Xerox Routing Problem 3

8/14/2019 Xerox Routing Problem 3

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Column Generation combined with Constraint

Programming to solve a Technician Dispatch

Problem

Andrs WeintraubIndustrial Engineering Department

University of Chile

Working with:

November 2005

Cristin E. CortsCivil Engineering Department

University of Chile

Michael GendreauCentre de recherche sur les transports,

Universit de Montral

Sebastin SouyrisIndustrial Engineering Department

University of Chile

Fernando OrdoezIndustrial Engineering DepartmentUniversity of Southern California


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Motivation

Real Problem: Dynamic dispatch of technicians of Xerox Chile to repair failuresof their machinesalong the day.

The proposed scheme is based upon the classic formulationof the Vehicle Routing Problem with Soft Time Windows(VRPSTW), and it is formulated and solved as a Dantzig-Wolfe decomposition by using column generation, with aconstraint programming approach.

Xerox strategic objective: Client satisfaction, sotechnical service becomes relevant


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The Problem: Characteristic

Service times are stochastic and depend on the type of

machine and the specific failure.

Requests for one day must be scheduled at the beginning of it.

There is a scheduled response times

0tClientj

calls

time

jTW

jbMax arrival

timeClientj

Max Response Time


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Master Problem

Set Partitioning

Model

COLUMN

GENERATOR

Sub Problem

: min

1

{0,1},

r r

xr R

r ri

r R

r

P c

a i I

r R

=

DualMultipliers

Columns: dual variablei

,

Reduced Cost Column

(1 )

: path followed by technician

k k k

k ki ij

i P i j P

i

i

k

P

c t

P k

= +

Reformulation

* 0?kc

Generate complete daily schedule for each technician

,

variable 1 if route served out by some technician

cost of the route

parameter 1 if client is in route

: pool of routes

:

= (1 ) ,

:

r

r

i ij

i r i j r r

i

r

i r

R

r R

c t

a

+

2I

1I+

1I


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{ } 1 2 2

2

0,1 ,

, 0

ij

i i

x i I I j I

w i I

UNature of Variables

OF: Minimize reduced cost

2 1 2

, ,, ,

min (1 )i ij ij i ijx w

i I i j I I k K i j I

t x x

+

{ }1 2 2

2

1

0ij jii I I i I I

j x x I

+

= U U

Flow Conservation

1 2

1iji I j I

x

= One technician follow the new path

1 2

1 2 2

2

2

(1 )* ,i i ij j ij

i ji

j I I

i i i

w s t w x M i I I j I

w L x i I

w b i I

+ +

U

UTemporal and

feasibility constraints

Path ends in the final

fictitious node

SP: Shortest Path Problem With Soft Time Windows

1 2

11

iI

i I I

x+

=U


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How to solve it?

Dynamic Programming:

The most used approach in literature.

Difficult to implement and solve it under Branch and

Price scheme. Constraint Programming:

Good performance for short routes (five clients).

Easy to implement and solve it under Branch and Price

scheme with Constraint Branching.


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Constraint Programming Model

MaqSeq[l] M: secuence of clients in path to generate, l = 0..L

w[l] [0..F] : time to begin service of client l = 0..L

d[l] [0..F] : time windows violation of client l =1..L

t[l] [0..tvmax]: travel

VARIABLES

time to clients in position l-1 to l

SETS

M : clients

MT M : clients with technicianMQ M : clients in queue

PARAMETERS

i

i

ij

i

s :reparation time of client i

b :goal time of client i

F : period end of evaluation

tv : travel time between clients i and j

L : lenght of the path to generate

c : dual variable of client i given by the LR of SSP

tvmax : maximum time of trip allowed in the path to generate


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)

OBJECTIVE FUNCTION

MaqSeq[l]

l l l

Min ReducedCost = * d[l]+(1- t[l] - c

1. Time to begin the service

SUBJECT TO

MaqSeq[l-1]w[l]= w[l-1]+s +t[l] , l =1..L

2. Violation of time windows

MaqSeq[l]

w[l]-d[l] b , l = 0..L

3. Travel time between clients in the path

MaqSeq[l-1],MaqSeq[l]

t[l]=tv , l =1..L

4. Path begin time (depends of the state of service of the first client)

MaqSeq[0]w[0]= b

5. All the clients in the path must be different

alldifferent[MaqSeq]

6. First client in the path must have technician

MaqSeq[0] MT

7. Other clients in the path be not have technician

MaqSeq[l] MQ , l = 1..L


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Other constraints:

Avoid routes created before.

