91Ó°ÊÓ

We're given a standard maximization linear programming problem: Objective function: \(P = a_1x_1 + a_2x_2 + \cdots + a_nx_n\) We want to maximize P subject to some constraints (inequalities). The statement suggests that if at least one of the coefficients \((a_1, a_2, ..., a_n)\) is positive, then the optimal solution can't be at the point where all the variables are equal to zero \((0, 0, ..., 0)\). We will test this statement.

Let's suppose we have a linear programming problem with the following setup: Objective function: \(P = x_1 + x_2\) Constraints: \(x_1 \geq 1\) \(x_2 \geq 1\) We are supposed to find the solution that maximizes the value of P, and at least one of the coefficients is positive. If we look at these constraints, we can see that the point \((0, 0)\) is not a feasible solution, as it doesn't satisfy the constraints. We'll try another example to test the statement: Objective function: \(P = -x_1 + x_2\) Constraints: \(x_1 \leq 1\) \(x_2 \geq 1\) In this case, the all-zero point is not an optimal solution because the constraints do not allow it. Also, it's crucial to note that even at the point \((1,1)\), which lies within the feasible region, it is still not an optimal solution, as increasing \(x_2\) will increase the value of P. Since we haven't found any counterexamples yet, we can conclude the statement is true.

The statement is true because, in a standard maximization linear programming problem, having at least one positive coefficient in the objective function will cause the objective function to increase when the corresponding variable increases. This means that starting from the point \((0, 0, ..., 0)\), we can always find at least one direction (by increasing the variable with a positive coefficient) that will increase the objective function's value, given that this movement stays within the feasible region. In conclusion, the statement is true: If at least one of the coefficients \(a_{1}, a_{2}, \ldots, a_{n}\) of the objective function \(P=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\) is positive, then the all-zero solution \((0,0, \ldots, 0)\) cannot be the optimal solution of the standard (maximization) linear programming problem.