SIG$718$ Real World Analytics

Assessment Task $3,2023$

Total Marks $=100,$ Weighting $-30\%$

Your final submission should consist of:

$1.``$name$-$report.pdf$$: A pdf file $($created in any word processor$)$

containing the solutions of the questions, appropriate graphs,

tables and references. The file should be labelled with your name.

This file should consist of up to $8$ pages, $9$ pages with title page$).$

$2.``$name$-$code.R$$: Two codes combined in one R file, labelled

with yourname.R$,$ with codes for the LP models for Question $2$

and Question $3.$

$$ Your assignment will not be assessed if we cannot reproduce

your results with your R codes, R codes are missing or not

working.

$$ Your assignment will not be assessed if your report is missing

from the submission.

$$ Reference style: Harvard

$1.$ A garment factory produces shirts and pants for Kmart chain. The contract is such that

quality control is done before shipping and all products supplied to Kmart satisfying the

quality requirements would be accepted by the chain. The factory employs $20$ workers

in the cutting department, $50$ workers in the sewing department, and $14$ workers in the

packaging department. The garment factory works $8$ productive hours a day $($no idle

time during these $8$ hours$).$ There is a daily demand for at most $180$ shirts. The demand

for pants is unlimited. Each worker can participate only in one activity$-$ the activity to

which they are assigned. The table below gives the time requirements $($in minutes$)$ and

profit per unit for the two garments.

Amount $($minutes$)$ per operation

Cutting Sewing Packaging Profit per unit$($$$)$

Shirts $40402010$

Pants $20100208$

a$)$ Explain why a Linear Programming $($LP$)$ model would be suitable for this case study.

$[5$ marks$]$

b$)$ Formulate a LP model to help the factory management to determine the optimal daily

production schedule that maximises the profit while satisfying all constraints.

$[10$ marks$]$

c$)$ Use the graphical method to find the optimal solution. Show the feasible region and

the optimal solution on the graph. Annotate all lines on your graph. What is the optimal

daily profit for the factory?

$[10$ marks$]$

Note: you can use graphical solvers available online but make sure that your graph is

clear, all variables involved are clearly represented and annotated, and each line is clearly

marked and related to the corresponding equation.

d$)$ Find the range for the profit $($$$)$ per shirt $($if any$)$ that can be obtained without affecting

the optimal point of part $($c$).$

$[5$ marks$]$

$2.$ A factory makes three products called Bloom, Amber, and Leaf, from three materials

containing Cotton, Wool and Nylon. The following table provides details on the sales

price, production cost and purchase cost per ton of products and materials respectively.

Sales price Production cost Purchase price

Bloom $$60$ $$5$ Cotton $$40$

Amber $$55$ $$4$ Wool $$45$

Leaf $$60$ $$5$ Nylon $$30$

The maximal demand $($in tons$)$ for each product, the minimum cotton and wool proportion

in each product are as follows:

Demand min Cotton proportion min Wool proportion

Bloom $420050\%40\%$

Amber $320060\%40\%$

Leaf $350050\%30\%$

a$)$ Formulate an LP model for the factory that maximises the profit, while satisfying the

demand and the cotton and wool proportion constraints.

$[20$ Marks$]$

b$)$ Solve the model using R$/$R Studio. Find the optimal profit and optimal values of the

decision variables.

$[20$ Marks$]$

Hints:

$1.$ Let xij $>=0$ be a decision variable that denotes the number of tons of products j for

j in $\{1=$ Bloom, $2=$ Amber, $3=$ Leaf$\}$ to be produced from Materials i in $\{$C$=$Cotton,

W$=$Wool, N$=$Nylon$\}.$

$2.$ The proportion of a particular type of Material in a particular type of Product can be

calculated as:

e$.$g$.,$ the proportion of Cotton in product Bloom is given by:

xC$1$

xC$1+$ xW$1+$ xN$1$

$.$

$3.$ Two construction companies, Giant and Sky$,$ bid for the right to build in a field. The

possible bids are $ $10$ Million, $ $20$ Million, $ $30$ Million, $ $35$ Million and $ $40$ Million.

The winner is the company with the higher bid.

The two companies decide that in the case of a tie $($equal bids$),$ Giant is the winner and

will get the field.

Giant has ordered a survey and, based on the report from the survey, concludes that

getting the field for more than $ $35$ Million is as bad as not getting it $($assume loss$),$

except in case of a tie $($assume win$).$ Sky is not aware of this survey.

$($a$)$ State reasons why$/$how this game can be described as a two$-$players$-$zero$-$sum game

$[5$ Marks$]$

$($b$)$ Considering all possible combinations of bids, formulate the payoff matrix for the

game.

$[5$ Marks$]$

$($c$)$ Explain what is a saddle point. Verify: does the game have a saddle point?

$[5$ Marks$]$

$($d$)$ Construct a linear programming model for Company Sky in this game.

$[5$ Marks$]$

$($e$)$ Produce an appropriate code to solve the linear programming model in part $($d$).$

$[5$ Marks$]$

$($f$)$ Solve the game for Sky using the linear programming model and the code you constructed

in parts $($d$)$ and $($e$).$ Interpret your solution.