Transcript Document

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.
Caller 1
Caller 2
Caller 3
Caller 4
Caller 5
Caller 6
Caller 7
Caller 8
Caller 9
Caller 10
Caller 11
Caller 12
Time
7:00
7:10
7:20
7:30
7:40
Figure 6.1: A somewhat odd service process
7:50
8:00
Caller 1
1
0
5
2
7
6
3
9
7
4
12
6
5
18
5
6
22
2
7
25
4
8
30
3
9
36
4
10
45
2
11
51
2
12
55
3
Caller 3
Caller 2
Caller 5
Caller 4
Caller 7
Caller 6
Caller 9
Caller 8
Caller 11
Caller 10
Caller 12
Time
7:00
7:10
7:20
7:30
7:40
7:50
3
Number of cases
Arrival Service
Time
Caller Time
2
1
0
2 min.
3 min.
4 min.
5 min.
6 min.
Service times
Figure 6.2.: Data gathered at a call center
7 min.
8:00
Caller 1
Caller 2
Service time
Caller 3
Caller 4
Caller 5
Caller 6
Caller 7
Caller 8
Wait time
Caller 9
Caller 10
Caller 11
Caller 12
7:00
7:10
7:20
7:30
7:40
7:50
8:00
7:00
7:10
7:20
7:30
7:40
7:50
8:00
Time
Inventory
(Callers on 5
hold)
4
3
2
1
0
Figure 6.3.: Detailed analysis of call center
Activity times:
• Inherent variation
• Lack of operating procedures
• Quality (scrap / rework)
Input:
• Random arrivals
(randomness is the rule,
not the exception)
• Incoming quality
• Product Mix
Buffer
Processing
Resources:
• Breakdowns / Maintenance
• Operator absence
• Set-up times
Routes:
• Variable routing
• Dedicated machines
Figure 6.4.: Variability and where it comes from
Call
Arrival
Time, ATi
1
6:00:29
2
6:00:52
00:23
3
6:02:16
01:24
4
6:02:50
00:34
5
6:05:14
02:24
6
6:05:50
00:36
7
6:06:28
00:38
Inter-Arrival
Time, IAi=ATi+1 -ATi
Call 1 Call 2
6:00
Call 3
6:01
IA1
6:02
IA2
Call 4
6:03
IA3
Call 5
6:04
IA4
Figure 6.5.: The concept of inter-arrival times
Call 6
6:05
Call 7
6:06
IA5
IA6
Time
Number of customers
Per 15 minutes
160
140
120
100
80
60
40
23:00
21:15
19:30
17:45
16:00
14:15
12:30
10:45
9:00
7:15
5:30
3:45
2:00
0
0:15
20
Time
Figure 6.6 : Seasonality over the course of a day
Cumulative
Customers
Cumulative
Customers
700
70
600
60
500
400
50
Expected arrivals
if stationary
40
300
30
200
Actual, cumulative
arrivals
20
100
10
0
6:00:00
7:00:00
8:00:00
9:00:00
10:00:00
Time
0
7:15:00 7:18:00 7:21:00 7:24:00 7:27:00 7:30:00
Time
Figure 6.7.:Test for stationary arrivals
0.8
0.6
0.4
0.2
0
100
Number of calls with given duration t
Probability{Interarrival time  t}
1
90
80
70
60
50
40
30
20
10
time
0
Duration t
Figure 6.8.: Distribution function of the exponential distribution (left) and
an example of a histogram (right)
Distribution Function
1
0.8
Exponential distribution
0.6
0.4
Empirical distribution
(individual points)
0.2
Inter-arrival time
0
0:00:00 0:00:09 0:00:17 0:00:26 0:00:35 0:00:43 0:00:52 0:01:00 0:01:09
Figure 6.9: Empirical vs. exponential distribution for inter-arrival times
Stationary
Arrivals?
YES
NO
Exponentially distributed
inter-arrival times?
YES
• Compute a: average interarrival time
• CVa=1
• All results of chapters 6 and 7 apply
Break arrival process up
into smaller time intervals
NO
• Compute a: average interarrival time
• CVa= St.dev. of interarrival times / a
• All results of chapter 6 apply
• Results of chapter 7 do not apply, require
simulation or more complicated models
Figure 6.10: How to analyze a demand / arrival process?
