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ISM 270ISM 270
Service Engineering and ManagementService Engineering and Management
Lecture 7: Forecasting and Managing Lecture 7: Forecasting and Managing Service CapacityService Capacity
AnnouncementsAnnouncements
Brenda Deitrich (Mathematical Sciences, IBM) visited Brenda Deitrich (Mathematical Sciences, IBM) visited UCSC todayUCSC today
Should be available to watch online at:Should be available to watch online at: http://ucsc.citris-uc.org/http://ucsc.citris-uc.org/
Project Proposal Due todayProject Proposal Due today Homework 4 due next weekHomework 4 due next week $15 check for ‘Responsive Learning Technologies’$15 check for ‘Responsive Learning Technologies’ Final four weeks:Final four weeks:
Forecasting and Capacity PlanningForecasting and Capacity Planning Supply Chains in ServicesSupply Chains in Services Capacity Management GameCapacity Management Game Project PresentationsProject Presentations
TodayToday
ForecastingForecasting Queueing ModelsQueueing Models
Forecasting Demand for Forecasting Demand for ServicesServices
Forecasting ModelsForecasting Models Subjective ModelsSubjective Models
Delphi MethodsDelphi Methods Causal ModelsCausal Models
Regression ModelsRegression Models Time Series ModelsTime Series Models
Moving AveragesMoving AveragesExponential SmoothingExponential Smoothing
Delphi ForecastingDelphi ForecastingQuestionQuestion: In what future election will a woman become president of the united states for : In what future election will a woman become president of the united states for
the first time?the first time?
YearYear 11stst Round Round Positive ArgumentsPositive Arguments 22ndnd Round Round Negative ArgumentsNegative Arguments 33rdrd Round Round
20082008
20122012
20162016
20202020
20242024
20282028
20322032
20362036
20402040
20442044
20482048
20522052
NeverNever
TotalTotal
N Period Moving Average N Period Moving Average
Let : MAT = The N period moving average at the end of period T AT = Actual observation for period T
Then: MAT = (AT + AT-1 + AT-2 + …..+ AT-N+1)/N
Characteristics: Need N observations to make a forecast Very inexpensive and easy to understand Gives equal weight to all observations Does not consider observations older than N periods
Moving Average ExampleMoving Average Example
Saturday Occupancy at a 100-room Hotel
Three-period Saturday Period Occupancy Moving Average Forecast
Aug. 1 1 79 8 2 84 15 3 83 82 22 4 81 83 82 29 5 98 87 83Sept. 5 6 100 93 87 12 7 93
Exponential SmoothingExponential Smoothing
Let : ST = Smoothed value at end of period T AT = Actual observation for period T FT+1 = Forecast for period T+1
Feedback control nature of exponential smoothing
New value (ST ) = Old value (ST-1 ) + [ observed error ]
S S A S
S A S
F S
T T- T T
T T T
T T
1 1
1
1
1
[ ]
( )or :
Exponential SmoothingExponential SmoothingHotel ExampleHotel Example
Saturday Hotel Occupancy ( =0.5) Actual Smoothed Forecast Period Occupancy Value Forecast ErrorSaturday t At St Ft |At - Ft|Aug. 1 1 79 79.00 8 2 84 81.50 79 5 15 3 83 82.25 82 1 22 4 81 81.63 82 1 29 5 98 89.81 82 16Sept. 5 6 100 94.91 90 10
Mean Absolute Deviation (MAD) = 6.6 Forecast Error (MAD) = ΣlAt – Ftl/n
Exponential SmoothingExponential SmoothingImplied Weights Given Past DemandImplied Weights Given Past Demand
S A S
S A A S
S A A S
S A A S
T T T
T T T T
T T T T
T T T T
( )
( )[ ( ) ]
( )[ ( ) ]
( ) ( )
1
1 1
1 1
1 1
1
1 1 2
1 2
12
2
Substitute for
If continued:
S A A A A ST T T TT T ( ) ( ) ..... ( ) ( )1 1 1 11
22
11 0
Exponential Smoothing Weight Exponential Smoothing Weight DistributionDistribution
0
0.1
0.2
0.3
0 1 2 3 4 5
Age of Observation (Period Old)
Wei
gh
t 0 3.
