i LlFi i Mdl Facility Location Models: An...
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F ili L i M d l Facility Location Models: An OverviewAn Overview
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LARRY SNYDERD E P T . O F I N D U S T R I A L A N D S Y S T E M S E N G I N E E R I N G
C E N T E R F O R V A L U E C H A I N R E S E A R C H
L E H I G H U N I V E R S I T Y
E W O S E M I N A R S E R I E S A P R I L E W O S E M I N A R S E R I E S – A P R I L 2 1 , 2 0 1 0
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Outline
Introduction
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Taxonomy of location models
IP formulations for some classical models
Algorithms
Extensions
Facility location / network design software
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Introduction3
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Overview4
Decide where to locate facilities(factories / warehouses / DCs / retail outlets / etc.)
To serve customersIn order to achieve some balance between In order to achieve some balance between
CostService
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Decisions5
Usually 2 decisions to make:yWhere to locate?
Which customers are assigned/allocated to which facilities?
Sometimes referred to as “location–allocation models”
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Applications of Facility Location Models
Widely applied in public and private sectors:
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y pp p pEmergency medical services (EMS) / fire stations
Airline hubs
Bl d b kBlood banks
Hazardous waste disposal sites
Fast-food restaurantsFast food restaurants
Public swimming pools
Schools
Vehicle inspection stations
Bus stops
etcetc.
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Uses for Facility Location Models7
Also applied to “virtual facilities”:Wildlife reserves
Satellite orbits
Apparel sizesApparel sizes
Flexible manufacturing system tool selection
Location of bank accounts
P li i l l fPolitical party platforms
Product positioning
etc.
Sometimes arise as subproblems for other OR problemsVehicle routing
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Taxonomy of Location Modelsy8
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Topology9
Continuous Discrete NetworkLocate anywhere on planeContinuous, non-linear optimization“Weber problem”
Locate at pre-defined pointsInteger programming
Locate anywhere on networkTravel along arcsInteger programmingH ki i t
We’ll consider discrete problems(Network problems are a special case)
pHakimi property: Optimal to locate at nodes
Holds for some (not all) problems
(Network problems are a special case)p
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Distance Metric10
Euclidean: 221
221 )()( yyxx −+−
Rectilinear / Manhattan:(travel along streets)
G t i l t f E th’ t
2121 yyxx −+−
Great circle: accounts for Earth’s curvatureHighway / network: shortest path within network
(e g U S highway network)(e.g., U.S. highway network)
Matrix: distance between each pair given explicitly
For sake of generality, we’ll assume matrix distancesAlso, “distance” = “transportation cost”
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Distance Objective11
Total distance: total distance between customers and h i i d f ili itheir assigned facilities
(distance is usually demand-weighted)
∑ f iliti dtdi t
Maximum distance: maximum distance between a d i i d f ili
∑customers
facility assignedtodistance
customer and its assigned facility(distance is usually unweighted)
facility}assignedto{distancemax
Coverage: cust. is “covered” if distance ≤ specified radius
facility} assignedto{distancemaxcustomers
Can appear in objective function or constraints
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Preventing Too Many Facilities12
Fixed cost: fixed (annual) cost to open/operate ( ) p / pfacility
Represents construction / leasing cost + overhead (lights, heat, security etc )security, etc.)
