Linear Programming. Math 4R. I - # 2 (set up table) II - # 2 (maybe take off), 3 (set up table. 3 - # 1, 3, 4 (maybe take off) HW#60:

Transcription

1 Math 4R HW#60: Text pp # 1,2, (Sketch only #13, #17, #18) HW#61: 1. Maximize R = 2x - y given the following constraints: xzo yzo x~2 y z -2x+4 y~x+4 2. Minimize M = x + y given the following constraints: xzo yzo y ~ 3x+5 Y z -4x+ 12 y~-x+13 HW#62: I - # 2 (set up table) II - # 2 (maybe take off), 3 (set up table HW#63: 11- # 4, 5, 6 HW#64: 3 - # 1, 3, 4 (maybe take off) Study!!! HW#65: 3 - # 2 A HW#66: WS - Review for Test on #1

2 MATH4H MRS WILLIAMS NAME LINEAR PROGRAMMING DATE, _ Problem Solving: is a procedure for finding the maximum or minimum value for a function in two variables subject to given constraints on the variables. CONSISTS OF: 1. linear function to be optimized (objective function) 2. set of inequalities - constraints representing the restrictions on resources Vertex Theorem: Objective need only be evaluated for the coordinates of the vertices of the polygonal boundary (convex set) to find the minimum or maximum values. Procedure: 1. Define variables. 2. Create a table. 3. Write the constraints as a system of inequalities. 4. Write an expression to be maximized or minimized. 5. Graph the system and find the coordinates of the vertices of the polygon formed. 6. Substitute values from the coordinates of the vertices into the objective function to be maximized or minimized. 7. Select the greatest or least result.

3 I 1. It takes one pin and one bolt to make a gadget. However, it takes one pin and two bolts to make a gimmick. We have only 5 pins and 6 bolts. If we could make $5 on each gadget and $7 on each gimmick, how many of each should we make to maximize our profit? 2. A meat packer wants to blend beef and pork to make two types of frankfurters, regular and deluxe. Suppose that each pound of regular frankfurters contains 0.2 pound of beef and 0.2 pound of pork, while each pound of deluxe frankfurters contains 0.4 pound of beef and 0.2 pound of pork. Also suppose that the profit on the regular is 10 cents and on the deluxe is 12 cents. If the meat packer has 30 pounds of beef in stock and 20 pounds of pork in stock, how many pounds of each type of frankfurter should he make in order to obtain the largest profit? 3. A nutritionist wants to design a breakfast menu for certain hospital patients. The menu is to include two items, A and B. Suppose that each ounce of A provides 1 unit of iron and 2 units of vitamin 0, while each ounce of 8 provides 2 units of iron and 2 units of vitamin D. Also, suppose that the calorie content per ounce for the items is 3 calories per ounce of A and 4 calories per ounce of B. If the breakfast must provide at least 8 units of iron and 10 units of vitamin 0, how many ounces of each item should be provided in order to meet the iron and vitamin 0 requirements with the smallest possible intake of calories? 4. The managers of a pension plan want to invest up to $5000 in two stocks, X and Y. Stock X is considered conservative, while stock Y is considered speculative. The managers agree that the investment in stock X should be at most $4000 and the investment in stock Y should be at least $600.Suppose also that for each $100 invested in stock X is expected to return $8 and in stock Y is expected to return $10. If the bylaws of the pension plan require that investment in speculative stock Y can be no greater than one-third of the investment in the conservative stock X. how much should be invested in stock X and how much in stock Y to maximize the return of the investments?

4 II 5. A diet is to include at least 140 mg of Vitamin A and at least 145 mg of Vitamin B. These requirements are to be obtained from two types of food. Type X contains 10 mg of Vitamin A and 20 mg of Vitamin B per pound. Type Y contains 30 mg of Vitamin A and 15 mg of Vitamin B per pound. If Type X food costs $12 and Type V food costs $8 per pound, how many pounds of each type of food should be purchased to satisfy the requirements at the minimum cost? z {O Type X, 9- Type Y} 3 6. A pharmaceutical company manufactures two drugs. Each case of drug one requires 3 hours of processing time and 1 hour of curing time per week. Each case of drug two requires 5 hours of processing time and 5 hours of curing time per week. The schedule allows 55 hours of processing time and 45 hours of curing time weekly. The company must produce no more than 10 cases of drug one and no more than 9 cases of drug two. If the company makes a profit of $320 on each case of drug one and $500 on each case of drug two, how many cases of each drug should be produced in order to maximize profit? {10 cases drug one, 5 cases drug two}

