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Facility
F
ili L
Location
i Models:
M d l
An Overview
1
LARRY SNYDER
DEPT. OF INDUSTRIAL AND SYSTEMS ENGINEERING
CENTER FOR VALUE CHAIN RESEARCH
LEHIGH UNIVERSITY
EWO SEMINAR SERIES – APRIL 21, 2010
Outline
2
y Introduction
y Taxonomy of location models
y IP formulations for some classical models
y Algorithms
y Extensions
y Facility location / network design software
Introduction
3
Overview
4
y Decide where to locate facilities
{
(factories / warehouses / DCs / retail outlets / etc.)
y To serve customers
y In order to achieve some balance between
{
{
Cost
Service
Decisions
5
y Usually
y 2 decisions to make:
{ Where to locate?
{ Which customers are assigned/allocated to which facilities?
y Sometimes referred to as “location–allocation
models”
Applications of Facility Location Models
6
y Widely
y applied
pp
in p
public and p
private sectors:
{ Emergency medical services (EMS) / fire stations
{ Airline hubs
{ Blood
Bl d banks
b k
{ Hazardous waste disposal sites
{ Fast
Fast-food
food restaurants
{ Public swimming pools
{ Schools
{ Vehicle inspection stations
{ Bus stops
{ etc.
etc
Uses for Facility Location Models
7
y Also applied to “virtual facilities”:
{
{
{
{
{
{
{
{
Wildlife reserves
Satellite orbits
Apparel sizes
Flexible manufacturing system tool selection
Location of bank accounts
P li i l party platforms
Political
l f
Product positioning
etc.
y Sometimes arise as subproblems for other OR problems
{
Vehicle routing
Taxonomyy of Location Models
8
Topology
9
Discrete
Continuous
{
{
{
Locate anywhere on plane
Continuous, non-linear
optimization
“Weber p
problem”
{
{
Locate at pre-defined
points
Integer programming
We’ll consider discrete problems
(Network problems are a special case)
Network
{
{
{
{
Locate anywhere on
network
Travel along arcs
Integer programming
H ki i property:
Hakimi
t
Optimal to locate at
nodes
Ù Holds for some (not
all) p
problems
Distance Metric
10
y Euclidean:
( x1 − x2 ) 2 + ( y1 − y2 ) 2
y Rectilinear / Manhattan:
{ (travel along streets)
x1 − x2 + y1 − y2
y Great
G t circle:
i l accounts
t ffor E
Earth’s
th’ curvature
t
y Highway / network: shortest path within network
{ (e.g.,
(e g U
U.S.
S highway network)
y Matrix: distance between each pair given explicitly
y For sake of generality, we’ll assume matrix distances
y Also, “distance” = “transportation cost”
Distance Objective
11
y Total distance: total distance between customers and
their
h i assigned
i
d ffacilities
ili i
{
(distance is usually demand-weighted)
di t
t assigned
to
i d facility
f ilit
∑ distance
customers
y Maximum distance: maximum distance between a
customer and
d its
i assigned
i
d ffacility
ili
{
(distance is usually unweighted)
max {distance to assigned facility}
customers
y Coverage: cust. is “covered” if distance ≤ specified radius
y Can appear in objective function or constraints
Preventing Too Many Facilities
12
y Fixed cost: fixed ((annual)) cost to open/operate
p / p
facility
{
{
Represents construction / leasing cost + overhead (lights, heat,
security etc
security,
etc.))
