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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Location Problems on Networks with Routing Elena Fernández Universitat Politècnica de Catalunya-Barcelona TECH (UPC) e.fernandez@upc.edu

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Ivan ContrerasElena Fernández I. Contreras, E. Fernández, General network design: A unified view of combined location and network design problems. European Journal of Operational Research 219 (2012)

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Which set of facilities to open ? Location How to satisfy the customers demands from open facilities ? From which facility does the customer receive service ? Allocation How is service provided ? Routing Are facilities somehow connected ? Routing Which are the possible (or preferable) connections between Network design customers or between customers and facilities ? Decisions related to Location NETWORK OPTIMIZATION PROBLEMS

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 where customers obtain service from where flows between pairs of customers are consolidated and rerouted connect customers and facilities Connect customers and facilities Connect facilities between them What are facilities used for? Routing Which are the possible connections ? Network design

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 A classication of Network Optimization problems based on the type of demand User-facility demand (UF): Service relates users and facilities User-User demand (UU): Service relates pairs of users; facilities are used as intermediate locations in the routes that connect pairs of users

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Shortest spanning tree problem (Borůvka, 1926) UF Routes between users and facilities UU Routes between users through facilities p-median problem (Hakimi, 1964) Facility Location –Network design problem (Melkote and Daskin, 2001a) Optimum communication spanning tree problem (Hu, 1974) Hub location problem (O’Kelly, 1986) Tree of hubs problem (Contreras, Fernández, Marín, 2009) Underlying network Facility location Network design Facility location- Network design

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Outline Optimization criteria: design costs / service costs UF: Facilities used to give service: Location + Network design UU: Facilities used to re-route flows between customers: Hub location (more general) HUB LOCATION

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 What optimization criteria are relevant? Design costs Facilities Sum of set-up costs Connections Service (routing) costs Sum of service costs of customers Maximum service cost

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 1 2 4 6 3 5 2 2 3 3 4 2 3 3 3 3 1 2 4 6 3 5 2 2 3 3 3 1 2 4 6 3 5 10 8 4 4 3 3 2 2 5 7 2 Per unit routing costs ( d ij ) Communication Requirements ( W ij ) Minimum Spanning Tree Optimum Communication Spanning Tree 1 2 4 6 3 5 2 4 3 3 2 The Optimum communication Spanning Tree Hu (1974); Ahuja and Murty (1987); Rothlauf (2009); C, F, Marín, (2010), F, Luna, Hildenbrandt, Reinelt, Wiesberg (2013) To find a tree that minimizes the sum of all routing costs 3(3+3+3)3(3+3+3) 3434

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Minimize service cost 1 1 UU: Routing costs 55 1 1+1

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Total set-up cost: 8+ 4 5 + 8 2 Total service cost: 10 + 4 5 + 8 2 Max service cost: 2 2 Total set-up cost: 16 Total service cost: 38 Max service cost: 4 UF: Routing (service) costs / Set-up costs Routing costs Set-up costs ● Euclidean distances ● One facility is located

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 How to evaluate service costs? Each arc is accounted for as many times as it is used Trace paths

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 G = (V, E); Vertices V={1, 2, …, n }; Edges e = (k,m), k, m V, k < m, c km 0: set-up cost for edge (k,m) d km 0: per unit routing cost from k to m Commodities: C ={ l =( i, j ): i, j V } W ij : Flow that must be sent from i to j, ( i, j ) C Modeling flows in Network Design

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 i j kmm ⋮ ⋮ ⋮ ⋮ x ij km x ij mk Modeling flows in network optimization: path formulation (4-index) x ij km : Fraction of flow from i to j routed via nodes k and m, y km =1 iff edge (k, m) activated (i, j) C Decision variables: For (k, m) E For each commodity (i,j) define a path from i to j

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 i j kmm ⋮ ⋮ ⋮ ⋮ x ij km x ij mk Modeling flows in network optimization: path formulation (4-index) For (i, j) C, for all k Routing costs Set-up costs for links For (i, j), (i, h) C

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 For all i Routing costs Set-up costs for links i j k m m ⋮ ⋮ ⋮ ⋮ h i km h i mk j j h i km : Fraction of flow emanating from i routed via arc ( k, m) Modeling flows in network optimization: flow formulation (3-index) For each vertex i, send a flow of value O i, with demand W ij at each j with (i, j) C

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Path formulation vs flow formulation For (i, j) C, for all k For i fixed, adding on j with (i, j) C

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 How to evaluate routing costs if location decisions are involved ? connecting customers and facilities Trace paths that use facilities as intermediate vertices But facilities are not known before hand !!!

