Facility LocationFacility LocationFacility Locationand Distributionand Distributionand Distribution System PlanningSystem PlanningSystem Planning

Thomas L. MagnantiThomas L. Magnanti

Today’s AgendaToday’s Agenda

� Why study facility location? � Issues to be modeled � Basic models

� Fixed charge problems � Core uncapacitated and capacitated

facility location models � Large-scale application (Hunt-

Wesson Foods)

� Why study facility location?� Issues to be modeled� Basic models

� Fixed charge problems� Core uncapacitated and capacitated

facility location models� Large-scale application (Hunt-

Wesson Foods)

Logistics IndustryLogistics Industry� U.S. logistics industry: $900 billion - almost

double the size of the high-tech industry: > 10percent of the U.S. gross domestic product

� 11 per cent of Singapore's GDP with a growth of9 per cent in year 2000

� Singapore Logistics Enhancement &Applications Programme (LEAP) 2001

� Global logistics: $3.43 trillion � 1998, U.S. trucking industry revenues just

under $200 billion � 7.7 million trucks carried over 1 trillion ton

miles of freight

� U.S. logistics industry: $900 billion - almost double the size of the high-tech industry: > 10percent of the U.S. gross domestic product

� 11 per cent of Singapore's GDP with a growth of9 per cent in year 2000

� Singapore Logistics Enhancement &Applications Programme (LEAP) 2001

� Global logistics: $3.43 trillion � 1998, U.S. trucking industry revenues just

under $200 billion� 7.7 million trucks carried over 1 trillion ton

miles of freight

Singapore Retail 21 PlanSingapore Retail 21 Plan

Basic IssueBasic Issue

� Where to locate and how to size facilities?

� How to meet customer demands from the facilities? � Which facility (facilities) serve each

customer? � How much customer demand is met by

each facility?

� Where to locate and how to size facilities?

� How to meet customer demands from the facilities?� Which facility (facilities) serve each

customer?� How much customer demand is met by

each facility?Facilities might be warehouses, retail outlets,

wireless bay stations, communication concentrators

Some Elements of Cost & ServiceSome Elements of Cost & Service

� Transportation Costs � Vehicles, Drivers, Fuel

� Warehousing � Facility Construction/Rental, Handling

Costs, Inventory � Customer Service

� Service Time, Single Sourcing

� Transportation Costs� Vehicles, Drivers, Fuel

� Warehousing � Facility Construction/Rental, Handling

Costs, Inventory� Customer Service

� Service Time, Single Sourcing

Effect of More FacilitiesEffect of More FacilitiesEffect of More Facilities

+++---

Fixed Costs FixedFixed CostsCosts

TransportationCostsTransportationTransportationCostsCosts

System Trade-offsSystem Trade-offs

Effect of “Individualized” Service (e.g., Direct Shipments) Effect of “Individualized”Effect of “Individualized” Service (e.g., Direct Shipments)Service (e.g., Direct Shipments)

+++---

ServiceServiceServiceCostsCostsCosts

System Trade-offsSystem Trade-offs

Nature of CostsNature of Costs

� Fixed Costs � Facility construction/rental � Vehicle purchases & rentals � Personnel (drivers, managers) � Fixed overhead

� Variable Costs � Inventory, handling, fuel

� Fixed Costs� Facility construction/rental� Vehicle purchases & rentals� Personnel (drivers, managers)� Fixed overhead

� Variable Costs� Inventory, handling, fuel

Optimization ApplicationsOptimization Applications

� Hunt-Wesson Foods saves over $1 million per year

� Restructuring North America operations, Proctor and Gamble reduces plants by 20%, saving $200 million/year

� Many, many others (e.g., supplying parts to plants)

� Hunt-Wesson Foods saves over $1 million per year

� Restructuring North America operations, Proctor and Gamble reduces plants by 20%, saving $200 million/year

� Many, many others (e.g., supplying parts to plants)

1919

Facility Location ChallengeFacility Location Challenge

Modeling IssueModeling Issue

� How do we model “lumpiness” of the costs (e.g., fixed costs)?