800
Frequency
600
400
200
Std. Dev = 141.46
Mean
127.2
Call=durations
N[seconds]
= 2061.00
0
A
Figure 6.11: Service times in call center
Call
duration
[minutes]
2.5
Week-end averages
2
1.5
1
Week-day averages
0.5
Time of
the day
0
0:00
23:00
Figure 6.12: Average call durations: week-day vs. week-end
Inflow
Entry to
system
Outflow
Begin
Service
Departure
Figure 6.13.: A simple process with one queue and one server
Inventory
waiting Iq
Inventory
in service Ip
Inflow
Outflow
Entry to
system
Begin
Service
Waiting Time Tq
Departure
Service Time p
Flow Time T=Tq+p
Figure 6.14.: A simple process with one queue and one server
Inflow
Entry to
system
Outflow
Begin
Service
Departure
Figure 6.15.: A process with one queue and multiple, parallel servers
Inventory in the system I=Iq+ Ip
Inventory
in service Ip
Inventory
waiting Iq
Outflow
Inflow
Entry to system
Begin Service
Waiting Time Tq
Departure
Service Time p
Flow Time T=Tq+p
Figure 6.16.: Summary of key performance measures
Fraction of
1
customers
who have
to wait x
0.8
seconds
or less
0.6
Waiting times for those customers who
do not get served immediately
0.4
Fraction of customers who get served
without waiting at all
0.2
0
0
50
100
150
200
Waiting time [seconds]
Figure 6.17: Empirical distribution of waiting times at Anser
Call center
Answered Calls
Incoming calls
Calls
on Hold
Blocked calls
(busy signal)
Lost throughput
Lost goodwill
Sales reps
processing
calls
Abandoned calls
(tired of waiting)
Holding cost (line charges)
Lost goodwill
Lost throughput (abandoned)
Financial consequences
Cost of capacity
Figure 6.18: Economic consequences of waiting
Revenue
Number of customers
Per 15 minutes
160
Number of
CSRs
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
140
120
100
80
60
40
Time
23:00
21:15
19:30
17:45
16:00
14:15
12:30
10:45
9:00
7:15
5:30
3:45
2:00
0
0:15
20
Number of
CSRs
Number of customers
Per 15 minutes
160
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
140
120
100
80
60
40
23:00
21:15
19:30
17:45
16:00
14:15
12:30
10:45
9:00
7:15
5:30
3:45
2:00
0
0:15
20
Time
Figure 6.19 : Staffing and incoming calls over the course of a day
Independent Resources
2x(m=1)
Pooled Resources
(m=2)
Figure 6.20.: The concept of pooling
Waiting 70.00
Time Tq
60.00
[sec]
m=1
50.00
40.00
m=2
30.00
20.00
m=5
10.00
m=10
0.00
60%
65%
70%
75%
80%
85%
90%
95%
Figure 6.21: How pooling can reduce waiting time
Utilization u
A
Service times:
A: 9 minutes
B: 10 minutes
C: 4 minutes
D: 8 minutes
C
9 min.
4 min.
B
19 min.
12 min.
C
23 min.
Total wait time: 9+19+23=51min
D
D
21 min.
A
B
Total wait time: 4+12+21=37 min
Figure 6.22: The shortest processing time (SPT) rule (used in the right case)
Call durations
2:00 min.
Operator KB
short
Operator NN
2:30 min.
Operator BJ
3:00 min.
Operator BK
3:30 min.
Operator NJ
4:00 min.
long
4:30 min.
Low
courtesy
High
courtesy
Courtesy / Friendliness
(qualitative information)
Figure 6.23: Operator performance concerning speed and courtesy
Responsiveness
High
Increase staff
(lower utilization)
Responsive
process with
high costs
Now
System improvement
(e.g. pooling of resources)
Reduce staff
(higher utilization)
Low cost process
with low
responsiveness
Frontier reflecting
current process
Low
High per
unit costs
(low utilization)
Low per
unit costs
(high utilization)
Figure 6.24: Balancing efficiency with responsiveness
Efficiency