final presentation

“LOGISTICS MODELS”

Andrés Weintraub P.Departament Industrial Engineering

University of Chile

PASISantiago

Agosto 2013

Final PresentationXerox Fleet Design

Student: Daniel LengProfessors: Andrés Weintraub

Cristian CortesMichel Gendreau

Pablo Rey

Problem Description◦ The Company ◦ Precedents ◦ The problem

Objectives◦ Primary Objective◦ Specific Objectives

Available Information

Information Analysis◦ Amount of Calls◦ Time of Call◦ Commune (County) ◦ Service Time (by product line)◦ Target Arrival Time

Obtaining Instances

Routing Weeks

Results Analyses

Economic Analyses

Agenda

The Company ◦ Xerox offers home or office technical support

to its clients for its different lines of products: Printers, copiers, plotters, etc.

◦ It is necessary to meet a target time of arrival from the client’s calling time, depending on the client’s priority.

◦ There are 16 areas of attention, divided by type of equipment or geographical area.

◦ There is a force of 102 technical personnel force assigned to the 16 areas.

Problem Description

Precedents ◦ There is a routing model for the

dispatch of technicians. It works as an "24-hour-in-advance service".

◦ The number of technicians assigned to each area has been calculated using only the judgment of the decision makers.

The Problem ◦ How to determine the suitable number of technicians for each

area, in order to fulfill the chosen performance indicators?

Obtaining Instances

Objectives

Problem Description


Information Analysis

Routing Weeks

Results Analyses

Economic Analyses

Primary Objective◦ “To obtain the minimal possible number of technical personnel, capable

of fulfilling the most accurate service level”.

Specific Objectives◦ “To estimate the demand for service and the parameters that conform

it”.◦ “To construct a master model that integrates the weekdays interactions

using the routing model”.◦ “To use the week model to simulate different scenarios”.◦ “To apply statistical studies on the results and use and economical

approach to determine the optimal number of technical personnel”.

Objectives

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

There is one complete year of information

(May, 2002- April, 2003)

Each call is defined according to :◦ Address◦ Commune (County) ◦ Attending technician◦ Date and time of call ◦ Target arrival time ◦ Area◦ Model ◦ Client ◦ Service Time ◦ Arrival Time

Available Information Two areas were used for the

pilot model, namely the 101 and 110 areas

For each of these areas we must analyze the following aspects: ◦ Amount of calls ◦ Time of call (for each call)◦ Commune (County) (for each call)◦ Service time (for each call)◦ Target arrival time (for each call)

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

010

110

310

510

711

011

411

611

812

212

414

014

215

020

220

420

630

130

330

530

730

931

140

141

043

050

070

088

095

002000400060008000

100001200014000

Area

Amou

nt o

f Ca

lls

Janua

ry

Februa

ry Marc

h Ap

ril May Jun

e July

Augu

st

Septe

mber

Octobe

r

Novem

ber

Decembe

r0

5000

10000

15000

20000

Month

Amou

nt o

f Ca

lls

Amount of Calls◦ By Area

◦ By Month (all areas)


Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Amount of Calls◦ By weekday


Mond

ay

Tues

day

Wed

nesd

ay

Thu

rsda

y

Frid

ay

200002200024000260002800030000320003400036000

Day of the Week

Amou

nt o

f Ca

lls We used the Chi Squared Test at

95 % of confidence to check if there is a relation between:◦ Month - Area – Weekday – Amount of

calls

The following results were obtained for each area

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

◦ Area 101

Information AnalysisGroup Months Chi Squared Freedom Degrees Asymptotic Significance

Group 1 January, March, April, June, August, October, November 6,894 6 0,331

Group 2 May, December 0,032 1 0,857Group 3 February, September 0,458 1 0,499Group 4 July 0 0 1

◦ Area 110

◦ Both areas

Group Months Chi Squared Freedom Degrees Asymptotic Significance

Group 1 January, March, June, November, August 6,89 4 0,142

Group 2 October, April, May, July 0,574 3 0,902Group 3 February, September, December 2,149 2 0,342

Group Months Chi Squared Freedom Degrees Asymptotic Significance

Group 1 January, March, June, August, November, May 6,844 5 0,233

Group 2 October, April, July 5,226 2 0,73Group 3 February, September, December 3,113 2 0,211

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

We cannot assume the same relation by month in both areas◦ As a consequence of having found significant differences among

areas, the monthly amount of calls were grouped in a different way for each area


January

February

March

April

May

June

July

August

SeptemberOctober

November

December

Area 101 Area 110 Both areas

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

In the same way, groups were made for the relation:

◦ Weekday – Amount of calls

◦ Analogous results were obtained for both areas, individually and as a

whole, so that the use of different criteria was not necessary. Both Monday and Friday show distinct results from the rest of the week,

and between them.

