Discrete facility location problems · 4.Each customer has to be serviced from one, and only one, open facility. Decisions: 1.Which potential facilities should be open in an optimal

Discrete facility location problems– Theory, Algorithms, and extensions to multiple objectives

Sune Lauth Gadegaard

Department of Economics and Business Economics, Aarhus University

June 22, 2016

CCORALAARHUS UNIVERSITY

Outline Introduction Single objective SSCFLP Bi–objective SSCFLP

Outline

I Overview of the PhD processI Introduction to discrete facility location problems

I MotivationI Branch–and–cut

I Single objective problemsI A cut–and–solve approach (First paper)I A semi–Lagrangean approach (Second paper)

I Bi–objective problemsI An ε–constrained approach for cost–bottleneck FLPs

(Third paper)I Bound set based branch–and–bound algorithms for

bi–objective combinatorial optimization (Fourth paper)


The process

2014 2015 2016

Start

(Feb

. 2013)

Stochasticprogramming

EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris

Convex optimizationEdinburgh

IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon

Researchprocessesin logistics

Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in

Four research projects

Super

vision


The process

2014 2015 2016Start

(Feb

. 2013)


EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris


IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon


Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in


Super

vision


The process

2014 2015 2016Start

(Feb

. 2013)


EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris


IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon


Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in


Super

vision


The process

2014 2015 2016Start

(Feb

. 2013)


EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris


IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon


Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in


Super

vision


The process

2014 2015 2016Start

(Feb

. 2013)


EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris


IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon


Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in


Super

vision


The process

2014 2015 2016Start

(Feb

. 2013)


EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris


IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon


Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in


Super

vision


The process

2014 2015 2016Start

(Feb

. 2013)


EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris


IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon


Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in


Super

vision


The process

2014 2015 2016Start

(Feb

. 2013)


EURO-INFORM

S

Rome

Teac

hing

School on

Colum

n

Gener

ation

– Paris


IFORS

Barce

lona

Lanca

ster

Ehrgott

NetOpt

Lisbon


Sum

mer

universi

ty

MCDM

Hamburg

Course onTeaching

INFORM

S

Philadelp

hia

NetOptLisbon

Hand

in


Super

vision


Motivation


Motivation

Location problems in the public sectorI Fire and police stations and

hospitalsI SchoolsI Emergency centersI Sirens


Motivation




Motivation




Motivation

Location problems in the private sectorI Storage facilitiesI Production facilitiesI Supplier selection


Motivation

Location problems in the private sectorI Storage facilitiesI Production facilitiesI Supplier selection


Description

Description of the Single Source Capacitated Facility LocationProblem (SSCFLP)

Assumptions:1. Finite sets of customers and potential facility

sites.2. Demands and capacities are fixed and known.3. Costs are fixed and known.4. Each customer has to be serviced from one, and

only one, open facility.

Decisions:1. Which potential facilities should be open in an

optimal solution (location–decision)2. Given a set of open facilities, which facility

should the individual customer be allocated to(allocation–decision).

Objective: Minimize the total cost: Cost of locating facilities +cost of allocating customers


Description










Description










Description










Illustration


Illustration


Illustration


Illustration


Illustration


Solution procedure

I Procedure, that guarantees to findoptimal solutions (one of minimumcost!).

I Uses optimistic and pessimisticestimates to squeeze optimalsolution value.

I Enumerates (implicitly) all solutionsto the optimization problem.

I Adds “cutting planes” to make theoptimistic estimate more accurate.

Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

Optimal

Pessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Solution procedure





Cost

Pessimistic

Optimistic

OptimalPessimistic

Optimistic


Tree structure

Suppose we know a solution that costs us 10.000 kroner.

Facility in Holstebroclosed

Facility in Holstebroopen

Optimisticestimate = 9.200

Optimisticestimate = 10.042

People from Struergoes to Holstebro

People from Struerdoes not go to Holstebro


The first paper

– Joint work withAndreas Klose & Lars Relund Nielsen


Goal and purpose

I Want to solve the SSCFLP to proven optimalityI Want to do it as fast as possibleI Want an algorithm which is robust


Outline of procedure

Make an initialoptimistic estimate

Make an initialpessimistic estimate

Stop, we are done

Close the remaining gap





Stop, we are done






Stop, we are done






Stop, we are done






Stop, we are done



An optimistic estimate

I Drop the single source requirementI Allow each customer to be serviced from multiple

facilities.

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

2 21 1




facilities.

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

2 21 1




facilities.

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

2 21 1




facilities.

