Arc hub location problems as network design problems
Arc hub location problems as network design problems with routing Elena Fernández- Dpt EIO-UPC Ivan Contreras- CIRRELT- Montréal Seminario de Geometría Tórica Jarandilla 12 -15 de noviembre 2010 Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Decisions in discrete location problems on networks Ø Which set of facilities to open ? Location Ø How to satisfy the customers demands from open facilities ? § From which facility does the customer receive service ? Allocation § How Routing is service provided ? Ø Are facilities somehow connected ? Routing Ø Which are the possible (or preferable) connections between Network design customers or between customers and facilities ? Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Routing What are facilities used for? where customers obtain service from connect customers and facilities where flows between pairs of customers are consolidated and rerouted Ø Connect customers and facilities Ø Connect facilities between them Network design Which are the possible connections ? Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Customers receive service from/at facilities If customers move to facilities to recieve service ⋮ the routing of each customer is trivial Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Customers receive service from facilities If several customers are visited in the same route ⋮ the design of the routes may become difficult Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
HUB LOCATION Facilities used to reroute flows between pairs of customers There exists communication between each pair of customers. Flows are consolidated and re-routed at facilities (which must be connected) Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
HUB LOCATION Facilities used to reroute flows between pairs of customers Connection between facilities by means of a tree Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
i HUB LOCATION G=(V, E) fi: set-up cost for facilities i V dij: per unit routing cost from i to j i, j V Wij: flow between i and j i, j V j A set of facilities (hubs) to open Hubs are used to consolidate and reroute flow between customers Location Assignment TO FIND Subset of edges to connect customers to their allocated hubs Network design Subset of edges to connect hubs among them MINIMUM TOTAL COST Set-up costs + Flow Routing costs Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
HUB LOCATION: Typical asumptions Discount factors to routing costs Transfer between hubs Collection k Distribution i m j Full interconnetion of hubs Triangle inequality Paths: i-k-m-j Hub location problems are NP-hard Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Hub arcs j i i j j Campbell JF, Ernst AT, Krishnamoorthy M (2005 a) Hub arc location problems: Part i-introduction and results. Manag Sci 51(10): 1540– 1555 Campbell JF, Ernst AT, Krishnamoorthy M (2005 b) Hub arc location problems: Part ii-formulations and optimal algorithms. Manag Sci 51(10): 1556– 1571 Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Index Ø Hub arc location problems Ø Formulation based on properties of supermodular functions Ø Comparison of formulations Ø Some computational results Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Hub arc location problems G=(V, E) complete undirected graph n=|V|, m=|E|=n(n-1)/2 K={(i, j) V V: there is demand between i y j}; k K commodity i § Commodities demand is routed via hub arcs § If an arc hub is set-up then hub nodes are also established at both endnodes i q: Maximum (exact) number of hub arcs p: Maximum (exact) number of hub nodes Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre j
Hub arc location problems dij: unit routing cost from i to j i, j V ge: set-up cost for hub arc e e E cu : set-up cost for hub vertex u u V Fek: routing cost for commodity k K via hub arc e=(u, v) k K, e E Fek =Wij( diu + duv + dvj) i To find: § § Hub arcs to set-up Assignment of commodities to hub arcs i Such that the overall cost is minimized Hubs set-up cost (both arcs and nodes) + Commodities routing costs Location problems on networks with routing ▪ E Fernández cu u ▪ TGS 2010 i ge v dij j ▪ Jarandilla 12 -15 Noviembre j
