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 & pre-

register ratio

Simulation Setting Register 8:00 Am ~ 11:30 (210min)

Consult 8:30 Am ~ 12:00 Noon (210min) # of patient per session (N):

20( =10.5 min), 40 ( =5.25 min), 60( =3.5 min) Service-time distribution:

exponentially, cv =1

uniformally , cv = 0.2 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)

s s s

s

s

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)

If

If

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

2 sA

2 2 2

0.52 ...........k k sA A

2 2 2

1

0.52 .........................

k k s

i i s

A A

A A

: 2 ~k N

: 2 ~ (1 )k N

: ~2 1 1i NN

N α ρ λ TIQ E(F) E(L) TIQa TIQw TIQ E(F) E(L) TIQa TIQw20 0.3 0.1 1.5 55.89 3.03 236.62 14.78 71.8 54.29 0.46 233.7 5.51 73.19

2 66.01 0.91 234.47 15.18 85.6 65.92 0.04 233.82 5.52 89.26

μ s 0.2 1.5 54.04 3.6 231.07 14.25 67.72 52.18 0.73 227.74 5.46 68.27= 2 64 1.11 228.1 14.33 81.05 63.65 0.06 227.48 5.47 83.62

10.5 0.5 0.1 1.5 34.27 25.86 254.05 15.07 51.67 29.83 11.61 240.3 4.98 52.332 40.14 24.46 253.33 14.93 62.85 37.07 10.43 239.84 4.9 65.98

0.2 1.5 32.86 30.2 248.77 13.71 48.27 28.11 18.44 236.82 4.49 47.162 38.63 28.72 247.31 13.42 58.83 35.16 17.4 236.39 4.43 59.75

0.7 0.1 1.5 20.62 37.2 261.58 16.58 29.21 12.49 20.58 245 5.27 27.872 22.34 34.72 260.15 16.11 35.45 14.15 17.99 243.13 4.32 34.84

0.2 1.5 18.91 45.05 254.98 14.16 27.88 11.78 31.4 241.46 4.29 25.972 20.38 43.18 253.64 13.49 33.27 13.51 29.57 240.17 3.43 32.36

40 0.3 0.1 1.5 55.33 0.57 235.04 9.19 73.12 54.71 0.03 233.54 2.87 74.712 66.84 0.06 233.71 9.08 89.1 66.83 0 233.69 2.88 91.5

μ s 0.2 1.5 52.39 0.89 227.61 8.19 67.55 52.37 0.06 227.4 2.78 69.39= 2 64.86 0.08 227.63 8.31 84.25 64.86 0 227.3 2.78 86.13

5.25 0.5 0.1 1.5 31.82 19.46 248.83 9.51 51.93 29.16 10.33 239.69 2.62 53.082 39.15 19.11 248.72 9.47 65.87 37.49 9.98 239.64 2.62 68.9

0.2 1.5 29.37 25.55 244.32 7.89 46.54 26.98 18.85 237.72 2.32 46.762 37.34 24.76 244.11 7.88 60.9 35.52 18.67 237.59 2.29 62.09

0.7 0.1 1.5 15.88 30.17 254.83 11.51 25.1 10 18.13 243.15 3.43 23.892 18 28.26 253.44 10.47 33.8 12.66 16.05 241.49 2.28 34.47

0.2 1.5 13.98 39.3 249.42 9.1 23.15 9.12 30.36 240.65 2.56 21.422 16.4 37.56 248.41 8.24 31.66 11.94 29.35 239.85 1.7 31.05

60 0.3 0.1 1.5 54.82 0.17 233.58 6.65 73.4 54.97 0 233.62 1.95 75.442 67.45 0 233.79 6.72 90.83 67.38 0 233.81 1.95 92.63

μ s 0.2 1.5 52.41 0.21 227.33 5.84 68.38 52.36 0.01 227.48 1.86 69.72= 2 64.79 0.02 226.88 5.83 85.02 65.17 0 227.46 1.86 86.85

3.5 0.5 0.1 1.5 30.31 17.31 246.29 7.01 51.3 28.95 10.03 239.59 1.78 53.442 38.81 16.65 246.03 6.98 67.45 37.91 9.99 239.6 1.78 70.45

0.2 1.5 28.25 23.47 242.77 5.62 46.38 26.46 19.58 238.34 1.52 46.382 36.75 23.44 242.61 5.56 61.71 35.65 19.35 238.25 1.52 62.94

0.7 0.1 1.5 13.41 26.77 251.31 8.82 23.07 8.69 16.95 242.08 2.42 21.892 16.18 25.09 250.34 7.96 33.43 11.95 15.66 240.91 1.45 33.99

0.2 1.5 11.89 36.48 246.97 6.77 21.47 8 29.75 240.35 1.76 19.672 14.63 35.68 245.91 5.89 30.92 11.36 29.26 239.87 1.09 30.51

UNIT=MINUTE CV=1.0 CV=0.2

Table Ⅰ . Experimental results using benchmark ASR.

Benchmark rule results

TIQ 23 min , for α=0.7

TIQ 52 min , for α=0.3

Leave 3~13 min before noon when α=0.3

Leave even 21 min after noon when α=0.7

0

10

20

30

40

50

60

70

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

Figure1. the average waiting time in queue per patient using Benchmark ASR

TIQ

(

min

utes

)

TIQ_N=20 TIQ_N=40 TIQ_N=60

0

2

4

6

8

10

12

14

16

18

20

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

Figure2. TIQ in terms of mean service time using Benchmark ASR

TIQ

/mea

n se

rvic

e tim

e

TIQ_N=20 TIQ_N=40 TIQ_N=60

0

10

20

30

40

50

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

) (

min

ute

s )

E(F)_N=20 E(F)_N=40 E(F)_N=60

0

2

4

6

8

10

12

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(F)

/mea

n se

rvic

e tim

e

E(F)_N=20 E(F)_N=40 E(F)_N=60

200

210

220

230

240

250

260

270

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

Figure5. the closing time using Benchmark ASR

E(L)

( m

inut

es )

E(L)_N=20 E(L)_N=40 E(L)_N=60

Improve ASR(1) kfirst =expected # of walk-in before consultation

na = # of pre-register

tlag=time lag between register and consult

2 sA tlag kfirst p

2int ( ) /v Noon A na

2 2 2 int .............. : 2 ~k kA A v k N cv α p

1 0.5 1

0.7 1

0.2 0.3 3

0.5 2.4

0.7 1.8

N

Improve ASR(4)

— when

0.3, 1.0, , ,cv N

2 sA tlag kfirst p

2 2 2 2 .............. : 2 ~k k sA A k N

cv α p

1.0 0.3 1.5

Benchmark vs. ASR(1) & (4)

0

10

20

30

40

50

60

70

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 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 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 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 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 α

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 (minutes) TIQ_Benchmark ASR TIQ_Improved ASR

0

2

4

6

8

10

12

14

16

18

20

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 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 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 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 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 α

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

TIQ/mean service time TIQ_Benchmark ASR TIQ_Improved ASR

210

220

230

240

250

260

270

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 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 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 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 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 α

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

E(L) (minutes) E(L)_Benchmark ASR E(L)_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.