![Page 1: Introduction to Management Science 1e. - file.upi.edufile.upi.edu/Direktori/FPEB/PRODI._MANAJEMEN_FPEB/... · Management Science with Spreadsheets McGraw-Hill/Irwin Copyright© 2007](https://reader031.vdocument.in/reader031/viewer/2022012914/5b8a1b727f8b9ae7298de277/html5/thumbnails/1.jpg)
Introduction to
Management Sciencewith Spreadsheets
McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Waiting-Line ModelsPart 3 Probabilistic Decision ModelsTEORI ANTRIAN
( WAITING LINES THEORY)
Oleh :
Rofi Rofaida,SP.,M.Si
Program Studi Manajemen
Fakultas Pendidikan Ekonomi dan Bisnis
Universitas Pendidikan Indonesia
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Penerapan Teori antrian ( waiting line) tidak hanya
terbatas pada industri manufaktur saja tetapi berlaku
juga pada industri jasa seperti antrian di bank, antrian
di loket, dsb.
First come, first served (FCFS)
Priority Classification
Sumber : Stevenson and Ozgur, 2007, The McGraww-Hill Companies
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Calling population = populasi yang akan memasuki
sistem antrian
arrivals = tingkat kedatangan
Waiting line = antrian
Processing order = sistem prosesing yang berlaku
service = pelayanan
exit = keluar dari sistem antrian
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◦ Tingkat kedatangan mengikuti distribusi
Poisson.
◦ Tingkat pelayanan mengikuti distribusi
eksponential negatif
◦ Sistem pelayanan adalah FCFO (first come
first order)
◦ Populasi yang akan memasuki antrian tidak
terbatas
◦ Steady state
◦ Panjang antrian tidak terbatas
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Sumber : Stevenson and Ozgur, 2007, The McGraww-Hill Companies
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α= the arrival rate
µ= the service rate
Lq = the average number waiting for service
L = the average number in the system (i.e.,
waiting for service and being served)
P0 = the probability of zero units in the system (%)
ρ = the system utilization (percentage of time servers
are busy serving customers)
Wa = the average time customers must wait for service
W = the average time customers spend in the system
(i.e., waiting for service and service time)
Pn = the probability of n units in the system (%)
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A Basic single-channel model digunakan jika :
◦ Hanya ada satu pusat pelayanan/server
◦ Asumsi tingkat kedatangan mengikuti distribusi
Poisson
◦ Asumsi tingkat pelayanan mengikuti distribusi
eksponential negatif
◦ First-come, first-served processing order.
◦ Populasi kedatangan tidak terbatas
◦ Panjang antrian tidak terbatas
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Performance measures Formula
System utilization
Average number in line
Average number in system
Average time in line
1
2
qL
1L
)(
qW
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Performance measures Formula
Average time in system
Probability of zero units in the system
Probability of n units in the system
1W
1oP
nPn )1(
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Penumpang kereta api datang pada sebuah loket mengikuti distribusi Poisson dengan rata-rata tk kedatangan 20 per jam. Misalkan rata-rata setiap penumpang dapat dilayani 2 menit dan tingkat pelayanan mengikuti distribusi eksponential negatif serta sistem dalam keadaan steady state, carilah :
(a).P4, (b). L, (c). Lq, (d). W, (e)Wq, (f) P0
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667.03/230
20
%6.6066.0)667.0)(667.01(4 4 P
orangL 2)667.01(
667.0
1
orangLq 33.1)667.01(
)667.0(
1
22
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menitjamW 6/10.0)2030(
11
menitWq 4)2030(30
20
)(
%3333.0)667.01(0 P
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Misalkan kepala stasiun memutuskan utkmemperbaiki kualitas pelayanan dengan menyewapenjaga yg lebih trampil. Dengan cara ini waktupelayanan diperkirakan akan berkurang dari 2 menitmenjadi 1.5 menit/penumpang. Konsekuensinyakepala stasiun harus membayar gaji penjaga trampilRp.12.000/jam sedangkan gaji penjaga sebelumnyaadalah Rp.6000/jam. Kepala stasiun jugamemperkirakan bahwa biaya menunggu/antri adalahRp.500/menit dan loket harus tetap dibuka 8jam/hari. Haruskah kepala stasiun menggantipenjaga yang ada dengan yg baru?
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Kondisi 1
Kondisi 2
menitWq 4)2030(30
20
)(
menitWq 5.1)2040(40
20
)(
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Karena rata-rata tk kedatangan 20 orang/jam danloket dibuka 8 jam/hari,maka banyaknya pengantridiperkirakan 160 orang.
• Jumlah waktu menunggu utk kondisi 1 :
160x4 menit = 640 menit
• Jumlah waktu menunggu utk kondisi 2 :
160 X1.5 menit =240 menit.
• Pelayan yg ada dibayar :
Rp.6000/jam X 8 jam = Rp 48.000
• Pelayan kondisi 2 :
Rp.12.000/jam X 8 jam =Rp.9.600,
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Kondisi 1 Kondisi 2
Biayatunggu
640 menit x Rp.500/menit
= Rp.320.000
240 menit XRp.500 /menit
= Rp.120.000
Biayapelayanan
Rp.6000/jam X 8 jam = Rp 48.000
Rp.12.000/jam X 8 jam =Rp.96.000,
Kesimpulan : dgn mengganti pelayan yang ada dengan pelayanan yg lebih trampil, kepala stasiun dapat menurunkan biaya total
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The multiple-channel model tepat untuk kondisi :
1. Tingkat kedatangan mengikuti distribusi Poisson
2. Tingkat pelayanan mengikuti distribusi
3. First-Come, first-served processing order.
4. Lebih dari satu pusat pelayanan/server
5. Populasi kedatangan tidak terbatas
6. Tingkat pelayanan/kemampuan melayani setiap
server diasumsikan sama
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Performance measures
Formula
System utilization
Average number in line
Average number in system
Average time in line
c
2
0
)1(!
)(
cc
cP
L
c
q
qLL
LW
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Performance measures
Formula
Average time in system
Probability of zero units in the system
Probability of n units in the system
jika n ≤ c
jika n > c
1
qWW
1
0 )1(!
)(
!
)(
1
cn
n
cno
ccn
P
XPon
Pnn
!
)(
XPocc
Pncn
n
!
)(