3 2;\n 4 2; \n 5 2;\n 5 1;\n 5 0;\n 4 0;\n 3 0;\n 2 0;\n 1 0;\n]\n\nn = size(pts, 1\ndistmat = zeros(n, n)\n\nfor i in 1:n\n for j in 1:n\n distmat[i, j] = sqrt(sum((pts[i, :] .- pts[j, :]).^2))\n end \nend\n\nresult = travelingsalesman(distmat, ngen = 1000, popsize = 100, pcross = 1.0, pmutate = 0.10)\n\n\n\n\n\n","category":"function"},{"location":"algorithms/#Simplex","page":"Algorithms","title":"Simplex","text":"","category":"section"},{"location":"algorithms/","page":"Algorithms","title":"Algorithms","text":"OperationsResearchModels.createsimplexproblem","category":"page"},{"location":"algorithms/#OperationsResearchModels.Simplex.createsimplexproblem","page":"Algorithms","title":"OperationsResearchModels.Simplex.createsimplexproblem","text":"createsimplexproblem(obj::Vector, amat::Matrix, rhs::Vector, dir::Vector, opttype::OptimizationType)::SimplexProblem\n\nDescription:\n\nThis function creates a SimplexProblem object from the given parameters.\n\nArguments:\n\nobj::Vector: The objective function coefficients.\namat::Matrix: The LHS of the constraints.\nrhs::Vector: The RHS of the constraints.\ndir::Vector: The directions of the constraints. Can be a vector of LE (<=), GE (>=), or EQ (==).\nopttype::OptimizationType: The type of the optimization. Can be Maximize or Minimize.\n\nReturns:\n\nA SimplexProblem object.\n\nExample:\n\nSuppose the linear programming problem is as follows:\n\nMaximize: 1.0x1 + 2.0x2 + 3.0x3\nSubject to:\n1.0x1 + 2.0x2 + 3.0x3 <= 10.0\n3.0x1 + 1.0x2 + 5.0x3 <= 15.0\nx1, x2 >= 0\n\nThe following code creates a SimplexProblem object for the above problem:\n\nusing OperationsResearch.Simplex\n\nobj = Float64[1.0, 2.0, 3.0]\namat = Float64[1.0 2.0 3.0; 3.0 1.0 5.0]\nrhs = Float64[10.0, 15.0]\ndir = [LE, LE]\nopttype = Maximize\n\ns = createsimplexproblem(obj, amat, rhs, dir, opttype)\niters = simplexiterations(s)\n\n\n\n\n\n","category":"function"}] }