The document discusses using linear programming and the simplex method in R to solve transportation and assignment problems. It presents a transportation problem with 5 origins and 5 destinations and shows the code to solve it in R, obtaining an optimal solution. It also presents an assignment problem with 8 workers and 8 work hours categories and the R code to solve it, minimizing costs.

1. Pesquisa Operacional – Programação Linear no R-project
O método simplex é utilizado na resolução de problemas de
transporte onde se deseja minimizar o custo de transporte, aplicamos
a técnica na minimização do custo de um problema de transporte com
Graziela oito destinos. O método foi desenvolvido no2012
oito origens e
M. Alves - PO - UFS software
estatístico R- Project, onde é possível aplicar o método simplex,
carregando inicialmente o pacote library(boot).

2. Pesquisa Operacional – Programação Linear no R-project
O problema de transporte é um tipo especial de
problema de programação linear que trata do envio de
produtos da origem para o destino. O problema de
Graziela M. Alves - PO e- também 2012
designação designa atividades, UFS é um
problema onde aplica-se programação linear, no
presente trabalho a origem são trabalhadores e o
destino a carga horária de trabalho.

3. Pesquisa Operacional – Programação Linear no R-project
No R- Project, carregamos o pacote
library(boot), escrevemos a função objetivo,
Graziela M. as restrições e -em seguida
escrevemos Alves - PO UFS 2012
usamos o algoritmo que realiza o cálculo de
minimização de custos através do SIMPLEX.

4. Pesquisa Operacional – Programação Linear no R-project
PROBLEMA DE TRANSPORTE
Uma empresa é contratada pelo MDS, para distribuir
laranjas em municípios do alto sertão sergipano. Os
Graziela M. Alves - laranja.-Deseja-se 2012
municípios de origem produzem PO UFS saber
a melhor forma de planejar a distribuição de laranja para
os municípios do alto sertão, de modo que o custo seja
mínimo. Cada caminhão tipo baú tem capacidade máxima
de 41.000 Kg de laranja e a demanda de cada município é
de 40.000 Kg de laranja.

5. Tabela c/ custos do transporte
Graziela M. Alves - PO - UFS 2012

6. Solução no R
> #Implementando no R o problema de transporte de 5 origens e 5 destinos
> library (boot) #Pacote com o comando SIMPLEX no R
> z=c(60,40,70,50,30,30,30,60,70,50,50,60,80,70,40,40,50,60,70,60,30,40,70,60,50)
> #Abaixo as restrições, considerando que o sistema é equilibrado
> res1=c(1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res2=c(0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res3=c(0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0)
> res4=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0)
> res5=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1)
> simplex(a=z,A3=rbind(res1,res2,res3,res4,res5),b3=c(40000),maxi=FALSE)
#as restrições são iguais, já que o sistema é equilibrado, sendo assim não repete-se 40000, 5 vezes.
> library(boot)
> z=c(60,40,70,50,30,30,30,60,70,50,50,60,80,70,40,40,50,60,70,60,30,40,70,60,50)
> length(z)
Graziela M. Alves - PO - UFS 2012
[1] 25
> res1=c(1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res2=c(0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res3=c(0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0)
> res4=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0)
> res5=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1)
> simplex(a=z,A3=rbind(res1,res2,res3,res4,res5),b3=c(40000),maxi=FALSE)
Linear Programming Results
Call : simplex(a = z, A3 = rbind(res1, res2, res3, res4, res5), b3 = c(40000),
maxi = FALSE)
Minimization Problem with Objective Function Coefficients
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25

7. Pesquisa Operacional – Programação Linear no R-project
60 40 70 50 30 30 30 60 70 50 50 60 80 70 40 40 50 60 70 60 30 40 70 60 50
Optimal solution has the following values
Graziela M. Alves - PO - UFS 2012
x1
x24 x25
x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23
0 0 0 0 40000 40000 0 0 0 0 0 0 0 0 40000 40000 0 0 0 0 40000 0 0 0
0
The optimal value of the objective function is 6800000.

8. Pesquisa Operacional – Programação Linear no R-project
PROBLEMA DE DESIGNAÇÃO
Uma empresa em convesão coletiva de trabalho decidiu
pagar seus trabalhadores por horas de trabalho, foi pactuado
que as horas-extras também seguiriam a tabela de hora de
Graziela M. por oito categorias em acordo coletivo de
trabalho, decidida
Alves - PO - UFS 2012
trabalho. Porém notando aumento dos gastos com
pagamento de funcionários a empresa deseja determinar uma
carga horária fixa para cada trabalhador, de modo a minimizar
os gastos com pagamento de salário de funcionários.

9. Pesquisa Operacional – Programação Linear no R-project
Graziela M. Alves - PO - UFS 2012

10. Problema de designação com 8 origens e 8 destinos. Resolução no R
> library(boot)
>
z=c(40,42,44,46,40,41,38,46,46,41,40,42,45,43,45,43,43,45,42,41,30,44,38,30,44,30,41,45,38,42,44,40,42,40,45,38,46,38,39,41,41,46,4
6,44,43,40,46,45,45,43,48,39,44,42,30,39,30,44,41,46,46,30,44,42)
>res1=c(1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res2=c(0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res3=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res4=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
Graziela M. Alves - PO - UFS 2012
> res5=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res6=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
> res7=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0)
> res8=c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1)
> simplex(a=z,A3=rbind(res1,res2,res3,res4,res5,res6,res7,res8),b3=c(1,1,1,1,1,1,1,1),maxi=FALSE)

11. Linear Programming Results
Call : simplex(a = z, A3 = rbind(res1, res2, res3, res4, res5, res6,
res7, res8), b3 = c(1, 1, 1, 1, 1, 1, 1, 1), maxi = FALSE)
Graziela M. Alves - PO - UFS 2012
Minimization Problem with Objective Function Coefficients
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27
x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 x41
40 42 44 46 40 41 38 46 46 41 40 42 45 43 45 43 43 45 42 41 30 44 38 30 44 30 41 45 38
42 44 40 42 40 45 38 46 38 39 41 41
x42 x43 x44 x45 x46 x47 x48 x49 x50 x51 x52 x53 x54 x55 x56 x57 x58 x59 x60 x61 x62 x63 x64
46 46 44 43 40 46 45 45 43 48 39 44 42 30 39 30 44 41 46 46 30 44 42

12. Optimal solution has the following values
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22
x23 x24 x25 x26 x27 x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 x41
0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
Graziela M. Alves - PO - UFS 2012
0 0 0 0 0 0 0 1 0 0 0 0 0
x42 x43 x44 x45 x46 x47 x48 x49 x50 x51 x52 x53 x54 x55 x56 x57 x58 x59 x60 x61 x62
x63 x64
0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0
The optimal value of the objective function is 276.

13. Pesquisa Operacional – Programação Linear no R-project
REFERÊNCIAS BIBLIOGRÁFICAS
[1] Ajuda do software R, acesso em 31 de outubro de 2012 às 20 horas
< http://127.0.0.1:25218/library/boot/html/simplex.html >
Graziela M. Alves - PO - UFS 2012
[2] AGOSTI, Cristiano. Apostila de Pesquisa Operacional. Universidade do
Oeste de Santa Catarina. Xânxere Santa Catarina, agosto de 2003.
[3] Aplicação de Programação Linear no Software Estatístico R-gui.
Acesso em 25 de outubro de 2012 às 21 horas. < http://goo.gl/e13z6 >
[4] PRADO, Santos Darci. PERT/ CPM volume 4. INDG Tecnologia e
Serviços LTDA, Nova Lima – MG 2004.