12. The Elixer Drug /company produces a drug from two ingredients. Each ingredient contains the same three antibiotics, in different proportions. One gram of ingredient 1 contributes 3 units, and 1 gram of ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required, and the ingredients each contribute 1 unit per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient 2 is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost. a. Formulate a linear programming model for this problem. b. Solve this problem by using graphical analysis. Answer: Drugs Antibiotics Cost 1 2 3 Ingredient 1 (x) 3 1 2 $80 Ingredient 2 (y) 1 1 6 $50 Total 6 4 12 a. From the question, we can write it mathematically to this table. Let the ingredient 1 be and the ingredient as . Then, we formulate it as: And also the cost, b. We need to picture the graphic. Here it is.
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12. The Elixer Drug /company produces a drug from two ingredients.Each ingredient contains the same three antibiotics, in differentproportions. One gram of ingredient 1 contributes 3 units, and 1gram of ingredient 2 contributes 1 unit of antibiotic 1; the drugrequires 6 units. At least 4 units of antibiotic 2 are required, andthe ingredients each contribute 1 unit per gram. At least 12 unitsof antibiotic 3 are required; a gram of ingredient 1 contributes 2units, and a gram of ingredient 2 contributes 6 units. The cost fora gram of ingredient 1 is $80, and the cost for a gram of ingredient2 is $50. The company wants to formulate a linear programming modelto determine the number of grams of each ingredient that must gointo the drug in order to meet the antibiotic requirements at theminimum cost.
a. Formulate a linear programming model for this problem.b. Solve this problem by using graphical analysis.
Answer:
Drugs Antibiotics Cost1 2 3Ingredient 1(x) 3 1 2 $80
Ingredient 2(y) 1 1 6 $50
Total 6 4 12a. From the question, we can write it mathematically to this table.
Let the ingredient 1 be and the ingredient as .
Then, we formulate it as:
And also the cost, b. We need to picture the graphic. Here it is.
Elimination point between and is :
The graph tells us that the unhatched area is the solved sets. The
solved set is passing some points. The points are: , ,.
After we got the points, now we can count the cost and find theminimum cost from the cost equation,
1.) For
2.) For 3.) For
From the accounting above, we can see that the minimum cost willhappen when the point is . So, to minimize the production cost,The Eixer Drug company should produce Ingredient 2 only.
57. Solve the following linear programming model graphically andexplain the solution result:
Maximize
Subject to
Answer:
The question want us find the maximum score. First we graph the
equation.
Then, we can see that the solved set is passing one point. It is an
elimination point between and . Now,we find the elimination point by eliminating those equations.
From the accounting, we got elimination point: .
Because the elimination point for , it against the .So that, the equation has no solved set and we can’t find themaximum score.
32. Solve the following linear programming model graphically:
Maximize
Subject to
Answer:
The question want us find the maximum score. First we graph theequation.
The graphic shows us that it has no solved subsets. So that, theequation has no solved subsets and we can’t find the maximum score.
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