Exam 6: Linear Programming

You are thinking of combining the three supplements in the table that gives you the maximum possible amount of creatine, but no more than1,000 milligrams of alpha lipoic acid and 225 grams of carbohydrates. How many servings of each supplement should you combine to meet your specifications, and how much creatine will you get Creatine ( ) Carbohydrates () Taurine ( ) Alpha Lipoic Acid () RiboForce HP (EAS) 5 15 1 0 Creatine Transport (Kaizen) 5 35 1 100 Pre-Load Creatine (Optimum) 6 35 1 25 Use __________ servings of RiboForce HP, __________ servings of Creatine Transport and __________ servings of Pre-Load Creatine for a maximum of __________ g creatine.

The Enormous State University History Department offers three courses, Ancient, Medieval, and Modern History, and the department chairperson is trying to decide how many sections of each to offer this semester. They may offer up to 50 sections total, up to 6,000 students would like to take a course, and there are 60 professors to teach them (no student will take more than one history course, and no professor will teach more than one section). Sections of Ancient History have 100 students each, sections of Medieval History have 50 students each, and sections of Modern History have 200 students each. Modern History sections are taught by a team of two professors, while Ancient and Medieval History need only one professor per section. Ancient History nets the university $10,000 per section, Medieval nets $20,000, and Modern History nets $30,000 per section. How many sections of each course should the department offer in order to generate the largest profit What is the largest profit possible Will there be any unused time slots, any students who did not get into classes, or any professors without anything to teach ?

Your portfolio manager has suggested two high-yielding stocks: The Altria Group (MO) and Reynolds America (RAI). MO shares cost $50, yield 4.5% in dividends, and have a risk index of 2.0 per share. RAI shares cost $55, yield 5% in dividends, and have a risk index of 3.0 per share. You have up to $7,200 to invest and would like to earn at least $335 in dividends. How many shares (to the nearest whole number) of each stock should you purchase to meet your requirements and minimize the total risk index for your portfolio What is the minimum total risk index ?

We are to minimize $p = 3 a + 2 b$ subject to the following constraints. $\left\{ \begin{array} { c } a + b \geq 4 \\3 a + b \geq 6 \\a , b \geq 0\end{array} \right\}$ What is the dual problem in matrix form

Solve the standard minimization problem using duality. Minimize $c = 4 s + 7 t + 9 u$ Subject to s+2t+2u\geq60 2s+t+3u\geq60 s\geq0,t\geq0,u\geq0

Solve the standard minimization problem using duality. Minimize $c = 6 s + 6 t$ subject to s+4t\geq20 4s+t\geq20 s\geq0,t\geq0 ​ $s =$ __________ ​ $t =$ __________ ​ $c =$ __________

Maximize ​ $p = 2 x + 3 y + 1.1 z + 4 w$ ​ subject to ​ 1.2x+y+z+w \leq42 2.2x+y-z-w \geq15 1.2x+y+z+1.2w \geq11.5 x\geq0,y\geq0,z\geq0,w \geq0 ​ Round answers to two decimal places, if necessary. ​ If there is no solution, do not fill the fields. ​ $p =$ __________ ​ $x =$ __________ ​ $y =$ __________ ​ $z =$ __________ ​ $w ^ { \prime } =$ __________

If x dollars is invested in a company that controls, say, 30 percent of the market with 5 brand-names, then 0.3x is a measure of market exposure and 5?x is a measure of brand-name exposure. Now suppose you are a broker at a large securities firm, and one of your clients would like to invest up to $100,000 in recording industry stocks. You decide to recommend a combination of stocks in four of the world's largest companies: Warner Music, Universal Music, Sony, and EMI. Warner Music Universal Music Sony EMI Market Share 12\% 20\% 20\% 15\% Number of brand-names 8 20 10 15 You wanted your client to maximize his total market exposure but limit his brand-name exposure to 3.5 million or less (representing an average of 35 labels or fewer per company), and still invest at least 20 percent of the total in Universal. How much should you advise your client to invest in each company $__________ in Warner Music, $__________ in Universal Music, $__________ in Sony, and $__________ in EMI.

Minimize $c = 50 x + 50 y + 11 z$ Subject to 2x+z \geq8 2x+y-z \geq4 3x+y-z \leq6 x\geq0,y\geq0,z \geq0

My friends: I, the mighty Brutus, have decided to prepare for retirement by instructing young warriors in the arts of battle and diplomacy. For each hour spent in battle instruction, I have decided to charge 90 ducats. For each hour in diplomacy instruction I shall charge 50 ducats. Due to my advancing years, I can spend no more than 60 hours per week instructing the youths, although the great Jove knows that they are sorely in need of instruction! Due to my fondness for physical pursuits, I have decided to spend no more than half of the total time in diplomatic instruction. However, the present border crisis with the Gauls is a sore indication of our poor abilities as diplomats. As a result, I have decided to spend 15 hours per week instructing in diplomacy. Finally, to complicate things further, there is the matter of Scarlet Brew: I have estimated that each hour of battle instruction will require at least 10 gallons of Scarlet Brew to quench my students' thirst, and that each hour of diplomacy instruction, being less physically demanding, requires half that amount. Because my harvest of red berries has far exceeded my expectations, I estimate that I'll have to use at least 625 gallons per week in order to avoid storing the fine brew at great expense. Given all these restrictions, how many hours per week should I spend in each type of instruction to maximize my income ? If no optimal solution exists enter no solution in each box. ? Number of hours in battle = __________ ? Number of hours in diplomacy = __________

