1. Dynamic Workforce Planning Models
Example 4.3
Dynamic Workforce Planning Models

2. Background Information
CyberLinks is a chain of computer stores.
During the next 5 months the following number of skilled repair hours will be needed: 6000 in January; in February; 8000 in March; 9500 in April; 11,000 in May.
At the beginning of January, CyberLinks employees 50 skilled technicians.
Each skilled technician can work up to 160 hours per month. To meet future demands, new technicians must be trained. It takes one month to train a new technician.

3. Background Information -- continued
During the month of training, a trainee must be supervised for 50 hours by an experienced technician.
Experienced technicians are paid $2000 per month and trainees are paid $1000 per month during their month of training.
Historical data indicate that 5% of the company’s skilled technicians quit each month.
CyberLinks wants to use LP to determine a training schedule that minimizes the cost of meeting the demands for the next five months.

4. Solution
To model the company’s situation, we must keep track of the following:
The number of workers trained each month
The number of experiences workers on hand each month
The number of skilled hours available to meet each month’s needs
The monthly and total costs

5. CYBERLINKS.XLS This file shows the spreadsheet model for this problem.
The spreadsheet figure on the next slide shows the model.

7. Developing the Model To develop this model, proceed as follows.
Inputs. Enter all the input data in the range B4:B9 and in the HrsReqd range.
Number of trainees each month. The only decision CyberLinks must make is the number of trainees to hire each month. This will automatically determine the number of skilled technicians needed each month. Therefore, enter any trial values for trainees in the Trainees range.
Skilled technicians available each month. In cell B14 enter the formula =InitTechs This corresponds to the 50 workers available at the beginning of January. Now let St, Qt, and Tt be, respectively, the number of skilled workers available in month t, the number of quitters at the end of month t, and the number of trainees in month t.

8. Developing the Model -- continued
Then we have the following balance equation: St = St-1 – Qt-1 +Tt-1 This follows because all of the trainees in month t – 1 are skilled by month t, and all but the quitters come back the following month. Use this relationship to computer the number of skilled technicians available during February by entering the following formula =B13+B14-B15 in cell C14. Then copy this formula to the range D14:F14 for the other months.
Quitters each month. Calculate the number who quit at the end of month 1 in cell B15 with the formula =PctQuitters*B14 and copy this ti the range C15:F15 for the quitters in other months.

9. Developing the Model -- continued
Hours for repair each month. Let Ht be the number of hours available for repair in month t. Then the following equation relates the number of hours available for repair to the number of skilled workers and the number of trainees Ht = 160St – 50Tt. Use this relationship to computer the available repair hours for January in cell B18 with the formula =HrsPerTech*B14 – HrsPerTrainee*B13. Then copy this formula to the range C18:F18 for the other months.
Payroll costs. Computer the total payroll costs for skilled technicians and trainees in cells B23 and B24 with the formulas =UnitTechCost*SUM(Technicians) and =UnitTraineeCost*SUM(Trainees). Then sum these to get the total cost in the TotCost cell.

10. Developing the Model -- continued
Using Solver: The Solver dialog box should appear as shown here.

11. Developing the Model -- continued
Objective. Select the TotCost cell as the target cell to minimize.
Changing cells. Select the Trainees range as changing cells.
Skilled technician hour constraint. Enter the constraint HrsAvailable>=HrsReqd. This ensures that enough skilled technician hours are available each month to meet requirements.
Specify nonnegativity and optimize. Under Solver Options, check the nonnegativity box, and use the LP algorithm to obtain the optimal solution shown.

12. Developing the Model -- continued
We see that a minimum cost of $593,777 is obtained by the following training schedule: train 0 in January, in February, in March, 9.52 in April and 0 in May.
If the number of trainees each month is required to be an integer, then we add the constraint that the Trainee range be integers.
Unfortunately, there is still a problem – the number of skilled technicians during each month is still fractional. This is because of quitters.

13. Developing the Model -- continued
When we multiply the number of skilled trainees by 0.05, we typically do not get integers.
One possible remedy is to assume that approximately 5% quit, and implement this with Excel’s ROUND function. Specifically enter the formula =ROUND(PctQuitters*B14,0) in cell B15 and copy it across row 15.
This rounds the first argument to 0 decimals.
This model is now nonlinear, as Solver will remind you if you try to use the LP algorithm.

14. Developing the Model -- continued
The logical solution is to use the GRG nonlinear algorithm.
Although it is not guaranteed to work with models using the ROUND function, it does work here, giving the solution shown on the next slide.
Note that this solution costs a little over $2,000 more than the solution without the integer constraints. Extra constraints cost money!

15. Sensitivity Analysis
One interesting sensitivity analysis is to see how the number of trainees and the total cost vary with the percentage of quitters.
We tried this by using SolverTable on the model that includes the ROUND function and integer constraints, letting the percentage of quitters vary from 0% to 10% in increments of 1%.
The results appear in the top half of the following table.

17. Sensitivity Analysis -- continued
The values in column H are the increases in total cost as the percentage of quitters increases.
Do you believe these “bumpy” results? We didn’t.
We thought it strange that total cost would first increase by $4000, then by $0, then by $4000, and so on.
Therefore, we ran Solver “manually” on each of these problems, inputting each percentage of quitters in the PctQuitters cell.
The results in the bottom half of the previous figure are similar,, but not identical, to the results from the SolverTable. Why the inconsistency?

18. Sensitivity Analysis -- continued
Sooner or later, inconsistent results such as these will probably happen to you, so be ready and willing to do some detective work.
We thought the problem was probably with Solver’s nonlinear algorithm not liking the ROUND function. However, this was not it.
Upon careful inspection, we found that SolverTable uses input values that are slightly off from those specified, at least when the specified values involve decimals.

19. Sensitivity Analysis -- continued
Inconsistencies like these can be a source of frustration or a learning experience. We favor the latter.
Specifically, we learn how sensitive integer models can be to slight changes in inputs.
Evidently, the model is on the edge of rounding up or down. In one case it evidently rounds down, and in the other it evidently rounds up.