p-Hub median problems

The model proposed by O’Kelly (1987) forms the basis of all problems in this group. All problems under this class are characterised by at least 4 features; 1. Every origin-destination path must visit at least one hub, 2. Exactly p number of hubs must be installed on the network, 3. All hubs are assumed to be connected to each other and inter-hub cost per unit flow is discounted by a factor 𝛼 to reflect the economies of scale benefits from concentration of flows and 4. The objective function is often to minimise the weighted total transport cost of all flow movements. The economies of scale factor 𝛼 as noted in Campbell (1994) plays an important role in determining the best locations of the p hubs and the assignment of demand nodes to the located hubs. It is expected and demonstrated by O’Kelly (1987), O’Kelly et al. (1996) and Campbell (1994) that as 𝛼 decreases, hubs tend to spread farther apart and the number of spokes (The links or arcs connecting the demand nodes to hubs) decreases, since a lower

inter-hub transport cost favours allocation to the nearest inter-hub. In the extreme case where 𝛼 = 0 the inter-hub cost will reduce to zero and each demand point will be allocated to exactly one hub (least cost hub) and the p-HMP collapses to the classical p-median problem (Hakimi 1965).

For large 𝛼 (𝛼 > 1) hub interactions are expensive and hubs are drawn closer together to reduce inter-hub transport costs.

Several variants of the p-HMP exist in the literature with the probably the most noticeable one being how demands are allocated to hubs. Two types of allocations are discussed in the literature, the single allocation and the multiple allocations p-hub median problems. Single allocation p-hub median problems (SApHMP) assign each demand node to exactly one hub. The model by O’Kelly (1987) is a classic example of the SApHMP. Multiple allocation p-hub median problems (MApHMP) on the other hand allows the assignment of each demand node to more than one hub. The difference between these two problems is illustrated through Figure 2.1. In Figure 2.1a demand node D1 is assigned to only hub Y1 and it implies that flows to and from node D1 can only go through hub Y1. Under Figure 2.1b, demand node D4 is assigned to hubs Y1 and Y2 allowing flows to and from this node to go through either hub. This example illustrates the restrictive nature of the SApHMP and may be unrealistic in many real applications especially in transport applications.

Nevertheless, research on SApHMP have been pursued by several authors including the work by Campbell (1994) who presented the linear version of the problem reducing the quadratic integer programming formulation (O’Kelly 1987) to integer linear programming formulation. Skorin-Kapov et al. (1996) show that the linear formulation by Campbell (1994) is not tight enough as it produces highly fractional solutions. They then presented a tighter formulation of the problem and demonstrated using the CAB data set (Ernst and Krishnamoorthy 1996) that their formulation almost always yields integral solutions using the CPLEX software. Ernst and Krishnamoorthy (1996) presented a new mixed integer linear programing (MILP) formulation of the problem also in an attempt to reduce the number of variables and constraints, which are directly linked to the computational time required to solve the problem. In their formulation, they treated the inter-hub transfers as a multicomodity flow problem where each commodity represents the traffic flow from a demand node. They showed the computational benefits of their new formulation using the AP (Australian Post) data set which uses different discount factors for collecting and distributing flows.

Table 2.1 presents a summary of other relevant work on this class of problem and includes the work by O’Kelly and Bryan (1998) who focussed on the effects of the economies of scale factor 𝛼 on hub locations and potential usage. Sohn and Park (1998) presented a new formulation of the problem using fewer decision variables and constraints. The formulation by Ebery (2001) reduced both the number of decision variables and constraints to the order of O(n²), making it theoretically the most computationally efficient model on the subject, with n being the number of nodes on the network. However, it was shown that in practice the formulation by Ernst and Krishnamoorthy (1996) is computationally more efficient (Alumur and Kara 2008).

