Question $1$: $(10$ Points$)$

An FMCG$,$ specialized in producing refreshments, produces two types of soda; black and orange soda. Formulate the following constraints if:

$($Clearly$,$ define variables first$)$

The daily demand of black soda exceeds that of orange soda by at least $100$ cans.

The maximum amount that can be produced of both black and orange soda is $10000$ cans.

The daily usage of sugar is at most $8$ tons and at least $5$ tons.

The amount of black soda cans should exceed that of orange soda.

The proportion of black soda to orange soda is at least $1\mathrm{.}3$

The proportion of black soda to the total production of both black and orange soda must not exceed $0\mathrm{.}7$

Question $2$: $(10$ Points$)$

A leather manufacturing company produces two types of products; shoes and bags. The sales volume for shoes represents at least $80\%$ of the total sales. However, due to logistics constraints the company can$$t sell more than $100$ units of shoes per day. Both products use the same type of leather. The usage availability of leather is $240$ Ib$.$ The usage rates are $2$ Ib per one shoe and $4$ units per one bag. The profit units for shoes is $20$$ and $50$$ for bags.

Formulate the problem

Determine the optimal number of units to be produced of each product using the graphical method

Question $3$: $(10$ Points$)$

A company produces two types detergents A and B$.$ A couple of raw materials are used in producing the detergents. The daily availability of raw material I is $150$ while that of raw material II is $145$ units. To produce one unit of detergent A$,0\mathrm{.}5$ units of raw material I and $0\mathrm{.}6$ units of raw material II is consumed. On the other hand, to produce one unit of detergent B$,0\mathrm{.}5$ units of raw material I and $0\mathrm{.}4$ units of raw material II is consumed. The unit profits of detergents A and B are $8$$ and $10$$ respectively. The daily demand for detergents A is between $30$ and $150$ units while that of B are between $40$ and $200$ units.

Formulate the problem

Find the optimal number of units to be produced of both A and B using the graphical method

Identify the status of each resource $($scarce$,$ abundant, etc..$)$

Question $4$: $(15$ Points$)$

A toy manufacturing company is planning to boost its sales by launching an advertising campaign. They are planning to launch the campaign using two mediums; radio and TV$.$ They figure out a radio ad costs $$300$ while a TV ad costs $$2000.$ The finance department states that the total budget the marketing department can consume is $$20,000.$ A marketer suggests that to ensure that each medium will have at least one radio ad and one TV ad$,$ the most that can be allocated to either medium cannot exceed $80\%$ of the total budget. Based on previous data, it is estimated that the first radio ad will reach $5000$ people, with each additional commercial reaching only $2000$ new ones. For TV$,$ the first ad will reach $4500$ people and each additional ad and each additional ad an additional $3000$ people.

Formulate the problem

Estimate how to allocate the budget $($Use the graphical method$)$

Question $5$: $(15$ Points$)$

A city is studying the feasibility of introducing a new transportation system. The aim is to minimize the number of buses that can handle the transportation needs. The executive Engineer noticed that the minimum number of buses needed varies according to the time of the day. She found out that the required number of buses could be approximated by constant values over successive $4-$hour intervals. Table$\mathrm{.}1$ summarizes her findings. Note that each bus can operate $8$ successive hours a day only. Formulate the problem to determine the number of operating buses in each shift that$$will meet the required minimum demand while minimizing the operating total number of buses.

Table $1$: Findings Table $1$ : Findings

$\begin{array}{cc}\text{T}\text{i}\text{m}\text{e}\text{i}\text{n}\text{t}\text{e}\text{r}\text{v}\text{a}\text{l}\text{s}& No.of\text{b}\text{u}\text{s}\text{e}\text{s}\\ \text{F}\text{r}\text{o}\text{m}12\text{:}00AMto4\text{:}00AM& 4\\ \text{F}\text{r}\text{o}\text{m}4\text{:}00AMto8\text{:}00AM& 8\end{array}$

$\begin{array}{cc}\text{F}\text{r}\text{o}\text{m}8\text{:}00AMto12\text{:}00PM& 10\\ \text{F}\text{r}\text{o}\text{m}12\text{:}00PMto4\text{:}00PM& 7\\ \text{F}\text{r}\text{o}\text{m}4\text{:}00PMto8\text{:}00PM& 12\\ \text{F}\text{r}\text{o}\text{m}8\text{:}00to12\text{:}00AM& 4\end{array}$