# Workforce Scheduling: Hotel Employees## Problem OverviewA hotel requires minimum number of employees on each day of the week (see table). Each employee works **five consecutive days** with **two consecutive days off**. Employees are paid **$120 per shift**, with a **$10 bonus** for working on Sundays. The objective is to **minimize total labor costs** while ensuring daily staffing requirements are met. | Day | Minimum Number of Employees | |------------|----------------------------| | Sunday | 330 | | Monday | 240 | | Tuesday | 255 | | Wednesday | 350 | | Thursday | 375 | | Friday | 335 | | Saturday | 360 | ## (a) Mathematical Formulation ### **Decision Variables** Let: - \( x_i \): Number of employees whose **first working day is day \( i \)** (where \( i = 1 \) to \( 7 \): 1=Sunday, 2=Monday, ..., 7=Saturday) ### **Objective Function** Minimize **total cost**: \[ \text{Total Cost} = 120 \times \text{Total shifts worked} + 10 \times \text{Total Sunday shifts} \] But since each employee works five shifts, with only those scheduled on Sunday receiving a $10 bonus for that day: \[ \text{Total Cost} = 120 \times 5 \times \sum_{i=1}^{7} x_i + 10 \times (\text{Number of employees working Sunday}) \] But since every employee works 5 days, and we only pay bonus for those working on Sunday, we need to count how many employees work on Sunday. Let’s note that anyone who **does not have Sunday or Saturday off** will work on Sunday. Let’s clarify with the schedule: - Employees starting on **Sunday**: work Sun, Mon, Tue, Wed, Thu - Employees starting on **Monday**: work Mon, Tue, Wed, Thu, Fri - Employees starting on **Tuesday**: work Tue, Wed, Thu, Fri, Sat - Employees starting on **Wednesday**: work Wed, Thu, Fri, Sat, Sun - Employees starting on **Thursday**: work Thu, Fri, Sat, Sun, Mon - Employees starting on **Friday**: work Fri, Sat, Sun, Mon, Tue - Employees starting on **Saturday**: work Sat, Sun, Mon, Tue, Wed #### Therefore, the number of employees working on Sunday = \( x_1 \) (start Sun) + \( x_4 \) (start Wed) + \( x_5 \) (start Thu) + \( x_6 \) (start Fri) + \(_7 \) (start Sat) So: \[ \text{Sunday Workers} = x_1 + x_4 + x_5 + x_6 + x_7 \] #### **Objective Function (Full):** \[ \text{Minimize } Z = 600 \sum_{i=1}^{7} x_i + 10(x_1 + x_4 + x_5 + x_6 + x_7) \] ### **Constraints** #### **Coverage Constraints** On each day, the sum of all employees scheduled to work that day must be at least the minimum required: Let’s define which variables cover each day: - **Sunday**: \( x_1 \) (start Sun), \( x_4 \) (start Wed), \( x_5 \) (start Thu), \( x_6 \) (start Fri), \( x_7 \) (start Sat) - **Monday**: \( x_1 \) (Sun), \( x_2 \) (Mon), \( x_5 \) (Thu), \( x_6 \) (Fri), \( x_7 \) (Sat) - **Tuesday**: \( x_1 \) (Sun), \( x_2 \) (Mon), \( x_3 \) (Tue), \( x_6 \) (Fri), \( x_7 \) (Sat) - **Wednesday**: \( x_1 \) (Sun), \( x_2 \) (Mon), \( x_3 \) (Tue), \( x_4 \) (Wed), \( x_7 \) (Sat) - **Thursday**: \( x_1 \) (Sun), \( x_2 \) (Mon), \( x_3 \) (Tue), \( x_4 \) (Wed), \( x_5 \) (Thu) - **Friday**: \( x_2 \) (Mon), \( x_3 \) (Tue), \( x_4 \) (Wed), \( x_5 \) (Thu), \( x_6 \) (Fri) - **Saturday**: \( x_3 \) (Tue), \( x_4 \) (Wed), \( x_5 \) (Thu), \( x_6 \) (Fri), \( x_7 \) (Sat) So, the constraints are: \[ \begin{align*} x_1 + x_4 + x_5 + x_6 + x_7 &\geq 330 \quad \text{(Sunday)} \\ x_1 + x_2 + x_5 + x_6 + x_7 &\geq 240 \quad \text{(Monday)} \\ x_1 + x_2 + x_3 + x_6 + x_7 &\geq 255 \quad \text{(Tuesday)} \\ x_1 + x_2 + x_3 + x_4 + x_7 &\geq 350 \quad \text{(Wednesday)} \\ x_1 + x_2 + x_3 + x_4 + x_5 &\geq 375 \quad \text{(Thursday)} \\ x_2 + x_3 + x_4 + x_5 + x_6 &\geq 335 \quad \text{(Friday)} \\ x_3 + x_4 + x_5 + x_6 + x_7 &\geq 360 \quad \text{(Saturday)} \\ \end{align*} \] #### **Non-negativity** \[ x_i \geq \quad \text{and integer, for } i = 1,...,7 \] --- ## (b) **Excel Model** Set up variables \( x_1 \) through \( x_7 \) in Excel, and use **Solver**: - **Objective:** Minimize \( 600 \times \text{SUM}(x_1:x_7) + 10 \times (x_1+x_4+x_5+x_6+x_7) \) - **Constraints:** As above (one for each day), and \( x_i \geq \), integer. --- ## (c) **LINGO Model** LINGO code: ```LINGO MODEL: SETS: DAYS /1..7/: x, req, sun_bonus; ENDSETS DATA: req = @TABLE( 1, 330, 2, 240, 3, 255, 4, 350, 5, 375, 6, 335, 7, 360 ); sun_bonus(1) = 1; sun_bonus(2) = ; sun_bonus(3) = ; sun_bonus(4) = 1; sun_bonus(5) = 1; sun_bonus(6) = 1; sun_bonus(7) = 1; ! Coverage matrix: work[i][j] = 1 if shift i covers day j; work(i,1) = @OR(i=1,i=4,i=5,i=6,i=7); work(i,2) = @OR(i=1,i=2,i=5,i=6,i=7); work(i,3) = @OR(i=1,i=2,i=3,i=6,i=7); work(i,4) = @OR(i=1,i=2,i=3,i=4,i=7); work(i,5) = @OR(i=1,i=2,i=3,i=4,i=5); work(i,6) = @OR(i=2,i=3,i=4,i=5,i=6); work(i,7) = @OR(i=3,i=4,i=5,i=6,i=7); ENDSETS ! Objective MIN = 600 * @SUM(DAYS(i): x(i)) + 10 * @SUM(DAYS(i): sun_bonus(i) * x(i)); ! Coverage constraints @FOR(DAYS(j): @SUM(DAYS(i): work(i,j) * x(i)) >= req(j) ); ! Integrality @FOR(DAYS(i): @GIN(x(i))); END ``` --- ## (d) **Recommendation** - **Interpret the solution:** The optimal values for \( x_1, ..., x_7 \) will give you the **number of employees to start on each day**. - This ensures minimum cost while meeting staffing requirements. - Employees should be scheduled according to the optimal start days as determined by the model. --- ## **Final Summary** - **Formulate as an integer linear program**: variables represent number of employees starting each day, objective is to minimize total labor + Sunday bonus. - **Implement in Excel or LINGO** using the model above. - **Recommendation:** Hire and schedule employees so that their start days match the optimal solution to minimize costs and ensure daily coverage. **(Actual numeric solution requires running the model in Excel or LINGO with real solvers.)**