Question

If the isoprofit line moves away from the origin, then the value of the objective function (1) increases (2) decreases (3) does not change (4) becomes zero

Short answer

Answer: The value of the objective function increases.

Step-by-step solution

Understanding Isoprofit Line and Objective Function

An isoprofit line represents all the points where the profit(or any other objective function) is the same. In linear programming problems, the objective function is a linear function of the variables, and it can be presented as Z = c1 * x1 + c2 * x2 + ... + cn * xn. In an economic context, Z could represent profit, and c1,...,cn are coefficients that dictate how the values of variables x1,...,xn contribute to the profit. When we graph the objective function, we need to find the combination of x1, x2,..., xn that maximizes (or minimizes) the value of Z, subject to given constraints.

Isoprofit Line Movement Away from the Origin

If the isoprofit line moves away from the origin, it means that the objective function's value is increasing or decreasing with respect to the current point (x1, x2,...,xn) in the feasible region. This movement can be explained by either increasing or decreasing the coefficients c1,...,cn, or by changing the constraint conditions.

Effect on the Value of the Objective Function

If the isoprofit line moves away from the origin, then the value of the objective function (1) increases. This is because the objective function's value along the isoprofit line becomes higher as it moves away from the origin, implying a higher profit (or lower cost, depending on the context). Therefore, the correct answer is (1) increases.

Key concepts

Objective Function

In linear programming, the objective function is a key element that defines what you're trying to achieve. It is usually expressed as a linear equation, like \( Z = c_1 \times x_1 + c_2 \times x_2 + \ldots + c_n \times x_n \). Here, \( Z \) represents the quantity you aim to maximize or minimize, such as profit or cost.

The coefficients \( c_1, c_2, \ldots, c_n \) represent the contribution of each variable \( x_1, x_2, \ldots, x_n \) to this goal. For instance, in a profit scenario, \( c_1, c_2, \ldots, c_n \) can be the profit margins for selling products \( x_1, x_2, \ldots, x_n \).

Finding the optimal solution involves identifying the combination of variables \( x_1, x_2, \ldots, x_n \) that maximizes or minimizes \( Z \), within certain constraints. These constraints ensure the solution is practical and feasible for real-life applications.

Isoprofit Line

The concept of an isoprofit line is integral in understanding how objectives are met in linear programming. An isoprofit line indicates all the possible combinations of variables \( x_1, x_2, \ldots, x_n \) that yield the same level of profit, represented by \( Z \).

When graphed, this line helps visualize where different profit levels lie. The farther these lines are from the origin, the greater the profit. Movement of an isoprofit line outward, away from the origin, signifies an increase in the objective function value. This could result from increasing coefficients \( c_1, c_2, \ldots, c_n \) or changing constraints that allow the line to shift, thus affecting overall profitability.

Understanding these movements is crucial, as they directly relate to how changes in variables or constraints can impact the achieved profit or cost.

Feasible Region

The feasible region is a fundamental concept in linear programming, defining the set of all possible solutions that satisfy all constraints. Graphically, it is the area where all the constraint inequalities overlap.

This region is crucial because it encompasses every potential combination of variable values \( x_1, x_2, \ldots, x_n \) that meets the criteria set by constraints. Every point in this region represents a possible solution to the problem, with only one being the optimal solution which maximizes or minimizes the objective function.

The boundaries of the feasible region are determined by the constraints, which can include limits on resources or other economic conditions. The optimal solution usually lies at a vertex or along an edge of this region, making understanding and visualizing the feasible region essential for solving any linear programming problem effectively.