advertisement

Facility Location Location and Distribution Distribution System Planning Planning Thomas Thomas L. L. Magnanti Magnanti Today’s Today’s Agenda Agenda �� �� �� Why Why study study facility facility location? location? Issues Issues to to be be modeled modeled Basic Basic models models � � Fixed Fixed charge charge problems problems � � Core Core uncapacitated uncapacitated and and capacitated capacitated facility facility location location models models �� Large-scale Large-scale application application (Hunt(HuntWesson Wesson Foods) Foods) Logistics Logistics Industry Industry �� �� �� �� �� �� U.S. U.S. logistics logistics industry: industry: $900 $900 billion billion -- almost almost double double the the size size of of the the high-tech high-tech industry: industry: >> 10 10 percent percent of of the the U.S. U.S. gross gross domestic domestic product product 11 11 per per cent cent of of Singapore's Singapore's GDP GDP with with aa growth growth of of 99 per per cent cent in in year year 2000 2000 Singapore Singapore Logistics Logistics Enhancement Enhancement & & Applications Applications Programme Programme (LEAP) (LEAP) 2001 2001 Global Global logistics: logistics: $3.43 $3.43 trillion trillion 1998, 1998, U.S. U.S. trucking trucking industry industry revenues revenues just just under under $200 $200 billion billion 7.7 7.7 million million trucks trucks carried carried over over 11 trillion trillion ton ton miles miles of of freight freight Singapore Singapore Retail Retail 21 21 Plan Plan Basic Basic Issue Issue �� �� Where Where to to locate locate and and how how to to size size facilities? facilities? How How to to meet meet customer customer demands demands from from the the facilities? facilities? � � Which Which facility facility (facilities) (facilities) serve serve each each customer? customer? � � How How much much customer customer demand demand is is met met by by each each facility? facility? Facilities might be warehouses, retail outlets, wireless bay stations, communication concentrators Some Some Elements Elements of of Cost Cost & & Service Service �� Transportation Transportation Costs Costs � � Vehicles, Vehicles, Drivers, Drivers, Fuel Fuel �� Warehousing Warehousing � � Facility Facility Construction/Rental, Construction/Rental, Handling Handling Costs, Costs, Inventory Inventory �� Customer Customer Service Service � � Service Service Time, Time, Single Single Sourcing Sourcing System System Trade-offs Trade-offs Transportation Costs - Fixed Costs + Effect of More Facilities System System Trade-offs Trade-offs Costs - Service + Effect of “Individualized” Service (e.g., Direct Shipments) Nature Nature of of Costs Costs �� Fixed Fixed Costs Costs � � Facility Facility construction/rental construction/rental � � Vehicle Vehicle purchases purchases & & rentals rentals � � Personnel Personnel (drivers, (drivers, managers) managers) � � Fixed Fixed overhead overhead �� Variable Variable Costs Costs � � Inventory, Inventory, handling, handling, fuel fuel Optimization Optimization Applications Applications �� �� �� Hunt-Wesson Hunt-Wesson Foods Foods saves saves over over $1 $1 million million per per year year Restructuring Restructuring North North America America operations, operations, Proctor Proctor and and Gamble Gamble reduces reduces plants plants by by 20%, 20%, saving saving $200 $200 million/year million/year Many, Many, many many others others (e.g., (e.g., supplying supplying parts parts to to plants) plants) Facility Facility Location Location Challenge Challenge 19 19 Modeling Modeling Issue Issue �� �� How How do do we we model model “lumpiness” “lumpiness” of of the the costs costs (e.g., (e.g., fixed fixed costs)? costs)? How How do do we we model model logical logical conditions conditions (e.g., (e.g., choice choice of of warehouse warehouse locations)? locations)? Modeling Modeling Fixed Fixed Costs Costs Flow x > 0 Flow z > 0 �� �� Incur Incur fixed fixed cost cost F F if if either either xx >> 00 or or zz >> 00 Suppose 3/2 (demand (demand limitation) limitation) Suppose xx ++ zz << 3/2 Model Minimize Fy + other terms subject to y = 1 if either x > 0 or z > 0 Three Three Models Models (LP (LP Relaxations) Relaxations) Model 1 x + z < 3/2 x < 1, z < 1 x + z < 2y x > 0, z > 0 0<y<1 Forcing Constraints Weak Strong Model 3 x + z < 3y/2 x < 1, z < 1 x < y, z < y x > 0, z > 0 0<y<1 Model 2 x + z < 3/2 x < 1, z < 1 x < y, z < y x > 0, z > 0 0<y<1 Geometry Geometry (Weak (Weak Model) Model) y