Scheduling of the Central Security Checkpoint

GAMS and Excel
This report has been generated using GAMS as the solver and then Excel to process and visualize the results.

In GAMS, two sets has been used. One set for the shifts, and another for the timeslots. Since in GAMS it is not allowed to have “.” or “,” in a set name, the timeslots has been denoted 4:30=45. A file which holds all the parameters has been generated, named parameters.gms. In this file all the data such as a shift cost, the time shift table etc. is present.

Part 1 – Basic model for decision support

Question 1
Which benefits and drawbacks of the two scenarios would you expect considering PlanAir, the employees and their unions?

Scenario one:
Benefits
If the work schedule is produced for every single day of the year according to the forecast, the number of employees needed, will be more accurate than with a “general day” scenario.

Drawbacks
PlanAir will also need a very flexible workforce since it could be hard to have all the employees working full time. It takes a lot of effort to create this schedule because of the many work plans that has to be generated and taken into consideration. Scenario two:

Benefits
It’s easier to calculate the work plan mathematically and requires a less flexible workforce.

Drawbacks
If only the average number of employees are in the schedule, the company
might be some employees short or too many at work on a given day. If there are too many employees working on a shift, then an overstaffing problem arises and that results in a loss of money for PlanAir. If there is understaffing, then it has a negative impact on an employee’s working environment during that particular shift, because they will have to do more work leading to stress.

Question 2
Taking into account the data given for shifts, employees and demand, which rules from page two must be included in a mathematical formulation to ensure a feasible solution? All employees must be assigned a shift.

There are two rules which are necessary to have from page two: 1. The forecasted demand must be satisfied
2. 75 employees are available

The first equation is the objective function where the total cost (z) is aimed for a minimization. Here xs is the number of employees working in shift s, cs is the total cost of one employee working in the same shift, calculated from the prices given within the assignment.

The second equation “Meet demand” ensures that the demand is met in all timeslots (t). TBs,t is the time shift table which indicates whether an employee from shift s is working in timeslot t. “1” will indicate that the employee is working and “0” will indicate that the employee is not working (see Appendix A for the table) and Dt is the demand in timeslot t.

The last equation ensures that all 75 employees are used.

Question 3
What is the best use of the employees? (Use a solver of your choice to generate a solution)

The best use of the employees is shown in Table (if the shift is not mentioned then it has zero employees working it)

Shift
A
F
H
I
J
N
R
S
Employees
12
4
21
7
6
10
9
6
Table Best schedule with 75 employees

Question 4
What is the total cost?

The total cost is € 12674. This has been calculated in Excel using the following formula:

Part 2 – Investigating staff size
Question 5
Which rule(s) from Question 2 must be changed such that only a subset of the 75 employees works?

The second rule is changed so it no longer states that use of exactly 75 employees is required but instead states less than or equal to 75 employees.

Question 6
What is the optimal use of the employees? (Use a solver of your choice to generate a solution). How large is the saving?

The optimal use of the employees is: (again a shift that is not mentioned is not used) Shift
A
D
F
H
I
J
N
R
S
Employees
8
2
6
17
7
6
12
7
6
Table Best schedule with < 75 employees
The total cost is calculated as in question 4 to € 12190 which is a saving of € 484 from the schedule with exactly 75 employees. This reduction is a result of only using 71 employees. Question 7
Which challenges would you anticipate if PlanAir wishes to implement the solution using fewer than 75 employees?

PlanAir would have to fire employees since they won’t be needed anymore. PlanAir also have to consider that by having less people in their workforce they won’t be as flexible. PlanAir would be more sensitive to deviations
from the forecasted demand. Also the Union might intervene if contracts are not followed Part 3 – Over and Understaffing

Question 8
Change the formulation from Question 5 to include over- and understaffing.

In the above, cot and cut are the costs of over- or understaffing in timeslot t. xt+ (xp(t) in GAMS file) and xt- (xm(t) in GAMS file) are the deviational variables that represents the amount of employees over- or understaffed.

Question 9
Suggest appropriate values for cot and cut. Why did you decide on these values?

The formula already attempts to minimize the total cost and as such, increased salary costs are already penalized. In order to find the correct price for understaffing it was decided to look at what effects the staff at work had on the queue. By taking the demand and subtracting the staff at work during the specific timeframes we generate a visual representation of how the queue changes during the day (see Error: Reference source not found). A period with a larger demand than staff at work will have a positive value on the queue flux and vice versa. By taking the sum of the queue flux during the whole day it shows whether the staff meets the demands or not, and thus if the queue will grow or diminish.

Making the iterations of Table , it shows that that 2.2 and 2.1 both had the same staffing and resulted in a sum queue flux of -2. The optimal solution would have a sum of zero and since this number represents how many customers are in queue when the dayshift ends and it is better to have it negative. In Error: Reference source not found it is also possible to notice that the queue fluctuates and that potential queues are diminished within a reasonable timeframe.

Understaffing coefficient
Sum of queue flux
1.5 under
191
3 under
-75
2.2 and 2.1 under
-2
2.0 under
22
Table Results from our iterations

Question 10
What is the optimal work plan? What is the total cost?

The optimal work plan is now calculated in GAMS and shown in Table

Shift
A
D
G
H
I
J
N
O
P
Q
S
Employees
3
3
10
1
15
4
9
2
2
2
3
Table Best schedule with understaffing

This work plan results in a staff of 54 employees and the total cost being € 11119.40, including penalties for over- and understaffing. Question 11
Which benefits and drawbacks do you see with this approach and with your choice of values for cot and cut?

Benefits
The company saves money by allowing understaffing and gain some flexibility in the work plan regarding planning rotational shifts within the workforce.

Drawbacks
By allowing understaffing and penalizing overstaffing the quality of service might go down since customers will have to wait more than necessary.

The airport might impose penalties if queues are too long or result in complaints.

With the selected values for cot and cut it is stated that understaffing is worse than overstaffing. In conclusion the plan of reducing expenses for PlanAir has resulted in (as shown in Figure ) a removal of the headway in the initial plan with 75 employees and a lowered cost by our calculations.

Figure Showing the forecasted demand and people at work in question 6 and 8

Appendix A
Time shift table

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
4:30
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5:00
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5:30
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6:00
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
6:30
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
7:00
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
7:30
1
1
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
8:00
1
0
1
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
8:30
1
1
1
1
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
9:00
1
1
1
1
1
0
1
1
1
1
1
0
0
0
0
0
0
0
0
0
9:30
1
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
0
0
0
0
10:00
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
0
0
0
0
0
10:30
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
11:00
0
1
1
1
1
1
1
1
1
0
1
1
1
0
0
0
0
0
0
0
11:30
0
1
1
1
1
1
1
1
1
1
0
1
1
0
0
0
0
0
0
0
12:00
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
12:30
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
13:00
0
0
1
1
1
1
1
1
1
1
1
0
1
0
0
0
0
0
0
0
13:30
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
14:00
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
14:30
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
15:00
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
15:30
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
16:00
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
16:30
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
1
1
1
1
1
17:00
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
1
1
1
1
1
17:30
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
1
1
1
1
18:00
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
1
1
18:30
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
19:00
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
19:30
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
20:00
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
20:30
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
21:00
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
21:30
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
22:00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
22:30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
23:00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1