The rest of the week shows homogeneous results.


Weekday Chi Squared Freedom Degrees Asymptotic Significance Monday 0 0 1

Tuesday, Wednesday, Thursday 0,25 2 0,883

Friday 0 0 1

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Adjusting distributions to amount of calls◦ Several discrete distributions of probabilities were adjusted ◦ Poisson’s distribution ended up fitting all combinations of Groups

and Weekdays


Lambda estimation [calls/day]:◦ Area 101

Group Monday Rest of the Week Friday

Group 1 33,75 31,5 25,38Group 2 29,88 29,06 25,63Group 3 25,56 30,41 21,14Group 4 34,2 33,43 26

◦ Area 110 Group Monday Rest of the

Week Friday

Group 1 32,8 29,64 23,05Group 2 35,88 33,22 23,31Group 3 28,57 23,69 21,19

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Time of call◦ Histograms were constructed to check relations of the type:

Amount of calls - Hour of the day - Monthly Groups

◦ There were no significant differences among Areas or among Monthly

Groups. Therefore analogous histograms will be used for both areas.


7:00 - 8:00

8:00 - 9:00

9:00 - 10:00

10:00 - 11:00

11:00 - 12:00

12:00 - 13:00

13:00 - 14:00

14:00 - 15:00

15:00 - 16:00

16:00 - 17:00

17:00 - 18:00

18:00 - 19:00

19:00 - 20:00

0%2%4%6%8%

10%12%14%16%18%

Group 1Group 2Group 3Group 4General

Hour Interval

Perc

enta

ge o

f ca

lls

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Commune (County) ◦ Histograms were constructed to check relations of the type:

Amount of calls - Commune (County) Given that each Area has different communes (counties) assigned, it is

necessary to use different histograms for each Area


Santiago Communes (Counties) map and Xerox service areasObtaining

Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

◦ Area 101


VITACURA LAS CONDES PROVIDENCIA SANTIAGO0%

10%

20%

30%

40%

50%

Communes

Perc

enta

ge o

f ca

lls

◦ Area 110

MAIPU

CERRILL

OS

LO ES

PEJO

PEDRO AGUIRRE C

ERDA

SAN MIGUEL

SAN JO

AQUIN

LA CIST

ERNA

SAN RAMON

LA GRANJA

LA FL

ORIDA

EL BOSQ

UE

LA PI

NTANA

PUEN

TE ALT

O

SAN BER

NARDO

PADRE H

URTADO

0%2%4%6%8%

10%12%14%16%18%20%

Communes

Perc

enta

ge o

f ca

lls

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Service Time◦ The different products are grouped by lines of products


CH-COP. PERSO-

NAL

ODP-CNV DIG

ODP-CNV L&L

ODP-DCS WG

ODP-WG Otros0%

20%40%60%80%

Area 101Area 110Lines of Products

Perc

enta

ge o

f c

alls

◦ There is no known probabilities distribution function that can be adjusted to the service time at any confidence level.

◦ A lot of observations are concentrated around a few points.◦ Cluster analyses were made to study this behavior◦ The result of the cluster study was used to generate a discrete

probabilities distribution.

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Target Service Time (TST)◦ The TST is measured as the elapsed time between the calling

time and the maximum allowed arrival time◦ This parameter depends on the client’s priority◦ Because of the absence of a priorities’ list, we use empiric

probabilities to assign a TST to each call


Time [min] Probability15 0,5

30 0,13

45 0,25

60 0,12

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

An instance is defined as the simulation of an entire week of calls

The algorithm to create an instance works as follows:

◦ Define Group and Area

◦ Use Poisson distribution to generate amount of calls for each day

◦ For each call determine: Commune (County) Time of call TST Service Time

Obtaining Instances

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

The construction of the week model is based on a set of policies of interaction◦ The clients that couldn’t be attended to on the previous day have priority one on the following day

routing◦ If there are not enough technicians to look after all the left-outs at the beginning of the day, the one

with the longest delay has priority◦ The rest of none-attended-clients will be included on the routing model with a TST equals cero.