( ) .1 0 21
( ) .1 01472 ( ) .1 01033
( ) .1 0 0724
( ) .1 0 0505
Relationship Between and N
(exponential smoothing constant) : 0.05 0.1 0.2 0.3 0.4 0.5 0.67 N (periods in moving average) : 39 19 9 5.7 4 3 2
Saturday Hotel OccupancySaturday Hotel Occupancy
Effect of Alpha ( =0.1 vs. =0.5)
7580859095
100105
0 1 2 3 4 5 6Period
Occ
upan
cy
Actual
Forecast
Forecast
( . ) 05
( . ) 01
Recall from Charles Ng:Recall from Charles Ng:Start with historical volume: Start with historical volume:
What explains changes over time?What explains changes over time?
pos_units
Pull out the Influence of Pull out the Influence of Seasonality and Trend Seasonality and Trend
pos_units Intercept Seasonality
Estimate the relationship of price Estimate the relationship of price and promotion changes to volumeand promotion changes to volume
pos_units Intercept Seasonality Holiday Price Change Ad Display
Ad Flag
Display Flag
Base Price
Once estimated separately, all Once estimated separately, all these effects can be combined to these effects can be combined to
predict volume. This is the predict volume. This is the modelmodel. .
POS Price Change Display Ad Holiday Seasonality Intercept
Exponential Smoothing With Exponential Smoothing With Trend AdjustmentTrend Adjustment
S A S T
T S S T
F S T
t t t t
t t t t
t t t
( ) ( )( )
( ) ( )
1
11 1
1 1
1
Commuter Airline Load Factor
Week Actual load factor Smoothed value Smoothed trend Forecast Forecast error t At St Tt Ft | At - Ft|
1 31 31.00 0.00 2 40 35.50 1.35 31 9 3 43 39.93 2.27 37 6 4 52 47.10 3.74 42 10 5 49 49.92 3.47 51 2 6 64 58.69 5.06 53 11 7 58 60.88 4.20 64 6 8 68 66.54 4.63 65 3 MAD = 6.7
( . , . ) 05 0 3
Exponential Smoothing with Exponential Smoothing with Seasonal AdjustmentSeasonal Adjustment
S A I S
F S I
IA
SI
t t t L t
t t t L
tt
tt L
( / ) ( )
( )( )
( )
1
1
1
1 1
Ferry Passengers taken to a Resort Island Actual Smoothed Index Forecast ErrorPeriod t At value St It Ft | At - Ft| 2003January 1 1651 ….. 0.837 ….. February 2 1305 ….. 0.662 ….. March 3 1617 ….. 0.820 …..April 4 1721 ….. 0.873 ….. May 5 2015 ….. 1.022 …..June 6 2297 ….. 1.165 ….. July 7 2606 ….. 1.322 ….. August 8 2687 ….. 1.363 ….. September 9 2292 ….. 1.162 …..October 10 1981 ….. 1.005 …..November 11 1696 ….. 0.860 …..December 12 1794 1794.00 0.910 ….. 2004January 13 1806 1866.74 0.876 - - February 14 1731 2016.35 0.721 1236
495March 15 1733 2035.76 0.829 1653 80
( . , . ) 0 2 0 3
More sophisticated forecasting More sophisticated forecasting techniquestechniques
Nonlinear RegressionNonlinear Regression Data miningData mining Machine LearningMachine Learning Simulation-basedSimulation-based
Managing Waiting Lines Managing Waiting Lines – Queueing Models– Queueing Models
Essential Features of Queuing SystemsEssential Features of Queuing Systems
DepartureQueue
discipline
Arrival process
Queueconfiguration
Serviceprocess
Renege
Balk
Callingpopulation
No futureneed for service
Arrival ProcessArrival Process
Static Dynamic
AppointmentsPriceAccept/Reject BalkingReneging
Randomarrivals withconstant rate
Random arrivalrate varying
with time
Facility-controlled
Customer-exercised
control
Arrival process
Distribution of Patient Interarrival TimesDistribution of Patient Interarrival Times
0
10
20
30
40
1 3 5 7 9 11 13 15 17 19
Patient interarrival time, minutes
Rel
ativ
e fr
eque
ncy,
%
Temporal Variation in Arrival RatesTemporal Variation in Arrival Rates
0
0.5
1
1.5
2
2.5
3
3.5
1 3 5 7 9 11 13 15 17 19 21 23
Hour of day
Avera
ge ca
lls pe
r hou
r
60708090
100
110120130140
1 2 3 4 5
Day of week
Perc
enta
ge o
f ave
rage
dail
y ph
ysici
an vi
sits
Poisson and Exponential EquivalencePoisson and Exponential Equivalence
Poisson distribution for number of arrivals per hour (top view)Poisson distribution for number of arrivals per hour (top view)
One-hourOne-hour
1 2 0 1 interval1 2 0 1 interval
Arrival Arrivals Arrivals ArrivalArrival Arrivals Arrivals Arrival
62 min.40 min.