Independent of volume of demand served by facility
Restriction on # facilities: require # of facilities ≤ PRestriction on # facilities: require # of facilities ≤ Pin constraints
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Classical Models13
P-median problem: minimize demand-weighted distance l P f ili is.t. locate ≤ P facilities
Uncapacitated fixed-charge location problem (UFLP):minimize fixed cost + DWD
P-center problem: minimize maximum distances.t. locate ≤ P facilities
Set covering location problem (SCLP):Set covering location problem (SCLP):minimize # of facilitiess.t. cover all customers
Maximum covering location problem (MCLP):Maximum covering location problem (MCLP):maximize covered demandss.t. locate ≤ P facilities
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Capacity14
Most models also have a capacitated versionpFacilities have fixed throughput capacity
Capacity is usually an input
But sometimes a decision variableDiscrete choices (50,000 sq ft / 100,000 sq ft / 200,000 sq ft)
C i i bl ( i f i f i )Continuous variable (cost is a function of capacity)
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IP F l ti f S IP Formulations for Some Classical Models
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Notation16
SetsI = {customers}
J = {potential facility sites}
Parametershi = annual demand of customer i ∈ I
c = cost to transport one unit from j ∈ J to i ∈ Icij = cost to transport one unit from j ∈ J to i ∈ I
fj = fixed (annual) cost to open a facility at site j ∈ J
Decision variablesDecision variablesxj = 1 if facility j ∈ J is opened, 0 otherwise
yij = 1 if facility j ∈ J serves customer i ∈ I, 0 otherwise
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P-Median Formulation17
∑∑∑∈ ∈Ii Jj
ijiji ychmin Min demand-weighted distance (transportation cost)
jixy
iy
jij
Jjij
,
1 s.t.
∀≤
∀=∑∈
Satisfy all demands
Don’t assign cust to closed facility
Px
jy
Jjj
jij ,
=∑∈
g y
Locate P facilities
jiyjx
ij
j
, }1,0{ }1,0{∀∈
∀∈Integrality
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UFLP Formulation18
∑∑∑∑∈ ∈∈
+Ii Jj
ijijiJj
jj ychxfmin Min fixed + transportation cost
jixy
iy
jij
Jjij
,
1 s.t.
∀≤
∀=∑∈
Satisfy all demands
Don’t assign cust to closed facility
jiyjx
jy
j
jij
}10{ }1,0{
,
∀∈
∀∈
g y
Integrality
jiyij , }1,0{ ∀∈
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Maximal Covering Formulation19
P∑∑∈Ii
ii zhmax Maximize covered demand
ixz
Px
ji
Jjj
∀≤
=
∑
∑∈
s.t. Locate P facilities
Definition of coverage
izjx j
Vjji
i
∀∈
∀∈
∑∈
}10{
}1,0{ Integrality
izi ∀∈ }1,0{
where Vi = set of facilities that can cover customer iz = 1 if customer i is covered 0 otherwisezi = 1 if customer i is covered, 0 otherwise
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Algorithmsg20
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Algorithms21
Most facility location problems are NP-hardy p
But many classical problems are “easy” computationally
LP relaxations are often extremely tight
Sometimes integer solutions “for free”
Vi t ll t f l ith f di t Virtually every type of algorithm for discrete optimization has been applied to facility location
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Algorithms22
Heuristics:Greedy add/dropSwapNeighborhood searchMetaheuristics
Genetic algorithms, tabu search, variable neighborhood search, simulated annealing, ant algorithms, bee algorithms, …
Exact Algorithms:Branch and boundCutting planesg pBenders decompositionColumn generation / Dantzig-Wolfe decompositionLagrangian relaxationag a g a e a at o
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Lagrangian Relaxation for P-Median23
∑∑∑∈ ∈Ii Jj
ijiji ychmin
jixy
iy
jij
Jjij
,
1 s.t.
