Kindly complete the following assignment individually and submit your assignment as an electronic

submission on SunLearn before the due date. The final deliverable of this assignment is a report in

PDF format providing and explaining your results.

Make sure that you submit your own work and provide references where appropriate. Plagiarism will

not be tolerated. Furthermore, no artificial intelligence tools $($e$.$g$.$ LLMs$)$ may be used in the

completion of this assignment and only SunLearn submissions will be accepted.

Question $1($For Optimisation $774($postgraduate diploma$)$ AND $874($Master$$s$)$ students$)[25]$

Extru $($Pty$)$ Ltd is a firm that manufactures world class extruders that are exported to Asia, Europe and

North America. In the planning of the monthly manufacturing for the next six months, Extru $($Pty$)$ Ltd

financial analysts must take all market and company information into consideration, to come up with

the most cost$-$effective solution. This solution should govern both the number of extruders that need

to be manufactured per month, as well as machine set$-$up configurations.

The machines used in the manufacturing process can, for simplicity, be assumed to comprise two

possible configuration set$-$ups. Configuration $1$ allows for production of up to $55$ units per month at a

fixed cost of R$2100000\mathrm{.}00.$ This cost is therefore not influenced by the number of units produced

under configuration $1.$

Configuration $2$ allows for production of up to $80$ units per month at a fixed cost of R$3000000\mathrm{.}00.$

Part of the increased cost is due to additional equipment that needs to be hired. This cost is also not

influenced by the number of units produced under configuration $2.$

A change$-$over from configuration $1$ to configuration $2$ will always result in raw$-$ and semi processed

materials in the system that cannot be used anymore. Increased maintenance costs over time, as well

as a shorter cycle lifetime expectancy of the machines owned by Extru $($Pty$)$ Ltd will result. To model

for these, each time when configuration $1$ is changed to configuration $2,$ a fixed additional cost of

R$280000\mathrm{.}00$ is assumed to be incurred.

The cost of holding stock is R$25000\mathrm{.}00$ per unit per month $($based on the stock held at the end of each

month$)$ and the initial stock is $25$ units $($produced by configuration $1).$ The number of stock units at the

end of month $6$ should be at least $15.$ The estimated demand for the company's extruders in each of

the next six months is provided in the following table:

Month $123456$

Demand $605575605085$

Page $2$ of $5$

Production constraints are such that if the company produces anything in a particular month it must

produce at least $25$ units.

Each unit is sold for R$100000\mathrm{.}00.$

Formulate an integer linear programming $($ILP$)$ model that solves for the production schedule over the

following six months. It is required to know how much units should be manufactured each month, and

at what configuration setting.

Following your formulation, solve your ILP in the mathematical programming code of your choosing.

Re$-$solve your model where the cost of holding stock is increased to R$60000\mathrm{.}00.$ This is to test how

sensitive your schedule is towards a large fluctuation in holding cost.

To be submitted:

Write up and submit a concise, but sufficient report that contains your ILP model formulation. The

objective function, as well as constraint or set of constraints must be numbered and explained. The

explanation must lay out the rationale of the formulation, in order to understand what each constraint

$($together with the objective function$)$ contributes towards solving for optimality. Also explain why the

constraint will hold and provide the intended outcome.

Provide your solved results for both the base case, as well as the scenario where holding costs are

increased to R$60000\mathrm{.}00$ per unit. Comment on your solved answers