123 min.
Exponential distribution of time between arrivals in minutes (bottom view)
Queue ConfigurationsQueue Configurations
Multiple Queue Single queue
Take a Number Enter
3 4
8
2
6 10
1211
5
79
Queue DisciplineQueue Discipline
Queuediscipline
Static(FCFS rule)
Dynamic
selectionbased on status
of queue
Selection basedon individual
customerattributes
Number of customers
waitingRound robin Priority Preemptive
Processing timeof customers
(SPT rule)
Queuing FormulasQueuing Formulas
Single Server Model with Poisson Arrival and Service Rates: M/M/1
1. Mean arrival rate:
2. Mean service rate:
3. Mean number in service:
4. Probability of exactly “n” customers in the system:
5. Probability of “k” or more customers in the system:
6. Mean number of customers in the system:
7. Mean number of customers in queue:
8. Mean time in system:
9. Mean time in queue:
Pn
n ( )1
P n k k( )
sL
qL
1sW
qW
Queuing Formulas (cont.)Queuing Formulas (cont.)
Single Server General Service Distribution Model: M/G/1
Mean number of customers in queue for two servers: M/M/2
Relationships among system characteristics (Little’s Law for ALL queues):
)1(2
222
qL
2
3
4
qL
ss
qs
qs
LW
LW
WW
LL
1
1
1
Congestion as Congestion as 10.
0 1.0
100
10
8
6
4
2 0
With:
Ls 1Then:
Ls
0 00.2 0.250.5 10.8 40.9 90.99 99
Single Server General Service Single Server General Service Distribution Model : M/G/1Distribution Model : M/G/1
)1(2
222
qL
1. For Exponential Distribution:
22
1
)1()1(2
2
)1(2
/ 22222
qL
2. For Constant Service Time: 2 0
)1(2
2
qL
3. Conclusion:
Congestion measured by Lq is accounted for equally by variability in arrivals and service times.
Queuing System Cost TradeoffQueuing System Cost Tradeoff
Let: CLet: Cww = Cost of one customer waiting in = Cost of one customer waiting in queue for an hour queue for an hour
CCss = Hourly cost per server = Hourly cost per server
C = Number of serversC = Number of serversTotal Cost/hour = Hourly Service Cost + Total Cost/hour = Hourly Service Cost +
Hourly Customer Waiting CostHourly Customer Waiting Cost
Total Cost/hour = CTotal Cost/hour = Css C + C C + Cww L Lq q
NoteNote: Only consider systems where : Only consider systems where C
General Queuing ObservationsGeneral Queuing Observations
1. Variability in arrivals and service times contribute equally to congestion as measured by Lq.
2. Service capacity must exceed demand.
3. Servers must be idle some of the time.
4. Single queue preferred to multiple queue unless jockeying is permitted.
5. Large single server (team) preferred to multiple-servers if minimizing mean time in system, WS.
6. Multiple-servers preferred to single large server (team) if minimizing mean time in queue, WQ.
Laws of ServiceLaws of Service
Maister’s First LawMaister’s First Law::Customers compare expectations with Customers compare expectations with perceptions.perceptions.
Maister’s Second LawMaister’s Second Law::Is hard to play catch-up ball.Is hard to play catch-up ball.
Skinner’s LawSkinner’s Law::The other line always moves faster.The other line always moves faster.
Jenkin’s CorollaryJenkin’s Corollary::However, when you switch to another other line, However, when you switch to another other line, the line you left moves faster.the line you left moves faster.
