Question

A) Formulate but do not solve the following exercise as a linear programming problem.

Madison Finance has a total of $18 million earmarked for homeowner loans and auto loans, where x is homeowner loans in millions of dollars and y is auto loans in millions of dollars. On the average, homeowner loans have a 10% annual rate of return, whereas auto loans yield a 12% annual rate of return. Management has also stipulated that the total amount of homeowner loans should be greater than or equal to 4 times the total amount of automobile loans. Determine the total amount of loans of each type Madison should extend to each category to maximize its returns P in millions of dollars.

Maximize		P	=		______ subject to the constraints
total loans			=	______
distribution of loans			=	_______
				x ≥ 0
				y ≥ 0

B)

Formulate but do not solve the following exercise as a linear programming problem.

A nutritionist at the Medical Center has been asked to prepare a special diet for certain patients. She has decided that the meals are to be prepared from Foods A and B and that the meals should contain a minimum of 420 mg of calcium, 15 mg of iron, and 45 mg of vitamin C. Each ounce of Food A contains 30 mg of calcium, 2 mg of iron, 1 mg of vitamin C, and 2 mg of cholesterol. Each ounce of Food B contains 25 mg of calcium, 0.5 mg of iron, 5 mg of vitamin C, and 5 mg of cholesterol. How many ounces of each type of food should be used in a meal so that the cholesterol content C (in mg) is minimized and the minimum requirements of calcium, iron, and vitamin C are met?

Minimize		C	=		______ subject to the constraints
calcium			=	_________
iron			=	________
vitamin C			=	__________
				x ≥ 0
				y ≥ 0

C)Formulate but do not solve the following exercise as a linear programming problem.

A financier plans to invest up to $400,000 in two projects. Project A yields a return of 11% on the investment of x dollars, whereas Project B yields a return of 13% on the investment of y dollars. Because the investment in Project B is riskier than the investment in Project A, the financier has decided that the investment in Project B should not exceed 35% of the total investment. How much should she invest in each project to maximize the return on her investment P in dollars?

Maximize		P	=		______ subject to the constraints
amount available for investment			=	_________
allocation of funds			=	__________
				x ≥ 0
				y ≥ 0

Answer 1

a.

Let's consider

x = Loan amount disbursed as home loans (in Millions)

y = Loan amount disbursed as auto loans (in Millions)

Objective Function:

Total return from loaning activity = 10%*x + y*12%

Total return from loaning activity = 0.1*x + 0.12*y

Maximize P= 0.1*x + 0.12*y ............ as Madison Finance wants to maximize return on loaning activity

Constraints:

x + y =18 .............Constraint on the total loan amount to be disbursed

x>=4*y ............. Contraint on the distribution of loan as homeowner loans should be greater than or equal to 4 times the total amount of automobile loans

x,y>=0 .......Non-negativity constraint as selected variables cannot be negative

b.

Let's consider

x = Ounces of Food A

y = Ounces of Food B

Objective Function:

Total cholesterol content = 2*x + 5*y as each ounce of A has 2 mg of cholesterol and each ounce of B has 5 mg of cholesterol

Minimize C = 2*x + 5*y ............as nutritionist wants to minimize cholesterol content

Constraints:

30*x + 25*y >= 420 ..........Constraint on minimum calcium requirement

2*x + 0.5*y >= 15 ..........Constraint on minimum iron requirement

1*x + 5*y >= 45 ..........Constraint on minimum vitamin C requirement

x,y >=0 .......Non-negativity constraint as selected variables cannot be negative

c.

Let's consider

x = Money invested in project A

y = Money invested in project B

Objective Function:

Total return from investment activity = 11%*x + y*13%

Total return from investment activity =0.11*x + 0.13*y

Maximize P= 0.11*x + 0.13*y ............ as financier wants to maximize return on investment activity

Constraints:

x+y = 400,000 ..........Constraint on the amount to be invested in projects A and B

y<= 35%*(x+y)

y <= 0.35*x + 0.35*y

0.65*y<=0.35*x .............Constraint on amount to be invested in project B as it is riskier than project A

x,y>=0 .......Non-negativity constraint as selected variable cannot be negative