Outpatient Appointment Scheduling with Different Arrival Rates of Walk-ins in Taiwan Fenghueih Huarng1 & Ming-Te Chen2 1. Dept. of Business Adm, Southern Taiwan Univ. of Technology 2. Dept. of Business Adm, National Chung Cheng Univ. Why Walk-in? Patient’s habit (Taiwan starts pre-register since 1980.) Lack of good appointment system— ‧pre-register given only sequence number ( no appointed time) ‧late penalty for pre-register ( every 10th, more 3,etc.) Different clinic nature ‧Fetter & Thompson (1996) — two air force hospitals. average 37% walk-in, pediatric 55~58% walk-in and call-in, urology 7~11%, dermatology 37.5% —clinic TV pediatric 15.2% walk-in, 42.7% call-in ‧Babes & Sarma (1991) — Algeria ‧Liu & Liu (1998) — Hong Kong Motivation Lack of good pre-registration system High percentage of walk-in Time lag between registration & consultation ‧accumulation of walk-in patients before consultation ‧schedule late arrival of first pre-registered Understand the impacts of walk-in arrival rate & preregister ratio Simulation Setting Register 8:00 Am ~ 11:30 (210min) Consult 8:30 Am ~ 12:00 Noon (210min) # of patient per session (N): 20( s =10.5 min), 40 ( s =5.25 min), 60( s =3.5 min) Service-time distribution: exponentially, cv =1 s uniformally , cv = 0.2 s No-show ratio: ρ= 0.1, 0.2 Pre-registration ratio: α = 0.3, 0.5, 0.7 Walk-in arrival rate: λ=1.5, 2.0 mean inter-arrival time depends on (N,α) λ =1.5 λ =2.0 λ =1.0 N\α 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 20 10 14 23.33 7.5 10.5 17.5 15 21 35 40 6.67 9.33 15.55 5 7 11.67 10 14 23.33 60 3.33 4.67 7.78 2.5 3.5 5.84 5 7 11.67 Benchmark ASR (given even # to pre-register) A2 s If 0.5 A2 k A2 k 2 2s ........... k : 2 ~ N If 0.5 A2 k A2 k 2 2 s ............ k : 2 ~ (1 ) N Ai Ai 1 s ............. i: 2 1 N 1 ~ N Note: (1) the patient with least sequence # has highest priority (2) no penalty for pre-register ( punctuality assumption) (3) the best rule has been used in practice in Taiwan Table Ⅰ . Experimental results using benchmark ASR. UNIT=MINUTE N α ρ λ 20 0.3 0.1 1.5 2 μs 0.2 1.5 = 2 10.5 0.5 0.1 1.5 2 0.2 1.5 2 0.7 0.1 1.5 2 0.2 1.5 2 40 0.3 0.1 1.5 2 μs 0.2 1.5 = 2 5.25 0.5 0.1 1.5 2 0.2 1.5 2 0.7 0.1 1.5 2 0.2 1.5 2 60 0.3 0.1 1.5 2 μs 0.2 1.5 = 2 3.5 0.5 0.1 1.5 2 0.2 1.5 2 0.7 0.1 1.5 2 0.2 1.5 2 TIQ 55.89 66.01 54.04 64 34.27 40.14 32.86 38.63 20.62 22.34 18.91 20.38 55.33 66.84 52.39 64.86 31.82 39.15 29.37 37.34 15.88 18 13.98 16.4 54.82 67.45 52.41 64.79 30.31 38.81 28.25 36.75 13.41 16.18 11.89 14.63 E(F) 3.03 0.91 3.6 1.11 25.86 24.46 30.2 28.72 37.2 34.72 45.05 43.18 0.57 0.06 0.89 0.08 19.46 19.11 25.55 24.76 30.17 28.26 39.3 37.56 0.17 0 0.21 0.02 17.31 16.65 23.47 23.44 26.77 25.09 36.48 35.68 CV=1.0 E(L) 236.62 234.47 231.07 228.1 254.05 253.33 248.77 247.31 261.58 260.15 254.98 253.64 235.04 233.71 227.61 227.63 248.83 248.72 244.32 244.11 254.83 253.44 249.42 248.41 233.58 233.79 227.33 226.88 246.29 246.03 242.77 242.61 251.31 250.34 246.97 245.91 TIQa 14.78 15.18 14.25 14.33 15.07 14.93 13.71 13.42 16.58 16.11 14.16 13.49 9.19 9.08 8.19 8.31 9.51 9.47 7.89 7.88 11.51 10.47 9.1 8.24 6.65 6.72 5.84 5.83 7.01 6.98 5.62 5.56 8.82 7.96 6.77 5.89 TIQw 71.8 85.6 67.72 81.05 51.67 62.85 48.27 58.83 29.21 35.45 27.88 33.27 73.12 89.1 67.55 84.25 51.93 65.87 46.54 60.9 25.1 33.8 23.15 31.66 73.4 90.83 68.38 85.02 51.3 67.45 46.38 61.71 23.07 33.43 21.47 30.92 TIQ 54.29 65.92 52.18 63.65 29.83 37.07 28.11 35.16 12.49 14.15 11.78 13.51 54.71 66.83 52.37 64.86 29.16 37.49 26.98 35.52 10 12.66 9.12 11.94 54.97 67.38 52.36 65.17 28.95 37.91 26.46 35.65 8.69 11.95 8 11.36 E(F) 0.46 0.04 0.73 0.06 11.61 10.43 18.44 17.4 20.58 17.99 31.4 29.57 0.03 0 0.06 0 10.33 9.98 18.85 18.67 18.13 16.05 30.36 29.35 0 0 0.01 0 10.03 9.99 19.58 19.35 16.95 15.66 29.75 29.26 CV=0.2 E(L) 