x + z < 3/2 Feasible Feasible region region with x<1, x<1, y<1, y<1, y with z<1, z<1, xx ++ z< z< 3/2 3/2 Feasible points z x z x x + z < 2y intersects at y = 3/4 Geometry Geometry (Improved (Improved Model) Model) x + z < 3/2 y z z x y x xx << y, y, zz << yy intersects intersects at at xx == zz == yy == 3/4 3/4 & & at at yy == 11 w/ w/ xx or or zz =1 =1 Geometry Geometry (Strong (Strong Model) Model) y y z z x x Exact Exact representation! representation! xx ++ zz << 3y/2, 3y/2, xx << y, y, zz << yy intersects intersects at at yy == 11 Core Core (Uncapacitated) (Uncapacitated) Facility Facility Location Location Minimize Fixed + Routing Costs Subject to � Meet customer demand from facilities � Assign customer only to open facility Parameters: Parameters: Core Core Facility Facility Location Location Model Model �� �� �� Demand Demand d dii for for each each customer customer ii Fixed Fixed cost cost F Fjj for for each each facility facility location location jj Cost of routing routing all all customer customer ii Cost ccijij of demand demand to to facility facility jj == per per unit unit cost cost times times demand demand d dii Decisions: Decisions: Core Core Facility Facility Location Location Model Model �� Where Where do do we we locate locate facilities? facilities? � � yyjj == 11 ifif we we locate locate facility facility at at location location jj �� Fraction Fraction of of service service that that customer customer ii receives receives from from facility facility jj (x (xijij )) Network Network Representation Representation 33 Customers, Customers, 44 Facilities Facilities d11 1 d22 2 d33 3 Customers xijij yjj 1 2 3 4 Facilities Facility Facility Location Location Costs Costs c11 x 11 + c12 x 12 11 11 12 12 + c13 x 13 + c14 x 14 13 13 14 14 +. . . + c31 x 31 + c32 x 32 31 31 32 32 + c33 x 33 + c34 x 34 33 33 34 34 + F11y11 + F22y22 + F33y33 + F44y44 Routing Routing Costs Costs Fixed Fixed Costs Costs Constraints: Constraints: Tabular Tabular Representation Representation C u s t o m e r s Facilities (Locations) x11 x12 x13 x14 11 12 13 14 x21 x22 x23 x24 21 22 23 24 x31 x32 x33 x34 31 32 33 34 =1 =1 =1 x11 x 12 ≤ y2,2, x13 x 14 ≤ y44 11 ≤ y1, 1, 12 13 ≤ y3, 3, 14 x21 x 22 ≤ y2,2, x23 x 24 ≤ y44 21 ≤ y1, 1, 22 23 ≤ y3, 3, 24 x31 x 32 ≤ y2,2, x33 x 34 ≤ y44 31 ≤ y1, 1, 32 33 ≤ y3, 3, 34 Model Model (Uncapacitated (Uncapacitated Facilities) Facilities) Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj Subject to Σjj xijij = 1 xijij ≤ yjj xijij ≥ 0 yjj = 0 or 1 } for all customers i for all customers i and facilities j for all facilities j Modeling Modeling Variations Variations �� Open Open at at most most three three of of facilities facilities 1, 1, 66 and and 8-11 8-11 � � yy11 ++ yy66 ++ yy88 ++ yy99 ++ yy10 + y 11 ≤≤ 33 10 + y11 �� Assign Assign each each customer customer to to aa single single facility facility � � xx11 integer, x 12 integer, integer, etc. etc. 11 integer, x12 Modeling Modeling Variations Variations �� Open Open aa facility facility at at location location 33 only only if if we we open open one one at at location location 77 � � yy33 ≤≤ yy77 Note: Power of using integer variables to model logical restrictions Modeling Modeling Enhancements Enhancements �� �� �� �� Multiple Multiple products products Facility Facility capacities capacities and and operating operating ranges ranges (min (min and and max max throughput throughput if if open) open) Multi-layered Multi-layered distribution distribution networks networks Service Service restrictions restrictions � � Single Single sourcing sourcing � � Timing Timing of of deliveries deliveries �� Inventory Inventory positioning positioning and and control control Alternate Alternate Model Model (Uncapacitated (Uncapacitated Facilities) Facilities) Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj Subject to Σjj xijij = 1 Σii xijij ≤ nyjj xijij ≥ 0 yjj = 0 or 1 for all customers i for all facilities j for all pairs i,j for all facilities j Alternate Alternate Model Model (Uncapacitated (Uncapacitated Facilities) Facilities) Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj Subject to Σjj xijij = dii Σii xijij ≤ ((Σ Σii dii)yjj xijij ≥ 0 yjj = 0 or 1 for all customers i for all facilities j for all pairs i,j for all facilities j Model Model (Capacitated (Capacitated Facilities) Facilities) Minimize ΣiiΣjj cijijxijij + Σjj Fjjyjj Subject to Kj Σjj xijij = dii for all customers i xijij ≤ diiyjj for all i, j pairs Σii xijij ≤ CAPjj yjj for all facilities j