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

d = 2 d = 2 d = 2

s = 3 s = 3 s = 3

2 21 1


Making the optimistic estimate more accurate

We can add more information by generating cutting planes

ExampleI Consider facility 1 with capacity of 5.I Four customers with demands d1 = 2, d2 = 2, d3 = 3 and

d4 = 4I xij = 1 iff facility i serves customer j

2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2y1

x1,1 + x1,4 ≤ 1y1

x1,2 + x1,4 ≤ 1y1

x1,3 + x1,4 ≤ 1y1






2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2y1

x1,1 + x1,4 ≤ 1y1

x1,2 + x1,4 ≤ 1y1

x1,3 + x1,4 ≤ 1y1






2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2y1

x1,1 + x1,4 ≤ 1y1

x1,2 + x1,4 ≤ 1y1

x1,3 + x1,4 ≤ 1y1






2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2

y1

x1,1 + x1,4 ≤ 1y1

x1,2 + x1,4 ≤ 1y1

x1,3 + x1,4 ≤ 1y1






2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2

y1

x1,1 + x1,4 ≤ 1

y1

x1,2 + x1,4 ≤ 1y1

x1,3 + x1,4 ≤ 1y1






2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2

y1

x1,1 + x1,4 ≤ 1

y1

x1,2 + x1,4 ≤ 1

y1

x1,3 + x1,4 ≤ 1y1






2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2

y1

x1,1 + x1,4 ≤ 1

y1

x1,2 + x1,4 ≤ 1

y1

x1,3 + x1,4 ≤ 1

y1






2x1,1 + 2x1,2 + 3x1,3 + 4x1,4≤ 5y1

x1,1 + x1,2 + x1,3 + x1,4 ≤ 2y1

x1,1 + x1,4 ≤ 1y1

x1,2 + x1,4 ≤ 1y1

x1,3 + x1,4 ≤ 1y1



TheoremAll (interesting) strong inequalities (facets) for the capacityconstraints can be obtained from strong inequalities (facets) of asimilar knapsack constraint.


Making a (not too) pessimistic estimate

1. Obtain a feasible solution.2. Solve a much reduced problem centered at the current

solution.3. Obtain a new solution and make that the current.4. Remove the neighborhood around previous solution and

go to 2.


Closing the gap: Cut–and–solve

1. Solve the full problem with single source requirementremoved (optimistic estimate).

2. Partition the set of facilities into two sets: open and closedones.

3. Solve the problem with all closed facilities fixed(pessimistic estimate).

4. Require that at least one of the closed facilities should beopen in the next iteration. Repeat from 1.


Cut–and–solve: Illustration








Results

Cost

opt

Pessimistic

Optimistic

0.03 to 0.3 percent offon average



Results

Cost

opt

Pessimistic

Optimistic




Results

Cost

opt

Pessimistic

Optimistic




Results

I Closing the gap (comparing with CPLEX)I For easy instances (solved within 2 seconds) CPLEX is

slightly faster than cut–and–solve.I For larger and harder instances cut–and–solve performs

between 5 and 135 times better.

I Comparing with customized algorithm (Yang et. al (2012))I Our improved algorithm runs 12 to 85 times faster than

Yang’s on average.I Reason: Smart way of generating cutting planes, definition

of subproblems, stopping optimization early, quality of theinitial “pessimistic estimate”.


Results

I Closing the gap (comparing with CPLEX)I For easy instances (solved within 2 seconds) CPLEX is

slightly faster than cut–and–solve.I For larger and harder instances cut–and–solve performs

between 5 and 135 times better.I Comparing with customized algorithm (Yang et. al (2012))

I Our improved algorithm runs 12 to 85 times faster thanYang’s on average.

I Reason: Smart way of generating cutting planes, definitionof subproblems, stopping optimization early, quality of theinitial “pessimistic estimate”.


The second paper


Motivation

I Too many possible assignments!I 80 potential facility sites and 400 customers leads to 32.000

variables

I Hard to find feasible solutions in the subproblems


Motivation

I Too many possible assignments!I 80 potential facility sites and 400 customers leads to 32.000

variables

I Hard to find feasible solutions in the subproblems


Semi–Lagrangean relaxation

We have a tough constraint:

All customers have to be serviced by exactly one facility

mAll customers have to be serviced by at most one facilityANDAll customers have to be serviced by at least one facility



We have a tough constraint:

All customers have to be serviced by exactly one facilitym

All customers have to be serviced by at most one facilityANDAll customers have to be serviced by at least one facility



I We keep the “serviced by at most one facility”

I Remove the “serviced by at least one facility”

I Easy to find a solution!

I If a customer is not serviced, we add cost of λ–kroner perunserviced customer.

I Thin out by removing all possible assignments that costsus more than λ–kroner.



I We keep the “serviced by at most one facility”I Remove the “serviced by at least one facility”























A dual–ascent algorithm

1. Set an initial penalty price λ.2. Remove all assignments that costs more than λ.3. Solve the SSCFLP with only “service by at most one

facility” constraints (semi–Lagrangean subproblem).I If all customers are serviced, then the solution is optimal.I Else, the penalty price was too low, increase it and go to 2.