� How do we model logical conditions (e.g., choice of warehouse locations)?

� How do we model “lumpiness” of the costs (e.g., fixed costs)?

� How do we model logical conditions (e.g., choice of warehouse locations)?

Modeling Fixed CostsModeling Fixed Costs

� Incur fixed cost F if either x > 0 or z > 0 � Suppose x + z < 3/2 (demand limitation) � Incur fixed cost F if either x > 0 or z > 0� Suppose x + z < 3/2 (demand limitation)

Flow x > 0 Flow z > 0

Minimize Fy + other terms subject to y = 1 if either x > 0 or z > 0

Model

Three Models (LP Relaxations)Three Models (LP Relaxations)

Forcing Constraints

Weak Strong

x + z < 3/2 x < 1, z < 1 x + z < 2y x > 0, z > 0 0 < y < 1

Model 1 x + z < 3/2 x < 1, z < 1 x < y, z < y x > 0, z > 0 0 < y < 1

Model 2

x + z < 3y/2 x < 1, z < 1 x < y, z < y x > 0, z > 0 0 < y < 1

Model 3

Geometry (Weak Model)Geometry (Weak Model)

x + z < 2y intersects at y = 3/4

Feasible region with x<1, y<1, z<1, x + z< 3/2

Feasible region with x<1, y<1, z<1, x + z< 3/2

x + z < 3/2 y

x

z

x

y

z

Feasible points

y

x

z

Geometry (Improved Model)Geometry (Improved Model)

x < y, z < y intersects at

x = z = y = 3/4 & at y = 1 w/ x or z =1

x < y, z < yintersects at

x = z = y = 3/4 & at y = 1 w/ x or z =1

x + z < 3/2 y

x

z

y

x

z

Geometry (Strong Model)Geometry (Strong Model)

x + z < 3y/2, x < y, z < y intersects at y = 1

x + z < 3y/2,x < y, z < yintersects at y = 1

Exact representation!Exact representation!

y

x

z

Minimize Fixed + Routing CostsMinimize Fixed + Routing CostsMinimize Fixed + Routing CostsSubject toSubject toSubject to

� Meet customer demand from facilities

� Assign customer only to open facility

�� Meet customerMeet customer demand from facilitiesdemand from facilities

�� Assign customer only toAssign customer only to open facilityopen facility

Core (Uncapacitated) Facility LocationCore (Uncapacitated) Facility Location

Parameters:

Core Facility Location Model

Parameters:

Core Facility Location Model

� Demand di for each customer i � Fixed cost Fj for each facility location j � Cost cij of routing all customer i

demand to facility j = per unit cost times demand di

� Demand di for each customer i� Fixed cost Fj for each facility location j� Cost cij of routing all customer i

demand to facility j = per unit cost times demand di

Decisions: Core Facility Location Model

Decisions: Core Facility Location Model

� Where do we locate facilities? � yj = 1 if we locate facility at location j

� Fraction of service that customer i receives from facility j (xij )

� Where do we locate facilities?� yj = 1 if we locate facility at location j

� Fraction of service that customer i receives from facility j (xij )

CustomersCustomersCustomers FacilitiesFacilitiesFacilities

xijxxijijyjyyjj

Network Representation 3 Customers, 4 Facilities Network Representation 3 Customers, 4 Facilities

d1dd11

d2dd22

d3dd33

111

222

333 444

111

222

333

Facility Location CostsFacility Location Costs

c11x11 + c12x12 + c13x13 + c14x14 + . . . + c31x31 + c32x32 + c33x33 + c34x34

+ F1y1 + F2y2 + F3y3 + F4y4

cc1111xx1111 + c+ c1212xx1212 + c+ c1313xx1313 + c+ c1414xx1414 + . . .+ . . .+ c+ c3131xx3131 + c+ c3232xx3232 + c+ c3333xx3333 + c+ c3434xx3434

