Consider the following constraints from a linear programming problem:

2X + Y ≤ 200
X + 2Y ≤ 200
X, Y ≥ 0

If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

Question

Consider the following constraints from a linear programming problem:

2X + Y ≤ 200
X + 2Y ≤ 200
X, Y ≥ 0

If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

BlackTom AI · Accepted Answer

We begin by listing and reading every option against the given constraints so we can test feasibility first.
Option 1: (0, 0). Check: 2X + Y = 0 ≤ 200 and X + 2Y = 0 ≤ 200, with X ≥ 0, Y ≥ 0. This point satisfies all constraints, so it is feasible. 
Option 2: (0, 200). Check: 2X + Y = 0 + 200 = 200 ≤ 200, which satisfies the first constraint. However, X + 2Y = 0 + 2×200 = 400 ≤ 200 is false. Therefore this point violates the second constraint and is not feasible. 
Option 3: (0, 100). Check: 2X + Y = 0 + 100 = 100 ≤ 200; X + 2Y = 0 + 2×100 = 200 ≤ 200. Both constraints hold, so this point is feasible. 
Option 4: (100, 0). Check: 2X + Y = 2×100 + 0 = 200 ≤ 200; X + 2Y = 100 + 0 = 100 ≤ 200. Both constraints hold, so this point is feasible.

Now, interpreting the question: which of these points cannot be the optimal solution under the given constraints? A point that is infeasible cannot be optimal by definition, since it does not lie in the feasible region the LP is optimizing over. Among the options, only (0, 200) fails to satisfy the constraints, specifically the second one, and thus cannot be the optimal solution.

The other three points are feasible (they satisfy all constraints), so depending on the (unspecified) objective function, any of them could potentially be the optimal solution; there is nothing restricting them from being optimal solely based on feasibility.

In summary, the key reason (0, 200) cannot be the optimal solution is that it violates the constraint X + 2Y ≤ 200, rendering it infeasible, whereas the remaining points satisfy all constraints and therefore remain candidates for optimality given an objective function.

QANT_620-VA1-2025SP-VR Quiz_2_LP

Consider the following constraints from a linear programming problem: 2X + Y ≤ 200 X + 2Y ≤ 200 X, Y ≥ 0 If these are the only constraints, which of the following points (X,Y) cannot be the optimal solution?

查看解析

登录即可查看完整答案

类似问题

Solving a linear program can never result in integer values for the decision variables.

Consider the following optimization problem and the constraint boundary lines given below. Maximize profit = 4X + 4Y Constraints 3X + 2Y ≤ 150 X - 2Y ≤ 10 2X + 3Y ≤150 X, Y ≥ 0 When the constraint coefficient of x in the blue constraint changes from 1 to 3, the optimal solution changes.

A constraint is binding if it left hand side of a constraint is equal to the right hand side after the optimal solution is plugged into the constraint function.

A zero in the allowable increase or decrease columns of the variable cells table in the sensitivity report indicates that

It is not possible to have more than one optimal solution to a linear programming problem.

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单