Robust Wait Time Estimation in the US Kidney Allocation System

Transcription

1 Robust Wait Time Estimation in the US Kidney Allocation System CMS 2017 Phebe Vayanos (joint with Bandi and Trichakis) Epstein Department of Industrial & Systems Engineering Viterbi School of Engineering, USC

2 End-Stage Renal Disease source: Number of Patients (in thousands) terminal disease affecting >600,000 patients in U.S. dialysis vs. kidney transplant (preferred) living donors vs. deceased donors Year 2

3 Organ Shortage 100k patients waiting 36k additions per year 19k transplants/year 13.4k (70%) from deceased donors 5.6k (30%) from living donors 3

4 Organ Shortage 3-yr trend 100k patients waiting 36k additions per year 19k transplants/year 13.4k (70%) from deceased donors 5.6k (30%) from living donors 3

5 U.S. Kidney Allocation System 4

6 U.S. Kidney Allocation System

7 U.S. Kidney Allocation System medical compatibility: blood group, weight, etc. geographic proximity (24-36 hours to transplant) point based: wait time, blood antigens: ~FCFS 4

8 Wait Time Estimation 5

9 Wait Time Estimation important for: - dialysis management - planning of daily life activities - accept/reject decisions 5

10 Challenges 6

11 Challenges predicting accept/decline decisions is already hard Kim et al 15: use all available historical data, build series of prediction models (log. reg., SVM, CART, RF); error rates vary 22-47% 6

12 Challenges predicting accept/decline decisions is already hard Kim et al 15: use all available historical data, build series of prediction models (log. reg., SVM, CART, RF); error rates vary 22-47% in practice: incomplete information: other patients preferences unstable/ non-stationary system 6

13 Resource Allocation System multiclass, multiserver (MCMS) queuing system servers: resource types customer classes/queues: preferences 7

14 Resource Allocation System multiclass, multiserver (MCMS) queuing system servers: resource types customer classes/queues: preferences 7

15 MCMS under FCFS 8

16 MCMS under FCFS arrival order of customer ( ) 2 {1,..., P i N i} 8

17 MCMS under FCFS arrival order of customer ( ) 2 {1,..., P i N i} W i (N 1,...,N K,, X 1,...,X M ) clearing time of queue i 8

18 Model of Uncertainty service times: ( X j = x j 2 R `j : `X k=1 ) x k j apple ` + X j (`) 1/ j, ` =1,..., `j µ j 9

19 Model of Uncertainty service times: X j = ( x j 2 R `j : `X k=1 population vector n 2 P \ N K ) x k j apple ` + X j (`) 1/ j, ` =1,..., `j µ j 9

20 Model of Uncertainty service times: ( X j = x j 2 R `j : population vector `X k=1 ) x k j apple ` + X j (`) 1/ j, ` =1,..., `j µ j n 2 P \ N K arrival order 2 (n) 9

21 Robust Wait Times W i : maximize W i (n 1,...,n K,,x 1,...,x M ) subject to n 2 P \ N K 2 (n) x j 2 X j, j =1,...,M 10

22 Robust Wait Times W i : maximize W i (n 1,...,n K,,x 1,...,x M ) subject to n 2 P \ N K 2 (n) x j 2 X j, j =1,...,M robust wait time estimation problem is NP-hard 10

23 Robust Wait Times W i : maximize W i (n 1,...,n K,,x 1,...,x M ) subject to n 2 P \ N K 2 (n) x j 2 X j, j =1,...,M robust wait time estimation problem is NP-hard no tractable expression for W i - Lindley equations break down 10

24 Robust Wait Times W i : maximize W i (n 1,...,n K,,x 1,...,x M ) subject to n 2 P \ N K 2 (n) x j 2 X j, j =1,...,M robust wait time estimation problem is NP-hard no tractable expression for W i - Lindley equations break down key idea: model assignment of servers to customers - : th service from server assigned to class y`kj ` j k 10

25 Robust Wait Times assignment-style formulation maximize subject to w X i k X y`kj apple 1, X `,j k 0 w k apple c`j + f `kj y`k 0 j f `kj w k c`j 1 y`kj y`kj apple n k (c, n) 2 uncertainty sets, (y, f) binary 11

26 Performance: Accuracy estimation error vs simulation statistics avg. 95-%ile 97-%ile 99-%ile avg. abs. rel. error 6.52% 2.64% 2.55% 3.41% 12

27 Performance: Accuracy estimation error vs simulation statistics avg. 95-%ile 97-%ile 99-%ile avg. abs. rel. error 6.52% 2.64% 2.55% 3.41% estimation error when true distribution = assumed avg. queue population simulation avg. abs. rel. error 21% 15% 12% our avg. abs. rel. error 13% 9% 7.5% 12

28 Hierarchical MCMS hierarchy across resource types e.g., different quality service level radiation therapy organ quality server j provides jth ranked service induces threshold-type customer preferences 13

29 Hierarchical MCMS nested structure enables to strengthen formulations robust wait time for service of any rank W K problem remains NP-hard 14

30 Wait Time for Service in HMCMS Lemma. For HMCMS systems: W K increasing in completion times completion times can be fixed to their worst-case values: c`j = x 1 j x`j = ` + X µ j (`) 1/ j j 15

