Optimization procedure

1 if W_t,r > τ, 0 otherwise .

The probability of excessive waiting at time t then can be estimated as:

Pr(Wt> τ ) = 1

The ISA(τ ) algorithm starts with an initial staffing solution s^init that is not necessarily feasible. In our experiments, the capacity in each staffing interval is set equal to the staffing solution obtained by applying the SRS rule² to each interval, using the lagged SIPP arrival rate and β that results from the Garnett Delay function for M/M/s + M systems [88].

A first phase in the algorithm (Phase I or “exploration phase”, sum-marized in Algorithm 4.1) aims at quickly finding a staffing vector for which

2We also obtained good results if the capacity in each staffing interval is set equal to

⌈¯λ/µ⌉, with ¯λ the average arrival rate, especially for small-scale systems.

performance is close to the target (but not necessarily below the target value at all times). To this purpose, the current staffing level function s_is,k is altered iteratively, based on the simulation output.

During each iteration k of the algorithm, s_is,k is updated as follows:

s_is,k+1=

{ ⌈sis,kA_is,k⌉ if A_is,k≥ 1, ∀ is∈ Is

⌊sis,kAis,k⌋ if Ais,k< 1, ∀ is∈ Is

(4.5) where A_is,k refers to an amplification factor, which is determined based on the deviation between P_i^max

s,k and the target α (in percent):

A_is,k = 1 +P_i^max_s,k − α

αk ∀ is∈ Is, (4.6)

with P_i^max

s,k derived from the simulation results, as described in Section 4.2.2.

Values P_i^max

s,k (above) target will result in an A_is,k below (above) 1 and thus a decrease (increase) in capacity in the corresponding interval (note that due to the rounding in Expression 4.5, capacity is always increased or decreased with at least one unit).

The use of the scaling factor k in the denominator of Expression 4.6 ensures that A_is,k approaches 1 (for all i_s) as the number of iterations in-creases. This forces the algorithm to decrease the size of the staffing changes as it progresses, eventually switching to unit-size changes and converging to a final staffing vector (despite the fact that possible deviations from the target may still remain). In fact, the choice of the scaling factor in the denominator of Expression 4.6 is rather arbitrary; other factors may be con-sidered, such as k/2 or k². Especially in large-scale systems, high values for the factor should be used with caution: they may lead the algorithm to switch to unit-size capacity changes too soon. This issue is illustrated numerically in Section 4.3.1.

We allow the algorithm to stop exploring when cycling occurs (meaning that the staffing vector put forward in the current iteration has already been assessed during a previous iteration)³. This stop criterion usually yields

3Recall that the original ISA uses a different terminating condition: it stops when the staffing level changes at most with one unit in each interval, compared to the previous iteration. We opt not to use this stop criterion due to the focus on small system sizes, where one unit capacity changes may occur more frequently (causing the algorithm to stop prematurely).

Algorithm 4.1 ISA(τ ): Phase I : Exploration

Initial staffing vector: s0= s^init Initialize stop criterion: stop← false Initialize iteration counter: k← 0 while stop = false do

k← k + 1

Simulate staffing vector sk(result: performance vector P_i^max_s,k) Update capacity in all is∈ Is:

Ai_s,k← 1 +^Pis,kmax_αk^−α s_is,k+1←

{ ⌈sis,kAis,k)⌉ if Ais,k≥ 1, ∀ i^s∈ I^s

⌊si_s,kAi_s,k⌋ if Ai_s,k< 1, ∀ i^s∈ I^s Determine ¯Pkand MAk

if∃j < (k + 1)| ∀ is: si_s,j= s_is,k+1then

stop← true; repetition in staffing levels, so proceed to Phase II else if ∀ j = k, k − 1, . . . , k − 4 : ¯Pj∈ [MAk− 0.025, MAk+ 0.025] then

stop← true; Performance is stabilizing, so proceed to Phase II

good results for small-scale systems.

For large-scale systems, we propose to use an additional stop criterion.

We observed that in these systems, many iterations may be needed before cycling in the staffing levels occurs, while the excess wait probability usually stabilizes far more quickly. We thus suggest to keep track of the average probability of excessive waiting over the time horizon in each iteration ( ¯Pk) and stop the algorithm if the most recent values of ¯P_k consistently are close to the moving average of ¯P_k. Formally, the algorithm terminates if ∀ j = k, k− 1, ..., k − 4 : ¯Pj ∈ [MAk− 0.025, MAk+ 0.025], with MAk the moving average of ¯P_k over the past 10 iterations. An illustration of the typical evolution of MA_kand ¯P_k throughout the algorithm in a large-scale problem setting is given in Figure 4.2. The ranges around MA7and MA13are plotted;

the algorithm terminates after iteration 13 because ¯P₉to ¯P₁₃do not deviate more than 0.025 from MA13 (this criterion is not yet met in the previous iterations, e.g., in iteration 7). For large-scale systems, this additional stop criterion can substantially lower the computational time in Phase I.

Note that the exploration phase does not necessarily result in a feasible staffing vector. Consequently, an additional fine-tuning procedure (Phase

0

Average probability of excessive waiting

STOP

ITERATION 13 Last 5 observations fall within

[MA13  0.025,MA13  0.025]

ITERATION 7 No observation falls within [MA7  0.025,MA7 + 0.025]

Figure 4.2: Additional stop criterion

II or “exploitation phase”, see Algorithm 4.2) is needed. Phase II derives feasible solutions from the infeasible staffing levels obtained in Phase I, in hope of finding a solution which outperforms the best feasible solution found so far (if any) in terms of labor cost. To that end, all infeasible solutions encountered during Phase I are first sorted based on increasing maximum excess wait probability over the time horizon (i.e., increasing val-ues of max_is{

P_i^max_s,j }

, where index j denotes an infeasible solution). A lower value indicates smaller deviations from the target, which makes the solution more promising to fine-tune. In case of a tie, the number of intervals with a performance constraint violation is examined. If the target is exceeded in just a limited number of intervals, a small number of capacity increases may be sufficient to obtain a feasible staffing level vector, which is appealing from a labor cost perspective.

We calculate the staffing cost that results when adding one unit of capacity to each staffing interval that causes the excess wait probability to surpass the target (we call this the projected staffing cost). Thus, in case of a tie in the maximum excess wait probability, infeasible solutions are sorted based on increasing projected staffing cost.

Next, all infeasible solutions are improved one by one (in sorted or-der). For a given infeasible solution j, an improved staffing vector s^′_j (with

corresponding cost c^′_s,j) is constructed by adding one unit of capacity in all staffing intervals where performance is unsatisfactory. The new staffing vector’s performance is evaluated through simulation, provided that its cost is strictly lower than c^∗_s (further exploiting the current infeasible solution is futile if c^′_s,j ≥ c^∗s). Based on the simulation results, two cases can be distinguished:

• The performance constraint is not yet met, in which case the exploita-tion continues. Vector s^′_j is improved further (unit size capacity in-creases are made) and its performance is simulated (if the cost is lower than c^∗_s).

• The performance constraint is satisfied at all times; in this case a new feasible solution is found, which is stored if it is less costly than the current best feasible solution in terms of labor cost. The exploitation of solution j is terminated and the algorithm then proceeds to the next infeasible solution in the sorted list.

Note that the procedures described in Phase I and Phase II are suitable for small-scale as well as large-scale systems, largely avoid cyclic behavior and moreover guarantee that the algorithm yields a staffing vector meeting the performance constraint.