1. **Problem Statement:** We have two optimization problems: (a) a maximization problem and (b) a minimization problem, each with objective functions and constraints. 2. **Maximization Problem (a):** Maximize $$Z = 5x_1 + 8x_2$$ subject to $$2x_1 + 3x_2 \leq 12,$$ $$x_1 + x_2 \leq 5,$$ $$x_1, x_2 \geq 0.$$ 3. **Interpretations for (a):** - i) Objective function coefficients: The coefficients 5 and 8 represent the contribution to the objective value per unit increase in $x_1$ and $x_2$ respectively. Economically, they can be seen as the profit or value gained from one unit of $x_1$ and $x_2$. - ii) First constraint $$2x_1 + 3x_2 \leq 12$$: This represents a resource limitation where each unit of $x_1$ consumes 2 units of a resource and each unit of $x_2$ consumes 3 units, with a total resource availability of 12 units. - iii) Second constraint $$x_1 + x_2 \leq 5$$: This can represent a capacity or limit on the total quantity of $x_1$ and $x_2$ combined, such as a maximum production capacity of 5 units. 4. **Minimization Problem (b):** Minimize $$C = 4y_1 + 6y_2$$ subject to $$3y_1 + 2y_2 \geq 18,$$ $$y_1 + 4y_2 \geq 16,$$ $$y_1, y_2 \geq 0.$$ 5. **Interpretations for (b):** - i) Objective function coefficients: The coefficients 4 and 6 represent the cost per unit of $y_1$ and $y_2$ respectively. Economically, they are the unit costs to be minimized. - ii) First constraint $$3y_1 + 2y_2 \geq 18$$: This represents a minimum requirement or demand that must be met, where $y_1$ and $y_2$ contribute 3 and 2 units respectively towards satisfying this requirement. - iii) Second constraint $$y_1 + 4y_2 \geq 16$$: Similarly, this is another minimum requirement constraint with different weights for $y_1$ and $y_2$. 6. **Summary:** - Objective coefficients represent unit profit (maximization) or unit cost (minimization). - Constraints represent resource limits (\leq) or demand requirements (\geq).