xijij ≥ 0 for all i, j pairs yjj = 0 or 1 for all facilities j Tabular Tabular Representation Representation C u s t o m e r s Locations x11 x12 x13 x14 = d11 11 12 13 14 x21 x22 x23 x24 = d22 21 22 23 24 x31 x32 x33 x34 = d33 31 32 33 34 ≤ ≤ ≤ ≤ K11y11 K22y22 K33y33 K44y44 plus cell constraints xijij ≤ diiyjj Network Network Representation Representation yjj xijij d11 1 d22 2 d33 3 Customers 1 1 K11 2 2 K22 3 3 K33 4 4 Facilities K44 Solution Solution Approaches Approaches Heuristics Heuristics � � Add, Add, drop, drop, and/or and/or exchange exchange � � Linear Linear programming programming relaxation relaxation � � Bounding Bounding (Lagrangian (Lagrangian relaxation) relaxation) �� Optimization Optimization methods methods � � Large-scale Large-scale mixed mixed integer integer programming programming � � Benders Benders decomposition decomposition � � Lagrangian Lagrangian relaxation relaxation (e.g., (e.g., dualize dualize capacity capacity constraints constraints to to give give uncapacitated uncapacitated facility facility location location subproblem subproblem Hunt-Wesson Foods Ingredients Ingredients �� �� �� �� �� �� Multiple Multiple products products Multiple Multiple plants plants Many Many DCs, DCs, many many customers customers Site Site selection selection and and sizing sizing Customer Customer service service levels levels Complex Complex costs costs Flows � i � � � � j � � � � 14 plants � k � � 45 DC Choices 121 Customer Zones 17 Product Groups p Data Data Preprocessing Preprocessing �� 49 49 product-plant product-plant combinations combinations � � (from (from 14x17 14x17 == 238) 238) �� 682 682 DC-customer DC-customer zone zone combinations combinations � � (from (from 45x121 45x121 == 5,445 5,445 possibilities) possibilities) Data Data Preprocessing Preprocessing �� 23,513 23,513 product-plant-DC-customer product-plant-DC-customer combinations combinations � � (from (from 49x682 49x682 == 33,418 33,418 possibilities) possibilities) System System Requirements Requirements �� �� �� �� �� �� Data Data easy easy to to acquire acquire Inexpensive/quick Inexpensive/quick to to run run Easily Easily updated updated User-oriented User-oriented Flexible Flexible (what (what if if capabilities) capabilities) Measurable Measurable benefits benefits Indices Indices �� �� �� �� p p == products products ii == plants plants jj == distribution distribution centers centers kk == customer customer zones zones Plant DC Customer Zone Decision Decision Variables Variables �� �� �� xxpijk = amount of product p shipped pijk = amount of product p shipped from from plant plant ii to to customer customer zone zone kk through through DC DC jj zzjj == 11 if if DC DC jj open open yyjkjk == 11 if if DC DC is is sole sole source source of of customer customer zone zone kk Constraints Σjkjk xpijk pijk ≤ Spi pi Σii xpijk = Dpk y jk pijk pk jk Σjj yjkjk = 1 Vjjzjj ≤ Σpk D pkyjkjk ≤ Vjjzjj pk pk xpijk pijk ≥ 0 zjj,yjkjk = 0 or 1 + Configuration Constraints on y,z Objective Objective Σpijk c pijkxpijk pijk pijk pijk Transportation Cost + Σjj fjjzjj Fixed DC Cost + Σjj vjjΣpk D pkyjkjk Variable DC Cost pk pk Model Model Development Development �� �� Aggregation Aggregation of of Data Data Preselection Preselection of of Certain Certain Decisions Decisions in in Large Large Applications Applications Choice Choice of of Models Models Why integer instead of linear programming? Power Power of of Integer Integer Programming Programming �� �� �� �� Fixed Fixed costs costs Bounding Bounding ## of of facilities facilities Precedence Precedence constraints constraints Mandatory Mandatory service service constraints constraints � � Sole Sole sourcing sourcing �� Service Service timing timing Stages Stages in in Model Model Development Development �� �� Probationary Probationary analysis analysis Analyzing Analyzing results results � � Sensitivity Sensitivity analysis analysis � � What What ifif analysis analysis � � Priority Priority analysis analysis Today’s Today’s Lessons Lessons �� �� �� Facility Facility location location and and distribution distribution important important in in practice practice Geometry Geometry of of fixed fixed cost cost modeling modeling Model Model choice choice is is important important in in problem problem solving solving � � Strong Strong vs. vs. weak weak forcing forcing constraints constraints �� Optimization Optimization models models are are able able to to solve solve large-scale large-scale practical practical problems problems