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Outline Optimization criteria: design costs / service costs UF: Facilities used to give service: Location + Network design UU: Facilities used to re-route flows between customers: Hub location (more general) HUB LOCATION

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 LOCATION + ROUTING CONNECTING CUSTOMERS & FACILITIES (UF) Facility Location /Network Design Problem

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 f i 0 : set up cost for facility i. c ij 0: set-up cost for edge (i,j) d ij 0 : per unit routing cost k to m Data To find A subset of vertices to locate facilities An allocation of non-facilites to facilities A subset of arcs to connect each vertex to its allocated facility That minimizes some objective function Facility Location /Network Design Problem Total set-up cost Total routing cost (total service cost) Maximum routing cost from any vertex to its allocated facility Any combination of the above

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Set-up cost + routing cost Melkote and Daskin: (01, 01a) Routing cost subject to maximum budget constraint Cocking, Fleßa Reinelt (05), Cocking (08). Maximum routing cost subject to budget constraint C, F & Reinelt (2010) Maximum coverage within a given distance Murawski & Church (09): Facility Location /Network Design Problem

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Facility Location /Network Design Problem

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 f i =1, i ; c ij =0 i, j ; i, j ; B = p p -median or p -center (depending on the objective) NP-hard Facility Location /Network Design Problem Set-up cost + routing cost c ij =0 i, j ; Network design trivial. Alternative formulation of UFLP Routing cost subject to budget constraint NP-hard Some particular cases

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Facility Location /Network Design Problem There is an optimal solution to FLNNDP which is a rooted forest (If there are no capacity constraints on the facilities) Decisions Vertices that locate facilities Paths connecting non facilities and facilities Arcs of the rooted forests k i m j

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Vertices that locate facilities: z jj vertex j is the root of a tree Non-facility vertices: z ij vertex i allocated to facility j Paths connecting non x km ij facilities and facilities: arc ( k, m ) in the path from i to its allocated facility j Arcs of the rooted forests y km arc ( k, m ) is in the rooted forest Path formulation Facility Location /Network Design Problem:

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 i is a facility or assigned to one Equilibrium on path from i to j Path formulation for FL/ND Budget AB C

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Outline Optimization criteria: design costs / service costs UF: Facilities used to give service: Location + Network design UU: Facilities used to re-route flows between customers: Hub location (more general) HUB LOCATION

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 LOCATION + ROUTING FOR CONNECTING FACILITIES HUB LOCATION

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Location problems on networks with routing E Fernández EUROXXIV MINIMUM TOTAL COST Set-up costs + Flow Routing costs HUB LOCATION Network design A set of facilities (hubs) to open Subset of edges to connect hubs among them Edges to connect customers to their allocated hubs Location Assignment i j

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 i j m k HUB LOCATION: Typical asumptions Transfer between hubs Collection Distribution Discount factors to routing costs Full interconnetion of hubs Paths: i-k-m-j Triangle inequality Hub location problems are NP-hard

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 HUB LOCATION Alumur and Kara. 2008. Campbell (1994), (1996) Campbell, Ernst, Krishnamoorthy (2005a, 2005b) Ernst and M Krishnamoorthy (1998), (1996), (1999) O’Kelly (1986), (1987), (1986), (1992), (1992), (1994). Skorin-Kapov, Skorin-Kapov, O’Kelly. (1996) Cánovas, García, Marín (2007) CPhD. Thesis (2009) C, Cordeau, Laporte (2011), (2012) C, Díaz, F. (2010), (2008) C, F (2012), (2014) C, F, Marín, (2008), (2009) Ernst, Hamacher, Jiang, Krishnamoorthy, Woeginger. (2009) García, Landete, Marín (2012) Hamacher, Labbé, Nickel, Sonneborn. (2004). Labbé, Yaman. (2004). Labbé, Yaman, Gourdin. (2005). Marín. C&OR, 2005, Top, (2007). Marín, Cánovas, Landete. EJOR, (2006). Campbell, O’Kelly (2012) Twenty-five years of hub location research. Tr. Sci.