Routing Weeks

The routing model is applied to each instance with different numbers of available technicians.

A set of performance indicators will be obtained for every pair: Instance(i) - Number of technicians (k)

Utilization Average delay per client Average travel time Extra time

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Routing Weeks

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Definition of interval of the number of technicians Estimation of the calculation time

◦ Seven groups of months (three for area 110 and four for area 101)

◦ 220 minutes per instance◦ One instance for every technicians quantity◦ It takes 0,6 days for area 101 and 0,45 for area 110◦ Maximum of 640 days unfeasible of calculation time in the

laboratory (80 days on nine computers)

Routing Weeks

Length of the interval\ Number of

instances50

Instances100

Instances150

Instances200

Instances300

Instances

1 53 105 158 210 263

2 105 210 315 420 525

3 158 315 473 630 Unfeasible

4 210 420 630 Infactible Unfeasible

5 263 525 Unfeasible Unfeasible Unfeasible

6 315 Unfeasible Unfeasible Unfeasible Unfeasible

7 368 Unfeasible Unfeasible Unfeasible Unfeasible

◦ Area 101: ◦ 11, 10, 9, 8 Technicians

◦ Area 110: ◦ 10, 9, 8 ,7 Technicians

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

The average delay indicator will determine the service level

Log-normal distribution showed to be the best-adjusted probabilities distribution◦ If

Result analyses

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

0 3 5 8 11131619212427303235384043464851545759626567700

0.2

0.4

0.6

0.8

1

1.2

8 Técnicos 9 Técnicos 10 Técnicos 11 Técnicos

X [min]

Fx(X

)

Economic Analyses

Results analyses X is defined as the Maximum Average Delay feasible

for a given confidence level The confidence level selected is 98% (there are not significant

differences between 95% and the selected level) Then and X is defined as the Service Level

Group 8 Technicians

9 Technicians

10 Technicians

11 Technicians

1 38 16 10 8

2 24 10 6 4

3 29 19 8 6

4 48 26 15 10

Service Level (X) at 98% of confidence [min] Area 101 Area 110

Grupo 7 Technicians

8 Technicians

9 Technicians

10 Technicians

1 90 36 15 8

2 97 36 18 13

3 62 23 12 6

The other three indicators will be estimated using the sample mean

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Results analyses

0123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

7

8

9

10

11

12

Grupo 1Grupo 2Grupo 3Grupo 4

Service level [min]

Tech

nici

ans

[uds

]

0234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586878889909192939495969798991001011021031041051061071081091106

7

8

9

10

11

Grupo 1Grupo 2Grupo 3

Service level [min]

Tech

nici

ans

[uds

]

◦ Area 101

◦ Area 110

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Economic analyses Fixed costs

o Technicians salary ($600.000 a month, $3.333 per hour) Variable costs

o Real costs• Overtime cost (4.000 per hour)

o Opportunity costs• Travel time (3.333 per hour)• Unused time (3.333 per hour)

Economic modelo Variables

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Economic analyses Economic model

o Parameters

o Relations

o Cost Function

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Economic analyses ◦ Area 101

◦ Area 110

0 10 20 30 40 50 60 70$ 80,000,000$ 85,000,000$ 90,000,000$ 95,000,000

$ 100,000,000$ 105,000,000$ 110,000,000$ 115,000,000$ 120,000,000$ 125,000,000

Service level [min]

Anu

al c

ost

0 20 40 60 80 100 120 $ 65,000,000

$ 70,000,000

$ 75,000,000

$ 80,000,000

$ 85,000,000

$ 90,000,000

$ 95,000,000

$ 100,000,000

Service level [min]

Anu

al c

ost

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Economic analyses ◦ Area 101

◦ Area 110

5 10 15 20 25 30 35 40 45 50 55 $ 80,000,000 $ 85,000,000 $ 90,000,000 $ 95,000,000

$ 100,000,000 $ 105,000,000 $ 110,000,000 $ 115,000,000 $ 120,000,000 $ 125,000,000

Service level [min]

Anu

al c

ost

10 20 30 40 50 60 70 80 90 100 110 $ 65,000,000

$ 70,000,000

$ 75,000,000

$ 80,000,000

$ 85,000,000

$ 90,000,000

$ 95,000,000

$ 100,000,000

Service level [min]

Anu

al c

ost

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Feasible and efficient policies

Economic analyses

Política Nivel de servicio

Enero, Marzo, Abril, Junio, Agosto, Octubre, Noviembre Mayo, Diciembre Febrero, Septiembre Julio Costo