Hub arc location problems General model: If we allow G to have loops, then we can locate both hub arcs and independent hub nodes. i i Location problems on networks with routing j ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Hub arc location problems § q < p(p-1)/2 unfeasible (if exactly p hub nodes must be open) § q= p(p-1)/2 y ge=0, e p-hub (nodes) location problem § q≥ min{m, p(p-1)/2} the constraint on the number of hub arcs is redundant § If cu=0 u, and p ≥ 2 q, problem of locating only hub arcs. Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation I Variables: |E| + |V| + |E||K| Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation I Location problems on networks with routing ▪ E Fernández § Extension of UFLP § Many variables (xek 4 -index variables) § (|K|+2)(1+|E|) constraints ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation II (based on propertirs of supermodular functions) Hub arc minimization problem Minimization of supermodular function Maximization of submodular functions Nemhauser, Wolsey, Maximizing submodular set functions: formulations and analysis of algorithms, in P. Hansen, ed. , Studies on Graphs and Discrete Programming, N-H (1981) Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Supermodular functions Let E be a finite set, and Definition: f : P(E) ℝ f is supermodular if f(S T) + f(S T) ≥ f(S) + f(T) S, T E Characterization: f supermodular f(S {e}) - f(S) f(S {e’, e}) - f(S {e’}) Characterization: f supermodular and non-increasing f(T) ≥ f(S) + e TS [ f(S {e}) - S, T E f(S)] The maximization of supermodular functions is “easy” The minimization of supermodular functions is “difficult” Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Supermodular functions For S⊆ E Corollary: Proposition: assignment cost associated with k K hk (S) = Min e’ S Fe’k k (S) + k(S) hhkk (T) ≥ h e TS e is supermodular and non-increasing for , forallallk K k K, S, T E hk (S {e})- hk (S )= (Fek - Min e’ S Fe’k )(a)- =min {a, 0} where Optimization problem: Find T* such that hk (T*) =Min S⊆ E hk (S) Min k = hk (T*) Find k ≥ hk (S) Find (ze)e E, ze {0, 1} s. t. Min k k ≥ hk (S) Find (ze)e E, ze {0, 1} s. t. + e T*S ek(S) + e S ek(S) ze for all S E Min k Location problems on networks with routing k ≥(Mine’ S Fe’k)+ e S(Fek-Mine’ SFe’k)-ze for all S E ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation II Min k k ≥ (Mine’ S Fe’k) + e S (Fek - Min e’ S Fe’k )- ze for all S E Remark: Even if there is an exponential number of constraints (subsets S) there is a small number of possible values of Min e’ S Fe’k. The candidate values are Fe k for eh E. h Find (ze)e E, ze {0, 1} s. t. Min k k ≥ Fehk Location problems on networks with routing + e S (Fek - Fehk)- ze ▪ E Fernández ▪ TGS 2010 ▪ for all eh E Jarandilla 12 -15 Noviembre
Formulation II (based on propertirs of supermodular functions) Hub arc minimization problem Minimization of supermodular function S⊆ E, f(S) = g(S)+c(V(S))+ k K hk(S) supermodular g(S) = e E ge supermodular ĉ(S) = c(V(S))= u V(S) cu supermodular hk (S) =Min e S Fek supermodular and non-increasing Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation II “Saving” in allocation cost for using additional hub arc e Location problems on networks with routing ▪ E Fernández § |K|+|E|+|V| variables (variables with 1 -2 indices) § (|K|+2)(1+|E|) constraints ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation I vs Formulation II FI FII Variables § |E||K|+|E|+|V| § |K|+|E|+|V| Constraints § (|K|+2)(1+|E|) Theorem: (LP bounds ) Location problems on networks with routing v. LPF 1=v. LPF 2 ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation I vs Formulation II: APset ge = (cu+ cv)coeff , e=(u, v); coeff = 0. 15 p=3, q=9, Xpress, CPU limit: 3 hours Instance Natural Formulation Supermodular Formulation |V| LP bound LP % gap Final % gap Time (sec) nodes 10 0. 2 167708 5. 21 0. 00 4. 80 119 167708 5. 21 0. 00 20. 00 101 10 0. 5 173104 3. 84 0. 00 2. 00 57 173104 3. 84 0. 00 9. 50 53 10 0. 8 176158 2. 15 0. 00 1. 00 21 176158 2. 15 0. 00 5. 80 29 20 0. 2 188501 0. 80 0. 00 40. 00 13 188501 0. 80 0. 00 1136. 00 35 20 0. 5 194095 0. 33 0. 00 9. 90 5 194095 0. 33 0. 00 90. 60 9 20 0. 8 194737 0. 00 3. 10 0 194737 0. 00 17. 90 0 25 0. 2 191717 0. 52 0. 00 303. 40 21 191717 0. 52 0. 35 10800. 00 32 25 0. 5 196165 0. 66 0. 00 88. 30 11 196165 0. 66 0. 00 811. 00 13 25 0. 8 197387 0. 31 0. 00 37. 10 7 197387 0. 31 0. 00 258. 60 5 40 0. 2 196449 2. 68 2. 30 10800. 00 28 memory 40 0. 5 200711. 45 0. 00 828. 70 0 memory 40 0. 8 200711. 45 0. 00 407. 60 0 memory 50 0. 2 - - - 10800. 00 - memory 50 0. 5 200436. 8 0. 38 0. 27 10800. 00 53 memory 50 0. 8 201074. 02 0. 06 0. 00 3156. 30 5 memory Location problems on networks with routing ▪ E Fernández LP bound LP % gap Final % gap Time (sec) nodes ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation I Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Formulation II Separate Given ¿ k K, h t. q. Brute force: |K||E| (O(|V 4|) Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre ?