Managing Capacity and Managing Capacity and DemandDemand
Segmenting Demand at a Health Segmenting Demand at a Health ClinicClinic
60
70
80
90
100
110
120
130
140
1 2 3 4 5
Day of week
Perc
enta
ge o
f ave
rage
dail
y ph
ysici
an vi
sits
Smoothing Demand by AppointmentScheduling
Day Appointments
Monday 84Tuesday 89Wednesday 124Thursday 129Friday 114
Hotel Overbooking Loss TableHotel Overbooking Loss Table
Number of Reservations OverbookedNumber of Reservations Overbooked
No- Prob-No- Prob-
shows ability 0 1 2 3 4 5 6 7 8 9shows ability 0 1 2 3 4 5 6 7 8 9
0 .07 0 100 200 300 400 500 600 700 800 9000 .07 0 100 200 300 400 500 600 700 800 900
1 .19 40 0 100 200 300 400 500 600 700 8001 .19 40 0 100 200 300 400 500 600 700 800
2 .22 80 40 0 100 200 300 400 500 600 7002 .22 80 40 0 100 200 300 400 500 600 700
3 .16 120 80 40 0 100 200 300 400 500 6003 .16 120 80 40 0 100 200 300 400 500 600
4 .12 160 120 80 40 0 100 200 300 400 5004 .12 160 120 80 40 0 100 200 300 400 500
5 .10 200 160 120 80 40 0 100 200 300 400 5 .10 200 160 120 80 40 0 100 200 300 400
6 .07 240 200 160 120 80 40 0 100 200 300 6 .07 240 200 160 120 80 40 0 100 200 300
7 .04 280 240 200 160 120 80 40 0 100 200 7 .04 280 240 200 160 120 80 40 0 100 200
8 .02 320 280 240 200 160 120 80 40 0 100 8 .02 320 280 240 200 160 120 80 40 0 100
9 .01 360 320 280 240 200 160 120 80 40 0 9 .01 360 320 280 240 200 160 120 80 40 0
Expected loss, $ 121.60 91.40 87.80 115.00 164.60 231.00 311.40 401.60 497.40 560.00 Expected loss, $ 121.60 91.40 87.80 115.00 164.60 231.00 311.40 401.60 497.40 560.00
Daily Scheduling ofDaily Scheduling of Telephone Operator Workshifts Telephone Operator Workshifts
0
5
10
15
20
25
30
Time
Num
ber o
f ope
rato
rs
Scheduler program assigns tours so that the number of operators present each half hour adds up to the
number required
Topline profile
12 2 4 6 8 10 12 2 4 6 8 10 12
Tour
0
500
1000
1500
2000
2500
Time
Cal
ls
12 2 4 6 8 10 12 2 4 6 8 10 12
LP Model for Weekly Workshift LP Model for Weekly Workshift Schedule with Two Days-off ConstraintSchedule with Two Days-off Constraint
Objective function: Minimize x1 + x2 + x3 + x4 + x5 + x6 + x7
Constraints: Sunday x2 + x3 + x4 + x5 + x6
3 Monday x3 + x4 + x5 + x6 + x7 6
Tuesday x1 + x4 + x5 + x6 + x7 5
Wednesday x1 + x2 + x5 + x6 + x7 6 Thursday x1 + x2 + x3 + x6 + x7 5 Friday x1 + x2 + x3 + x4 + x7
5 Saturday x1 + x2 + x3 + x4 + x5 5
xi 0 and integer
Schedule matrix, x = day offOperator Su M Tu W Th F Sa 1 x x … … … … ... 2 … x x … … … … 3 … ... x x … … … 4 … ... x x … … … 5 … … … … x x … 6 … … … … x x … 7 … … … … x x … 8 x … … … … … xTotal 6 6 5 6 5 5 7Required 3 6 5 6 5 5 5Excess 3 0 0 0 0 0 2
Seasonal Allocation of Rooms by Seasonal Allocation of Rooms by Service Class for Resort HotelService Class for Resort Hotel
First class
Standard
Budget
Per
cent
age
of c
apac
ity a
lloca
ted
to d
iffer
ent s
ervi
ce c
lass
es
60%
50%30%
20%
50%
Peak Shoulder Off-peak Shoulder (30%) (20%) (40%) (10%)Summer Fall Winter Spring
Percentage of capacity allocated to different seasons
30%20% 20%
10% 30%
50% 30%
Demand Control Chart for a HotelDemand Control Chart for a Hotel
0
50
100
150
200
250
300
350
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89
Days before arrival
Rese
rvat
ions
Expected Reservation Accumulation
2 standard deviation control limits
Yield Management Using the Yield Management Using the Critical Fractile Model Critical Fractile Model
P d x
C
C C
F D
p Fu
u o
( )( )
Where x = seats reserved for full-fare passengers d = demand for full-fare tickets p = proportion of economizing (discount) passengers Cu = lost revenue associated with reserving one too few seatsat full fare (underestimating demand). The lost opportunity is the difference between the fares (F-D) assuming a passenger, willingto pay full-fare (F), purchased a seat at the discount (D) price. Co = cost of reserving one to many seats for sale at full-fare(overestimating demand). Assume the empty full-fare seat wouldhave been sold at the discount price. However, Co takes on twovalues, depending on the buying behavior of the passenger whowould have purchased the seat if not reserved for full-fare. if an economizing passenger if a full fare passenger (marginal gain)Expected value of Co = pD-(1-p)(F-D) = pF - (F-D)
CD
F Do
( )
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