Integrating cut–and–solve and dual–ascent

There are two obvious ways to integrate

Solve the semi–Lagrangeansubproblems usingthe cut–and–solve algorithm

Solve the cut–and–solvesubproblems usingthe dual–ascent algorithm












Results

Small

Tight

Less tig

ht

Loose0

1,000

2,000

3,000

4,000

5,000

CPU

seco

nds

DA-CS CS-DA CS


The third paper

– Joint work withAndreas Klose & Lars Relund Nielsen


Motivation

I There is almost always more than one objective whenmaking decisions

1. Minimize cost.2. Minimize environmental impact.3. Minimize violation of some constraints (laws).4. Maximizing equity/fairness of a solution.

I What is better? Profit of 10 kroner and a variance of 7 or aprofit of 5 and variance of 0.5?

I We want to find all the rational compromises betweenconflicting objectives.


Motivation






Motivation






Pareto optimal/rational compromise

Definition (Pareto optimal)

Given two objectives a decision is said to be Pareto optimal ifnon of the objectives can be improved without deterioratingthe other.


Pareto optimal: Illustration

Cost

Optimal = 10

Environmental impact

(1000,10)

(100,55)

The non–dominated frontier.



Cost

Optimal = 10


(1000,10)

(100,55)




Cost

Optimal = 10


(1000,10)

(100,55)




Cost

Optimal = 10


(1000,10)

(100,55)




Cost

Optimal = 10


(1000,10)

(100,55)




Cost

Optimal = 10


(1000,10)

(100,55)



A solution procedure

ε–constrained method:1. Minimizes only one objective.2. Keeps the other constrained from above (ε is the upper

bound).3. Continuously lowers the value of ε.4. Builds the non–dominated frontier from one end to the

other.


Illustration of ε–constraineapproach

Cost

Optimal = 10




Cost

Optimal = 10




Cost

Optimal = 10



The bi–objective cost–bottleneck location problem(BOCBLP)

We add an additional objective function:1. Minimize total cost of opening facilities and servicing

customers.

2. Minimize the maximum distance from a customer to thefacility she is assigned to.


The bi–objective cost–bottleneck location problem(BOCBLP)

We add an additional objective function:1. Minimize total cost of opening facilities and servicing

customers.2. Minimize the maximum distance from a customer to the

facility she is assigned to.


ε–constrained method for the BOCBLP

Set ε = ∞

Removeassignments

Minimizecost

Solutionexists?

StopnoUpdate εyes



Set ε = ∞Removeassignments

Minimizecost

Solutionexists?

StopnoUpdate εyes




Minimizecost

Solutionexists?

StopnoUpdate εyes




Minimizecost

Solutionexists?

StopnoUpdate εyes




Minimizecost

Solutionexists?

Stopno

Update εyes




Minimizecost

Solutionexists?

StopnoUpdate εyes




Minimizecost

Solutionexists?

StopnoUpdate εyes


ε–constrained method illustrated


ε–constrained method illustrated


Results

We tested with three different cost structures:

cij

tij

cij

tij

cij

tij


Results

I For the single source capacitated BOCBLP, negativelycorrelated cost/time produced many morenon–dominated solutions.

I Generally very few redundant iterations.I Comparing with another standard method

I Two–phase method generally much slower: between 2 and100 times slower than ε–method.

I Two–phase method suffers from bad scaling and too–largeprograms.

I The shape of the frontier does not fit well to the two–phasemethod.


The fourth paper– Joint work with

Matthias Ehrgott & Lars Relund Nielsen


A direct approach

I The ε–constraint approach required effective solution ofthe subproblems.

I We have to know the problem type before a priori

I We want to be able to solve general problems, with noparticular structure.


A direct approach





A direct approach





Bound sets

Cost 1

Cost 2

optimal

optimal


Bound sets

Cost 1

Cost 2

optimal

optimal


Bound sets

Cost 1

Cost 2

optimal

optimal


Bound sets

Cost 1

Cost 2

optimal

optimal


Bound sets

Cost 1

Cost 2

optimal

optimal


Bound sets

Cost 1

Cost 2

optimal

optimal


Bound sets

Lower bound setI Optimistic estimate of the non–dominated frontier

I A set of points ensuring, that all solutions lie above!I Disregard integrality constraints

Upper bound setI Pessimistic estimate of the non–dominated frontierI We only need to look for Pareto solutions below the upper

bound setI Cost–points of feasible solutions

We only need to search between the upper and lower boundsets


Bound sets

Lower bound setI Optimistic estimate of the non–dominated frontierI A set of points ensuring, that all solutions lie above!