Separation problem Given a polyhedron P and a point x*, to identify if x* P. If it does not, to find a valid inequality for P, px p 0 such that px*>p 0. px p 0 x* x* Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Separation of constraints Given to know if there exists k K, h s. t. Concave, piecewise linear; with break values Fek (k fixed) Proposition: For k given, the maximum of Shk, is attained for h=rk First index such that the slope is no longer positive Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Preliminary Results Fi nodes alpha LP %gap Final % gap Times (sec) Node s LP % gap nal % gap Tim es (sec) no des 10 0. 2 5. 21 0 4. 8 119 5. 21 0 1. 03 43 10 0. 5 3. 84 0 2 57 3. 84 0 0. 88 33 10 0. 8 2. 15 0 1 21 2. 15 0 0. 55 21 20 0. 2 0. 80 0 40 13 0. 80 0 6. 65 33 20 0. 5 0. 33 0 9. 9 5 0. 33 0 2. 46 9 20 0. 8 0. 00 0 3. 1 0 0. 00 0 1. 08 0 25 0. 2 0. 52 0 303. 4 21 0. 52 0 11. 05 7 25 0. 66 0 88. 3 11 0. 66 0 9. 117 11 25 0. 8 0. 31 0 37. 1 7 0. 31 0 4. 37 5 40 0. 2 2. 68 2. 3 10800 28 2. 68 0 923. 4 124 40 0. 5 0. 00 0 828. 7 0 0. 00 0 62. 36 0 40 0. 8 0. 00 0 407. 6 0 0. 00 0 29. 37 0 50 0. 2 1. 24 0 9438. 8 265 50 0. 5 0. 38 0. 27 10800 53 0. 38 0 810. 59 59 50 0. 8 0. 06 0 3156. 3 5 0. 06 0 145. 23 5 - - 10800 Location problems on networks with routing ▪ E Fernández - ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
Preliminary Results nodes alpha LP % gap Final % gap Times(sec) nodes 60 0. 2 memory 2. 10 0 6887. 43 183 60 0. 5 memory 1. 56 0 1830 59 60 0. 8 memory 0. 96 0 931. 22 35 75 0. 2 memory 1. 72 1. 26 13483 74 75 0. 5 memory 1. 17 0 12216 105 75 0. 8 memory 0. 98 0 5922. 39 55 90 0. 2 memory 0. 80 0 30479. 6 95 90 0. 5 memory 8. 41 8. 27 21293 58 90 0. 8 memory 0. 69 0 5822. 45 33 100 0. 2 memory - - 10800 100 0. 5 memory 0. 63 0 27033. 7 55 100 0. 8 memory 9. 17 8. 81 38200 28 Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre -
Summary Ø Arc hub location problems (involve routing decisions) Ø General Problem Ø Two alternative formulacions Ø Minimization of supermodular function Ø Efficient solution of separation problem Ø Promising preliminary results Location problems on networks with routing ▪ E Fernández ▪ TGS 2010 ▪ Jarandilla 12 -15 Noviembre
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