Linear programming：練習問題

Which of the following points lies in the feasible region defined by the inequalities \(x \ge 0\), \(y \ge 0\), \(x + y \le 5\), and \(2x + y \le 8\)?

A company produces two types of toys, A and B. Toy A requires 2 hours of assembly and 1 hour of finishing. Toy B requires 3 hours of assembly and 1 hour of finishing. There are 120 hours available for assembly and 50 hours for finishing per week. If the profit on Toy A is £5 and on Toy B is £7, which of the following represents the objective function to maximize profit \(P\)?

Maximize \(P = 3x + 2y\) subject to the constraints:
\(x \ge 0\), \(y \ge 0\),
\(x + y \le 7\),
\(2x + y \le 10\).
What is the maximum value of \(P\)?

A linear programming problem seeks to maximize \(P = 4x + 5y\) subject to:
\(2x + y \le 10\)
\(x + 2y \le 8\)
\(x, y \ge 0\) and \(x, y\) must be integers.
What is the maximum value of \(P\)?

A farmer wants to maximize the number of chickens \(x\) and ducks \(y\) he can raise. He has space for at most 100 birds in total. He has a budget of £240 for feed, where chicken feed costs £2 per bird and duck feed costs £3 per bird. Due to market demand, he must raise at least 20 chickens. Assuming \(x\) and \(y\) must be integers, what is the maximum total number of birds he can raise?

A factory produces two items, X and Y. Each item X requires 3 kg of material A and 2 hours of labor. Each item Y requires 4 kg of material A and 1 hour of labor. The factory has 60 kg of material A and 25 hours of labor available. Write down the inequalities representing the constraints on material A and labor, where \(x\) is the number of item X and \(y\) is the number of item Y produced.

A company manufactures two types of bicycles, Mountain bikes and Road bikes. Each Mountain bike requires 2 hours of assembly time and 1 hour of painting time. Each Road bike requires 1 hour of assembly time and 2 hours of painting time. The company has a maximum of 400 hours of assembly time and 350 hours of painting time available per week. The profit on each Mountain bike is $150 and on each Road bike is $100.

Let \(x\) be the number of Mountain bikes and \(y\) be the number of Road bikes manufactured per week.

(a) Formulate the problem as a linear programming problem, stating the objective function and all constraints.

(b) On a graph, draw the feasible region for this problem. Shade the unwanted region.

(c) Using the vertex method, find the number of Mountain bikes and Road bikes that should be manufactured each week to maximize the total profit. State the maximum profit.