Question

Consider a problem with two decision variables, x1 and x2 , which represent the levels of activities 1 and 2, respectively. For each variable, the permissible values are 0, 1, and 2, where the feasible combinations of these values for the two variables are determined from a variety of constraints. The objective is to maximize a certain measure of performance denoted by Z. The values of Z for the possibly feasible values of (x1 , x2 ) are estimated to be those given in the following table: Based on this information, indicate whether this problem completely satisfies each of the four assumptions of linear programming. Justify your answers.

Answer 1

This problem satisfies the objective function and feasible region assumptions of linear programming, but the linearity and rationality assumptions cannot be determined without further information.

Step-by-step explanation:

Linear programming is an optimization technique that makes certain assumptions. Let's examine if this problem satisfies each assumption:

Objective function: This problem has a measurable objective of maximizing the performance measure Z. Therefore, it satisfies the objective function assumption.

Feasible region: The feasible combinations of values for the decision variables x1 and x2 are limited to 0, 1, and 2. These values are within the permissible range determined by the constraints. So, it satisfies the feasible region assumption.

Linearity: The problem does not mention any non-linear terms or constraints. If all the expressions in the constraints and objective function are linear, then this problem satisfies the linearity assumption.

Rationality: The problem does not provide information about rationality. Without further information, we cannot determine if it satisfies the rationality assumption.

Answer 2

This problem satisfies the assumptions of proportionality, additivity, and divisibility in linear programming.

Step-by-step explanation:

This problem presents a linear programming scenario with two decision variables and constraints. Linear programming assumes four main assumptions: proportionality, additivity, divisibility, and certainty. We can determine whether this problem satisfies each assumption:

1. Proportionality: The problem assumes that the measure of performance, Z, is linearly related to the levels of activities x1 and x2. This assumption is satisfied.
2. Additivity: The problem defines the objective as maximizing Z, which implies an additive relationship between the decision variables. This assumption is satisfied.
3. Divisibility: The problem allows for fractional values of the decision variables, which satisfies the divisibility assumption.
4. Certainty: The problem assumes that the Z values are estimated accurately. If there is uncertainty involved, this assumption may not be satisfied.

In summary, this problem satisfies the assumptions of proportionality, additivity, and divisibility. The certainty assumption depends on the accuracy of the Z value estimations.