ISM 270 Service Engineering and Management Lecture 7: Forecasting and Managing Service Capacity.

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

qq

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

( )