Bus Company Believes It Will Need The Following Numbers

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The bus company project involves determining optimal strategies for hiring, firing, and managing drivers over a five-year period to minimize costs. The key components include calculating yearly driver requirements, hiring and firing costs, and salary expenses, with the initial number of drivers known. Additionally, the project explores how adjustments in hiring and firing costs impact total costs through sensitivity analysis using SolverTable.

Paper For Above instruction

Effective management of workforce planning and cost minimization is crucial for transportation companies such as bus operators. The challenge involves determining optimal hiring and firing strategies over a multi-year horizon, considering various costs and constraints. Using formal optimization techniques, the question seeks to establish a cost-efficient approach that balances salary expenses with the costs associated with hiring new drivers and firing existing ones. This problem can be approached as a mixed-integer linear programming (MILP) problem, where decision variables include the number of drivers to hire or fire in each period, subject to the constraint of meeting the projected driver requirements each year.

Initially, the problem provides the data: driver requirements for five consecutive years—60 in year 1, 70 in year 2, 50 in year 3, 65 in year 4, and 75 in year 5. The company starts with 50 drivers at the beginning of year 1. The costs associated with hiring a driver are $4,000, firing costs are $2,000, and annual salary per driver is $45,000. The goal is to determine the number of drivers to hire or fire annually that minimizes total costs, which include salaries, hiring, and firing expenses.

Cost Components and Decision Variables

The total cost per year comprises three parts: salaries for active drivers, hiring costs for new drivers, and firing costs for layoffs. Formally, if Ht is the number of drivers hired in year t, Ft is the number of drivers fired in year t, and Dt is the number of drivers at the start of year t, then:

• Salaries cost in year t: 45,000 * Dt
• Hiring cost in year t: 4,000 * Ht
• Firing cost in year t: 2,000 * Ft

The driver count updates as: Dt+1 = Dt + Ht - Ft, with D1 = 50. The constraints also include meeting yearly demand: Dt+1 ≥ required number of drivers in year t+1.

Mathematical Modeling and Optimization

Formulating the problem involves defining decision variables for each period: number of drivers hired (Ht), fired (Ft), and drivers remaining (Dt). The objective function aims to minimize the sum of salary costs, hiring, and firing costs over the five-year period:

Minimize: ∑ (over t=1 to 5) 45,000 Dt + 4,000 Ht + 2,000 * Ft

Subject to the constraints:

• Dt+1 = Dt + Ht - Ft for t = 1 to 5, with D1 = 50
• Dt+1 ≥ required drivers for year t+1
• Ht, Ft ≥ 0 and integer

One can solve this using an algorithmic approach like dynamic programming or commercial solvers such as Microsoft Excel Solver, setting the variables and constraints accordingly.

Sensitivity Analysis with SolverTable

To explore how changes in hiring and firing costs influence the total outcome, SolverTable can be used. The analysis involves gradually increasing hiring and firing costs by the same percentage and observing the resultant total costs, total number of drivers hired, and fired. This provides insight into the cost sensitivity and helps in strategic decision-making regarding cost fluctuations.

Implementation and Results

Implementing this model within Excel's Solver involves setting decision variables for each year, defining the objective and constraints, and running solutions iteratively. Results typically reveal the optimal hiring and firing pattern, often with surplus drivers in some years to reduce firing costs or minimized hiring spreads to limit expenses. Sensitivity analysis may show that as hiring and firing costs rise proportionally, the optimal strategy shifts towards less hiring or firing to avoid high costs, leading to more stable staffing levels.

Conclusion

By formulating the problem as an optimization model, the bus company can effectively minimize costs associated with staff management over a five-year horizon. Sensitivity analysis via SolverTable enhances understanding of how cost parameters influence staffing policies, enabling better strategic planning. Deploying these tools ensures cost-efficient operations while meeting operational demands across multiple years.

References

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• Microsoft Excel Developer Resources. (2022). Using Solver and SolverTable. Microsoft Support.
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• Green, S., & Keller, G. (2019). Cost Minimization Techniques in Transportation Management. Transportation Research Part E, 125, 78-89.
• Gass, S. I. (2017). Linear Programming: Methods and Applications. Dover Publications.
• Hiller, F. S., & Lieberman, G. J. (2020). Introduction to Operations Research. McGraw-Hill Education.