1. **State the problem:** We want to minimize the objective function $$Z = 5x_1 + 3x_2$$ subject to the constraints: $$2x_1 + 4x_2 \leq 12$$ $$2x_1 + 2x_2 = 10$$ $$5x_1 + 2x_2 \geq 10$$ $$x_1, x_2 \geq 0$$ 2. **Penalty (Big-M) method overview:** This method converts inequalities and equalities into equations by adding slack, surplus, and artificial variables. Artificial variables are penalized in the objective function with a large positive constant $M$ to force them out of the solution. 3. **Convert constraints:** - For $$2x_1 + 4x_2 \leq 12$$ add slack variable $s_1 \geq 0$: $$2x_1 + 4x_2 + s_1 = 12$$ - For $$2x_1 + 2x_2 = 10$$ add artificial variable $a_1 \geq 0$: $$2x_1 + 2x_2 + a_1 = 10$$ - For $$5x_1 + 2x_2 \geq 10$$ subtract surplus variable $s_2 \geq 0$ and add artificial variable $a_2 \geq 0$: $$5x_1 + 2x_2 - s_2 + a_2 = 10$$ 4. **Formulate the modified objective function:** Since we minimize $Z$, add penalty terms for artificial variables: $$Z = 5x_1 + 3x_2 + M a_1 + M a_2$$ where $M$ is a very large positive number. 5. **Set up the initial simplex tableau:** Variables: $x_1, x_2, s_1, s_2, a_1, a_2$ 6. **Solve using simplex method:** - Start with basic variables $s_1, a_1, a_2$ - Perform pivot operations to remove artificial variables from basis - Continue iterations until no artificial variables remain or their values are zero 7. **Interpret the solution:** - If artificial variables are zero in the optimal solution, the original problem is feasible. - The values of $x_1$ and $x_2$ at optimality minimize $Z$. **Summary:** - Introduce slack, surplus, and artificial variables. - Modify objective with big penalty $M$ for artificial variables. - Use simplex to solve. - Check artificial variables to confirm feasibility. This method ensures constraints are met while minimizing $Z$.