Incorporating Data Ambiguity via Distributionally Robust Model

We conduct numerical experiments to illustrate the value of considering distribution ambiguity and solvingDR-CCSP. Section2.5.5.1investigates the sensitivity of CCSPsolutions to distribu- tional assumptions. Section2.5.5.2designs the parameters forDR-CCSPand compares the results with CCSP. Section 2.5.5.3 shows how the data size Nobs affects the performance ofDR-CCSP

2.5.5.1 CCSP solution sensitivity to the distribution type

In addition to Log-Normal, we consider two other distributions, Normal and Gamma, that are commonly used for modeling random service durations. We solve CCSPon the three randomly generated samples {ξω : ω ∈ΩNor(NMC)}, {ξω : ω ∈ΩLog(NMC)}, and {ξω : ω ∈ΩGam(NMC)}, all

with NMC = 100. We set β to 0.95, 0.90 and 0.85. Table 2.6 presents the simulation results of

CCSPsolutions in the simulation sampleΘ.

Table 2.6: Performance of CCSP solutions under different distribution assumptions

Distribution β C

open _overtime

(# of openORs) Prob mean stdev 50% 95% skew

Log-Normal 0.95 2.2 (2) 0.12 13 39 0 56 5.2 0.90 2.1 (2) 0.21 18 46 0 66 4.5 0.85 2.1 (2) 0.32 21 48 0 82 4.0 Gamma 0.95 2.8 (3) 0.18 8 32 0 49 7.1 0.90 2.8 (3) 0.19 12 37 0 53 5.3 0.85 2.7 (3) 0.28 19 42 0 64 4.5 Normal 0.95 2.8 (3) 0.19 9 32 0 50 7.1 0.90 2.7 (3) 0.20 10 34 0 51 6.8 0.85 2.2 (2) 0.31 20 45 0 71 5.3

Distribution β waiting time idle time

mean stdev skew mean stdev skew

Log-Normal 0.95 205.1 60.1 2.3 94 55 0.2 0.90 208.2 71.8 1.5 43 34 0.8 0.85 231.1 89.0 0.7 71 66 0.3 Gamma 0.95 141.5 37.9 1.5 273 63 0.0 0.90 181.4 65.9 1.7 478 48 -0.3 0.85 195.6 71 1.1 456 78 0.0 Normal 0.95 145.7 49.2 2.1 259 57 -0.5 0.90 62.8 23.8 3.4 799 76 -0.3 0.85 94.5 45.1 1.9 540 112 0.0

In Table2.6, both Gamma and Normal distributions yield more conservativeCCSPsolutions than the Log-Normal distribution. In terms of overtime probability and statistical measures (i.e., mean, standard deviation, 50%, 95% quantiles) reported by using the simulation sampleΘ, solu- tions based on Gamma and Normal distributions perform very close to those based on the Log- Normal distribution, but have 10% ∼ 30% higherOR-opening cost. Moreover, the waiting time is shorter but the idling is longer. The results suggest that decision makers need to assume a precise distribution in situations where waiting restrictions are not binding.

We also observe in some cases that under the same Monte Carlo samples but different values of β, the solutions have the same cost of opening ORs. However, the performance of an optimal

Table 2.7: Probability of waiting > iamong all 25 surgeries given by CCSP

Distribution β Avg Max Min Log-Normal 0.95 0.05 0.11 0 0.9 0.06 0.12 0 0.85 0.07 0.14 0.02 Gamma 0.95 0.03 0.06 0 0.9 0.04 0.07 0 0.85 0.04 0.09 0 Normal 0.95 0.03 0.06 0 0.9 0.02 0.04 0 0.85 0.02 0.05 0

CCSPsolution associated with higher β is better. For example, using ΩLog(NMC), both β= 0.90

and β= 0.85 suggest to openOR4 andOR7 at optimum, but the solution provided by β= 0.85 has longerORovertime, waiting, and idleness. We observe similar results when usingΩGamma(NMC)

for β= 0.95 and β = 0.90. These observations suggest that raising β (without changing optimal

ORs to open) may generate operating schedules with better performance.

