An investigation of shift and break flexibility with real-time break assignments using a rolling horizon approach

Abstract

The purpose of this paper is to investigate the benefits that flexibility offers in daily shift scheduling, especially when demand is uncertain. The different forms of flexibility considered include shift start times, the number of breaks, break lengths, and break placement. Five related mixed-integer programming models are developed and used to compare break scheduling in advance and either sequentially or in real time for various shift and break profiles. The first model integrates the shift and break assignments while the second model assigns shifts only. The third and fourth models assume an optimal set of shifts. In the former case, the goal is to assign as many breaks as possible without violating demand requirements; in the latter case, the objective is to minimize undercoverage while ensuring that all shifts receive their required breaks. The fifth model takes a rolling horizon approach and provides the greatest advantage when responding to unexpected demand fluctuations over the day. Tests were performed using work requirements for ground handlers at a major European airport and showed in part that when demand is known and the most flexible break type is used, the total uncovered periods in a day can be decreased by up to 94% (down from 270 to 16). When demand is uncertain, we found that using our rolling horizon procedure reduces undercoverage by up to 41% compared to scheduling breaks in advance.

Appendices

Appendix A: flight delay update procedure

For the first forecast for flight f, we use the derived distributions to get a delay d_f (which can be positive or negative); for each subsequent forecast, we draw another sample but from a modified distribution with a mean and variance that shrinks geometrically from one iteration to the next. However, we always maintain the “direction” of the delay. If the first delay is positive meaning that the flight will be late, all updated samples are restricted to be positive. For example, if we draw + 30 as the first sample, this means this flight will be delayed 30 min. An hour later at the first update, it is not likely that the forecast would indicate that the flight is now expected to be only, say, 5 min late. This means that we don’t want a sample value of − 25, or more generally, that we want to preserve the sign of subsequent updates for each flight f. The rational for this restriction is empirically based. Once the flight arrival time is updated within its scheduled 4-h window, the historical data indicate that few if any delays shrink. For those rare cases in which the reverse is true, the reductions (or increases) are typically minimal.

Flight_Delay_Update_Procedure

Input :

Solution timespan Δ_s, current period t, arrival distributions for all points of origin

Output :

New arrival times for flights scheduled to arrive in the interval [t, t + Δ_s−1], updated workforce demand

Step 1:

Determine the set of flights that are scheduled to arrival in the interval [t, t + Δ_s−1]; call the set A(t)

Step 2:

For each f ∈ A(t), get its corresponding delay distribution, call it G_f(d), and modify it according to the number of times it has been called previously

Step 3:

Find a sample point d_f in G_f(d)

Step 4:

If this the first update for f, then

add d_f to the scheduled arrival time

Else

add |d_f| to the most recent arrival time

Go to Step 2

Step 5:

Recalculate staffing demand for the new arrival times

The sampling procedure mentioned in Step 2 depends on the particular distribution and its parameters. The details are given in “Appendix B”. Once the new arrival times are found, the staffing demand is recalculated in Step 5 and the next subproblem is solved. That is, we solve the current subproblem, fix the solution for the interval [t, t + Δ_r−1], and move to next iteration [t + Δ_r, t + Δ_r +Δ_s−1]. At this point, Flight_Delay_Update_Procedure is called again.

Appendix B: sampling procedure for arrival time delays

The procedures and modifications for sampling flight delays are a function of the delay distribution, which depends on the point of origin. We begin with the log-logistic distribution F(α, β) for a given scale parameter α and a given shape parameter β. For sampling purposes, we define a minimum value l (< 0), random rational number p, and the update iteration number n with the goal of obtaining a sample delay d. Since the log-logistic distribution is continuous on the set of positive real numbers, we add the minimum value l when we sample to allow for negative delays (i.e., early arrivals).

Sampling_Procedure_Log-logistisc_Distribution

Input :

Log-logistic distribution with known parameters α, β, l, n

Output :

Sample delay d

Step 1:

Generate a random rational number p from U[0, 1]

Step 2:

If n = 1, then

Generate a sample value d such that $d = F^{ - 1} \left( {p;\alpha ,\beta } \right) \, = \alpha (p/(1 - p))^{1/\beta } + l$

Else

Modify parameters based on the number of updates n by putting α ← α/2ⁿ, β ← β·2ⁿ, l ← l/2ⁿ

Generate a sample value d where $d = F^{ - 1} \left( {p;\alpha ,\beta } \right) \, = \alpha (p/(1 - p))^{1/\beta } + l$

Rounding a sample delay d and Return

In the analysis, we used the following parameter values.

Asia:

α = 51.6, β = 3.84, l = −50

America:

α = 74.7, β = 4.73, l = −86

Other:

α = 52.76, β = 4.47, l = −47

For flights originating in Germany and the remaining European countries as a whole, we constructed empirical piecewise linear delay distributions based on the original flight arrival data. Let m be the number of flights in either of the corresponding data sets. The delays associated with each of the m flights are arranged in nondecreasing d₁, d₂,…,d_m, and then grouped into k adjacent intervals [a_j−1, a_j), such that the jth interval contains m_j delays, j = 1,…,k. We define a minimum value l (< 0), maximum value u (> 0), random rational number p, and the update iteration number n. Note that a₀ is the minimum value l and a_k is the maximum value u.

Our piecewise linear empirical distribution function G(d) for delay d is specified as follows: G(a₀) = 0 and G(a_j) = (m₁ + m₂ +···+ m_j)/m. Then, interpolating linearly between the a_j’s, we define

$$G(d) = \left\{ {\begin{array}{ll} 0 & \quad{{\text{if }}d < a_{0} } \\ {G(a_{j - 1} ) + \frac{{d - a_{j - 1} }}{{a_{j} - a_{j - 1} }}[G(a_{j} ) - G(a_{j - 1} )]} & \quad{{\text{if }}a_{j - 1} \le d < a_{j} {\text{ for }}j = 1,2,\ldots,k} \\ 1 & \quad{{\text{if }}a_{k} \le d} \\ \end{array} } \right.$$

(4a)

$$d = a_{j - 1} + \frac{{G(d) - G(a_{j - 1} )}}{{G(a_{j} ) - G(a_{j - 1} )}}[a_{j} - a_{j - 1} ] \,$$

(4b)

Sampling_Procedure_Empirical_Distribution

Input :

Empirical distribution G(d) from Eq. (4a) with known parameters l, u, n along with the intervals [a₀, a₁), [a₁, a₂),…, [a_k-1, a_k]

Output :

A sample delay d

Step 1:

If n > 1, then put l ← l/2ⁿ, u ← u/2ⁿ and a_j ← a_j/2ⁿ, j = 1,…,k \\ first update when n = 1

Step 2:

Draw a random rational number p from U[0,1]

Step 3:

Find the interval a_j-1 and a_j such that G(a_j-1) ≤ p ≤ G(a_j); solve Eq. (4b) after setting G(d) = p to find a sample delay d

In the analysis, we used the following parameter values.

Germany:

l = − 35, u = 97, k = 84 with m = 6579

Europe:

l = − 67, u = 136, k = 98 with m = 16,186

In Sect. 6, we give the Flight_Delay_Update_Procedure which, at Step 3, calls one of the above sampling procedures. For the log-logistic distribution, we multiply the shape parameter β by two and divide both the scale parameter α and minimum value l by two at each update. For the empirical distribution, we divide the minimum value l, the maximum value u and the interval boundary values a_j, j = 1,…,k, by two at each update These modifications have the effect of reducing the variance of the selected distributions so that as n increases the sample value approaches zero.