Independent of volume of demand served by facility
y Restriction on # facilities: require # of facilities ≤ P
in constraints
Classical Models
13
y P-median problem:
minimize demand-weighted distance
s.t. locate
l
≤ P facilities
f ili i
y Uncapacitated fixed-charge location problem (UFLP):
minimize fixed cost + DWD
y P-center problem:
minimize maximum distance
s.t. locate ≤ P facilities
y Set covering location problem (SCLP):
minimize # of facilities
s.t. cover all customers
y Maximum covering location problem (MCLP):
maximize covered demands
s.t. locate ≤ P facilities
Capacity
14
y Most models also have a capacitated
p
version
{ Facilities have fixed throughput capacity
y Capacity is usually an input
y But sometimes a decision variable
{ Discrete choices (50,000 sq ft / 100,000 sq ft / 200,000 sq ft)
{ Continuous
C i
variable
i bl ((cost iis a ffunction
i off capacity)
i )
IP F
Formulations
l ti
ffor S
Some
Classical Models
15
Notation
16
y Sets
{ I = {customers}
{ J = {potential facility sites}
y Parameters
{ hi = annual demand of customer i ∈ I
{ cij = cost to transport one unit from j ∈ J to i ∈ I
{ fj = fixed (annual) cost to open a facility at site j ∈ J
y Decision variables
{ xj = 1 if facility j ∈ J is opened, 0 otherwise
{ yij = 1 if facility j ∈ J serves customer i ∈ I, 0 otherwise
P-Median Formulation
17
min ∑∑ hi cij yij
Min demand-weighted distance
(transportation cost)
i∈I j∈J
s.t.
∑y
j∈J
ij
=1
yij ≤ x j
∑x
j∈J
j
∀i
Satisfy all demands
∀i, j
Don’t assign
g cust to closed facilityy
=P
x j ∈ {0,1} ∀j
yij ∈ {0,1} ∀i, j
Locate P facilities
Integrality
UFLP Formulation
18
min ∑ f j x j + ∑∑ hi cij yij
j∈J
s.t.
∑y
j∈J
Min fixed + transportation cost
i∈I j∈J
ij
=1
yij ≤ x j
∀i
Satisfy all demands
∀i, j
g cust to closed facilityy
Don’t assign
x j ∈ {0,1} ∀j
yij ∈ {0,1} ∀i, j
Integrality
Maximal Covering Formulation
19
max ∑ hi zi
Maximize covered demand
i∈I
s.t.
∑x
j∈J
j
=P
Locate P facilities
zi ≤ ∑ x j
∀i
Definition of coverage
x j ∈ {0,1}
∀j
Integrality
zi ∈ {0,1}
∀i
j∈Vi
where
Vi = set of facilities that can cover customer i
zi = 1 if customer i is covered,
covered 0 otherwise
Algorithms
g
20
Algorithms
21
y Most facility
y location p
problems are NP-hard
y But many classical problems are “easy”
computationally
{
{
LP relaxations are often extremely tight
Sometimes integer solutions “for free”
y Virtually
Vi t ll every ttype off algorithm
l ith ffor di
discrete
t
optimization has been applied to facility location
Algorithms
22
y Heuristics:
{
{
{
{
Greedy add/drop
Swap
Neighborhood search
Metaheuristics
Ù Genetic algorithms, tabu search, variable neighborhood search,
simulated annealing, ant algorithms, bee algorithms, …
y Exact Algorithms:
{
{
{
{
{
Branch and bound
Cutting
gp
planes
Benders decomposition
Column generation / Dantzig-Wolfe decomposition
Lagrangian
ag a g a relaxation
e a at o
Lagrangian Relaxation for P-Median
23
min ∑∑ hi cij yij
i∈I j∈J
s.t.
∑y
j∈J
ij
=1
yij ≤ x j
∑x
j∈J
j
∀i
∀i, j
=P
x j ∈ {0,1} ∀j
yij ∈ {0,1} ∀i, j
RELAX
Lagrangian Subproblem
24
⎛
⎞
min ∑∑ hi cij yij + ∑ λi ⎜⎜1 − ∑ yij ⎟⎟
i∈I j∈J
i∈I
⎝ j∈J ⎠
= ∑∑ (hi cij − λi ) yij + ∑ λi
i∈I j∈J
s.t. yij ≤ x j
∑x
j∈J
j
∀i, j
=P
x j ∈ {0,1} ∀j
yij ∈ {0,1} ∀i, j
i∈I
Facility-j Subproblem
25
y Subproblem is separable by j
y Suppose we open j; need to solve
y Easy—solve by inspection:
{
{
Would set yij = 1 iff hicij – λi < 0
“Benefit” of opening
p
g j is
β j = ∑ min{0, hi cij − λi }
i∈I
{
Open
p P facilities with smallest βj
y This
gives lower bound
y Obtain upper bound from heuristic
y Update
U d t λ and
d repeatt
min ∑ (hi cij − λi ) yij
i∈I
s t yij ∈ {0,1}
s.t.
