Reference no: EM132830348

1) A retail store in Surrey, BC, receives shipments of a particular product from Hong Kong and Macau. Let x y = units of product received from Honk Kong and = units of product received from Macau

a. expression for the total units of product received by the retail store in Surrey.

b. Shipments from Hong Kong cost $0.20 per unit, and shipments from Macau cost $0.30 per unit. Develop an objective function representing the total cost of shipments to Surrey.

c. Assuming the monthly demand at the retail store is 8500 units, develop a constraint that requires 8500 units to be shipped to Surrey.

d. No more than 5500 units can be shipped from Hong Kong and no more than 4200 units can be shipped from Macau in a month. Develop constraints to model this situation.

e. Of course, negative amounts cannot be shipped. Combine the objective function and constraints developed to state a mathematical model for satisfying the demand at the Surrey retail store at minimum cost.

2) Preliminary plans are underway for construction of a new stadium for a major league baseball team. City officials question the number and profitability of the luxury corporate boxes planned for the upper deck of the stadium. Corporations and selected individuals may purchase a box for $300,000. The fixed construction cost for the upper-deck area is estimated to be $4,500,000, with a variable cost of $150,000 for each box constructed.

a. What is the breakeven point for the number of luxury boxes in the new stadium?

b. Preliminary drawings for the stadium show that space is available for the construction of up to 50 luxury boxes. Promoters indicate that buyers are available and that all 50 could be sold if constructed. What is your recommendation concerning the construction of luxury boxes? What profit is anticipated?

3) Financial Analysts, Inc., is an investment firm that manages stock portfolios for a number of clients. A new client has requested that the firm handle an $800,000 portfolio. As an initial investment strategy, the client would like to restrict the portfolio to a mix of the following two stocks:

Let

x = number of shares of Oil Alaska

y = number of shares of Southwest Petroleum

Stock

Price per share

Estimated Annual return per Share

Oil Alaska

$65

$8.5

Southwest Petroleum

$45

$4.0

a. Develop the objective function, assuming that the client desires to maximize the total annual return.

b. Show the mathematical expression for each of the following three constraints:

(1) Total investment funds available are $750,000.

(2) Maximum Oil Alaska investment is $600,000.

(3) Maximum Southwest Petroleum investment is $550,000.

c. Could either of x or y be negative? Why? What mathematical expression(s) would guarantee that?

4) Develop a mathematical model to be solved for each of the following examples:

(Write down the function to be maximized/minimized and write down all the constraints/limitations associated with each case)

a. A company manufactures two products, A and B, on two machines, I and II. It has been determined that the company will realize a profit of $3/unit of product "A" and a profit of $4/unit of product "B". To manufacture a unit of product "A" requires 6 min on machine I and 5 min on machine II. To manufacture a unit of product "B" requires 9 min on machine I and 4 min on machine II. There are 5 hr of machine time available on machine I and 3 hr of machine time available on machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?

b. A farmer plans to plant two crops, A and B. The cost of cultivating crop A is $40/acre whereas that of crop B is $60/acre. The farmer has a maximum of $7400 available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. If she expects to make a profit of $150/acre on crop A and $200/acre on crop B, how many acres of each crop should she plant in order to maximize her profit?

A veterinarian has been asked to make diet for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than 8 oz and must contain at least 29 units of nutrient I and 20 units of nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: brand A and brand B. Each ounce of brand A contains 3 units of nutrient I and 4 units of nutrient II. Each ounce of brand B contains 5 units of nutrient I and 2 units of nutrient II. Brand A costs 3 cents/ounce and brand B costs 4 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost.