Question 1 A Innis Investments manages funds for a number of companies and wealt

Question

Question 1

A Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%. The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis’s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis’s client also specified that at least $300,000 be invested in the money market fund.

a. Formulate a linear programming model and determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio (5 marks)

b. Graph the feasible region.(4 marks)

c. Determine the coordinates of each extreme point.(3 marks)

d. What is the optimal solution.(3 marks)

e. Solve in excel using the template provided. (5 marks)

Question 1 B

A company produces 2 (two) different grades of gasoline – regular and premium from 3(three) components (1,2,3). The company wants to determine the optimal mix of the three components in each grade of the gasoline that will maximize profit. The maximum quantities available for each component and their cost per gallon are as follows:

Component Cost per gallon maximum gallons available per day 1 2.50 5,000 2 2.60 10,000 3 2.84 10,000 In order to ensure the appropriate blend, each grade has certain specifications as follows: Grade Component Specs Selling Price per gallon Regular At most 30% component 1 At least 40% component 2 At most 20% component 3 2.90 Premium At least 25% component 1 At most 45% component 2 At Least 30% component 3 3.00

The company wants to produce at least 10,000 gallons of each grade of gasoline. Formulate a LP model including the ratio constraints, production constraints and the objective function for this question. (20 marks)

Explanation / Answer

As given in the question, decision variables are : S representing number of units of stock and M representing number of units of money market.

a. Objective function is to Minimize risk given as 8S +3M (given risk index 8/ stock unit, risk index 3/ money unit)

Constraint in the form of total investment as 50S + 100M = 1,200,000

Constraint in the form return is as (0.10*(50S)) + (0.04*(100M)) >= 60,000

Constraint in the form of minimum invested in money market as     100M >= 300,000

b. Solution as per solver in excel is as follows:

Sensitivity report

Limits report

Annual income (as per required) is $60,000 and Annual rate of return is 5%

Dual variable for Total investment is .0567approx, therefore with $1000 the minimum risk will change by 56.7

4000 10000 Value Decision Variables S M Objective function 8 3 62000 Constraints RHS Total Investment 50 100 1200000 1200000 Annual return 5 4 60000 60000 Invested Market 0 100 1000000 300000

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