Start with a technician already serving a client.

Solver Search:

Strategic dichotomic.

Procedure: depth first search.


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General ProcessGenerate a set of initial routes.

One Route for each client. Feasible problem.

Solve a Linear Relaxation of the Master

Problem (MP) with the actual pool of routes

Ad new route to

the MP

From the Master Dual Variables generate a

route with Minimum Reduced Cost

Solve MP

END

0?r

c

i

rc

YES

NO


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Example Real Instance

Real Operation: Normal working day, 36 clients,9

technicians

Optimization total time 500 sec. The following maps show observed routing versus

optimized routing.


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Real instance


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Path used by the firm


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3

Path in Column Generation


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Technician Path Arrival time b Violation Travel time Technician Path Arrival time b Violation Travel time

1 m1 755 755 0 0 2 m2 660 1002 0 19

m20 838 765 73 23 m10 810 1015 0 30

m17 892 929 0 24 m27 1680 2040 0 30

m34 993 1059 0 21 m34 1800 1059 741 0

m36 1109 1890 0 26 4 m4 1670 1022 648 20

2 m2 1002 1002 0 0 m36 2085 2032 53 22

m15 1152 1716 0 30 m32 1820 1890 0 26m32 1290 2032 0 18 1 m1 870 970 0 0

m26 1371 1652 0 11 m21 930 947 0 0

m16 1581 1718 0 30 9 m9 1020 1068 0 30

4 m4 1022 1022 0 0 m15 1710 1716 0 60

m10 1152 845 307 30 m16 1860 1718 142 17

m21 1262 947 315 25 5 m5 1665 1662 3 15

m30 1354 1893 0 32 m17 1770 1834 0 20

5 m5 1024 1024 0 0 m29 910 929 0 20

m31 1083 1780 0 14 m12 765 765 0 0

m22 1190 1832 0 17 m14 1020 1955 0 20

m11 1325 1779 0 15 m31 1980 1780 200 406 m6 1980 1980 0 0 3 m3 1665 1024 641 15

m23 2068 2250 0 28 m20 720 845 0 30

m33 2113 2214 0 25 m28 1980 1779 201 60

8 m8 943 943 0 0 m22 1800 1832 0 15

m19 1101 985 116 23 m11 1740 1701 39 30

m13 1244 999 245 23 m33 2130 2214 0 42

m28 1352 1701 0 48 7 m7 900 755 145 41

m35 1425 2010 0 43 m18 600 805 0 14

9 m9 1068 1068 0 0 m25 990 1651 0 18

m14 1143 1834 0 15 m26 1650 1652 0 0

m25 1244 1651 0 11 6 m6 990 1980 0 203 m3 1662 1662 0 0 m23 990 2250 0 30

m12 1761 2040 0 24 m30 1680 1893 0 30

m29 1812 1955 0 21 8 m8 705 943 0 10

7 m7 970 970 0 0 m13 930 999 0 20

m12 1045 1015 30 25 m19 1800 985 815 150

m18 1249 805 444 24 m24 1995 1758 237 15

m24 1474 1758 0 30 m35 2110 2010 100 10

TOTAL 1530 656 TOTAL 3965 918

B =0.7 Z* 1268 B =0.7 Z* 3051

Path used by the firmPath in Column Generation


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CG Firm CG Firm CG Firm

1530 3965 656 918 1268 3051

Z*Travel timeViolation

= 0.7


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0

100

200

300

400

500

0.1 0.3 0.4 0.6 0.8 0.9

B

[min]

Violation(B)

Travel Time(B)

But this procedure does not assure


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But, this procedure does not assureoptimality

Improve CG methodology: Branch and Price.

Along the B&B tree, new routes with negative reduced cost could be found.

B&P: Generate columns at each node of the Branch & Bound

First Approach: Maestro Library for B&P (ILOG).

Same Master Problem.

