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Grand Canyon COURSE BUS 660, Spring 2013




Question;LINEAR PROGRAMMING FORMULATION PROBLEMS AND SOLUTIONS;7-14;The Electrocomp Corporation;manufactures two electrical products: air conditioners and large fans. The;assembly process for each is similar in that both require a certain amount of;wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours;of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling.;During the next production period, 240 hours of wiring time are available and;up to 140 hours of drilling time maybe used. Each air conditioner sold yields a;profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and;solve this LP production mix situation to find the best combination of air;conditioners and fans that yields the highest profit. Use the corner point;graphical approach.;7-15;Electrocomp?s management realizes that;it forgot to include two critical constraints (see Problem 7-14). In;particular, management decides that to ensure an adequate supply of air;conditioners for a contract, at least 20 air conditioners should be;manufactured. Because Electrocomp incurred an oversupply of fans in the preceding;period, management also insists that no more than 80 fans be produced during;this production period. Resolve this product mix problem to find the new;optimal solution.;7-16;A candidate for mayor in a small town;has allocated $40,000 for last-minute advertising in the days preceding the;election. Two types of ads will be used: radio and television. Each radio ad;costs $200 and reaches an estimated 3,000 people. Each television ad costs $500;and reaches an estimated 7,000 people. In planning the advertising campaign;the campaign manager would like to reach as many people as possible, but she;has stipulated that at least 10 ads of each type must be used. Also, the number;of radio ads must be at least as great as the number of television ads. How;many ads of each type should be used? How many people will this reach?;7-17;The Outdoor Furniture Corporation;manufactures two products, benches and picnic tables, for use in yards and;parks. The firm has two main resources: its carpenters (labor force) and a;supply of redwood for use in the furniture. During the next production cycle;1,200 hours of labor are available under a union agreement. The firm also has a;stock of 3500 feet of good-quality redwood. Each bench that Outdoor Furniture;produces requires 4 labor hours and 10 feet of redwood, each picnic table takes;6 labor hours and 35 feet of redwood. Completed benches will yield a profit of;$9 each, and tables will result in a profit of $20 each. How many benches and;tables should Outdoor Furniture produce to obtain the largest possible profit?;Use the graphical LP approach.;7-18;The dean of the Western College of;Business must plan the school?s course offerings for the fall semester. Student;demands make it necessary to offer at least 30 undergraduate and 20 graduate;courses in the term. Faculty contracts also dictate that at least 60 courses be;offered in total. Each undergraduate course taught costs the college an average;of $2,500 in faculty wages, and each graduate course costs $3,000. How many;undergraduate and graduate courses should be taught in the fall so that total;faculty salaries are kept to a minimum?;7-19;MSA Computer Corporation manufactures;two models of minicomputers, the Alpha 4 and the Beta 5. The firm employs five;technicians, working 160 hours each per month, on its assembly line. Management;insists that full employment (i.e., all 160 hours of time) be maintained for;each worker during next month?s operations. It requires 20 labor hours to assemble;each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA;wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the;production period. Alpha 4s generate $1,200 profit per unit, and Beta 5s yield;$1,800 each. Determine the most profitable number of each model of minicomputer;to produce during the coming month.;7-20;A winner of the Texas Lotto has;decided to invest $50,000 per year in the stock market. Under consideration are;stocks for a petrochemical firm and a public utility. Although a long-range;goal is to get the highest possible return, some consideration is given to the;risk involved with the stocks. A risk index on a scale of 1?10 (with 10 being;the most risky) is assigned to each of the two stocks. The total risk of the;portfolio is found by multiplying the risk of each stock by the dollars;invested in that stock. The following table provides a summary of the return;and risk;Stock;Estimated Return;Risk Index;Petrochemical;12%;9;Utility;6%;4;The investor would like to maximize the;return on the investment, but the average risk index of the investment should;not be higher than 6. How much should be invested in each stock? What is the;average risk for this investment? What is the estimated return for this;investment?;7-21;Referring to the Texas Lotto situation;in Problem 7-20, suppose the investor has changed his attitude about the;investment and wishes to give greater emphasis to the risk of the investment.;Now the investor wishes to minimize the risk of the investment as long as a;return of at least 8% is generated. Formulate this as an LP problem and find;the optimal solution. How much should be invested in each stock? What is the;average risk for this investment? What is the estimated return for this;investment?;7-24 The