Transcript Session 01 - Introduction

Matching Supply with Demand:
An Introduction to Operations Management
Gérard Cachon
ChristianTerwiesch
All slides in this file are copyrighted by Gerard Cachon and Christian
Terwiesch. Any instructor that adopts Matching Supply with
Demand: An Introduction to Operations Management as a required
text for their course is free to use and modify these slides as desired.
All others must obtain explicit written permission from the authors to
use these slides.
Slide ‹#›
Variability – Throughput Loss
Slide ‹#›
Lessons from Call Center Case: The Three Enemies of Operations
Additional costs due to variability in
demand and activity times
Is associated with longer wait times
and / or customer loss
Variability
Waste
Use of resources beyond what is
needed to meet customer requirements
• Not adding value to the product, but
adding cost
• Reducing the performance of the
production system
• 7 different types of waste
Requires process to hold excess
capacity (idle time)
Inflexibility
Work Waste Value- Work Waste Valueadding
adding
Customer
demand
Capacity
Additional costs incurred because of supply demand
mismatches
• Waiting customers or
• Waiting (idle capacity)
Slide ‹#›
From Theory to Practice: How Many Patients will be Cured?
Accidents Happen
Ambulance Arrives Emergency Room
Care Process
Actual flows are determined by rolling the dice (1…6)
Round 1: patients die if they are not taken care of in the same round
Round 2: patients can be moved forward in the next round
How many patients will be cured after 10 rounds?
Slide ‹#›
Different Models of Variability
Waiting problems
Utilization has to be
less than 100%
Impact of variability is
on Flow Time
Pure waiting
problem, all customers
are perfectly patient.
All customers
enter the process,
some leave due to
their impatience
Loss problems
Demand can be bigger
than capacity
Impact of variability is on
Flow Rate
Customers do not
enter the process once
buffer has reached a
certain limit
Same if customers are patient
Customers are lost
once all servers are
busy
Same if buffer size=0
Same if buffer size is extremely large
Variability is always bad – you pay through lower flow
rate and/or longer flow time
Slide ‹#›
Macro Economic Trends Driving Emergency Room
Crowding and Ambulance Diversion
• Increase in ER visits
• 40% of patients admitted through the ER
• Decrease in number of emergency departments
• Consequences:
–Long wait times (see waiting time analysis)
–Loss of throughput (requires new analysis)
20% of US hospitals are on diversion status
for more than 2.4 hours per day
Data from L. Green; general accounting office
Slide ‹#›
Analyzing Loss Systems
Demand Process
One trauma case comes
in every 3 hours
(a=3 hours)
a is the interarrival time
Trauma center
moves to
diversion status
once all servers
are busy
Resources
3 trauma bays (m=3)
Service Process
Patient stays in trauma bay
for an average of 2 hours
(p=2 hours)
p is the service time
incoming
Can have any distribution
patients
What is Pm, the probability that all m resources are utilized?
are directed
to
Slide ‹#›
Exponential interarrival times
Analyzing Loss Systems: Finding Pm(r)
m
• Define r = p / a
• Example: r= 2 hours/ 3 hours
r=0.67
• Recall m=3
• Use Erlang Loss Table
• Find that P3 (0.67)=0.0255
r=p/a
0.10
0.20
0.25
0.30
0.33
0.40
0.50
0.60
0.67
0.70
0.75
0.80
0.90
1.00
1
0.0909
0.1667
0.2000
0.2308
0.2500
0.2857
0.3333
0.3750
0.4000
0.4118
0.4286
0.4444
0.4737
0.5000
2
0.0045
0.0164
0.0244
0.0335
0.0400
0.0541
0.0769
0.1011
0.1176
0.1260
0.1385
0.1509
0.1757
0.2000
Given Pm(r) we can compute:
• Time per day that system has to deny access
• Flow units lost = 1/a * Pm (r)
Slide ‹#›
3
0.0002
0.0011
0.0020
0.0033
0.0044
0.0072
0.0127
0.0198
0.0255
0.0286
0.0335
0.0387
0.0501
0.0625
4
0.0000
0.0001
0.0001
0.0003
0.0004
0.0007
0.0016
0.0030
0.0042
0.0050
0.0062
0.0077
0.0111
0.0154
5
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0002
0.0004
0.0006
0.0007
0.0009
0.0012
0.0020
0.0031
Erlang Loss Table
m
r = p/a
0.10
0.20
0.25
0.30
0.33
0.40
0.50
0.60
0.67
0.70
0.75
0.80
0.90
1.00
1.10
1.20
1.25