We also compare the waiting time of each surgery i in every scenario of the simulation sampleΘ with the maximum tolerable waiting i, and calculate the probability of having significant waiting

(i.e., surgery i’s waiting time being longer than i). Table2.7shows that the average and maximum

waiting probabilities of the 25 surgeries for different β-values are well below 0.05 for both Gamma and Normal distributed training samples, but higher when surgery durations of a training sample follow Log-Normal distributions. The results also show zero risk of having significant waiting for most of the surgeries. Therefore, if we require at most 0.10 probability of having significant waiting time, the deterministic upper bound constraints on waiting time in CCSP can produce feasible solutions to the individual chance constraints.

2.5.5.2 Results of DR-CCSP

We test two specific φ-divergences: (i) the modified χ2-distance divergence φ_χ2(t)= (t − 1)2 for a

scalar t ≥ 0, and (ii) the Kullback-Leibler (KL) divergence, where φKL(t)= tlnt − t + 1 for t ≥ 0.

Note that their φ-divergence distances Dφ( f k f0)= Dφ(t) considered as functions of t > 0 satisfy

D_φ

χ2(t) ≥ DφKL(t). We solveDR-CCSPunder these two divergence measures in order to compare the results with the results ofCCSPabove.

According to Theorem 2.1, we can reformulate a DR chance constraint (2.31) as a regular chance constraint (2.32) where the probability is measured based on the empirical distribution function f0, and bounded by Vφ(β, d) which is dependent on the choice of φ-divergence measure,

P

i∈Sdi as described in Section 2.4.3, where di yields an asymptotically 1 − α= 90%-confidence

uncertainty set for each marginalPMF f_i. For φ= φKL and φ= φχ2, we compute

Vφ_χ2(β, d)= β + p d2+ 4d(β − β2_{) − (2β − 1)d} 2d+ 2 VφKL(β, d)= inf t∈(0,1) ( e−d_tβ_{− 1} t −1 ) ,

both of which are obtained based on the general form provided in Theorem2.1. (In particular, the latter is attained through a bisection search procedure.) We obtain the discrete supportΞiof each ξi

by partitioning its range based on intervals of∆i= 5 minutes. Following Section2.5.1, we recover

the empirical marginal distributions f₀i for every i ∈ S and subsequently generate NMC scenarios

with respect to the empirical joint distribution f0.

We then solve the CCSPrepresentation of theDR-CCSP model by using the branch-and-cut algorithm. Table2.8comparesDR-CCSPandCCSPfor β= 0.90 and Ω = ΩLog(NMC). We simulate

the results in the previous simulation sample {ξω: ω ∈Θ}, and also two other simulation samples that involve random surgery cancellations, i.e., {ξω: ω ∈Θ0.005} and {ξω: ω ∈Θ0.01}.

The solutions of DR-CCSPshow better performance in ORovertime and surgery waiting; in most instances, overtime probability, overtime mean, and waiting time mean all decrease by 50%. The DR-CCSP solutions are also more reliable and robust when we introduce random surgery cancellations into the simulation samples. The deteriorations in the overtime probability, and in the means of overtime, waiting time, and idle time are smaller when we implement the solutions ofDR-CCSPthan those when implementing the solutions ofCCSP.

Comparing the two DR-CCSP models, the φKL-divergence produces more conservative so-

lutions. The simulated overtime probabilities by applying those solutions all meet (i.e., being no less than) the probability target β= 0.90, without too much extra cost for opening ORs (i.e.,

2.9−2.7

2.7 × 100%= 7.41% cost increase). As a result, the mean values of ORovertime and surgery

waiting time are shorter, but the idle time means are much longer given by DR-CCSP. In Ta- ble2.6note that theCCSPsolutions with inaccurate distribution assumptions (i.e., the Normal and Gamma distributions) yield OR-opening cost (i.e., Copen) as high as the cost of DR-CCSPsolu- tions. TheCCSPsolutions, however, perform worse in terms of overtime probability and length of

ORovertime.