stock brokerage firm of Blank;Leibowitz, and Weinberger has analyzed and recommended two stocks to an;investors? club of college professors. The professors were interested in;factors such as short term growth, intermediate growth, and dividend rates.;These data on each stock are as follows;Stock;Factor;Louisiana Gas;and Power;Trimex Insulation Company;Short term growth potential, per dollar;invested;.36;.24;Intermediate growth potential (over next;three years), per dollar invested;1.67;1.5;Dividend rate potential;4%;8%;Each member of the club has an investment;goal of (1) an appreciation of no less than $720 in the short term, (2) an;appreciation of at least $5,000 in the next three years, and (3) a dividend;income of at least $200 per year. What is the smallest investment that a;professor can make to meet these three goals?;7-25;Woofer Pet Foods produces a;low-calorie dog food for overweight dogs. This product is made from beef;products and grain. Each pound of beef costs $0.90, and each pound of grain;costs $0.60. A pound of the dog food must contain at least 9 units of Vitamin 1;and 10 units of Vitamin 2. A pound of beef contains 10 units of Vitamin 1 and;12 units of Vitamin 2. A pound of grain contains 6 units of Vitamin 1 and 9;units of Vitamin 2. Formulate this as an LP problem to minimize the cost of the;dog food. How many pounds of beef and grain should be included in each pound of;dog food? What is the cost and vitamin content of the final product?;SOLUTIONS;TO SELECT PROBLEMS FROM CHAPTER 8;8-1;(Production problem) Winkler;Furniture manufactures two different types of china cabinets: a French;Provincial model and a Danish Modern model. Each cabinet produced must go;through three departments: carpentry, painting, and finishing. The table below contains;all relevant information concerning production times per cabinet produced and;production capacities for each operation per day, along with net revenue per;unit produced. The firm has a contract with an Indiana distributor to produce a minimum of;300 of each cabinet per week (or 60 cabinets per day). Owner Bob Winkler would;like to determine a product mix to maximize his daily revenue.;(a) Formulate as an LP problem.;(b) Solve using an LP software program or;spreadsheet.;Cabinet;Style;Carpentry (Hours/Cabinet);Painting;(Hours/Cabinet);Finishing;(Hours/Cabinet);Net Revenue per Cabinet ($);French;Provincial;3;1.5;.75;28;Danish;Modern;2;1;.75;25;Dept.;capacity (hrs);360;200;125;8-2;(Investment decision problem) The;Heinlein and Krarnpf Brokerage firm has just been instructed by one of its;clients to invest $250,000 for her money obtained recently through the sale of;land holdings in Ohio. The client has a good deal of trust in the investment;house, but she also has her own ideas about the distribution of the funds being;invested. In particular, she requests that the firm select whatever stocks and;bonds they believe are well rated, but within the following guidelines;(a) Municipal bonds should constitute at least 20%;of the investment.;(b) At least 40%;of the funds should be placed in a combination of electronic firms, aerospace;firms, and drug manufacturers.;(c) No more than;50% of the amount invested in municipal bonds should be placed in a high-risk;high-yield nursing home stock.;Subject to these restraints, the client?s;goal is to maximize projected return on investments. The analysts at Heinlein;and Krampf, aware of these guidelines, prepare a list of high-quality stocks;and bonds and their corresponding rates of return.;Investment;Projected Rate of Return (%);Los;Angeles municipal bonds;5.3;Thompson Electronics, Inc.;6.8;United Aerospace Corp.;4.9;Palmer Drugs;8.4;Happy Days Nursing Homes;11.8;(a) Formulate this portfolio selection;problem using LP. (b) Solve this problem.;8-3;(Restaurant work scheduling;problem). The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and;busboys report for duty at 3AM., 7 AM., 11 AM., 3 P.M., 7 P.M., or 11 P.M., and;each works an 8-hour shift. The following table shows the minimum number of;workers needed during the six periods into which the day is divided. Chang?s;scheduling problem is to determine how many waiters and busboys should report;for work at the start of each time period to minimize the total staff required;for one day?s operation. (Hint: Let Xi equal the;number of waiters and busboys beginning work in time period i, where i = 1;2,3,4,5,6.);Period;Time;Number;of Waiters and Busboys Required;1;3;A.M?7 A.M.;3;2;7;A.M?11 A.M.;12;3;11;A.M?3 P.M.;16;4;3;P.M?7 P.M.;9;5;7;P.M?11 P.M.;11;6;11;P.M?3 A.M.;4;8-4;(Animal feed mix problem) The;Battery Park Stable feeds and houses the horses used to pull tourist-filled;carriages through the streets of Charleston?s;historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes;the need to set a nutritional diet for the horses in his care. At the same;time, he would like to keep the overall daily cost of feed to a minimum.;The feed;mixes available for the horses? diet are an oat product, a highly enriched;grain, and a mineral product. Each of these mixes contains a certain amount of;five ingredients needed daily to keep the average horse healthy. The table;below shows these minimum requirements, units of each ingredient per pound of;feed mix, and costs for the three mixes.;In addition, the stable owner is aware;that an overfed horse is a sluggish worker. Consequently, he determines that 6;pounds of feed per day are the