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MAT 540 Week 8 Homework Problems




Question;MAT540;Week 8 Homework;Chapter 4;14.;Grafton Metalworks;Company produces metal alloys from six different ores it mines. The company has an order from a customer to;produce an alloy that contains four metals according to the following;specifications: at least 21% of metal A;no more than 12% of metal B, no more than 7% of metal C and between 30% and 65%;of metal D. The proportion of the four;metals in each of the six ores and the level of impurities in each ore are provided;in the following table;Ore;Metal (%);Impurities (%);Cost/Ton;A;B;C;D;1;19;15;12;14;40;27;2;43;10;25;7;15;25;3;17;0;0;53;30;32;4;20;12;0;18;50;22;5;0;24;10;31;35;20;6;12;18;16;25;29;24;When the;metals are processed and refined, the impurities are removed.;The company;wants to know the amount of each ore to use per ton of the alloy that will;minimize the cost per ton of the alloy.;a.;Formulate a linear programming model for this;problem.;b.;Solve the model by;using the computer.;19.;As a result of a;recently passed bill, a congressman?s district has been allocated $4 million;for programs and projects. It is up to;the congressman to decide how to distribute the money. The congressman has decided to allocate the;money to four ongoing programs because of their importance to his district ? a;job training program, a parks project, a sanitation project, and a mobile;library. However, the congressman wants;to distribute the money in a manner that will please the most voters, or, in;other words, gain him the most votes in the upcoming election. His staff?s estimates of the number of votes;gained per dollar spent for the various programs are as follows.;Program;Votes/ Dollar;Job training;0.02;Parks;0.09;Sanitation;0.06;Mobile library;0.04;In order also to;satisfy several local influential citizens who financed his election, he is;obligated to observe the following guidelines;?;None of the;programs can receive more than 40% of the total allocation.;?;The amount;allocated to parks cannot exceed the total allocated to both the sanitation project and the mobile library;?;The amount;allocated to job training must at least equal the amount spent on the;sanitation project.;Any money not spent;in the district will be returned to the government, therefore, the congressman;wants to spend it all. The congressman;wants to know the amount to allocate to each program to maximize his;votes.;a. Formulate a linear programming model for this problem.;b. Solve the model by using the computer.;20.;Anna Broderick is;the dietician for the State University football team, and she is attempting to;determine a nutritious lunch menu for the team.;She has set the following nutritional guidelines for each lunch serving;?;Between 1,500 and;2,000 calories;?;At least 5 mg of;iron;?;At least 20 but no;more than 60 g of fat;?;At least 30 g of;protein;?;At least 40 g of;carbohydrates;?;No more than 30 mg;of cholesterol;She selects the;menu from seven basic food items, as follows, with the nutritional;contributions per pound and the cost as given;Calories;(per lb.);Iron;(mg/lb.);Protein;(g/lb.);Carbo-hydrates;(g/lb.);Fat (g/lb.);Chol-esterol;(mg/lb.);Cost;$/lb.;Chicken;520;4.4;17;0;30;180;0.80;Fish;500;3.3;85;0;5;90;3.70;Ground beef;860;0.3;82;0;75;350;2.30;Dried beans;600;3.4;10;30;3;0;0.90;Lettuce;50;0.5;6;0;0;0;0.75;Potatoes;460;2.2;10;70;0;0;0.40;Milk (2%);240;0.2;16;22;10;20;0.83;The dietician wants;to select a menu to meet the nutritional guidelines while minimizing the total;cost per serving.;a.;Formulate a linear;programming model for this problem.;b.;Solve the model by;using the computer;c. If a serving of each of the food items (other than milk) was limited;to no more than a half pound, what effect would this have on the solution?;22.;The Cabin Creek;Coal (CCC) Company operates three mines in Kentucky and West Virginia, and it;supplies coal to four utility power plants along the East Coast. The cost of shipping coal from each mine to;each plant, the capacity at each of the three mines and the demand at each;plant are shown in the following table;Plant;Mine;1;2;3;4;Mine Capacity (tons);1;$ 7;$ 9;$10;$12;220;2;9;7;8;12;170;3;11;14;5;7;280;Demand (tons);110;160;90;180;The cost of mining;and processing coal is $62 per ton at mine 1;$67 per ton at mine 2, and $75;per ton at mine 3. The percentage of ash;and sulfur content per ton of coal at each mine is as follows;Mine;% Ash;% Sulfur;1;9;6;2;5;4;3;4;3;Each plant has;different cleaning equipment. Plant 1;requires that the coal it receives have no more than 6% ash and 5% sulfur;plant 2 coal can have no more than 5% ash and sulfur combined, plant 3 can have;no more than 5% ash and 7% sulfur, and plant 4 can have no more than 6% ash and;sulfur combined. CCC wabts to;determine the amount of coal to produce at each mine and ship to its customers;that will minimize its total cost.;a.;Formulate a linear;programming model for this problem.;b.;Solve this model by;using the computer.;36.;Joe Henderson runs;a small metal parts shop. The shop contains three machines ? a drill press, a;lathe, and a grinder. Joe has three operators, each certified to;work on all three machines. However;each operator performs better on some machines than on others. The shop has contracted to do a big job that;requires all three machines. The times;required by the various operators to perform the required operations on each;machine are summarized as follows;Operator;Drill Press (min);Lathe (min);Grinder (min);1;23;18;35;2;41;30;28;3;25;36;18;Joe Henderson wants;to assign one operator to each machine so that the topal operating time for all;three operators is minimized.;a.;Formulate a linear;programming model for this problem.;b.;Solve the model by;using the computer;c.;Joe?s brother;Fred, has asked him to hire his wife, Kelly, who is a machine operator. Kelly can perform each of the three required;machine operations in 20 minutes. Should;Joe hire his sister-in-law?;43.;The Cash and Carry;Building Supply Company has received the following order for boards in three;lengths;Length;Order (quantity);7 ft.;700;9 ft.;1,200;10 ft.;300;The company has;25-foot standard-length boards in stock.;Therefore, the standard-length boards must be cut into the lengths;necessary to meet order requirements.;Naturally, the company wishes to minimize the number of standard-length;boards used.;a.;Formulate a linear;programming model for this problem.;b.;Solve the model by;using the computer;c.;When a board is cut;in a specific pattern, the amount of board left over is referred to as;?trim-loss.? Reformulate the linear programming model for this problem;assuming that the objective is to minimize trim loss rather than to minimize;the total number of boards used, and solve the model. How does this affect the solution?


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