Evening shift doctors of Awassa Referral Hospital work five consecutive days and have two consecutive days off. Their five days of work can start on any day of the week and the schedule rotates indefinitely. The hospital requires the following number of doctors working: No more 10than 100 doctors can start their five working days duty on the same day. Formulate the general linear Programming problem model to minimize the number of doctors employed by the hospital

Question

Evening shift doctors of Awassa Referral Hospital work five consecutive days and have two consecutive days off. Their five days of work can start on any day of the week and the schedule rotates indefinitely. The hospital requires the following number of doctors working: No more 10than 100 doctors can start their five working days duty on the same day. Formulate the general linear Programming problem model to minimize the number of doctors employed by the hospital 
Mon  Tues. wed.  Thurs.  Fri.    Sat.   Sun
110   115     120    100      112   108    120

GPT-4o mini · Answer

To formulate a linear programming model for minimizing the number of doctors employed at the Awassa Referral Hospital, we need to define variables, constraints, and the objective function based on the given data.

Decision Variables

Let:

\( x_i \) be the number of doctors starting their five consecutive days of duty on day \( i \), where \( i \) represents the days of the week (1 for Monday, 2 for Tuesday, 3 for Wednesday, 4 for Thursday, 5 for Friday, 6 for Saturday, 7 for Sunday).

Objective Function

We want to minimize the total number of doctors employed: \[ \text{Minimize } Z = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 \]

Constraints

The hospital requires a specific number of doctors to be present on each day of the week. Given the need for doctors on each day, our constraints will reflect these requirements, considering the rotations of the doctors.

Monday Requirement: Doctors starting on Sunday, Monday, and Tuesday cover the Monday shifts: \[ x_7 + x_1 + x_2 \geq 110 \]
Tuesday Requirement: Doctors starting on Monday, Tuesday, and Wednesday cover the Tuesday shifts: \[ x_1 + x_2 + x_3 \geq 115 \]
Wednesday Requirement: Doctors starting on Tuesday, Wednesday, and Thursday cover the Wednesday shifts: \[ x_2 + x_3 + x_4 \geq 120 \]
Thursday Requirement: Doctors starting on Wednesday, Thursday, and Friday cover the Thursday shifts: \[ x_3 + x_4 + x_5 \geq 100 \]
Friday Requirement: Doctors starting on Thursday, Friday, and Saturday cover the Friday shifts: \[ x_4 + x_5 + x_6 \geq 112 \]
Saturday Requirement: Doctors starting on Friday, Saturday, and Sunday cover the Saturday shifts: \[ x_5 + x_6 + x_7 \geq 108 \]
Sunday Requirement: Doctors starting on Saturday, Sunday, and Monday cover the Sunday shifts: \[ x_6 + x_7 + x_1 \geq 120 \]
Non-Negativity Constraints: The number of doctors cannot be negative: \[ x_i \geq 0 \quad \forall i \in {1, 2, 3, 4, 5, 6, 7} \]

Summary of the Linear Programming Model

Objective: \[ \text{Minimize } Z = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 \]

Subject to the constraints: \[ \begin{align*} x_7 + x_1 + x_2 & \geq 110 \ x_1 + x_2 + x_3 & \geq 115 \ x_2 + x_3 + x_4 & \geq 120 \ x_3 + x_4 + x_5 & \geq 100 \ x_4 + x_5 + x_6 & \geq 112 \ x_5 + x_6 + x_7 & \geq 108 \ x_6 + x_7 + x_1 & \geq 120 \ x_i & \geq 0 \quad \forall i \in {1, 2, 3, 4, 5, 6, 7} \end{align*} \]

This linear programming model can be solved using an optimization technique (such as the Simplex method or software tools like LINDO, LINGO, or Excel Solver) to determine the minimum number of doctors the hospital needs to employ while meeting the daily requirements.