Studies on MApHMP were first conducted by Campbell (1992). He presented an integer linear programming (ILP) formulation of the problem with a total of O(n⁴) binary variables and O(n⁴) linear constraints. He noted that an optimal solution to the problem exists if the capacity constraints on links are relaxed. Skorin-Kapov et al. (1996) again presented a tighter formulation of the problem with the required number of constraints reducing to the order O(n³). The model has been reported to return optimal solutions to many instances of the CAB data set and those non-optimal solutions they found were within 1% of the optimal solutions. Extending their work on the SApHMP to the MApHMP, Ernst and Krishnmoorthy (1996) also proposed a new formulation of the problem with significant improvements in computational efficiency. The number of binary variables reduce to the order O(n³) and required O(n²) constraints. Other noticeable work on the subject are summarised in Table 2.2, including Sohn and Park in (1998) formulation of the uncapacitated version of the problem and Sasaki et al. (1999) who considered a special case of problem where each route in the network uses only one of the located hubs.

Other important variants of the hub location problem include whether or not the amount of flows through the located hubs are restricted or unrestricted. The restricted versions are called capacitated HLP and work in the area includes Lin and Chen (2012), Stanimirovic (2010) and Ernst and Krishnamoorthy (1999), Aykin (1994). Most of the work described above including O’Kelly (1987) and Campbell (1994) are uncapacitated or unrestricted variants of the problem. Another variant of the HLP is the inclusion of the fixed costs of hub location in the objective function to account for the differential cost of land, labour and other factors in each candidate hub location. HLP with fixed cost of hub locations can be found in

Chen (2007), Topcuoglu et al. (2005), Klincewicz (1996), Aykin (1995) and Ernst and Krishnamoorthy (1998a).

Table 2.1: Summary of work on p-HMP (Single allocation)

Reference

Capacitated (Y/N)

With Fixed

Cost (Y/N) Comments

O' Kelly (1987) N N Quadratic integer program and heuristics

Aykin (1990) N N Exact algorithm

Klincewicz (1991) N N Exchange heuristic

Campbell (1994) N N First integer formulation

O' Kelly et al. (1995) N N Lower bounding technique

Klincewicz (1992) N N Tabu search and GRASP heuristics

Skorin-Kapov and Skorin-Kapov (1994) N N Tabu search heuristics

Campbell (1996) N N Heuristics

Ernst and Krishnamoorthy (1996). N N New formulation, SA and B&B

algorithms

O' Kelly et al. (1996) N N New formulation

Smith et al. (1996). N N Heuristics

Sohn and Park (1997) N N Problem complexity

Ernst and Krishnamoorthy (1998a). N N Shortest path based B&B

Pirkul and Schilling (1998). N N Lagrangian relaxation heuristic

Sohn and Park (1998) N N New formulation

Sohn and Park (2000) N N Problem Complexity

Abdinnour-Helm (2001) N N Simulated annealing

Ebery (2001) N N New formulation

Elhedhli and Hu (2005) N N Congestion cost function with Lagrangian heuristic

Stanimirovic (2010) Y Y New formulation with Genetic algorithm

Alumur et al. (2009) N Y Hub location in incomplete hub

networks

Table 2.2: Summary of work on p-HMP (Multiple allocation)

Reference Capacitated (Y/N)

With Fixed

Cost (Y/N) Comments

Campbell (1992) N N First integer program

Campbell (1994) N N New formulations with fixed costs

Campbell (1996) N N Greedy interchange heuristic

Skorin-Kapov and O' Kelly (1996) N N New formulation

Ernst and Krishnamoorthy (1998a) N N New formulation, B&B algorithms and heuristics Ernst and Krishnamoorthy (1998b) N N Shortest path-based B&B algorithms

Sasaki et al. (1999) N N 1-stop problem with B&B algorithm & heuristic

Boland et al. (2004) N N Preprocessing and tightening constraints

O' Kelly (1992) N Y Single allocation hub location with fixed costs

Abdinnour-Helm (1998) N Y Hybrid genetic and Tabu search heuristics