5 III 1. A merchant plans to sell two models of home computers at costs of $250 and $400 respectively. The $250 model yields a profit of $45 and the $400 model yields a profit of $50. The merchant estimates that the total monthly demand will not exceed 250 units. Find the number of units of each model that should be stocked in order to maximize profit. Assume that the merchant does not want to invest more than $70,000 in computer inventory. 2. A farmer mixes two brands of cattle feed. Brand X costs $25 per bag and contains 2 units of nutritional element A, 2 units of element B, and 2 units of element C. Brand Y costs $20 per bag and contains 1 unit of nutritional element A. 9 units of element B, and 3 units of element C. Find the number of bags of each brand that should be mixed to produce a mixture having the minimum cast per bag. The minimum requirements of nutrients A, 8, and Care 12 units, 36 units, and 24 units, respectively. 3. The Elite Tweet Pottery Shoppe budgets a maximum of $1000/month for newspaper and radio advertising. The newspaper charges $50 per ad and requires at least 4 ads per month. The radio station charges $100/min and requires a minimum of 5 minutes of advertising per month. It is estimated that each newspaper ad reaches 8,000 people and that each minute of radio advertising reaches 15,000. What combination of newspaper and radio advertising should the business use in order to reach the maximum number of people? 4. A small South American country grows coffee and cocoa for export. The country has 500,000 hectares of land available for the crops. It has contracts that require at least 100,000 hectares be devoted to coffee and at least 200,000 hectares to cocoa. Available equipment and labor limit cocoa production to 270,000 hectares. Coffee requires two workers while cocoa requires five workers per hectare. No more than 1,750,000 workers are available. Coffee provides a profit of $22 per hectare and cocoa provides a profit of $310 per hectare. How many hectares should the country devote to each crop to maximize profit? Find the maximum profit. 5. The International Canine Academy raises and trains Siberian sled dogs (huskies) and dancing French poodles. Breeders can supply lea with at most 20 poodles and 15 huskies each year. Each poodle eats 2 Ib of food a day and each sled dog will eat 6 Ib a day. ICA food supplies are restrained to at most 100 Ib of food each day. Poodles require 1000 hr of training per year, while a sled dog requires 250 hr/yr. The academy restricts training time to no more than 10,000 hr each year. How many of each kind of dog should the ICA breed in order to maximize their profit? Find the maximum yearly profit if each poodle will sell for a profit of $200 and each sled dog will sell for a profit of $80.

6 . A An automobile manufacturer makes hardtops and sports cars. Each hardtop requires 8 man-hours to q~~embl~l2,..,m~n-:q~~ ~9";p~irt~ 2 m~n::;hpw~t,9 lip!wlst{9[,t~nd is sold for a profit of $90. Eachsp'()ttsJcar requites '18 'rnan-heurs to.assemble,,2:,man-hours to paint, 1 man-hour to upholster, and is sold for a profit of $100. Ouring each day 360 man-hours,qr~ availai:>le fora,~seqlble,.50man-ljours forp~int, and 40t.m:m-ho.uf&,tB."",," upholster automobiles.. Howma'l1y'nardtops and, sports. cars should6e' produeed"ri 'a"':' day in order to maximize profrt? ' '.., ~-.",", 'f~:<?,<~',

7 Review for Test on #1 1. The AC Telephone Company manufactures two styles of cordless phones, deluxe and standard. Each deluxe telephone nets the company $9 in profit. and each standard telephone nets $6. Machines A and 8 are used to make both styles of telephones. Each deluxe telephone requires three hours of machine A time and one hour of machine B time. Each standard telephone requires two hours of machine A time and two hours of machine 8 time. An employee has an idea that frees twelve hours of machine A time and eight hours of machine B time. Determine the mix of telephones that can be made during the free time that most effectively generates profit for the company within the given constraints. 2. Minimize and Maximize W = 5y + 3x subject to : x~o y>o y<x+3 2x + y:5 8 Study the examples I assigned for homework and the examples we did in class!

8 Review for Test on #2 1. The following nutritional values are based on 10z of cereal and % cup of whole milk. Kellogg's Corn Flakes contains 23g of starch and carbohydrates and 7g of sucrose. Post Honeycombs contains 14g of starch and carbohydrates and 17g of sucrose. What is the minimum cost in order to receive at least 322g starch and 119g of sucrose by consuming these two cereals if corn flakes cost 7cents per ounce and Honecombs cost 19cents per ounce? 2. Maximize W = 5x + 3y subject to : xzo yzo 2x+ Y z 8 Y ~ 5 x-y~2 3x-2y z 5 Study the examples I assigned fer homework and the examples we did in class!