+ F+ F11yy11 + F+ F22yy22 + F+ F33yy33 + F+ F44yy44

Routing Costs Routing Costs

Fixed Costs Fixed Costs

x11 x12 x13 x14 = 1 x21 x22 x23 x24 = 1 x31 x32 x33 x34 = 1

x11 ≤ y1, x12 ≤ y2, x13 ≤ y3, x14 ≤ y4 x21 ≤ y1, x22 ≤ y2, x23 ≤ y3, x24 ≤ y4 x31 ≤ y1, x32 ≤ y2, x33 ≤ y3, x34 ≤ y4

xx1111 xx1212 xx1313 xx1414 = 1= 1xx2121 xx2222 xx2323 xx2424 = 1= 1xx3131 xx3232 xx3333 xx3434 = 1= 1

xx1111 ≤ yy1,1, xx1212 ≤ yy2,2, xx1313 ≤ yy3,3, xx1414 ≤ yy44xx2121 ≤ yy1,1, xx2222 ≤ yy2,2, xx2323 ≤ yy3,3, xx2424 ≤ yy44xx3131 ≤ yy1,1, xx3232 ≤ yy2,2, xx3333 ≤ yy3,3, xx3434 ≤ yy44

Facilities (Locations)Facilities (Locations)Facilities (Locations)C u s t o m e r s

CCuussttoommeerrss

Constraints: Tabular RepresentationConstraints: Tabular Representation

Σj xij = 1 for all customers i xij ≤ yj for all customers i xij ≥ 0 and facilities j yj = 0 or 1 for all facilities j

ΣΣjj xxijij = 1= 1 for all customers ifor all customers ixxijij ≤ yyjj for all customers ifor all customers ixxijij ≥ 00 and facilities jand facilities jyyjj = 0 or 1= 0 or 1 for all facilities jfor all facilities j

Minimize ΣiΣj cijxij + Σj FjyjMinimizeMinimize ΣΣiiΣΣjj ccijijxxijij ++ ΣΣjj FFjjyyjj

Subject toSubject toSubject to

Model (Uncapacitated Facilities)

Model (Uncapacitated Facilities)

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Modeling VariationsModeling Variations

� Open at most three of facilities 1, 6 and 8-11 � y1 + y6 + y8 + y9 + y10 + y11 ≤ 3

� Assign each customer to a single facility � x11 integer, x12 integer, etc.

� Open at most three of facilities 1, 6 and 8-11� y1 + y6 + y8 + y9 + y10 + y11 ≤ 3

� Assign each customer to a single facility� x11 integer, x12 integer, etc.

Modeling VariationsModeling Variations

� Open a facility at location 3 only if we open one at location 7 � y3 ≤ y7

� Open a facility at location 3 only if we open one at location 7� y3 ≤ y7

Note: Power of using integer variables to model logical restrictions

Note: Power of using integer variables to model logical restrictions

Modeling EnhancementsModeling Enhancements

� Multiple products � Facility capacities and operating

ranges (min and max throughput ifopen)

� Multi-layered distribution networks � Service restrictions

� Single sourcing � Timing of deliveries

� Inventory positioning and control

� Multiple products� Facility capacities and operating

ranges (min and max throughput ifopen)

� Multi-layered distribution networks� Service restrictions

� Single sourcing� Timing of deliveries

� Inventory positioning and control

Σj xij = 1 for all customers i Σi xij ≤ nyj for all facilities j

xij ≥ 0 for all pairs i,j yj = 0 or 1 for all facilities j

ΣΣjj xxijij = 1= 1 for all customers ifor all customers iΣΣii xxijij ≤ nynyjj for all facilities jfor all facilities j

xxijij ≥ 00 for all pairs i,jfor all pairs i,jyyjj = 0 or 1= 0 or 1 for all facilities jfor all facilities j

Minimize ΣiΣj cijxij + Σj FjyjMinimizeMinimize ΣΣiiΣΣjj ccijijxxijij ++ ΣΣjj FFjjyyjj

Subject toSubject toSubject to

Alternate Model (Uncapacitated Facilities)Alternate Model (Uncapacitated Facilities)