I Disregard integrality constraintsUpper bound set

I Pessimistic estimate of the non–dominated frontierI We only need to look for Pareto solutions below the upper




Bound sets

Lower bound setI Optimistic estimate of the non–dominated frontierI A set of points ensuring, that all solutions lie above!I Disregard integrality constraints





Bound sets


Upper bound setI Pessimistic estimate of the non–dominated frontier

I We only need to look for Pareto solutions below the upperbound set

I Cost–points of feasible solutionsWe only need to search between the upper and lower boundsets


Bound sets



bound set

I Cost–points of feasible solutionsWe only need to search between the upper and lower boundsets


Bound sets






Bound sets






Problem splitting

We can split the problem into smaller problems

1. Facility in Holstebro is open↔ Facility in Holstebro is notopen

2. Objective 1 must improve↔ Objective 2 must improve3. Solutions must be Pareto optimal


Problem splitting

We can split the problem into smaller problems1. Facility in Holstebro is open↔ Facility in Holstebro is not

open

2. Objective 1 must improve↔ Objective 2 must improve3. Solutions must be Pareto optimal


Problem splitting


open2. Objective 1 must improve↔ Objective 2 must improve

3. Solutions must be Pareto optimal


Problem splitting


open2. Objective 1 must improve↔ Objective 2 must improve3. Solutions must be Pareto optimal


Illustration

Cost2

Cost1


Illustration

Cost2

Cost1


Illustration

Cost2

Cost1


Illustration

Cost2

Cost1


The procedure

Choose asubproblem

Comparebounds

Roombetweenbounds?

Splitsubproblem

yes

Add newto list

no


The procedure

Choose asubproblem

Comparebounds

Roombetweenbounds?

Splitsubproblem

yes

Add newto list

no


The procedure

Choose asubproblem

Comparebounds

Roombetweenbounds?

Splitsubproblem

yes

Add newto list

no


The procedure

Choose asubproblem

Comparebounds

Roombetweenbounds?

Splitsubproblem

yes

Add newto list

no


The procedure

Choose asubproblem

Comparebounds

Roombetweenbounds?

Splitsubproblem

yes

Add newto list

no


The procedure

Choose asubproblem

Comparebounds

Roombetweenbounds?

Splitsubproblem

yes

Add newto list

no


The procedure

Choose asubproblem

Comparebounds

Roombetweenbounds?

Splitsubproblem

yes

Add newto list

no


Results

I Tested different ways of comparing lower and upperbound sets

1. When stating the lower bound sets explicitly, LP based testworse than point–in–polytope test.

2. When stating the lower bound sets implicitly, new splittingscheme is not improving the performance


Results





Results





Results

I Tested if a bi–objective approach to cutting planes works

I It does! The algorithm becomes much more robust andalso faster.

I Tested an updating strategy of the lower bound setI Works very well. Lower bounds are worse, but we can

check many more subproblems.


Results

I Tested if a bi–objective approach to cutting planes worksI It does! The algorithm becomes much more robust and

also faster.

I Tested an updating strategy of the lower bound setI Works very well. Lower bounds are worse, but we can

check many more subproblems.


Results

I Tested if a bi–objective approach to cutting planes worksI It does! The algorithm becomes much more robust and

also faster.I Tested an updating strategy of the lower bound set

I Works very well. Lower bounds are worse, but we cancheck many more subproblems.


Results

I Compared with a two phase method

1. Ranking based two phase method works very bad on ourproblems

2. PSM based two phase method works better, and even beston smaller problems

3. Our best algorithm, outperforms two phase methods onlarger problems


Results

I Compared with a two phase method1. Ranking based two phase method works very bad on our

problems

2. PSM based two phase method works better, and even beston smaller problems



Results


problems2. PSM based two phase method works better, and even best

on smaller problems



Results


problems2. PSM based two phase method works better, and even best

on smaller problems3. Our best algorithm, outperforms two phase methods on

larger problems


Thanks

I Thanks for listening!I Thanks to the committeeI Thanks to Matthias EhrgottI Thanks to Lars, Andreas, and Kim


Thanks

I Thanks for listening!

I Thanks to the committeeI Thanks to Matthias EhrgottI Thanks to Lars, Andreas, and Kim


Thanks

I Thanks for listening!I Thanks to the committee

I Thanks to Matthias EhrgottI Thanks to Lars, Andreas, and Kim


Thanks

I Thanks for listening!I Thanks to the committeeI Thanks to Matthias Ehrgott

I Thanks to Lars, Andreas, and Kim


Thanks

I Thanks for listening!I Thanks to the committeeI Thanks to Matthias EhrgottI Thanks to Lars, Andreas, and Kim

Documents

Discrete facility location problems · 4.Each customer has to be serviced from one, and only one, open facility. Decisions: 1.Which potential facilities should be open in an optimal