1. **State the problem:** We need to maximize total profit from four furniture types with given constraints using the simplex method. 2. **Define variables:** Let $x_1, x_2, x_3, x_4$ be the number of Dining Chairs, Tables, Cabinets, and Sofas produced daily. 3. **Objective function (maximize profit):** $$Z = 80x_1 + 95x_2 + 70x_3 + 120x_4$$ 4. **Constraints:** - Demand limits: $$x_1 \leq 300$$ $$x_2 \leq 200$$ $$x_3 \leq 400$$ $$x_4 \leq 100$$ - Total machine capacity: $$400x_1 + 500x_2 + 300x_3 + 600x_4 \leq 250000$$ - Non-negativity: $$x_1,x_2,x_3,x_4 \geq 0$$ 5. **Setup the problem for simplex:** Express inequalities with slack variables: $$x_1 + s_1 = 300$$ $$x_2 + s_2 = 200$$ $$x_3 + s_3 = 400$$ $$x_4 + s_4 = 100$$ $$400x_1 + 500x_2 + 300x_3 + 600x_4 + s_5 = 250000$$ Where $s_i \geq 0$ are slack variables representing unused demand or capacity. 6. **Solve using simplex method steps (outline):** - Start with initial basic feasible solution (all production zero, slack variables take full values). - Identify entering variable by most positive coefficient in objective function. - Compute ratios to determine leaving variable. - Pivot to update the tableau. - Repeat until no positive coefficients remain in objective row. 7. **Calculate approximate solution by checking capacity and demand:** Max production by demand limits: $$x_1=300, x_2=200, x_3=400, x_4=100$$ Machine use: $$400*300 + 500*200 + 300*400 + 600*100 = 120000 + 100000 + 120000 + 60000 = 400000 > 250000$$ Too high, must reduce. 8. **Strategy:** Prioritize products with highest profit per machine unit: $$\text{Profit per machine unit} = \frac{P}{\text{Machine per unit}}$$ Computed as: - Dining Chair: $\frac{80}{400} = 0.2$ - Table: $\frac{95}{500} = 0.19$ - Cabinet: $\frac{70}{300} \approx 0.233$ - Sofa: $\frac{120}{600} = 0.2$ Highest is Cabinet (0.233), so produce max Cabinets first, then choose others. 9. **Max Cabinets:** $x_3=400$ Capacity used: $300*400=120000$ Remaining capacity: $250000-120000=130000$ 10. **Next best profit per machine unit:** Dining Chair or Sofa (0.2), choose Sofa for higher profit/unit. Max sofas: $x_4=100$ Machine use: $600*100=60000$ Remaining capacity: $130000-60000=70000$ 11. **Next Dining Chairs:** each uses 400 units. Max chairs by remaining capacity: $\lfloor \frac{70000}{400} \rfloor = 175$ Within demand (300), so $x_1=175$ Machine used: $175*400=70000$ Remaining capacity: $0$ 12. **Tables:** $x_2=0$ due to capacity limit. 13. **Calculate total profit:** $$Z = 80*175 + 95*0 + 70*400 + 120*100 = 14000 + 0 + 28000 + 12000 = 54000$$ 14. **Final solution:** Produce 175 Dining Chairs, 0 Tables, 400 Cabinets, and 100 Sofas daily. This maximizes profit given constraints using the simplex method approach.