A 9 11 10 10 11 $ 121.258.883

B 10 10 10 10 11 $ 112.665.826

C 11 10 9 10 11 $ 110.300.408

D 15 10 9 10 10 $ 109.215.879

E 16 9 9 10 10 $ 101.048.499

F 19 9 9 9 10 $ 98.636.470

G 25 9 8 9 10 $ 96.358.384

H 26 9 8 9 9 $ 95.190.756

I 30 9 8 8 9 $ 92.853.603

J 39 8 8 8 9 $ 84.676.682

K 48 8 8 8 8 $ 83.520.106

o Area 101

o Area 110Política Nivel de

servicioEnero, Marzo, Junio, Noviembre, Agosto

Octubre, Abril, Mayo, Julio

Febrero, Septiembre, Diciembre Costo

L 15 9 10 9 $ 98.284.266

M 18 9 9 9 $ 93.666.644

N 24 9 9 8 $ 90.650.798

O 36 8 8 8 $ 78.722.648

P 62 8 8 7 $ 74.958.308

Q 90 7 8 7 $ 71.193.299

R 97 7 7 7 $ 66.205.289

Obtaining Instances

Objectives

Problem Description



Routing Weeks

Results Analyses

Economic Analyses

Conclusions

Final PresentationXerox Fleet Design

Student: Daniel LengProfessors: Andrés Weintraub

Cristian CortesMichel Gendreau

Pablo Rey

March 26, 2009

Cluster analyses

Backup

Obtaining Instances

Objectives

Problem Description



Obtaining Indicators

The Model

Marca de Clase [min]

Cantidad de observaciones Probabilidad

349 6 0,0046

207 8 0,0061

60 344 0,263

256 17 0,013

150 35 0,0268

177 19 0,0145

35 221 0,169

91 145 0,1109

118 116 0,0887

136 26 0,0199

1 192 0,1468

76 91 0,0696

15 88 0,0673

Total 1308

Modelo Intervalo recomendado

Cantidad utilizada

CH-COP. PERSONAL 12-16 13

ODP-CNV DIG 14 - 17 16

ODP-CNV L&L 11-15 13

ODP-DCS WG 10-16 15

ODP-WG 12-15 14

CH-COP. PERSONAL Results

An instance is defined as follows:

Backup

Call ID Commune (County)

Date Time of call Target Arrival Time [Hrs]

Service Time [Hrs]

1 5 14-Mar-08 8:00 4 4,4

2 46 14-Mar-08 6:00 5 3,8

3 8 14-Mar-08 7:00 3 4,2

4 6 15-Mar-08 8:00 7 9,7

5 45 15-Mar-08 8:30 9 9,4

6 57 16-Mar-08 9:15 2 3,8

7 34 16-Mar-08 10:30 24 3,5

8 34 17-Mar-08 14:00 6 4,8

9 87 17-Mar-08 18:00 8 7,8

10 25 17-Mar-08 18:00 4 3,3

11 362 18-Mar-08 18:15 72 9,7

Obtaining Instances

Objectives

Problem Description




The Model

The model was implemented on Java language and uses the following Pseudo code

Backup

Obtaining Instances

Objectives

Problem Description




The Model

Cost composition area 101

Backup

Obtaining Instances

Objectives

Problem Description




The Model

Nivel de servicio Costo Anual Costo Fijo Tiempo de viaje Tiempo extra Ineficiencia

9 $ 121.258.883 63,30% 3,50% 1,00% 32,10%

10 $ 112.665.826 64,40% 3,90% 1,10% 30,60%

11 $ 110.300.408 64,70% 4,00% 1,10% 30,20%

15 $ 109.215.879 64,80% 4,10% 1,20% 29,90%

16 $ 101.048.499 65,90% 4,60% 1,30% 28,10%

19 $ 98.636.470 66,30% 4,80% 1,40% 27,60%

25 $ 96.358.384 66,60% 4,90% 1,50% 27,00%

26 $ 95.190.756 66,80% 5,00% 1,50% 26,70%

30 $ 92.853.603 67,20% 5,20% 1,50% 26,10%

39 $ 84.676.682 68,70% 5,80% 1,90% 23,60%

48 $ 83.520.106 69,00% 5,90% 1,90% 23,20%

final presentation

Documents

calls time

target time of arrival

callservice time

calltarget arrival time

economic analyses agenda

areas information analysis

routing model

callsby area