Quantitative Models for Decision-Makers Problems Management Coursework

Quantitative Models for Decision-Makers
Please answer all the problems(5) in the attachment.

Problem 1:

Brooklyn Cabinets is a manufacturer of kitchen cabinets. The two cabinetry styles manufactured by Brooklyn are contemporary and farmhouse. Contemporary style cabinets sell for $90 and farmhouse style cabinets sell for $85. Each cabinet produced must go through carpentry, painting, and finishing processes. The following table summarizes how much time in each process must be devoted to each style of cabinet.

Carpentry costs $15 per hour, painting costs $18 per hour, and finishing costs $12 per hour, and the weekly number of hours available in the processes is 3000 in carpentry, 1500 in painting, and 1500 in finishing. Brooklyn also has a contract that requires the company to supply one of its customers with 500 contemporary cabinets and 650 farmhouse style cabinets each week.

Let

a)    Develop the objective function, assuming that Brooklyn Cabinets wants to maximize the total weekly profit.

b)   Show the mathematical expression for each of the constraints on the three processes.

c)     Show the mathematical expression for each of Brooklyn Cabinets’ contractual agreements.

Problem 2:

Solve the following linear program to find the:

Min    5X + 6Y

s.t.                  3X + Y                       ≥ 15

X +2Y                        ≥ 12

3X + 2Y        ≥ 24

                       X, Y     ≥ 0

a)    Optimal values for X and Y?

b)   What is the optimal solution value?

Problem 3:

Embassy Motorcycles (EM) manufactures two lightweight motorcycles designed for easy handling and safety. The EZ-Rider model has a new engine and a low profile that make it easy to balance. The Lady-Sport model is slightly larger, uses a more traditional engine, and is specifically designed to appeal to women riders. Embassy produces the engines for both models at its Des Moines, Iowa, plant. Each EZ-Rider engine requires 6 hours of manufacturing time and each Lady-Sport engine requires 3 hours of manufacturing time. The Des Moines plant has 2100 hours of engine manufacturing time available for the next production period. Embassy’s motorcycle frame supplier can supply as many EZ-Rider frames as needed. However, the Lady-Sport frame is more complex and the supplier can only provide up to 280 Lady-Sport frames for the next production period. Final assembly and testing requires 2 hours for each EZ-Rider model and 2.5 hours for each Lady-Sport model. A maximum of 1000 hours of assembly and testing time are available for the next production period. The company’s accounting department projects a profit contribution of $1600 for each EZ-Rider produced and $2600 for each Lady-Sport produced.

a)    Formulate a linear programming model that can be used to determine the number of units of each model that should be produced in order to maximize the total contribution to profit.

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b)   What is the optimal solution?

Problem 4:

Consider the following linear program:

Min    225X + 350Y

s.t.      1X       + 3Y   ≥ 18

2X      + 2Y   ≥ 20

6X      + 2Y   ≥ 36

           X, Y     ≥ 0

a)    What is the optimal solution?

b)   Assume that the objective function coefficient for X changes from 225 to 125. What is the new optimal solution value?

c)     Assume that the objective function coefficient for X remains 225, but the objective function coefficient for Y changes from 350 to 195. What is the new optimal solution for X, Y and value Z?

Problem 5:

Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $75 per share and the annual return for Huber Steel is $90 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $90,000 to be invested. The portfolio risk index (0.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 1250. In addition, the portfolio is limited to a maximum of 2000 shares of U.S. Oil.

a)    Formulate a linear programming model that can be used to determine the number of units of each model that should be produced in order to maximize the total annual return.

b)   What is the optimal solution, and what is the value of the total annual return?

c)     Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or why not?

Problem 6:

Grant Energy Producers supplies electrical power to residential customers for many U.S. cities. Its main power generation plants are located in New York, Virginia, and Kentucky. The following table shows Grant Energy Producers’ major residential markets, the annual demand in each market (in megawatts or MW), and the cost to supply electricity to each market from each power generation plant (prices are in $/MW).

a)    If there are no restrictions on the amount of power that can be supplied by any of the power plants, what is the optimal solution to this problem?

b)   If at most 2000 MW of power can be supplied by any one of the power plants, what is the optimal solution?

c)     What is the annual increase in power distribution cost that results from adding this constraint to the original formulation?