Σj xij = di for all customers i Σi xij ≤ (Σi di)yj for all facilities j

xij ≥ 0 for all pairs i,j yj = 0 or 1 for all facilities j

ΣΣjj xxijij = d= dii for all customers ifor all customers iΣΣii xxijij ≤ ((ΣΣii ddii)y)yjj for all facilities jfor all facilities j

xxijij ≥ 00 for all pairs i,jfor all pairs i,jyyjj = 0 or 1= 0 or 1 for all facilities jfor all facilities j

Minimize ΣiΣj cijxij + Σj FjyjMinimizeMinimize ΣΣiiΣΣjj ccijijxxijij ++ ΣΣjj FFjjyyjj

Subject toSubject toSubject to

Alternate Model (Uncapacitated Facilities)Alternate Model (Uncapacitated Facilities)

Σj xij = di for all customers i xij ≤ diyj for all i, j pairs

Σi xij ≤ CAPj yj for all facilities j xij ≥ 0 for all i, j pairs yj = 0 or 1 for all facilities j

ΣΣjj xxijij = d= dii for all customers ifor all customers ixxijij ≤ ddiiyyjj for all i, j pairsfor all i, j pairs

ΣΣii xxijij ≤ CAPCAPjj yyjj for all facilities jfor all facilities jxxijij ≥ 00 for all i, j pairsfor all i, j pairsyyjj = 0 or 1= 0 or 1 for all facilities jfor all facilities j

Minimize ΣiΣj cijxij + Σj FjyjMinimizeMinimize ΣΣiiΣΣjj ccijijxxijij ++ ΣΣjj FFjjyyjj

Subject toSubject toSubject to

Model (Capacitated Facilities)Model (Capacitated Facilities)

KKjj

x11 x12 x13 x14 = d1 x21 x22 x23 x24 = d2 x31 x32 x33 x34 = d3 ≤ ≤ ≤ ≤

K1y1 K2y2 K3y3 K4y4

xx1111 xx1212 xx1313 xx1414 = d= d11xx2121 xx2222 xx2323 xx2424 = d= d22xx3131 xx3232 xx3333 xx3434 = d= d33≤ ≤ ≤ ≤KK11yy11 KK22yy22 KK33yy33 KK44yy44

LocationsLocationsLocationsC u s t o m e r s

CCuussttoommeerrss

Tabular RepresentationTabular Representation

plus cell constraints xij ≤ diyjplus cell constraints xxijij ≤ ddiiyyjj

d1dd11

d2dd22

d3dd33

111

222

333 444 444

K1KK11

K2KK22

K3KK33

K4KK44CustomersCustomersCustomers FacilitiesFacilitiesFacilities

xijxxijij yjyyjj

Network RepresentationNetwork Representation

111

222

333

111

222

333

Solution ApproachesSolution Approaches

� Heuristics � Add, drop, and/or exchange � Linear programming relaxation � Bounding (Lagrangian relaxation)

� Optimization methods � Large-scale mixed integer programming � Benders decomposition � Lagrangian relaxation (e.g., dualize capacity

constraints to give uncapacitated facilitylocation subproblem

� Heuristics� Add, drop, and/or exchange� Linear programming relaxation� Bounding (Lagrangian relaxation)

� Optimization methods� Large-scale mixed integer programming� Benders decomposition� Lagrangian relaxation (e.g., dualize capacity

constraints to give uncapacitated facilitylocation subproblem

HuntHunt--Wesson FoodsWesson Foods

IngredientsIngredients

� Multiple products � Multiple plants � Many DCs, many customers � Site selection and sizing � Customer service levels � Complex costs

� Multiple products� Multiple plants� Many DCs, many customers� Site selection and sizing� Customer service levels� Complex costs

14 plants14 plants14 plants45 DC Choices45 DC Choices45 DC Choices

121 Customer Zones121 Customer Zones121 Customer Zones

FlowsFlowsFlows

17 Product Groups p17 Product Groups17 Product Groups pp

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Data PreprocessingData Preprocessing

