C ARACTERÍSTICAS GENERALES DE LOS PARTICIPANTES DE LAS ESCUELAS

 Division of the original problem

Three decisions are involved in the original problem, i.e. the hub location decision, the allocation decision and the air service selection decision. Actually our original problem is essentially a hub location problem, which is embedded with an allocation problem and an air service selection problem. In the first place, we would like to divide the original problem into three hierarchical sub-problems and clarify the relationship between the three sub-problems and the three decisions in original problem (see Fig. 4-2)

Figure 4-2: Hierarchical sub-problems of the original problem

(1) Hub location problem (upper problem )

The hub location problem takes the objective of minimizing the total cost under the constraint that the dis- tance between all demand nodes and their nearest hub is within the hub maximum coverage. The input of hub location problem includes the location of all demand nodes, potential hub set, demand volume between all O-D pairs, hub fixed cost, transportation cost rate for both backbone and tributary network, and number and capac- ity of aircraft. In order to calculate the total cost of the network, the hub location problem involves all the three decisions, i.e. hub location decision, allocation decision and air service selection decision.

(2) Demand allocation problem (median problem)

When the hubs are determined, all the demand nodes are singly allocated to “home” hubs with the objective of minimizing travel cost. The input of allocation problem includes the location of hubs and other demand nodes, demand volume between all O-D pairs, transportation cost rate for both backbone and tributary network, and

Service selection problem Air service selection

decision Allocation decision Allocation problem Hub location decision Hub location problem

number and capacity of aircraft. The corresponding decisions involve allocation of demand nodes to predeter- mined hubs and service selection for backbone link.

(3) Service selection problem (lower problem)

Once the hubs are determined and all the demand nodes are allocated to “home” hubs, the optimal air service can be determined for each hub link directly in Ext.1, while in Ext.2 the service selection problem is actually a flow problem with the objective of minimizing air cost and under the constraint of self-owned aircraft number and capacity. So it is exclusive for Ext.2. The input of the flow problem is outcome of allocation problem, air cost function, self-owned aircraft number and capacity.

The description of the three sub-problems manifests that they are “hierarchical” or “nested”. The lower prob- lem includes only air service decision, while the median problem includes extra decision on allocation besides air selection decision. The upper problem-hub location problem- is actually original problem and involves all the decisions.

 Corresponding algorithms

As a matter of fact, the original problem is divided according to the three decisions, i.e. hub location, allocation and service selection decisions. We propose for each decision specific algorithms. One works incorporated in another, so that the original problem can be solved near optimally in hierarchical approach with optimal and near-optimal solution of lower and median problems.

(1) Air service selection decision: integer programming

When the hub location and demand allocation decisions are made, the volume through each hub link is deter- mined. So service selection decision in Ext.2 can be easily made with an integer programming. This decision is passed over in the basic model for there is only one type of air service without capacity constraint, and also is passed in Ext.1 for there is no numerical constraint on each service type.

(2) Allocation decision: LS heuristics

It has been proved that the allocation decision is NP-hard even when hub locations are determined354_{. The} travel cost of each O-D pair of demand consists of three components: (a) the travel cost from the origin to the hub, (b) the cost between hubs (if necessary) and (c) the travel cost from the hub to the destination. We resort to LS heuristics which will be discussed in detail in Sec.4.1.5.

(3) Hub location decision: GAs

The hub location decision problem is NP-hard so that we resort to GAs which will be discussed in details in Sec.4.1.4.

 Overall solution process

354 _{This has been proved by Kara. See Kara (1999).}

We propose three specific algorithms for the three decisions, which have different roles in different sub- problems. As sub-problems are hierarchical or nested, the algorithms also work in hierarchical order. In per- spective of solution process, the integer programming for air service selection is incorporated into LS for allo- cation decision, while the LS is again embedded in GAs for hub location decision. Therefore, GAs serves as the master algorithm for the original problem.

We apply the word “stage” to describe the solution process. A uniform hierarchical framework of solution pro- cess is proposed in Fig.4-3 for both basic and extension models in this dissertation. The upper, median and lower solution stages are highlighted with light, median and dark background respectively. We divide the whole solution process into three hierarchical and iterative stages according to the three sub-problems. Each stage solves the corresponding sub-problem described above.

Figure 4-3: Overall solution process

In the whole solution process the three stages are interrelated with two hierarchical feedback cycles. In this sense the upper algorithms do not simultaneously solve all the underlying problems, but work cooperatively with lower algorithms. The two feedback cycles are repeated until certain stop criteria are met.

Specifically, Feedback cycle 2 is impelled by LS for allocation decision. First, the hub number and locations are fixed by GAs. On the basis of the selected hubs, demand nodes are allocated to the “home” hubs according to certain criterion. The volume through each hub link is thus determined, and the service on each hub link can be optimally determined with simplex algorithm. However, the allocation decision at allocation stage (median stage) restricts the optimal solution of service selection stage. Therefore LS of allocation decision serves as Feedback cycle 2 to seek better solution under the determined hub location.

Meanwhile, Feedback cycle 1 is impelled by GAs for hub location decision. Hub location decision at location stage restricts the optimal decision of allocation stage. We can also expect sub-optimization of the original problem if there is no iteration between allocation stage and hub location stage. GAs process takes up the role of Feedback cycle 1.

Feedback 1 (Gas)

Hub location decicion (GAs)

Final solution Feedback 2 (local search) M edi an sta ge L ow er stag e Upp er stage Allocation decison (local search) Service selection Decision (Simplex algorithm)

In sum, LS at allocation stage, i.e. Feedback cycle 2, attempts to minimize the travel cost based on determined hubs, while GAs at hub location stage, i.e. Feedback cycle 1, attempts to minimize the total cost of the network. With the hierarchical iteration of the subordinate algorithms the original problem is solved to near optimal finally.