$1$ Linear Programming

Question $1$

An investor is going to invest $$100M.$ The following investments are available: Bonds, home loans, hedge funds and stock market with annual rate of returns $8\%,9\%,7\%$ and $11\%$ respectively. There are several restrictions on investments:

The amount invested in hedge funds should be at least equal to the amount invested in bonds.

The amount invested in home loans should be less than or equal to the amount invested in stock market and the amount invested in bonds.

No more than $30\%$ of the total amount invested should be on stock markets.

Formulate an LP that will maximize the annual rate of return of the investor.

Question $2$

A soft drink manufacturer produces a soda drink, which requires ingredients and labor. Two production processes are available: Process $1$ transforms $1$ unit of labor and $2$ units of ingredients into $3oz$ of drink. Process $2$ transforms $3$ units of labor and $7$ units of ingredients into $50oz$ of drink. It costs the company $$3$ to purchase a unit of labor and $$2$ to purchase a unit of ingredients. Each year, up to $20,000$ units of labor and $35,000$ units of ingredients can be purchased. In the absence of advertising, the company believes it can sell $10000oz$ of drink. To stimulate demand for the drink, the company can hire a celebrity, say Charlie Harper. Charlie is paid $$\frac{100}{?}$ hour. Each hour Charlie works for the company is estimated to increase the demand for soda drink by $2000oz.$ Each ounce of drink sells for $$3.$ Use LP to determine how the company can maximize profits.

Question $3$

Sailco Corporation must determine how many sailboats should be produced during each of the next four quarters. The demand during each of the next four quarters is as follows: $40,60,75$ and $25.$ Sailco must meet demands on time. At the beginning Sailco has an inventory of $10$ boats. During each quarter Sailco can produce up to $40$ sailboats with regular time with a cost of $$400$ per boat. Sailco can produce additional sailboats with overtime at a cost of $$450$ per boat. At the end of each quarter a holding cost of $$20$ per sailboat is incurred. Formulate the problem as a linear program that minimizes the total cost. How would you modify the model if integer values are required?

Question $4$

Cornco produces two products: PS and QT$.$ The sales price for each product and the maximum quantity of each that can be sold during each of the next three months are given in Table $8$ and $8.$ Each product must be processed through two assembly lines: $1$ and $2.$ The number of hours required by each product on each assembly line are given in Table $7.$ The number of hours available on each assembly line during each month are given in Table $8.$

Each unit of PS requires $4$ pounds of raw material while each unit of QT requires $3$ pounds. Up to $710$ units of raw material can be purchased at $$3$ per pound. At the beginning of month $1,10$ units of PS and $5$ units of QT are available. It costs $$10$ to hold a unit of either product in inventory for a month. Formulate this problem as an LP that maximizes the total profit.