Your farm encompasses 500 acres, and you are planning to grow soy beans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: $5 for soy beans, $2 for corn, and $1 for wheat.You estimate that each acre of soy beans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of $3,000 for each acre of soy beans, $2,000 for each acre of corn, and $1,000 for each acre of wheat. You can afford to spend no more than $3,000 on fertilizer, and your farm laborers can supply 3,000 hours per week. How many acres of each crop should you plant to maximize total profits You should plant __________ acre(s) of soy beans, __________ acre(s) of corn and __________ acre(s) of wheat. In this event, will you be using all the available labor

You are thinking of making your mansion more energy efficient by replacing some of the light bulbs with compact fluorescent bulbs, and insulating part or all of your exterior walls. Each compact fluorescent light bulb costs $4 and saves you an average of $2 per year in energy costs, and each square foot of wall insulation costs $1 and saves you an average of $0.20 per year in energy costs. Your mansion has 252 light fittings and 3,000 sq. ft. of un-insulated exterior wall. To impress your friends, you would like to spend as much as possible, but save no more than $880 per year in energy costs (you are proud of your large utility bills). If no optimal solution exists enter no solution. How many fluorescent light bulbs should you purchase Number of light bulbs = __________ How many square feet of insulation should you purchase Number of sq.ft. of insulation = __________ How much will you save in energy costs per year Saving = $ __________

We are to minimize $p = 3 a + 2 b$ subject to the following constraints. a+b\geq4 3a+b\geq6 a,b\geq0 After using the simplex method for dual problem to arrive at the tableau below, what is the maximum value for $p$ X y s t p $\left[ \begin{array} { c c c c c } 0 & 1 & \frac { 1 } { 2 } & - \frac { 1 } { 2 } & 0 \\\frac { 1 } { 2 } & & & \\1 & 0 & - \frac { 1 } { 2 } & \frac { 3 } { 2 } & 0 \\\frac { 3 } { 2 } & & & & \\0 & 0 & 1 & 3 & 1 \\9 & & & &\end{array} \right]$

Select the correct graph of the given inequality. $x \leq 4$

Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 260 grams of protein and at least 210 grams of fat. If chicken costs 10 cents per ounce and grain costs 1 cent per ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost ? If no optimal solution exists enter no solution. ? Number of ounces of chicken = __________ ? Number of ounces of grain = __________

Fancy Pineapple produces pineapple juice and canned pineapple rings. This year the company anticipates a demand of at least 15,000 pints of pineapple juice and 2,500 cans of pineapple rings. Each pint of pineapple juice requires 2 pineapples, and each can of pineapple rings requires 1 pineapple. The company anticipates using at least 10,000 pineapples for these products. Each pint of pineapple juice costs the company 20 cents to produce, and each can of pineapple rings costs 50 cents to produce. How many pints of pineapple juice and cans of pineapple rings should Fancy Pineapple produce to meet the demand and minimize total costs __________ pints of pineapple juice and __________ cans of pineapple rings.

You are thinking of combining Cell-Tech and Riboforce HP to obtain a 10-day supply that provides at least 110 grams of creatine and at least 16 grams of taurine, but no more than 1,050 grams of carbohydrates and no more than 1,600 milligrams of alpha lipoic acid. Use the data on bodybuilding supplements. (Figures shown correspond to a single serving.) Creatine (g) Carbohydrates (g) Taurine (g) Alpha Lipoic Acid ( ) Cost (\ ) Cell-Tech MuscleTech) 10 75 2 200 2.20 RiboForce HP (EAS) 5 15 1 0 1.60 ? 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 Riboforce HP = __________

Minimize $c = 2.2 x + y + 3.3 z$ Subject to x+1.5y+1.2z \geq150 2x+1.5y \geq40 1.5y+1.1z \geq40 x\geq0,y\geq0,z \geq0 Round all answers to two decimal places.

Safety-Kleen operates the world's largest oil refinery at Elgin, Illinois. You have been hired by the company to determine how to allocate its intake of up to 60 million gallons of used oil to its three refinery processes: A, B, and C. You are told that electricity costs for process A amount to $150,000 per million gallons treated, while for processes B and C, the costs are respectively $100,000 and $50,000 per million gallons treated. Process A can recover 60 percent of the used oil, process B can recover 55 percent, and process C can recover only 50 percent. Assuming a revenue of $4 million per million gallons of recovered oil and an annual electrical budget of $3.6 million, how much used oil would you allocate to each process in order to maximize total revenues ?

The Pimco New York Municipal Bond Fund (PNF) and the Fidelity Spartan Mass Fund (FDMMX) are tax-exempt municipal bond funds. In 2003, the Pimco fund yielded 6% while the Fidelity fund yielded 5%. You would like to invest a total of up to $70,000 and earn at least $3,600 in interest in the coming year (based on the given yields). Draw the feasible region that shows how much money you can invest in each fund. Find the corner points of the region. Let x be the invested sum into FDMMX in thousand dollars, and y be the invested sum into PNF in thousand dollars.