233.7 233.82 227.74 227.48 240.3 239.84 236.82 236.39 245 243.13 241.46 240.17 233.54 233.69 227.4 227.3 239.69 239.64 237.72 237.59 243.15 241.49 240.65 239.85 233.62 233.81 227.48 227.46 239.59 239.6 238.34 238.25 242.08 240.91 240.35 239.87 TIQa 5.51 5.52 5.46 5.47 4.98 4.9 4.49 4.43 5.27 4.32 4.29 3.43 2.87 2.88 2.78 2.78 2.62 2.62 2.32 2.29 3.43 2.28 2.56 1.7 1.95 1.95 1.86 1.86 1.78 1.78 1.52 1.52 2.42 1.45 1.76 1.09 TIQw 73.19 89.26 68.27 83.62 52.33 65.98 47.16 59.75 27.87 34.84 25.97 32.36 74.71 91.5 69.39 86.13 53.08 68.9 46.76 62.09 23.89 34.47 21.42 31.05 75.44 92.63 69.72 86.85 53.44 70.45 46.38 62.94 21.89 33.99 19.67 30.51 Benchmark rule results 23 min , TIQ 52 min , TIQ for α=0.7 for α=0.3 Leave 3~13 min before noon when α=0.3 Leave even 21 min after noon when α=0.7 TIQ_N=20 TIQ_N=40 TIQ_N=60 70 TIQ ( minutes ) 60 50 40 30 20 10 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Figure1. the average waiting time in queue per patient using Benchmark ASR cv TIQ_N=20 TIQ_N=40 TIQ_N=60 TIQ/mean service time 20 18 16 14 12 10 8 6 4 2 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Figure2. TIQ in terms of mean service time using Benchmark ASR cv E(F)_N=20 E(F)_N=40 E(F)_N=60 50 E(F) ( minutes ) 40 30 20 10 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 ρ 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 λ 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 α 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv Figure3. physician average idle time per session using Benchmark ASR E(F)_N=20 E(F)_N=40 E(F)_N=60 E(F)/mean service time 12 10 8 6 4 2 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 ρ 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 λ 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 α 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv Figure4. physician idle time in terms of mean service time using Benchmark ASR E(L)_N=20 E(L)_N=40 E(L)_N=60 270 E(L) ( minutes ) 260 250 240 230 220 210 200 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Figure5. the closing time using Benchmark ASR cv Improve ASR(1) kfirst =expected # of walk-in before consultation na = # of pre-register N tlag=time lag between register and consult A2 tlag kfirst p s int v ( Noon A2 ) / na A2 k A2 k 2 int v..............k : 2 ~ N cv α p 1 0.5 1 0.7 1 0.3 3 0.5 2.4 0.7 1.8 0.2 Improve ASR(4) — when 0.3, cv 1.0, , , N A2 tlag kfirst p s A2 k A2 k 2 2 s ..............k : 2 ~ N cv α p 1.0 0.3 1.5 Benchmark vs. ASR(1) & (4) TIQ_Benchmark ASR TIQ (minutes) TIQ_Improved ASR 70 60 50 40 30 20 10 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv Figure1. Comparisons of TIQ between Benchmark ASR and Improved ASR TIQ/mean service time TIQ_Benchmark ASR TIQ_Improved ASR 20 18 16 14 12 10 8 6 4 2 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv Figure2. Comparisons of TIQ/mean service time between Benchmark ASR and Improved ASR E(L) (minutes) E(L)_Benchmark ASR E(L)_Improved ASR 270 260 250 240 230 220 210 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 1.5 1.5 2 2 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv Figure3. Comparisons of the closing time between Benchmark ASR and Improved ASR Conclusions Walk-in practice need academic research Improving Benchmark ASR → ”closing time” move toward noon. α is the most influential factor, α↑,TIQ↓ , IDLE↓ , Leave↓ walk-in arrival rate λ↑, TIQ↑ , IDLE↓ , Leave↓ Future Research Time lag ↑, more accumulation of walk-ins before consultation, the impact of delaying A2↑. “time lag=0” fits for other country Appointment problem with walk-in(Fetter & Thompson,1996) and for Taiwan with electronical records. Need to develop different ASR for different appointment ratio(α) May consider different arrival distribution for walk-ins. More ASRs should be tested and created.