1.30
1.33
1.40
1.50
1.60
1.67
1.70
1.75
1.80
1.90
2.00
2.10
2.20
2.25
2.30
2.33
2.40
2.50
2.60
2.67
2.70
2.75
2.80
2.90
3.00
3.10
3.20
3.25
3.30
3.33
3.40
3.50
3.60
3.67
3.70
3.75
3.80
3.90
4.00
1
0.0909
0.1667
0.2000
0.2308
0.2500
0.2857
0.3333
0.3750
0.4000
0.4118
0.4286
0.4444
0.4737
0.5000
0.5238
0.5455
0.5556
0.5652
0.5714
0.5833
0.6000
0.6154
0.6250
0.6296
0.6364
0.6429
0.6552
0.6667
0.6774
0.6875
0.6923
0.6970
0.7000
0.7059
0.7143
0.7222
0.7273
0.7297
0.7333
0.7368
0.7436
0.7500
0.7561
0.7619
0.7647
0.7674
0.7692
0.7727
0.7778
0.7826
0.7857
0.7872
0.7895
0.7917
0.7959
0.8000
2
0.0045
0.0164
0.0244
0.0335
0.0400
0.0541
0.0769
0.1011
0.1176
0.1260
0.1385
0.1509
0.1757
0.2000
0.2237
0.2466
0.2577
0.2687
0.2759
0.2899
0.3103
0.3299
0.3425
0.3486
0.3577
0.3665
0.3836
0.4000
0.4156
0.4306
0.4378
0.4449
0.4495
0.4586
0.4717
0.4842
0.4923
0.4963
0.5021
0.5078
0.5188
0.5294
0.5396
0.5494
0.5541
0.5587
0.5618
0.5678
0.5765
0.5848
0.5902
0.5929
0.5968
0.6007
0.6082
0.6154
3
0.0002
0.0011
0.0020
0.0033
0.0044
0.0072
0.0127
0.0198
0.0255
0.0286
0.0335
0.0387
0.0501
0.0625
0.0758
0.0898
0.0970
0.1043
0.1092
0.1192
0.1343
0.1496
0.1598
0.1650
0.1726
0.1803
0.1955
0.2105
0.2254
0.2400
0.2472
0.2543
0.2591
0.2684
0.2822
0.2956
0.3044
0.3087
0.3152
0.3215
0.3340
0.3462
0.3580
0.3695
0.3751
0.3807
0.3843
0.3915
0.4021
0.4124
0.4191
0.4224
0.4273
0.4321
0.4415
0.4507
4
0.0000
0.0001
0.0001
0.0003
0.0004
0.0007
0.0016
0.0030
0.0042
0.0050
0.0062
0.0077
0.0111
0.0154
0.0204
0.0262
0.0294
0.0328
0.0351
0.0400
0.0480
0.0565
0.0624
0.0655
0.0702
0.0750
0.0850
0.0952
0.1058
0.1166
0.1221
0.1276
0.1313
0.1387
0.1499
0.1612
0.1687
0.1725
0.1781
0.1837
0.1949
0.2061
0.2172
0.2281
0.2336
0.2390
0.2426
0.2497
0.2603
0.2707
0.2775
0.2809
0.2860
0.2910
0.3009
0.3107
5
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0002
0.0004
0.0006
0.0007
0.0009
0.0012
0.0020
0.0031
0.0045
0.0063
0.0073
0.0085
0.0093
0.0111
0.0142
0.0177
0.0204
0.0218
0.0240
0.0263
0.0313
0.0367
0.0425
0.0488
0.0521
0.0554
0.0577
0.0624
0.0697
0.0773
0.0825
0.0852
0.0892
0.0933
0.1016
0.1101
0.1187
0.1274
0.1318
0.1362
0.1392
0.1452
0.1541
0.1631
0.1691
0.1721
0.1766
0.1811
0.1901
0.1991
6
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0005
0.0008
0.0012
0.0015
0.0018
0.0021
0.0026
0.0035
0.0047
0.0056
0.0061
0.0069
0.0078
0.0098
0.0121
0.0147
0.0176
0.0192
0.0208
0.0220
0.0244
0.0282
0.0324
0.0354
0.0369
0.0393
0.0417
0.0468
0.0522
0.0578
0.0636
0.0666
0.0697
0.0718
0.0760
0.0825
0.0891
0.0937
0.0960
0.0994
0.1029
0.1100
0.1172
7
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0001
0.0002
0.0003
0.0003
0.0004
0.0005
0.0008
0.0011
0.0013
0.0015
0.0017
0.0020
0.0027
0.0034
0.0044
0.0055
0.0061
0.0068
0.0073
0.0083
0.0100
0.0119
0.0133
0.0140
0.0152
0.0164
0.0190
0.0219
0.0249
0.0283
0.0300
0.0318
0.0331
0.0356
0.0396
0.0438
0.0468
0.0483
0.0506
0.0529
0.0577
0.0627
8
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0001
0.0001
0.0001
0.0002
0.0003
0.0003
0.0004
0.0005
0.0006
0.0009
0.0011
0.0015
0.0017
0.0019
0.0021
0.0025
0.0031
0.0039
0.0044
0.0047
0.0052
0.0057
0.0068
0.0081
0.0096
0.0112
0.0120
0.0130
0.0136
0.0149
0.0170
0.0193
0.0210
0.0218
0.0232
0.0245
0.0274
0.0304
9
10
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0000 0.0000
0.0001 0.0000
0.0001 0.0000
0.0001 0.0000
0.0001 0.0000
0.0001 0.0000
0.0002 0.0000
0.0003 0.0001
0.0004 0.0001
0.0004 0.0001
0.0005 0.0001
0.0005 0.0001
0.0007 0.0002
0.0009 0.0002
0.0011 0.0003
0.0013 0.0003
0.0014 0.0004
0.0016 0.0004
0.0018 0.0005
0.0022 0.0006
0.0027 0.0008
0.0033 0.0010
0.0040 0.0013
0.0043 0.0014
0.0047 0.0016
0.0050 0.0017
0.0056 0.0019
0.0066 0.0023
0.0077 0.0028
0.0085 0.0031
0.0089 0.0033
0.0096 0.0036
0.0102 0.0039
0.0117 0.0046
Slide
‹#›
0.0133
0.0053
Erlang Loss Table
Probability{all m servers busy}=
rm
m!
Pm (r ) 
r1 r 2
rm
1 
 ... 
1! 2!
m!
Implied utilization vs probability of having all servers
utilized: Pooling Revisited
0.6
Probability
that all servers
0.5
are utilized
0.4
0.3
m=1
0.2
m=3 m=5 m=10
0.1
0
m=2
m=20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Implied utilization
Slide ‹#›