1. **Problem Statement:** We want to minimize the total number of doctors employed by the hospital while meeting daily doctor requirements and ensuring no more than 100 doctors start their 5-day duty on the same day. 2. **Define Variables:** Let $x_1, x_2, x_3, x_4, x_5, x_6, x_7$ be the number of doctors starting their 5-day duty on Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday respectively. 3. **Duty Coverage:** Each doctor works 5 consecutive days starting from their start day. The coverage for each day is the sum of doctors who started on that day or the previous 4 days (mod 7 for wrap-around). 4. **Constraints:** - Daily requirements: $$ \begin{cases} x_1 + x_7 + x_6 + x_5 + x_4 \geq 110 \\ x_2 + x_1 + x_7 + x_6 + x_5 \geq 120 \\ x_3 + x_2 + x_1 + x_7 + x_6 \geq 115 \\ x_4 + x_3 + x_2 + x_1 + x_7 \geq 120 \\ x_5 + x_4 + x_3 + x_2 + x_1 \geq 100 \\ x_6 + x_5 + x_4 + x_3 + x_2 \geq 112 \\ x_7 + x_6 + x_5 + x_4 + x_3 \geq 108 \end{cases} $$ - Start limit: $$ x_i \leq 100 \quad \text{for } i=1,2,...,7 $$ - Non-negativity: $$ x_i \geq 0 \quad \text{for } i=1,2,...,7 $$ 5. **Objective Function:** Minimize total doctors: $$ \min Z = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 $$ 6. **Summary:** $$ \begin{aligned} &\min Z = \sum_{i=1}^7 x_i \\ &\text{s.t.} \\ &x_1 + x_7 + x_6 + x_5 + x_4 \geq 110 \\ &x_2 + x_1 + x_7 + x_6 + x_5 \geq 120 \\ &x_3 + x_2 + x_1 + x_7 + x_6 \geq 115 \\ &x_4 + x_3 + x_2 + x_1 + x_7 \geq 120 \\ &x_5 + x_4 + x_3 + x_2 + x_1 \geq 100 \\ &x_6 + x_5 + x_4 + x_3 + x_2 \geq 112 \\ &x_7 + x_6 + x_5 + x_4 + x_3 \geq 108 \\ &x_i \leq 100, \quad x_i \geq 0, \quad i=1,...,7 \end{aligned} $$ 7. **Solving Graphically:** This is a 7-variable LP, so graphing all variables is impossible in 2D or 3D. However, we can analyze constraints and use software or linear programming solvers. 8. **Interpretation:** The constraints ensure daily coverage by summing the appropriate $x_i$ variables. The limit $x_i \leq 100$ restricts the number of doctors starting on any day. 9. **Conclusion:** The problem is formulated as a linear program with 7 variables and constraints. Graphical solution is not feasible due to dimensionality. Use LP solvers (e.g., simplex method) to find the minimum total doctors.