Question 1170863: A famous restaurant chain decided to open three new branches in your city: one in downtown area (DT), one in a residential area (RA), and one in a touristic area (TA). All branches will open daily from 12pm to 10pm. According to a study done by the analyst team, the minimum number of employees needed in each branch at each hour is shown in table 1.
Table 1
Time Branch 1 (DT) Branch 2 (RA) Branch 3 (TA)
12pm-1pm 12 9 5
1pm-2pm 12 9 5
2pm-3pm 11 8 6
3pm-4pm 9 8 6
4pm-5pm 7 9 8
5pm-6pm 6 10 8
6pm-7pm 6 10 10
7pm-8pm 5 11 12
8pm-9pm 5 9 12
9pm-10pm 4 7 10
The restaurant chain management offered three types of jobs to hire employees for the three new branches: 2- hours part timers, 4-hours part timers, 8-hours full timers. The 2-hours part timers are only available to work after 4pm.
The average cost per employee hour is different for each type and each branch (Cost includes salaries, insurance, transportation, etc.). Table 2 shows the average cost per employee hour ($/hour) for each branch.
Table 2
Employee type Branch 1 Branch 2 Branch 3
A (2-hours) 16 10.5 12
B (4-hours) 14 9.5 11
C (8-hours) 12 8 9
The number of applicants who applied for each type of jobs is 30 for the 2-hours part time jobs, 35 for the 4-hours part time jobs, and 15 for the full-time jobs. All applicants agreed to work in any of the new branches if hired.
The restaurant chain management has a policy that there should be at least one full time employee available at each branch when the branch closes at the end of the day.
Your team was chosen by the restaurant chain management to consult them to find the best action for this case. The type of analysis that they are looking for is:
1- What would be the number of employees required from each type (i.e. 2-hour part timers, 4-hours part timers, and 8-hours full timers) in each branch that will minimize the total daily cost? What would be the daily cost in each branch in this case?
2- How many extra employee hours will be available in each branch at each hour after the required hours are fulfilled?
3- How much could the company saves daily if five more full-time employees were available to hire?
4- Which branch costs the most for satisfying the policy of having one full time employee available at the end of the day?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! This is a classic linear programming problem, but with some extra constraints and calculations. Let's break down each part of the analysis.
**1. Minimizing Total Daily Cost**
This part requires setting up and solving a linear programming model.
**Variables:**
* DT_A: Number of 2-hour part-timers in Downtown (DT)
* DT_B: Number of 4-hour part-timers in DT
* DT_C: Number of 8-hour full-timers in DT
* RA_A: Number of 2-hour part-timers in Residential Area (RA)
* RA_B: Number of 4-hour part-timers in RA
* RA_C: Number of 8-hour full-timers in RA
* TA_A: Number of 2-hour part-timers in Touristic Area (TA)
* TA_B: Number of 4-hour part-timers in TA
* TA_C: Number of 8-hour full-timers in TA
**Objective Function (Minimize Total Cost):**
Minimize Z = 16(DT_A) + 14(DT_B) + 12(DT_C) + 10.5(RA_A) + 9.5(RA_B) + 8(RA_C) + 12(TA_A) + 11(TA_B) + 9(TA_C)
**Constraints:**
* **Employee Availability:**
* DT_A + RA_A + TA_A <= 30 (2-hour part-timers)
* DT_B + RA_B + TA_B <= 35 (4-hour part-timers)
* DT_C + RA_C + TA_C <= 15 (8-hour full-timers)
* **Hourly Employee Requirements (for each branch and each hour):**
* **DT:**
* 12pm-1pm: DT_C >= 12
* 1pm-2pm: DT_C >= 12
* 2pm-3pm: DT_C >= 11
* 3pm-4pm: DT_C >= 9
* 4pm-5pm: DT_B + DT_C >= 7
* 5pm-6pm: DT_A + DT_B + DT_C >= 6
* 6pm-7pm: DT_A + DT_B + DT_C >= 6
* 7pm-8pm: DT_A + DT_B + DT_C >= 5
* 8pm-9pm: DT_A + DT_B + DT_C >= 5
* 9pm-10pm: DT_A + DT_B + DT_C >= 4
* **RA:**
* 12pm-1pm: RA_C >= 9
* 1pm-2pm: RA_C >= 9
* 2pm-3pm: RA_C >= 8
* 3pm-4pm: RA_C >= 8
* 4pm-5pm: RA_B + RA_C >= 9
* 5pm-6pm: RA_A + RA_B + RA_C >= 10
* 6pm-7pm: RA_A + RA_B + RA_C >= 10
* 7pm-8pm: RA_A + RA_B + RA_C >= 11
* 8pm-9pm: RA_A + RA_B + RA_C >= 9
* 9pm-10pm: RA_A + RA_B + RA_C >= 7
* **TA:**
* 12pm-1pm: TA_C >= 5
* 1pm-2pm: TA_C >= 5
* 2pm-3pm: TA_C >= 6
* 3pm-4pm: TA_C >= 6
* 4pm-5pm: TA_B + TA_C >= 8
* 5pm-6pm: TA_A + TA_B + TA_C >= 8
* 6pm-7pm: TA_A + TA_B + TA_C >= 10
* 7pm-8pm: TA_A + TA_B + TA_C >= 12
* 8pm-9pm: TA_A + TA_B + TA_C >= 12
* 9pm-10pm: TA_A + TA_B + TA_C >= 10
* **End of Day Full-Time Constraint:**
* DT_C >= 1
* RA_C >= 1
* TA_C >= 1
* **Non-negativity:**
* All variables >= 0
**Solving this Linear Program:**
This requires using a linear programming solver (like Excel Solver, Python's `scipy.optimize.linprog`, or a dedicated LP software).
**2. Extra Employee Hours**
Once the optimal solution is found, calculate the extra employee hours for each hour in each branch. For example, if the model says DT_C = 13, and the requirement for 12pm-1pm is 12, then there is one extra employee-hour.
**3. Savings with 5 More Full-Time Employees**
* Increase the constraint for full-time employees: DT_C + RA_C + TA_C <= 20
* Re-solve the linear program.
* Calculate the difference in the total cost between the original solution and the new solution.
**4. Branch Costing Most for End-of-Day Policy**
* Examine the optimal solution and determine which branch has the highest cost associated with its full-time employees (DT_C, RA_C, TA_C). Multiply the number of full time employees in each branch by the cost per hour, and then by 8. Then compare the results.
**Important Notes:**
* This problem requires a linear programming solver.
* The results will depend on the optimal solution found by the solver.
* The exact numbers will depend on the solver, but this process will provide the solution.
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