∀≤
∀=∑∈
RELAX
Px
jy
Jjj
jij ,
=∑∈
jiyjx
ij
j
, }1,0{ }1,0{∀∈
∀∈
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Lagrangian Subproblem24
⎞⎛∑ ∑∑∑∈ ∈∈ ∈
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
Ii Jjiji
Ii Jjijiji yych λ 1min
jixy jij , s.t. ∀≤
∑∑∑∈∈ ∈
+−=Ii
iIi Jj
ijiiji ych λλ )(
Px
jy
Jjj
jij ,
=∑∈
jiyjx
ij
j
, }1,0{
}1,0{
∀∈
∀∈
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Facility-j Subproblem25
Subproblem is separable by jSuppose we open j; need to solve
}10{s t
)(min
∈
−∑∈Ii
ijiiji
y
ych λ
Easy—solve by inspection:Would set yij = 1 iff hicij – λi < 0“Benefit” of opening j is
}1,0{ s.t. ∈ijy
p g j
Open P facilities with smallest βj
∑∈
−=Ii
iijij ch },0min{ λβ
p βj
This gives lower boundObtain upper bound from heuristicU d t λ d tUpdate λ and repeat
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Extensions26
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Other Flavors27
Obnoxious facilitiesObnoxious location: Maximize distance from facilities to customersDispersion: Maximize distance among facilities
Competitive locationpMultiple players try to capture demand by locating facilities
Multi-objective modelsAccount for multiple stakeholders’ objectivesAccount for multiple stakeholders objectives
Hub locationFlows from facilities to customers but also among facilitiesQuadratic objective
Dynamic locationFacilities are located over time, or move over timeFacilities are located over time, or move over time
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Uncertainty28
Types of randomness:ypDemand-side: Randomness in demands, costs, etc.
Supply-side: Randomness in supply (e.g., disruptions)
Approaches to uncertainty:Stochastic programming: min expected cost
Robust optimization: minimax cost minimax regret CVaR Robust optimization: minimax cost, minimax regret, CVaR, etc.
Modeling approaches:g ppScenario formulations
Interval uncertainty
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Integrated Models29
Incorporate tactical / operational costs into strategic (f ili l i ) d i i(facility location) decisions
InventoryRouting
Integrated location-inventory modelDaskin, Coullard, and Shen (Ann OR, 2002)Objective function includes two concave terms:Objective function includes two concave terms:
Inventory economies of scale (EOQ) Risk pooling (safety stock)
Constraints are same as UFLPConstraints are same as UFLPSolve via Lagrangian relaxation
Subproblem solved in O(|I| log |I|) time for each j(UFLP: O(|I|) time for each j)(UFLP: O(|I|) time for each j)
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Network Design30
Multi-echelon facility location modelsyMake open/close decisions for multiple tiers
Geoffrion and Graves (MS, 1974)
Generalization: network design problemsUsually locate arcs in the network
But locating nodes is equivalentBut locating nodes is equivalent
Rich literature on network designe.g., Magnanti and Wong (TS, 1984)g , g g ( , 9 4)
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F ilit L ti / Facility Location / Network Design Softwareg
31
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Software Packages32
LogicNet / LogicNet Plus g / gOriginally LogicTools
Now ILOG Supply Chain Applications, part of IBM Consulting
SAILSINSIGHT
SAP O l d lSAP, Oracle modules
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Capabilities33
Key decisions:yFacility locations, capacities, capabilities, volumes
Distribution lanes (yes/no, volumes)
M k bMake vs. buy
Key features:Data importData import
Output report export
GUI, GIS
Optimization solver
Rating engine
Wh t if iWhat-if scenarios
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Recent Developments34
New features:“Green”Risk managementTaxesSeasonalityCost vs. service level tradeoffs
Things most current software can’t do (very well):Things most current software can t do (very well):Non-linearities (e.g., quantity discounts)Open/close decisions on arcs for larger modelsClose integration with inventory modelingClose integration with inventory modelingUncertainty
Optimization under multiple scenariosRisk-averse objectivesRisk-averse objectives
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Further Reading35
Textbooks on facility locationDaskin (1995)Drezner (1995)Drezner and Hamacher (2001)Drezner and Hamacher (2001)
Review articles / book chaptersDiscrete location: Current, Daskin, and Schilling (D&H book,
)2001)Continuous location: Drezner, et al. (D&H book, 2001)Stochastic location: Snyder (IIE Trans, 2006)yLocation with disruptions: Snyder, et al. (INFORMS Tutorial, 2006)Location-inventory models: Shen (JIMO, 2007)Location inventory models: Shen (JIMO, 2007)
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Questions?
L A R R Y S N Y D E R @ L E H I G H E D U
36
Q
L A R R Y . S N Y D E R @ L E H I G H . E D U