Question

Consider the following LP problem developed at Zafar Malik's Carbondale, Illinois, cotical scanning firm: \[ \begin{array}{lll} \text { Maximize } \quad Z= & 1 x_{1}+1 x_{2} \\ \text { Subject to:

Answer 1

The objective function for the linear programming problem is
`\( Z = x_1 + x_2 \)`, subject to no explicit constraints. As there are no constraints provided, the maximum value of Z is unbounded.

Step-by-step explanation:

In this linear programming problem, the objective is to maximize the linear expression
`\( Z = x_1 + x_2 \),` where
`\( x_1 \)` and
`\( x_2 \)` are decision variables. However, the absence of any constraints means that there are no limitations or restrictions on the values that
`\( x_1 \)`and
`\( x_2 \)` can take. As a result, the solution space is unbounded, and the objective function Z can increase without any limit.

In a typical linear programming problem, constraints define a feasible region, and the optimization is performed within this region. Without constraints, the problem becomes an unconstrained optimization, and the solution space extends indefinitely. Mathematically, the absence of constraints means there are no restrictions on the values of
`\( x_1 \)` and
`\( x_2 \)`, allowing them to approach infinity. Consequently, the objective function \
`( Z = x_1 + x_2 \)` is unbounded, and there is no finite maximum value for Z .

In practical terms, this situation may not have a meaningful interpretation, as most real-world problems involve constraints that limit decision variables. The absence of constraints here simplifies the problem to a basic form, demonstrating the concept of an unbounded solution space in linear programming.