Table 2.9 shows the waiting probability in post-optimization simulation given by solutions of

CCSPandDR-CCSP. In general, bothDR-CCSP(φ_χ2) andDR-CCSP(φ_KL) produce more reliable

solutions than CCSP, yielding smaller risk of significant waiting of any surgery, on average and at maximum. TheDR-CCSP(φ_χ2) is slightly more conservative and yields smaller risk of waiting

Table 2.8: Comparison of CCSP and DR-CCSP for β= 0.90 and Ω = ΩLog(NMC)

Model Ref. overtime

Prob mean stdev 50% 95% skew

CCSP Θ 0.210 17.6 46 0 66.3 4.5 Copen= 2.1 Θ0.005 0.206 17.2 46 0 64.2 4.6 (2 openORs) Θ0.01 0.202 16.6 45 0 60.9 4.7 DR-CCSP(φ_χ2) Θ 0.131 7.2 30 0 27.1 8.4 Copen= 2.7 Θ0.005 0.129 7.1 30 0 26.8 8.5 (3 openORs) Θ0.01 0.126 6.8 29 0 25.9 8.7 DR-CCSP(φKL) Θ 0.100 5.7 26 0 21.2 7.6 Copen= 2.9 Θ0.005 0.098 5.5 26 0 20.8 7.6 (3 openORs) Θ0.01 0.094 5.3 25 0 20.5 7.8

Model Ref. waiting time idle time

mean stdev skew mean stdev skew

CCSP Θ 208.2 71.8 1.5 42.6 34 0.8 Copen= 2.1 Θ0.005 206.6 71.6 1.5 43.9 34 0.8 (2 openORs) Θ0.01 204.8 71.2 1.5 45.3 35 0.8 DR-CCSP(φ_χ2) Θ 102.6 36.3 2.5 198.9 62 0.0 Copen= 2.7 Θ0.005 101.9 36.3 2.5 201.3 63 0.0 (3 openORs) Θ0.01 101.2 36 2.5 203.4 64 0.0 DR-CCSP(φKL) Θ 130 43.3 1.8 225.2 66 -0.2 Copen= 2.9 Θ0.005 129.2 43.3 1.8 227.4 66 -0.2 (3 openORs) Θ0.01 128.2 43.1 1.8 229.4 67 -0.2 2.5.5.3 Value of data in DR-CCSP

The results of DR-CCSP in the previous section are based on Nobs= 50 observed samples. In-

tuitively, as the number Nobs of observations increases, we can depict the ambiguous probability

distribution of ξ more accurately with a smaller d for restricting the φ-divergence distance. We test data sizes Nobs from 50 to 200, and keep NMC = 500 for optimizing the equivalent CCSP

reformulations of the correspondingDR-CCSPmodels.

We set d = dM according to the procedures in Section 2.4.3, of which the values for

DR-CCSP(φKL) and for DR-CCSP(φχ2) only differ by 2, because φ00

KL(1)= 1 and φ 00

χ2(1) = 2.

Figure 2.2 maps the values of d against the choice of Nobs. Table 2.10 presents the post-

optimization simulation results of the attained solutions of DR-CCSP(φKL) and DR-CCSP(φχ2)

for Nobs= 50,100, and 200 in the simulation sample Θ.

The proposed value for d absorbs the differences in the divergence measures we use and in the size of data observations, so that when we vary Nobs, or switch from using φχ2 to φ_KL, we

have relatively small changes if implementing DR-CCSPsolutions. The results suggest that we can maintain good performance ofDR-CCSPsolutions by using bigger d for fewer observations.

Table 2.9: Probability of waiting > iamong all 25 surgeries given by CCSP and DR-CCSP models

Model Ref. Avg Max Min

CCSP Θ 0.06 0.12 0 Θ0.005 0.06 0.12 0 Θ0.01 0.07 0.14 0.02 DR-CCSP(φ_χ2) Θ 0.03 0.06 0 Θ0.005 0.03 0.06 0 Θ0.01 0.04 0.07 0.01 DR-CCSP(φKL) Θ 0.04 0.08 0 Θ0.005 0.04 0.09 0 Θ0.01 0.05 0.11 0.01 50 100 150 200 0 1 2 3 4

number of observed samples

divergence tolerance d

KL−divergence χ2−divergence

Figure 2.2: Distance tolerances d for KL- and χ2-divergences according to the number of observed samples (Nobs)

In turn, we can avoid over-conservative solutions by decreasing the value of d as the size of the observed data increases.

Table 2.11 confirms the low waiting probability of any solution given by the two DR-CCSP

models, verified via the simulation samples with 10,000 independent scenarios. By solving

DR-CCSPmodels, there is no more than a 0.05 probability of significant waiting on average among the 25 surgeries in post-optimization simulation. We observe no obvious trending for different N_obs.