Exam 6: Linear Programming

You have $600,000 and are considering investing in three municipal bond funds, Pimco, Fidelity Spartan, and Columbia Oregon. You have the following data: Company Yield PNF (Pimco NY) 7\% FDMMX (Fidelity Spartan Mass) 5\% CMBFX (Columbia Oregon) 4\% Your broker has made the following suggestions: At least 50 percent of your total investment should be in CMBFX. No more than 10 percent of your total investment should be in PNF. How much should you invest in each fund to maximize your anticipated returns while following your broker's advice

Solve the standard minimization problem using duality. ​ Minimize $c = s + t + 9 u$ ​ subject to s+2t+2u\geq60 2s+t+3u\geq60 s\geq0,t\geq0,u\geq0 ​ $c =$ __________ ​ $s =$ __________ ​ $t =$ __________ ​ $u =$ __________

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $\left\{ \begin{array} { c } - 6 x + y \leq 7 \\x + y \leq 2 \\3 y \geq x + 2\end{array} \right.$

Solve the games with the given payoff matrix. $P = \left[ \begin{array} { c c c } - 1 & 1 & 2 \\2 & - 1 & - 2 \\1 & 2 & - 1\end{array} \right]$

Minimize $p = 6 a + 4 b$ subject to the following constraints. a+b\geq10 6a+b\geq30 a\geq0,b\geq0

Use the simplex method to maximize $p = 6 x + 7 y$ subject to the following. Round your answers to two decimal place, if necessary. ​ 2x+2y\leq13 x+3y\leq13 $x \geq 0 , y \geq 0$ ​ $p =$ __________ ​ $x =$ __________ ​ $y =$ __________

The political pollster Canter is preparing for a national election. It would like to poll at least 900 Democrats and 900 Republicans. Each mailing to the East Coast gets responses from 100 Democrats and 50 Republicans. Each mailing to the Midwest gets responses from 100 Democrats and 100 Republicans. And each mailing to the West Coast gets responses from 50 Democrats and 100 Republicans. Mailings to the East Coast cost $40 each to produce and mail, mailings to the Midwest cost $60 each, and mailings to the West Coast cost $50 each. How many mailings should Canter send to each area of the country to get the responses it needs at the least possible cost __________ mailings to the East Coast, none to the Midwest, and __________ to the West Coast. Another solution resulting in the same cost is no mailings to the East Coast, __________ to the Midwest, and none to the West Coast. What will it cost $__________

Solve the linear programming problem. Minimize $C = - 7 x + y$ subject to $x + 5 y \leq 20$ $6 x + 5 y \leq 30$ $x \geq 0 , y \geq 0$

Your small farm encompasses 150 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $2,000 for each acre of tomatoes, $1,500 for each acre of lettuce and $500 for each acre of carrots. You would like to spend at least $600 on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 900 hours per week. How many acres of each crop should you plant to maximize total profits Plant __________ acres of tomatoes, __________ acres of lettuce, and __________ acres of carrots. In this event, will you be using all 150 acres of your farm

The Enormous State University's Business School is buying computers. The school has two models to choose from. Each first model comes with 400 MB of memory and 80 GB of disk space, while each second model has 300 MB of memory and 100 GB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least 48,000 MB of memory and at least 12,800 GB of disk space. If both models cost $4,400 each, how many of each should the school buy to keep the cost as low as possible What are the shadow costs of memory and disk space

Solve the linear programming problem. Maximize $P = 5 x + 8 y$ subject to x+y\leq4 2x+y\leq5 x\geq0,y\geq0 ​ $x =$ __________; $y =$ __________; $P =$ __________

Solve the linear programming problem. Minimize $C = - 7 x + y$ subject to x+4y\leq12 4x+4y\leq20 x\geq0,y\geq0 ​ $x =$ __________; $y =$ __________; $C =$ __________

The University's employee credit union has $8 million available for loans in the coming year. As VP in charge of finances, you must decide how much capital to allocate to each of four different kinds of loans, as shown in the table. Type of Loan Annual Rate of Return Automobile 8\% Furniture 10\% Signature 12\% Other secured 10\% State laws and credit union policies impose the following restrictions: Signature loans may not exceed 10 percent of the total investment of funds. Furniture loans plus other secured loans may not exceed automobile loans. Other secured loans may not exceed 200 percent of automobile loans. How much should you allocate to each type of loan to maximize the annual return Allocate $__________ to automobile loans, $__________ to signature loans, and $__________ to any combination of furniture loans and other secured loans.

You are thinking of combining Cell-Tech and Creatine Transport to obtain a 10-day supply that provides at least 60 grams of creatine and at least 10 grams of taurine, but no more than 450 grams of Carbohydrates and no more than 2,500 milligrams of alpha lipoic acid. Use the data on bodybuilding supplements. (Figures shown correspond to a single serving.) Creatine () Carbohydrates () Taurine () Alpha Lipoic Acid () Cost () Cell-Tech (MuscleTech) 10 75 2 200 2.20 Creatine Transport (Kaizen) 5 35 1 100 0.60 ? How many servings of each supplement should you combine to meet your specifications at the least cost If no optimal solution exists enter no solution in each box. Number of servings of Cell-Tech = __________ Number of servings of Creatine Transport = __________

Minimize ​ $c = 5 x + y + z + 2 w$ ​ subject to ​ 5x-y+w \geq1,000 z+w \leq1,500 x+y \leq600 x\geq0,y\geq0,z\geq0,w \geq0 ​ If there is no solution, do not fill the fields. ​ $c =$ __________ ​ $x =$ __________ ​ $y =$ __________ ​ $z =$ __________ ​ $w ^ { \prime } =$ __________

Your friend's portfolio manager has suggested two high-yielding stocks: Consolidated Edison (ED) and Royal Bank of Scotland (RBS-K). ED shares cost $40, yield 5.5% in dividends, and have a risk index of 1.0 per share. RBS-K shares cost $25, yield 7.5% in dividends, and have a risk index of 1.5 per share. You have up to $24,000 to invest and would like to earn at least $1,320 in dividends. How many shares (to the nearest whole number) of each stock should you purchase in order to meet your requirements and minimize the total risk index for your portfolio ?

Select the correct graph of the given inequality. $2 x + y \leq 6$

Write down (without solving) the dual linear programming problem. Minimize $C = 13 s + 35 t$ subject to the following. 7s-3t\geq15 s+4t\geq25 s\geq0,t\geq0

Given the set of constraints $\left\{ \begin{array} { c } 3 x + 2 y \leq 12 \\2 x + y \leq 7 \\x \geq 0 , y \geq 0\end{array} \right\}$ Where does the maximum of $p = 3 x + 2 y$ occur at

Write down (without solving) the dual LP problem. ​ Maximize $p = x + y + z + w$ subject to ​ x+y+z\leq7 y+z+w\leq9 x+z+w\leq6 x+y+w\leq8 x\geq0,y\geq0,z\geq0,w\geq0 ​ Use the letters s, t​, u, and v in the indicated order. ​ Minimize $c =$ __________ subject to the constraints. ​ Please enter the inequalities, separated by commas.