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Minimize set-up cost cost Single allocation / multiple allocation Does it really matter ? All flow leaving vertex i goes through the same hub Flow may leave vertex i trough different hubs

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 SINGLE ALLOCATION x ij km : Fraction of flow from vertex i to vertex j routed via hubs k and m O’Kelly 87, Campbell 94, Skorin-Karpov 96 Tight LP but many variables Routing cost of the path i-k-m-j C ij km z ik =1 iff vertex i is assigned to hub k x ij km {0,1} Capacity constraints HUB LOCATION: path formulation i j k m d ik d km d km Uncapacitated: n=500 C, Cordeau Laporte, 2010 Capacitated: n= 200 vertices C, Díaz, F, 2010 x ij km = z ik z jm Even if the set of hubs is known the allocation is still NP-HARD

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Multiple Allocation z ik 0 Fraction of flow with origin at i routed via hub k HUB LOCATION: path formulation x ij km = z ik z jm The logic no longer holds Can be substituted by the stronger If the set of hubs is known the allocation is easy (find best path for each commodity) x ij km : Fraction of flow from vertex i to vertex j routed via hubs k and m

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 h i km : Fraction of flow emanating from vertex i routed via hubs k and m i m k OiOi j i l HUB LOCATION: flow formulation Single allocation (Ernst and Krishnamoorthy 96) Routing costs Multiple allocation (Ernst and Krishnamoorthy 98, Marin 05) r i kj : Fraction of flow emanating from vertex i routed via hub k and vertex j i m k OiOi j l z ik r i kj h i km j k h i lk r i kj

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Optimization criteria: design costs / service costs UF: Facilities used to give service: Location + Network design UU: Facilities used to re-route flows between customers: Hub location (more general) HUB LOCATION Outline

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Hubs are not necessarily connected by means of a complete graph (more general) HUB LOCATION Tree of Hubs Hub arc location

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 p Number of hubs to open The Tree of Hubs Location Problem C, F, Marín, 08, 09. i j m k MINIMIZE: Flow Routing costs TO FIND A set of p hubs to open Subset of edges that define a tree to connect hubs among them (to route flow between customers) Subset of edges to connect customers to their allocated hubs Assignment

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 m1m1 im2m2 mrmr j z jm r z im 1 x ij m 1 m 2 x ij m r-1 m r i m1m1 j m2m2 mrmr x ij m r j z im 1 x ij m 1 m 2 x ij m r-1 m r i m1m1 jm2m2 mrmr z jm r x ij im 1 x ij m 1 m 2 x ij m r-1 m r i m1m1 j m2m2 mrmr x ij m r j x ij im 1 x ij m 1 m 2 x ij m r-1 m r Possible paths for sending the flow from i to j ( j i ) i non – hub j non- hub i non – hub j hub i hub j non - hub i hub j hub The Tree of Hubs Location Problem

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 1.There is an optimal solution such that x ij km {0,1} 2.The y variables define a spanning tree on subgraph induced the vertices s.t. z ii =1 3.It holds that z ik z kk, i k Every vertex is allocated Flow can only circulate in y edges Connectivity and flow equilibrium p hubs p-1 y edges The Tree of Hubs Location Problem

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Hub arc location problems i i j A hub node is set-up at each endnode of a hub arc Commodities are routed via hub arcs

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 g e : set-up cost for hub arc e c u : set-up cost for hub node u F ek : cost for sending commodity k=(i,j) via hub arc e Hub arc location problems: data i i j gege u v i j d ij F ek = W ij ( d iu + d uv + d vj ) cucu To find: the hub arcs to set-up The assignment of commodities to hub arcs Such that the total cost hubs set-up costs (arcs and nodes) + commodities distribution costs ARE MINIMIZED

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Further data q : Maximum number of hub arcs p : Maximum number of hub nodes implied by hub arcs Some particular cases q = p(p-1)/2 y g e =0, e (node) p -hub location problem c u =0 u, only hub arc location problem

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Formulation I: decision variables Hub arcs Hub nodes Allocation

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Formulation Extension of UFLP Many variables ( x ek 4 index) (|K|+2)(1+|E|) constraints Hub arc location NP-hard A better formulation is based on properties of supermodular functions C, F, (2014)

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 Network optimization problems involve location and routing decisions which, in turn, imply network design decisions. Objective function: Trade off between set-up costs and service (routing) costs Other objective functions (min-max, weighted average …) Tight (effective) formulations require (too) many variables. Strengthen formulations with fewer variables (valid inequalities) Column generation, Benders decomposition, … Best formulations exploit structure of the problem Research trend Forward: More general (complex) models Backward: Design and implement efficient algorithms for particular cases of network design problems. To conclude

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Location problems on networks with routing Elena Fernández CAPD Nov 4, 2014 THANK YOU FOR YOUR ATTENTION

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