1 Soal 1

Find the maximum solution

\[max \space Z=4x+3y\]

Suppose that the objective function is subject to the following constraints: \[\begin{align*} x≥0 \\ y≥2 \\ 2y≤25-x \\ 4y≤2x-8 \\ y≤2x-5 \\ \end{align*}\]

library(lpSolve)

max_z <- c(4,3)
LHS <- matrix(c(1,2,
                -2,4,
                -2,1,
                0,1)
              ,ncol=2, byrow=TRUE)
arah_kendala <- c("<=","<=","<=",">=")
RHS <- c(25,-8,-5,2)

# Penyelesaian
model <- lp ("max",
            max_z,
            LHS,
            arah_kendala,
            RHS,
            compute.sens = TRUE)

print(model)

## Success: the objective function is 90

model$solution

## [1] 21  2

provide value. There is one robot, 2 engineers and one detailer in the factory. The detailer has some holiday off, so only has 21 days available.

The 2 cars need different time with each resource:

2 Soal 2

Let say you are working as consultant for a boutique car manufacturer, producing luxury cars. They run on one-month (30 days) cycles, we have one cycle to show we can

Resource time	Car A	Car B
Robot	3 days	4 days
Engineer	5 days	6 days
Detailer	1.5 days	3 days

Car A provides $30,000 profit, whilst Car B offers $45,000 profit. At the moment, they produce 4 of each car per month, for $300,000 profit. Not bad at all, but we think we can do better for them.

library(lpSolve)

max_z <- c(30000,45000)
LHS <- matrix(c(3,4,
                5,6,
                1.5,3)
              ,ncol=2, byrow=TRUE)
arah_kendala <- c("<=","<=","<=")
RHS <- c(30,60,21)

# Penyelesaian
model <- lp ("max",
            max_z,
            LHS,
            arah_kendala,
            RHS,
            compute.sens = TRUE)

print(model)

## Success: the objective function is 330000

model$solution

## [1] 2 6

library(gMOIP)
p1 <- plotPolytope(
   LHS,
   RHS,
   max_z,
   type = rep("c", ncol(LHS)),
   crit = "max",
   faces = rep("c", ncol(LHS)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
) + ggplot2::ggtitle("Feasible region only")
p1

3 Soal 3

Let say you would like to make some sausages and you have the following ingredients available:

Ingredient	Cost ($/kg)	Availability (kg)
Chicken	4.32	30
Wheat	2.46	20
Starch	1.86	17

Assume that you will make 2 types of sausage:

Economy (>40% Chicken)
Premium (>60% Chicken)
One sausage is 50 grams (0.05 kg)

According to government regulations of Indonesia:

The most starch you can use in your sausages is 25%.
You have a contract with a butcher, and have already purchased 23 kg Chicken, that must go in your sausages.
You have a demand for 350 economy sausages and 500 premium sausages.

So, please figure out how to optimize the cost effectively to blend your sausages.

4 Soal 4

Please visualize a contour plot of the following function:

\[f(x,y)=x^2+y^2+3xy\]

4.1 How coordinate descent works

f <- function(x, y) {
        x^2 + y^2 + 3* x * y
}
n <- 30
xpts <- seq(-1.5, 1.5, len = n)
ypts <- seq(-1.5, 1.5, len = n)
gr <- expand.grid(x = xpts, y = ypts)
feval <- with(gr, matrix(f(x, y), nrow = n, ncol = n))
par(mar = c(5, 4, 1, 1))
contour(xpts, ypts, feval, nlevels = 20, xlab = "x", ylab = "y")
points(-1, -1, pch = 19, cex = 2)
abline(h = -1)

KOMPUTASI

MATERI 6.1

Brigita Tiara Elgityana Melantika (20204920001)

November 01, 2022

1 Soal 1

2 Soal 2

3 Soal 3

4 Soal 4

4.1 How coordinate descent works