Problem $2$: Freezer Truck Design Optimization Using Pareto Frontier $(4$ points$)$

You are tasked with optimizing the design of a freezer truck. The truck must balance three key factors: the

cost of building the truck, its fuel efficiency, and its cooling efficiency. To simplify, we assume that these

factors depend on just two variables: $x_1,$ the thickness of insulation $($in cm $),$ and $x_2,$ the size of the engine

$($in litres$).$ The objective functions are listed as follows:

Cost: $f_1(x_1,x_2)=3x_1+3x_2^2$

Fuel consumption: $f_2(x_1,x_2)=x_2^2-4x_1$

Cooling efficiency: $f_3(x_1,x_2)=2x_1{x}_2+2x_1$

The range of values for each variable are

$0x_{1}10$

$5x_{2}18$

Using the weighted average method ${?}^1,$ determine the Pareto frontier for the following pairs of objectives:

$($i$)$ Minimize cost, and Minimize fuel consumption

$($ii$)$ Maximize cooling efficiency and Minimize fuel consumption

$($iii$)$ Minimize cost and Maximize cooling efficiency

Clearly document your approach to finding the solution $($including any equations or code used$).$ Illustrate

the Pareto frontiers on $2$D plots.