MAC 2234 Homework 13

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1 : Consider the following linear programming problem.

Maximize $P = \sage {p1c1}x + \sage {p1c2}y$ subject to the constraints $\begin {array}{rcc} \sage {p1d1}x + \sage {p1d2}y & \le & \sage {p1f1} \\ \sage {p1e1}x + \sage {p1e2}y & \le & \sage {p1f2} \\ x \ge 0 & & y\ge 0 \end {array}$ Write the initial simplex tableau. $\begin {array}{ccccc|c} x & y & u & v & P & \text {constant} \\ \hline \answer {\sage {p1d1}} & \answer {\sage {p1d2}} & \answer {1} & \answer {0} & \answer {0} & \answer {\sage {p1f1}} \\ \answer {\sage {p1e1}} & \answer {\sage {p1e2}} & \answer {0} & \answer {1} & \answer {0} & \answer {\sage {p1f2}} \\ \hline \answer {\sage {-p1c1}} & \answer {\sage {-p1c2}} & \answer {0} & \answer {0} & \answer {1} & \answer {0} \end {array}$

2 : Consider the following linear programming problem.

Maximize $P = \sage {p2c1}x + \sage {p2c2}y + \sage {p2c3}z$ subject to the constraints $\begin {array}{rcc} \sage {p2d1}x + \sage {p2d2}y + \sage {p2d3}z & \le & \sage {p2f1} \\ \sage {p2e1}x + \sage {p2e2}y + \sage {p2e3}z & \le & \sage {p2f2} \\ \sage {p2k1}x + \sage {p2k2}y + \sage {p2k3}z & \le & \sage {p2f3} \\ x \ge 0 \quad y\ge 0 & & z\ge 0 \end {array}$ Write the initial simplex tableau. $\begin {array}{ccccccc|c} x & y & z & w & u & v & P & \text {constant} \\ \hline \answer {\sage {p2d1}} & \answer {\sage {p2d2}} & \answer {\sage {p2d3}} & \answer {1} & \answer {0} & \answer {0} & \answer {0} & \answer {\sage {p2f1}} \\ \answer {\sage {p2e1}} & \answer {\sage {p2e2}} & \answer {\sage {p2e3}} & \answer {0} & \answer {1} & \answer {0} & \answer {0} & \answer {\sage {p2f2}} \\ \answer {\sage {p2k1}} & \answer {\sage {p2k2}} & \answer {\sage {p2k3}}& \answer {0} & \answer {0} & \answer {1} & \answer {0} & \answer {\sage {p2f3}} \\ \hline \answer {\sage {-p2c1}} & \answer {\sage {-p2c2}} & \answer {\sage {-p2c3}} & \answer {0} & \answer {0} & \answer {0} & \answer {1} & \answer {0} \\ \end {array}$

3 : Use the simplex method to solve the linear programming problem.

Maximize $P = \sage {p3c1}x + \sage {p3c2}y$ subject to the constraints $\begin {array}{rcc} x + y & \le & 80 \\ 3x + y & \le & 90 \\ x \ge 0 & & y\ge 0 \end {array}$

The maximum is $P = \answer {\sage {p3ans}}$ at $(x,y) = (\answer {\sage {p3x}}, \answer {\sage {p3y}})$ .

4 : Use the simplex method to solve the linear programming problem.

Maximize $P = \sage {p4c1}x + \sage {p4c2}y$ subject to the constraints $\begin {array}{rcc} 3x + 5y & \le & 112 \\ 4x + y & \le & 70 \\ x \ge 0 & & y\ge 0 \end {array}$

The maximum is $P = \answer {\sage {p4ans}}$ at $(x,y) = (\answer {\sage {p4x}}, \answer {\sage {p4y}})$ .

5 : Use the simplex method to solve the linear programming problem.

Maximize $P = \sage {p5c1}x - \sage {p5c2}y + \sage {p5c3}z$ subject to the constraints $\begin {array}{rcc} 3x + y - z & \le & \sage {p5f1} \\ 2x + y - z & \le & \sage {p5f2} \\ -x + y + z & \le & \sage {p5f1} \\ x\ge 0 \; y\ge 0 & z\ge 0 \end {array}$

The maximum is $P = \answer {\sage {p5ans}}$ at $(x,y,z) = (\answer {\sage {p5x}}, \answer {\sage {p5y}}, \answer {\sage {p5z}})$ .

6 : Gator Office Supplies sells two models of fax machines. The wholesale cost of the Blue model is $ 100 and the cost of the Orange model is $ 150. The retail price of the Blue is $ 130 and of the Orange is $ 190. In any given month the store sells at most 2500 machines total and because of budget concerns management has earmarked no more than $ 600,000 per month to purchase inventory. How many units of each model should the manager order each month in order to maximize monthly profit? $\begin {array}{lc} \text {Blue} & \answer {\sage {p6x}} \\ \text {Orange} & \answer {\sage {p6y}} \end {array}$ What is the maximum profit? $\$ \answer {\sage {p6ans}}$

7 : Gator Furniture manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $ 45, and the profit for each chair is $ 20. In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many table and chairs should Gator Furniture manufacture in order to maximize its profit? $\begin {array}{lc} \text {tables} & \answer {\sage {p7x}} \\ \text {chairs} & \answer {\sage {p7y}} \end {array}$ What is the maximum profit? $\$ \answer {\sage {p7ans}}$

8 : Alberta has a total of $ 200,000 to invest in three types of mutual funds: growth, balanced, and income funds. Growth funds have a rate of return of 12% per year, balanced funds have a rate of return of 10% per year, and income funds have a return of 6% per year. The growth, balanced, and income funds are assigned risk factors of 0.1, 0.06, and 0.02, respectively. Alberta has decided that at least 50% of her total portfolio is to be in income funds and at least 10% in balanced funds. She has also decided that the average risk factor for her investment should not exceed 0.05. How should Alberta allocate her funds so as to maximize the return on her investment? (Hint: the constraint for the average risk factor for the investment is given by $0.1x + 0.06y + 0.02z \le 0.05(x+y+z)$ .) $\begin {array}{lc} \text {growth} & \$ \answer {\sage {p8x}} \\ \text {balanced} & \$ \answer {\sage {p8y}} \\ \text {income} & \$ \answer {\sage {p8z}} \end {array}$ What is the maximum return? $\$ \answer {\sage {p8ans}}$

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