Sub Problem: IP Shortest Path Model =>very inefficient.

Designed for hard time windows.

Second Approach:

Same Master Problem.

Branching Strategy: Constraint Branching (Ryan and Foster)

Sub Problem:1. Constraint Programming.2. Dynamic Programming very inefficient to satisfied constraint branching conditions.

B hi St t C t i t B hi (R


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Branching Strategy: Constraint Branching (Ryanand Foster)

If a basic solution is fractional then there existtwo clients i andj such that:

: 1, 1

0 1r r

i j

r

r a a

= =

<


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Branching Strategy: Constraint Branching (Ryanand Foster)

i

j

i

j

i

j

i

j

Two type of columns Three type of columns

i

j


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Ryan and Foster (1981) branching strategy:benefits

All constraints are imposed at the sub-problem directly

B&B terminates after a finite number of branches since

there are only a finite number of pairs of rows.

Each branching decision eliminates a large number of

variables from consideration .


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Stochastic Service Times (work with FernandoOrdez)

SVRP Literature: Formulate the problem either resource or

chance constraint, or both.

Robust Optimization:

Box uncertainty: completely independent uncertainty in

each sigiven by a box uncertainty set

{ }| ,= + i i i iU s s s s i I

Worst Case: all service times are the maximum, which is improbable


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Stochastic Service Times

Worst Case: for each path, first clients complete their repair time until Uk

Independent service time uncertainty per technician

robust parameter:

kk ii PU

=

( )

routing solution

( ) | , ,

:

kr

r i i i i i k i P x

r r

U x s s s s z i I z U

x

= +

Estimation of :1. From the real services time estimate each i.

2. From real routes estimate maximum over reparation time, max Uk.

3. From this data estimate statistically .

M d l f t h ti ti f l th t b i i f h th th t


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{ }2 1 2, 1

parameter for control uncertainty:

k

ik ij i

i I i I j I I

k

k Kz x

+

U

Max Over ReparationTime for technician

1 2

k

ik i ji

j I I

k Kz y

U

Max Over ReparationTime for client

{ }1 2, 1 ,k k

ij iji I j I k K y x I +U

Relation between variables xandy

{ } 1 2 2

, , 2

0,1 , ,

, 0 ,

k

ij

i k i k

x i I I j I k K

w i I k K

U

{ }1 2 2

2

1

,0k k

ij ji

i I I i I I

j k K x x I

+

= U U

1 21k

ij

k K j I

x i I I

=

U

2 2 2

end of the day

(1 )*

, ,

, , :

,

k

ik i i ij jk ij

k

ik ji

j I

ik ik i

w s z t w x M

i I I j I k K

w L x

i I k K L

w b i I k K

+ + +

U

1 2

1 1k

iI

i I I

x k K +

= U

,

,

min (1 )k

x ik ij ij

k K i I k K i j I

t x

+ Two new Variables:20

k

i z i I Upper Time of reparationfor client

{ }1 2 2

0,1,

ijyi I I j I

UIf previous client hasmax length reparation time

{ }1 2, 1 ,k

iiji

i I j I k Kz

y I +U

Relation between variables zandy

Model forces to have reparation time of max length at beginning of each path, the worst case.


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Robust Model Test

Robust Problem solved with Column Generation and ConstraintProgramming. New constraints are added to the SubProblem.

Test:

Comparison with deterministic model.

Artificial data.

Solve for same set of customers deterministic and robust model.

Obtain routes for technicians in each case. For each customer, assume Weibell distribution (which represents

well repair times). Parameters for Weibell, derived from data. Obtain2000 values of repair time for each customer, leading to 2000simulations.

For each simulation test which solution is better for a given


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P(rob


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Conclusions and Further Research

Routing Problems are mostly IP friendly, small LP/IP gaps.

Branch and Price does not improve much in these instances.

Constraint Programming works quite well and provides the shortestroutes: Better and simpler than DP?.

Results are promising, considering that Xerox is supposed to havegood quality logistics.

Consider less friendly problems for Branch and Price.

When incorporating stochastic nature of repair times, obtain numericalresults for Robust Optimization. Compare to deterministic model.

Develop algorithm for real time scheduling as customers appearduring the day.

Documents

Xerox Routing Problem 3