Hub and Spoke Network Design

Hub and Spoke Network Design

Outline • Motivation • Problem Description • Mathematical Model • Solution Method • Computational Analysis • Extension • Conclusion

Motivation 1 7 5 2 4 8 3 9

Spokes Hubs Motivation σ = 0.25 Spoke and Hub Network

Motivation • Hub and Spoke Network design: • Cited as “seventh in the American Marketing Association series of ‘Great Ideas in the Decade of Marketing’ (Marketing News, June 20, 1986) • Predominant architecture for airline route system since deregulation in 1978 • Finds applications in telecommunication network, express cargo

Problem Description • Given a network of nodes with given flows between each pair, determine: • Which nodes are set as hubs • Which hub is a node assigned to • So that: • Every flow is first routed through one or two hubs before being sent to its destination

Methodologies • Enumeration heuristics - O’Kelly (1986) • Meta-heuristics: • Tabu Search – Klincewicz (1991); Kapov & Kapov (1994) • Simulated Annealing – Ernst & Krishnamoorthy (1996) • Lagrangian relaxation – Pirkul & Schilling (1998); Aykin (1994); Elhedhli & Hu (2005)

m i j k Mathematical Model

(7) Mathematical Model Min (1) Subject to: for all i (2) for all i, k (3) (4) for all i, j > i, k (5) for all i, j > i, m (6)

Mathematical Model • Problem size: • For number of nodes = n: That’s too large! • For n = 15:

Solution Method • Lagrangian Relaxation • 31 different lagrangian relaxations possible • Review on Lagrangian Relaxation: Fisher (1981, 2005); Geoffrion (1974) • In current study, constriant sets (2), (5), (6) relaxed

Solution Method Min (1) Subject to: for all i (2) αi for all i, k (3) (4) for all i, j > i, k (5) βijk for all i, j > i, m (6) Gijm (7)

Solution Method Min (7) Subject to: for all i, k (3) (4) Sub problem 2 Where, Sub problem 1

Constrained added to improve bound Min for all i, j > i Subject to: [SUB1]: Min Subject to: for all i, k Solution Method [SUB2]:

Solution Method [MASTER]: Max for h = 1,2,…. Subject to: for h = 1,2,….

Solution Algorithm • [SUB1]: • For each i, j: • Find • Set

Solution Algorithm • [SUB2]:

Solution Algorithm • [Feasible Solution]:

Solution Algorithm • Issues: • Slow convergence as master problem grows too large • Could not converge in 30 minutes for 10 nodes • How to resolve???

Initialize α, β, γ; Initialize step size Solve SUB1; SUB2 and obtain LB Construct a feasible solution and obtain UB Is (UB-LB)/LB>ε? α, β, γ Adjust α, β, γ by the amount of infeasibility If no improvement in LB since long, decrease step size stop Solution Algorithm • Subgradient Optimization to find lagrang multipliers No Yes

Computational Analysis

Analysis Congested

Extended Model Min (1) Subject to: for all i (2) for all i, k (3) (4) for all i, j > i, k (5) for all i, j > i, m (6) Congestion Cost function

Extended Model cont.. Min Linear Approximation using tanget planes for congestion cost function Subject to: (2) – (7)

(8) Extended Model cont.. Min Subject to: MIP with an infinite number of constraints (2) – (7)

(8) (7) Solution Method (Langrangean Relaxation) Min Subject to: for all i, k (3) (4) Sub problem 2 Sub problem 1 Where,

In absence of this constraint, problem separates into k smaller problems; each can be solved using cutting plane method (8) (7) Solution Method contd.. [SUB1]: Min Subject to: for all i, k (3) (4)

Solution Method contd.. • Solution implemented in MATLAB 7.0 • [SUB1-k] solved using CPLEX 10 • CPLEX called from MATLAB

Computational Analysis

Discussion • Solution speed can be improved by using a compiled code (in C or Fortran). MATLAB is inefficient in executing loops as it is interpreted line by line.

Conclusion • A model for Hub and Spoke Network Design solved using lagrangean relaxation • Model extended to address the issue of congestion • Good solutions obtained in reasonable time • Solution speed can be further improved if implemented in a language that uses acompiler

Hub and Spoke Network Design