1 Hub and Spoke Network Design

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

3 Motivation

4 Spoke and Hub Network σ = 0.25 Spoke s Hubs

5 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

6 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

7 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)

8 Mathematical Model i j k m

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

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

11 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

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

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

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

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

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

17 Solution Algorithm [SUB2]:

18 Solution Algorithm [Feasible Solution]:

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

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

21 Computational Analysis Original Model (Cplex)Lagrangean Relaxation # Nodes# HubsTime (sec) % GapOptimal ? Y Y Y Y 153> 1 Hour

22 Analysis Congested

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

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

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

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

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

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

29 Computational Analysis # Nodes# HubsTime (sec)Hubs% Gap , , , ,3, ,14,152.81

30 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.

31 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 a compiler

32