Optimal Workforce Scheduling

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In various industries and organisations, there is often a need to schedule a specific number of employees on different days of the week to meet operational requirements. These requirements may vary from day to day, and different numbers of employees may be needed on different days. Additionally, there are often labour regulations or union rules in place that govern the work schedules of employees.

One common rule is that each employee must work for a consecutive number of days, followed by a specific number of days off. For instance, it may be required that employees work for a fixed number of consecutive days, such as five days, and then receive the remaining days of the week as days off.

To optimise workforce management and minimise costs, organisations seek to find the optimal scheduling solution that meets the daily requirements while minimising the number of employees that need to be hired. By efficiently allocating the available workforce, organisations can reduce labour expenses and ensure efficient utilisation of resources.

The objective is to develop a mathematical model, specifically a linear programming (LP) formulation, that allows organisations to determine the optimal scheduling of employees to meet the daily requirements while minimising the number of employees needed to be hired. This model can be applied across various industries, including retail, healthcare, manufacturing, and transportation, among others.

The LP formulation considers the specific requirements for each day of the week, as defined by the number of employees needed. It takes into account the labour regulations or union rules, such as the maximum consecutive working days and the corresponding number of days off. By formulating the problem as an LP, organisations can find an optimal solution that minimises the number of employees hired while ensuring the fulfilment of daily requirements.

The problem

You often require to hire different numbers of employees on m different days of the week. The number of full-time employees required on each day of the week is often given in the problem specifications. Union rules state that each full-time employee must work some consecutive days and then receive some days off depending on the country your business is operating in. Suppose that you want to meet your daily requirements using only full-time employees. You can formulate a Linear Program that you can use to minimise the number of full-time employees who must be hired.

Mathematical modelling:

To formulate the linear programming (LP) problem, we can define decision variables and constraints to minimise the number of employees hired. Let's define the decision variables and constraints step by step.

Decision Variables:

Let's define the decision variables as follows: x[i, j]: Number of employees starting work on day i and working for j consecutive days.

Objective: minimise the total number of employees hired. Since the goal is to minimise, we can write the objective function as:

minimise: Z = Σx[i, j]

where i represents the day (1 to 7), and j represents the consecutive working days (1 to 5).

Constraints:

Each day's requirement must be met: For each day, the sum of employees working on that day (considering those who have already started working and those who have just started) must be equal to or greater than the requirement. We can write this constraint for each day as: Σx[i, j] ≥ cnstr1[i]

Employee continuity constraint:

Suppose that an employee can work for a maximum of 5 consecutive days (cnstr2) and then must receive 7 - cnstr2 days off. We can enforce this constraint by ensuring that an employee cannot work more than 5 consecutive days. This can be written as: Σx[i, j] ≤ 5 for all i

Employee availability constraint:

Since you want to meet your daily requirements using only employees, we need to ensure that an employee cannot start working on a day unless they have already finished their off days. This can be written as: x[i, j] ≤ Σx[i - 7, k] for all i > 7 and j = 1, where k = j - 1

This constraint ensures that if an employee starts working on day i, they must have finished their off days by day i.

Non-negativity constraint:

The number of employees cannot be negative: x[i, j] ≥ 0 for all i and j

The program:

The complete linear program formulation to minimise the number of hired employees can be written as:

minimise: Z = Σx[i, j]

Subject to: Σx[i, j] ≥ cnstr1[i] for all i = 1 to 7

Σx[i, j] ≤ 5 for all i = 1 to 7, and j = 1 to 5

x[i, j] ≤ Σx[i - 7, k] for all i > 7 and j = 1,

where k = j - 1 x[i, j] ≥ 0 for all i and j

Conclusion:

This optimal scheduling problem has practical applications beyond just a specific scenario, as it can be tailored to suit the unique requirements of different organisations and industries. By using this mathematical model, organisations can make informed decisions regarding their workforce scheduling, leading to increased efficiency, cost savings, and improved overall operational performance.

In this article, we formulated a Linear Program that your company can use to minimise the number of full-time employees who must be hired to meet some operational requirements. Is your company struggling to solve more complex and challenging problems? We can help you find the most suitable mathematical model and the most efficient algorithm for your business problem. Get started now: https://www.paperecos.co.uk/get-started