SIT$718$ Real world Analytics

Assessment Task $3$

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

$1$

$1.$ A food factory is making a beverage for a customer from mixing two dierent existing

products A and B$.$ The compositions of A and B and prices $($$$/$L$)$ are given as follows,

Amount $($L$)$ in $/100$ L of A and B

Lime Orange Mango Cost $($$$/$L$)$

A $2645$

B $74815$

The customer requires that there must be at least $5$ Litres $($L$)$ Orange and at least $5$

Litres of Mango concentrate per $100$ Litres of the beverage respectively, but no more

than $6$ Litres of Lime concentrate per $100$ Litres of beverage. The customer needs at

least $150$ Litres of the beverage per week.

a$)$ Explain why a linear programming model would be suitable for this case study.

$[5$ marks$]$

b$)$ Formulate a Linear Programming $($LP$)$ model for the factory that minimises the total

cost of producing the beverage while satisfying all constraints.

$[5$ marks$]$

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

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

$[5$ 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$)$ What is the range for the cost $($$$)$ of A that can be changed without aecting the

optimum solution obtained above?

$[5$ marks$]$

$2.$ A factory makes three products called Spring, Autumn, and Winter, from three materials

containing Cotton, Wool and Silk. 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

Spring $$60$ $$5$ Cotton $$30$

Autumn $$55$ $$3$ Wool $$45$

Winter $$60$ $$5$ Silk $$50$

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

tion in each product is as follows:

Demand min Cotton proportion min Wool proportion

Spring $360055\%30\%$

Autumn $330045\%40\%$

Winter $400030\%50\%$

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

demand and the cotton and wool proportion constraints. There is no penalty for the

shortage.

$[20$ Marks$]$

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

decision variables.

$[20$ Marks$]$

Hints:

You may refer to Week $8\mathrm{.}7$ Example $-$ Blending Crude Oils into Gasolines. For ex$-$

ample, let xij $0$ be a decision variable that denotes the number of tons of products

j for j $2$ f$1=$ Spring; $2=$ Autumn; $3=$ Winterg to be produced from Materials

i $2$ fC$=$Cotton, W$=$Wool, S$=$Silkg.

$3.$ Consider the following parlor game to be played between two players. Each player begins

with three chips: one red, one white, and one blue. Each chip can be used only once.

To begin, each player selects one of her chips and places it on the table, concealed.

Both players then uncover the chips and determine the payo to the winning player.

In particular, if both players play the same kind of chip, it is a draw; otherwise, the

following table indicates the winner and how much she receives from the other player.

Next, each player selects one of her two remaining chips and repeats the procedure,

resulting in another payo according to the following table. Finally, each player plays

her one remaining chip, resulting in the third and nal payo.

Winning Chip Payo $($$$)$

Red beats white $250$

White beats blue $100$

Blue beats red $50$

Matching colors $0$

$($a$)$ Formulate the payo matrix for the game and identify possible saddle points.

$[10$ Marks$]$

$($b$)$ Construct a linear programming model for each player in this game.

$[10$ Marks$]$

$($c$)$ Produce an appropriate code to solve the linear programming model for this game.

$[10$ Marks$]$

$($d$)$ Solve the game for both players using the linear programming model.

$[10$ Marks$]$

$[$Hint: Each player has the same strategy set. A strategy must specify the rst chip

chosen, the second and third chips chosen. Denote the white, red and blue chips by W$,$

R and B respectively. For example, a strategy $\backslash$WRB$"$ indicates rst choosing the white

and then choosing the red, before choosing blue at the end.$]$