Extensions
26
Other Flavors
27
y Obnoxious facilities
{
{
Obnoxious location: Maximize distance from facilities to customers
Dispersion: Maximize distance among facilities
y Competitive
p
location
{
Multiple players try to capture demand by locating facilities
y Multi-objective models
{
Account for multiple stakeholders’
stakeholders objectives
y Hub location
{
{
Flows from facilities to customers but also among facilities
Quadratic objective
y Dynamic location
{
Facilities are located over time, or move over time
Uncertainty
28
y Types
yp of randomness:
{ Demand-side: Randomness in demands, costs, etc.
{ Supply-side: Randomness in supply (e.g., disruptions)
y Approaches to uncertainty:
{ Stochastic programming: min expected cost
{ Robust optimization: minimax cost,
cost minimax regret,
regret CVaR,
CVaR
etc.
y Modeling
g approaches:
pp
{ Scenario formulations
{ Interval uncertainty
Integrated Models
29
y Incorporate tactical / operational costs into strategic
(f ili llocation)
(facility
i )d
decisions
i i
{
{
Inventory
Routing
y Integrated location-inventory model
{
{
{
{
Daskin, Coullard, and Shen (Ann OR, 2002)
Objective function includes two concave terms:
Ù Inventory economies of scale (EOQ)
Ù Risk pooling (safety stock)
Constraints are same as UFLP
Solve via Lagrangian relaxation
Ù Subproblem solved in O(|I| log |I|) time for each j
Ù (UFLP: O(|I|) time for each j)
Network Design
30
y Multi-echelon facility
y location models
{ Make open/close decisions for multiple tiers
{ Geoffrion and Graves (MS, 1974)
y Generalization: network design problems
{ Usually locate arcs in the network
Ù
But locating nodes is equivalent
y Rich literature on network design
{ e.g.,
g , Magnanti
g
and Wong
g ((TS,, 1984)
9 4)
Facility
F
ilit L
Location
ti /
Network Design
g Software
31
Software Packages
32
y LogicNet
g
/ LogicNet
g
Plus
{ Originally LogicTools
{ Now ILOG Supply Chain Applications, part of IBM Consulting
y SAILS
{ INSIGHT
y SAP,
SAP Oracle
O l modules
d l
Capabilities
33
y Key
y decisions:
{ Facility locations, capacities, capabilities, volumes
{ Distribution lanes (yes/no, volumes)
{ Make
M k vs. b
buy
y Key features:
{ Data import
{ Output report export
{ GUI, GIS
{ Optimization solver
{ Rating engine
{ What-if
Wh t if scenarios
i
Recent Developments
34
y New features:
{
{
{
{
{
“Green”
Risk management
Taxes
Seasonality
Cost vs. service level tradeoffs
y Things most current software can’t
can t do (very well):
{
{
{
{
Non-linearities (e.g., quantity discounts)
Open/close decisions on arcs for larger models
Close integration with inventory modeling
Uncertainty
Ù Optimization under multiple scenarios
Ù Risk-averse objectives
Further Reading
35
y Textbooks on facility location
{ Daskin (1995)
{ Drezner (1995)
{ Drezner and Hamacher (2001)
y Review articles / book chapters
{ Discrete location: Current, Daskin, and Schilling (D&H book,
2001))
{ Continuous location: Drezner, et al. (D&H book, 2001)
{ Stochastic location: Snyder
y
(IIE Trans, 2006)
{ Location with disruptions: Snyder, et al. (INFORMS Tutorial,
2006)
{ Location
Location-inventory
inventory models: Shen (JIMO, 2007)
Questions?
Q
36
[email protected]