Example 15.1Daily Scheduling of Postal Employees

Workforce Scheduling Models

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Objective

To use LP, with integer variables, to find the optimal schedule of employees to handle daily staffing requirements.

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Background Information

A post office requires different numbers of full-time employees on different days of the week.

The number of full-time employees required each day is given in the table below.

Minimum Employees Required

Monday 17Tuesday 13Wednesday 15Thursday 19Friday 14Saturday 16Sunday 11

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Background Information -- continued Union rules state that each full-time employee must

work five consecutive days and then receive two days off.

The post office wants to meet its daily requirements using only full-time employees.

Its objective is to minimize the number of full-time employees that must be hired.

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Solution

To model this problem in Excel we must keep track of the following:

– number of employees starting work their 5-day ”shift” on each day of the week

– number of employees working each day

– total number of employees

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POSTAL1.XLS The spreadsheet model for this problem appears

below and can be developed by following these steps:

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Developing the Model– Daily requirements. Enter the number of employees needed on

each day of the week in the MinReqd range.

– Employees starting each day. Enter any trial values for the number of employees starting their 5-day shift on each day of the week in the Starting range.

– Employees on hand each day. The key to this solution is to realize that the numbers in the Starting range do not represent the number of workers who will show up each day. A simple way to see who shows up when, is to develop a table as in rows 14-20. We link the values in the Starting range to the appropriate places in this table. We enter the formula $B$4 in B14 and copy it across to F14. We proceed for the other starting dates in rows 15-20. Then we sum down each column to obtain the number of workers who work each day in row 21.

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Developing the Model

– Total employees. Calculate the total number of employees in cell B25 with the formula =SUM(Starting)

At this point, you might try rearranging the number in the Starting range to see if you can “guess” an optimal solution. It’s not that easy!

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Using Solver

Now that the model has been formulated, the Solver setup is straightforward.

We minimize the total number of employees, subject to having enough employees available to meet the minimal day requirements.

We also constrain the changing cells to be nonnegative and check the Assume Linear Model in the Solver Options dialog box.

Here is the solver dialog box selecting these options:

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Using Solver -- continued

The optimal solution is determined as follows.

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Using Solver -- continued To be realistic all values in the changing cells should

be integers. Fortunately, this can be accomplished quite easily with Solver.

To require any of the cells to be integers, we add an extra constraint in the main Solver dialog box. Now we select “int” instead of <=,=, or >= in the Add Constraint window.

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Using Solver -- continued With this extra constraint added, the optimal integer

solution appears next.

This solution indicates that the post office needs to hire 23 full-time employees, and the Starting range shows how to schedule them.

The solution reveals an interesting aspect common to many optimization problems.

Because of irregular daily requirements and two-consecutive-days-off constraint, no solution can exactly match available workers to daily requirements.

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Solution -- continued Sometimes the optimal solution to a modeling problem isn’t

the “perfect” solution we’re looking for, but it’s the best possible solution.

There is another interesting aspect to this problem. You may get a different solution that is still optimal - that is - it uses a total of 23 employees and meets all constraints. There are at least two optimal solutions and the one you obtain depends on the initial values you use in the changing cells.

This is the case of multiple optimal solutions and is not all uncommon in LP models.

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Solution -- continued

Our model can easily be expanded to handle part-time employees, the use of overtime, alternative objective functions such as maximizing the number of weekend days off received by employees, and other possibilities.

One simple alternative version of the model is illustrated on the next slide.

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POSTAL2.XLS The alternative spreadsheet model appears below .

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Alternative Model

Instead of minimizing the number of workers on the payroll, we now pay a slightly larger wage on the weekend days and minimize the total weekly payroll.

The only difference in this model is that the formula for the total weekly payroll in cell B29 is now

=B4*SUM(B25:F25) + B5*SUM(G25:H25)

Perhaps surprisingly, the optimal solution to this problem is the same as when we treated weekends and weekdays equally.