� 49 product-plant combinations � (from 14x17 = 238)

� 682 DC-customer zone combinations � (from 45x121 = 5,445 possibilities)

� 49 product-plant combinations� (from 14x17 = 238)

� 682 DC-customer zone combinations� (from 45x121 = 5,445 possibilities)

Data PreprocessingData Preprocessing

� 23,513 product-plant-DC-customer combinations � (from 49x682 = 33,418 possibilities)

� 23,513 product-plant-DC-customer combinations� (from 49x682 = 33,418 possibilities)

System RequirementsSystem Requirements

� Data easy to acquire � Inexpensive/quick to run � Easily updated � User-oriented � Flexible (what if capabilities) � Measurable benefits

� Data easy to acquire� Inexpensive/quick to run� Easily updated� User-oriented� Flexible (what if capabilities)� Measurable benefits

IndicesIndices

� p = products � i = plants � j = distribution centers � k = customer zones

� p = products� i = plants� j = distribution centers� k = customer zones

PlantPlant DCDC Customer ZoneCustomer Zone

Decision VariablesDecision Variables

� xpijk = amount of product p shipped from plant i to customer zone k through DC j

� zj = 1 if DC j open � yjk = 1 if DC is sole source of customer

zone k

� xpijk = amount of product p shipped from plant i to customer zone k through DC j

� zj = 1 if DC j open� yjk = 1 if DC is sole source of customer

zone k

Σjk xpijk ≤ SpiΣi xpijk = DpkyjkΣj yjk = 1

Vjzj ≤ ΣpkDpkyjk ≤ Vjzj xpijk ≥ 0 zj,yjk = 0 or 1

ΣΣjkjk xxpijkpijk ≤ SSpipiΣΣii xxpijkpijk = D= DpkpkyyjkjkΣΣjj yyjkjk = 1= 1

VVjjzzjj ≤ ΣΣpkpkDDpkpkyyjkjk ≤ VVjjzzjjxxpijkpijk ≥ 00zzjj,y,yjkjk = 0 or 1= 0 or 1

+ Configuration Constraints on y,z+ Configuration Constraints on y,z+ Configuration Constraints on y,z

ConstraintsConstraintsConstraints

Σpijk cpijkxpijk

+ Σj fjzj

+ Σj vjΣpk Dpkyjk

ΣΣpijkpijk ccpijkpijkxxpijkpijk

++ ΣΣjj ffjjzzjj

++ ΣΣjj vvjjΣΣpkpk DDpkpkyyjkjk

Transportation CostTransportation CostTransportation Cost

Fixed DC CostFixed DC CostFixed DC Cost

Variable DC CostVariable DC CostVariable DC Cost

ObjectiveObjective

Model DevelopmentModel Development

� Aggregation of Data � Preselection of Certain Decisions in

Large Applications

� Aggregation of Data� Preselection of Certain Decisions in

Large Applications

Why integer instead of linear programming?

Choice of ModelsChoice of Models

Power of Integer ProgrammingPower of Integer Programming

� Fixed costs � Bounding # of facilities � Precedence constraints � Mandatory service constraints

� Sole sourcing � Service timing

� Fixed costs� Bounding # of facilities� Precedence constraints� Mandatory service constraints

� Sole sourcing� Service timing

Stages in Model DevelopmentStages in Model Development

� Probationary analysis � Analyzing results

� Sensitivity analysis � What if analysis � Priority analysis

� Probationary analysis� Analyzing results

� Sensitivity analysis� What if analysis� Priority analysis

Today’s LessonsToday’s Lessons

� Facility location and distributionimportant in practice

� Geometry of fixed cost modeling � Model choice is important in problem

solving � Strong vs. weak forcing constraints

� Optimization models are able tosolve large-scale practical problems

� Facility location and distributionimportant in practice

� Geometry of fixed cost modeling� Model choice is important in problem

solving� Strong vs. weak forcing constraints

� Optimization models are able tosolve large-scale practical problems