31 Wait Time for Service in HMCMS Lemma. For HMCMS systems: W K increasing in completion times completion times can be fixed to their worst-case values: c`j = x 1 j x`j = ` + X µ j (`) 1/ j j assignment of servers to customers y`kj ` j k - : th service from server assigned to class - : time th service from server starts c`j ` j 15

32 Wait Time for Service in HMCMS Lemma. For HMCMS systems: W K increasing in completion times completion times can be fixed to their worst-case values: c`j = x 1 j x`j = ` + X µ j (`) 1/ j j assignment of servers to customers y`kj ` j k - : th service from server assigned to class - : time th service from server starts c`j ` j 15

33 Scalable Heuristic 16

34 Scalable Heuristic view so far: individual assignments y`kj - scales with n 16

35 Scalable Heuristic view so far: individual assignments y`kj - scales with n alternative view: - aggregate assignments m j - independent of n 16

36 Scalable Heuristic view so far: individual assignments y`kj - scales with n alternative view: - aggregate assignments m j - independent of n cw K maximize subject to w w apple m j + X j (m j ) 1/ j µ j KX KX m k apple n k + K j k=j n 2 P k=j 16

37 Scalable Heuristic view so far: individual assignments y`kj - scales with n alternative view: - aggregate assignments m j - independent of n cw K maximize subject to w w apple m j + X j (m j ) 1/ j µ j KX KX m k apple n k + K j k=j n 2 P k=j 16

38 Approximation Guarantee exact robust wait time W K cw K approximation let = max j 1 µ j + X j for a hierarchical MCMS system, W K apple c W K apple W K +2 17

39 Approximation Guarantee exact robust wait time W K cw K approximation let = max j 1 µ j + X j for a hierarchical MCMS system, W K apple c W K apple W K +2 n 17

40 Heuristic: Performance computation times for different HMCMS instances general MIP simpler MIP SOCP 100 customers 1 sec 0.8 sec 0.8 sec 1,000 customers < 1 min < 1/2 min 1.2 sec 10,000 customers 6 min 2 min 5.4 sec 100,000 customers 40 min 10 min < 1 min 18

41 Heuristic: Performance computation times for different HMCMS instances general MIP simpler MIP SOCP 100 customers 1 sec 0.8 sec 0.8 sec 1,000 customers < 1 min < 1/2 min 1.2 sec 10,000 customers 6 min 2 min 5.4 sec 100,000 customers 40 min 10 min < 1 min heuristic approximation errors 50 customers 1.9% 100 customers 0.85% 1,000 customers 0.08% 18

42 Application to the KAS 19

43 Application to the KAS PADV-OP1 Gift of Life Donor Program threshold type decisions model as HMCMS 19

44 Available Data well accepted kidney quality metric: KDPI historical kidney procurement rates (for each quality) historical patient accept/decline decisions training set testing set 20

45 Out-of-Sample Performance 2000 time to first offer (in days) percentile 97-percentile 95-percentile 68-percentile average patient rank 21

46 Out-of-Sample Performance relative prediction errors % for avg. and 11.73% for 68-percentile delay history estimator: - uses personalized info unavailable in practice - cannot estimate wait times for high ranks relative prediction errors of delay history estimator: % for avg. and 14.65% for 68-percentile 22

47 Summary modeling framework for MCMS systems under incomplete information MIP formulation - more structure: provably tight scalable heuristic FCFS, class priority application to U.S. kidney allocation system 23

48 Thank you!

49 Model Calibration cluster kidneys in quality levels service time uncertainty sets: for each quality level j ( ) historical procurement rate (std) 25

50 Model Calibration: Queue Populations 26

51 Model Calibration: Queue Populations patient is type, observes rank and historical accept/decline decisions 26

52 Model Calibration: Queue Populations patient is type, observes rank and historical accept/decline decisions probability of a patient being type 26

53 Model Calibration: Queue Populations patient is type, observes rank and historical accept/decline decisions probability of a patient being type fit to maximize likelihood of observed decisions 26

54 Model Calibration: Queue Populations patient is type, observes rank and historical accept/decline decisions probability of a patient being type fit to maximize likelihood of observed decisions CLT-based approach: r 1 X L apple (r 1)µ L + L p r 1 =1 class of th patient K X i=1 iq i K X i=1 i 2 q i µ 2 L 26

55 Model Calibration: Queue Populations patient is type, observes rank and historical accept/decline decisions probability of a patient being type fit to maximize likelihood of observed decisions CLT-based approach: P = ( n 2 R K : ) KX p in i k apple (r 1)µ L + L r 1 i=1 26

56 Extensions motivation: recent policy change: top 20% of healthier patients have priority for top 20% kidneys modeling implications: alternative priority rule: class priority customer arrivals all of our results can be extended!! 27

57 HMCMS under Class Priority 28

58 HMCMS under Class Priority model arrival times similar to service times 28

59 HMCMS under Class Priority model arrival times similar to service times arrival time of `th customer in class k 28

60 HMCMS under Class Priority model arrival times similar to service times arrival time of `th customer in class k y`kj constraints on assignments reflect priority rules 28

61 HMCMS under Class Priority model arrival times similar to service times arrival time of `th customer in class k y`kj constraints on assignments reflect priority rules e.g., if servers indexed in descending priority 28

62 HMCMS under Class Priority model arrival times similar to service times arrival time of `th customer in class k y`kj constraints on assignments reflect priority rules e.g., if servers indexed in descending priority all our results can be extended 28