1 Question:

Find the maximum solution to

\(Z=4x+3y\)

Suppose that the objective function is subject to the following constraints:

\(x≥0\) \(y≥2\) \(2y≤25−x\) \(4y≤2x−8\) \(y≤2x−5\)

1.1 Answer:

## Warning: package 'lpSolve' was built under R version 3.6.2
## Success: the objective function is 55
## [1] 7 9
## [1]  1.5 -2.0
## [1] 1e+30 8e+00
## [1]  2  0 -1  0  0
## [1]  2.500e+00 -1.000e+30 -2.375e+01 -1.000e+30 -1.000e+30
## [1] 1.00e+30 1.00e+30 1.25e+01 1.00e+30 1.00e+30

Solution: The maximum z value (and thus, the optimum) that can be obtained while satisfying the given constraints is 55, where x = 7 and y = 9. The sensitivity coefficients go from 1.5 and -2.0 to 1e+30 and 8e+00. The shadow/dual prices of the constraints are 2, 0 and -1, while for the decision variables are 0 and 0, respectively. The shadow/dual prices lower limits of the constraints are 2.500e+00, -1.000e+30 and -2.375e+01, while for the decision variables are -1.000e+30 and -1.000e+30, respectively. Finally, the shadow/dual prices upper limits of the constraints are 1.00e+30, 1.00e+30 and 1.25e+01, while for the decision variables are 1.00e+30 and 1.00e+30, respectively.

2 Question:

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 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:

\(CarA=x\) \(CarB=y\) \(Robot=3x+4y≥30\) \(Engineer=5x+6y≥60\) \(Detailer=1.5x+3y≥21\)

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.

2.1 Answer:

## List of 28
##  $ direction       : int 1
##  $ x.count         : int 2
##  $ objective       : num [1:2] 30000 45000
##  $ const.count     : int 3
##  $ constraints     : num [1:4, 1:3] 3 4 1 30 5 6 1 60 1.5 3 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:4] "" "" "const.dir.num" "const.rhs"
##   .. ..$ : NULL
##  $ int.count       : int 2
##  $ int.vec         : int [1:2] 1 2
##  $ bin.count       : int 0
##  $ binary.vec      : int 0
##  $ num.bin.solns   : int 1
##  $ objval          : num 330000
##  $ solution        : num [1:2] 2 6
##  $ presolve        : int 0
##  $ compute.sens    : int 0
##  $ sens.coef.from  : num 0
##  $ sens.coef.to    : num 0
##  $ duals           : num 0
##  $ duals.from      : num 0
##  $ duals.to        : num 0
##  $ scale           : int 196
##  $ use.dense       : int 0
##  $ dense.col       : int 0
##  $ dense.val       : num 0
##  $ dense.const.nrow: int 0
##  $ dense.ctr       : num 0
##  $ use.rw          : int 0
##  $ tmp             : chr "Nobody will ever look at this"
##  $ status          : int 0
##  - attr(*, "class")= chr "lp"
## [1] 0
## x y 
## 2 6
## [1] "Total Profit: 330000"

Solution : The total production of Car A = 2 and Car B = 6 to get maximum profit 330.000

3 Question:

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

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 Question:

Please visualize a contour plot of the following function:

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

How coordinate descent works How Steepest descent works How Newton Method works How Conjugate Gradient works