Consider the following examples of real$-$world optimisation problems. For each problem, define appropriate decision variables, and formulate the objective function and constraints mathematically. Finally, classify the optimisation problem according as belonging to one of the classes of optimization. You are not required to solve the problems.

$($a$)$ WooD$$ecor manufactures two types of wooden d$$ecor items: Floating shelves and rails. A floating shelf sells for R$500$ and is manufactured from R$200$ worth of raw materials. Each floating shelf manufactured increases WooD$$ecor$$s variable labour and overhead costs by R$250.$ A rail sells for R$450$ and is manufactured from R$220$ worth of raw materials. Each rail manufactured increases WooD$$ecor$$s variable labour and overhead costs by R$200.$ The manufacturing of wooden floating shelves and rails requires two types of skilled labour: Carpentry labour and finishing labour. A floating shelve requires $3$ hours of finishing labour and $1\mathrm{.}5$ hours of carpentry labour, while a rail requires $1\mathrm{.}5$ hours of finishing labour and $1\mathrm{.}5$ hours of carpentry labour. Each week, WooD$$ecor can obtain all the raw materials it requires, but only $150$ finishing hours and $120$ carpentry hours are available each week. Demand for rails is unlimited, but at most $30$ floating shelves are bought each week. WooD$$ecor wishes to maximise its weekly profit $($revenues less costs$).$ Formulate an optimisation problem that will inform of WooD$$ecor how to go about maximising its weekly profit.

$($b$)$ Investo is considering four investment options. Investment $1$ is expected to yield a net present value $($NPV$)$ of R$12000$; investment $2,$ an NPV of R$16500$; investment $3,$ an NPV of R$9000$; and investment $4,$ an NPV of R$6000.$ Investing in an investment requires a certain cash outflow at the present time:

Investment $1,$ R$3750$; investment $2,$ R$5250$; investment $3,$ R$3000$; and investment $4,$ R$2250.$ Investo

can either invest fully in an investment, or not at all $($fractions of investments are not possible$).$

Currently, R$10500$ is available for investment and Investo cannot invest in more than two options.

Formulate an optimisation problem whose solution will inform Investo how to maximize the NPV

obtained from the investments selected.

$($c$)$ Suppose you aim to firstly, find high$-$quality hyper$-$parameter values for a one layer perceptron, i$.$e$.$ standard feedforward neural network comprising one hidden layer. Secondly, you want to optimise the neural network architecture, specifically the hidden layer. Propose appropriate decision variables and one or more objective function to achieve this aim.