most that any horse needs to function properly.;Formulate this problem and solve for the optimal daily mix of the three feeds.;Feed;Mix;Diet Requirement;(Ingredients);Oat;Product;(units/lb);Enriched;Grain;(units/lb);Mineral;Product;(units/lb);Minimum Daily Requirement;(units);A;2;3;1;6;B;.5;1;.5;2;C;3;5;6;9;D;1;1.5;2;8;E;.5;.5;1.5;5;Cost/lb;$0.09;$0.14;$0.17;8-6;(Media selection problem) The;advertising director for Diversey Paint and Supply, a chain of four retail;stores on Chicago?s;North Side, is considering two media possibilities. One plan is for a series of;half- page ads in the Sunday Chicago Tribune newspaper, and the other is for;advertising time on Chicago TV. The stores are expanding their lines of;do-it-yourself tools, and the advertising director is interested in an exposure;level of at least 40% within the city?s neighborhoods and 60% in northwest;suburban areas.;The TV viewing time under consideration;has an exposure rating per spot of 5% in city homes and 3% in the northwest;suburbs. The Sunday newspaper has corresponding exposure rates of 4% and 3% per;ad. The cost of a half-page Tribune advertisement is $925, a television spot;costs $2,000.;Diversey Paint would like to select the;least costly advertising strategy that would meet desired exposure levels.;(a) Formulate using LP.;(b) Solve the problem.;8-11;(College meal selection problem) Kathy;Roniger, campus dietician for a small Idaho college, is responsible for;formulating a nutritious meal plan for students. For an evening meal, she feels;that the following five meal-content requirements should be met: (1) between;900 and 1,500 calories, (2) at least 4 milligrams of iron, (3) no more than 50;grams of fat, (4) at least 26 grams of protein, and (5) no more than 50 grams;of carbohydrates. On a particular day, Roniger?s food stock includes seven;items that can be prepared and served for supper to meet these requirements.;The cost per pound for each food item and the contribution to each of the five;nutritional requirements are given in the accompanying table;Table of Food Values and Costs;Food Item;Calories/;Pound;Iron;(mg/lb);Fat;(gm/lb);Protein;(gm/lb);Carbs.;(gm/lb);Cost/;Pound ($);Milk;295;0.2;16;16;22;0.60;Ground Meat;1216;0.2;96;81;0;2.35;Chicken;394;4.3;9;74;0;1.15;Fish;358;3.2;0.5;83;0;2.25;Beans;128;3.2;0.8;7;28;0.58;Spinach;118;14.1;1.4;14;19;1.17;Potatoes;279;2.2;0.5;8;63;0.33;What combination;and amounts of food items will provide the nutrition Roniger requires at the;least total food cost?;8-12 (High tech production;problem) Quitmeyer Electronics Incorporated manufactures the following six;microcomputer peripheral devices: internal modems, external modems, graphics;circuit boards, CD drives, hard disk drives, and memory expansion boards. Each;of these technical products requires time, in minutes, on three types of;electronic testing equipment, as shown in the table the following table;Internal;Modem;External;Modem;Circuit;Board;CD;Drive;Hard;Drive;Memory;Board;Test device 1;7;3;12;6;18;17;Test device 2;2;5;3;2;15;17;Test device 3;5;1;3;2;9;2;The first two test devices are available;120 hours per week. The third (device 3) requires more preventive maintenance;and may be used only 100 hours each week. The market for all six computer;components is vast, and Quitmeyer Electronics believes that it can sell as many;units of each product as it can manufacture. The table that follows summarizes;the revenues and material costs for each product;Device;Revenue;Per Unit Sold ($);Material;Cost Per Unit ($);Internal;modem;200;35;External;modem;120;25;Graphics;circuit board;180;40;CD drive;130;45;Hard;disk drive;430;170;Memory;expansion board;260;60;In addition, variable labor costs are $15;per hour for test device 1, $12 per hour for test device 2. and $18 per hour;for test device 3. Quitmeyer Electronics wants to maximize its profits.;(a) Formulate this problem as an LP model.;(b) Solve the problem by computer. What is;the best product mix?;(c) What is the value of an additional;minute of time per week on test device 1? Test device 2? Test device 3? Should;Quitmeyer Electronics add more test device time? If so, on which equipment?;8-15 (Material blending problem) Amalgamated Products has just received a;contract to construct steel body frames for automobiles that are to be produced;at the new Japanese factory in Tennessee.;The Japanese auto manufacturer has strict quality control standards for all of;its component subcontractors and has informed Amalgamated that each frame must;have the following steel content;Material;Minimum Percent;Maximum Percent;Manganese;2.1;2.3;Silicon;4.3;4.6;Carbon;5.05;5.35;Amalgamated mixes;batches of eight different available materials to produce one ton of steel used;in the body frames. The table below details these materials. Formulate and;solve the LP model that will indicate how much of each of the eight materials;should be blended into a 1-ton load of steel so that Amalgamated meets its;requirements while minimizing cost.;Material;Available;Manganese;(%);Silicon;(%);Carbon;(%);Pounds;Available;Cost Per Pound ($);Alloy 1;70.0;15.0;3.0;No limit;0.12;Alloy 2;55.0;30.0;1.0;300;0.13;Alloy 3;12.0;26.0;0;No limit;0.15;Iron 1;1.0;10.0;3.0;No limit;0.09;Iron 2;5.0;2.5;0;No limit;0.07;Carbide 1;0;24.0;18.0;50;0.10;Carbide 2;0;25.0;20.0;200;0.12;Carbide 3;0;23.00;25.0;100;0.09


Paper#60281 | Written in 18-Jul-2015

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