Improvement 4: initial solution for allocation local search

This improvement technique is aimed at providing a relatively “good” starting point for LS for allocation deci- sions at each generation, which is studied in Sec.4.1.5. Since the strength of LS is to exploit better solution in a relatively small solution space, a “good” starting point is essential to the result of LS400_.

398 _{For details, please see Sec.4.1.5.} 399 _{See Sec. 4.1.5.}

400 _{See Blum/ Roli (2008), p.7.}

Update Update Solution pool from GAs

FLS for hub location decision

FLS for allocation decision

Output final solution Final solution pool

According to the literature review on allocation decision in Sec.4.1.5, the most commonly used allocation pat- terns (or allocation criteria) include “distance-based” allocation, “multi-criteria” allocation, “maximum flow” allocation and “minimum cost” allocation. Except the first one, all the allocation patterns are constructive heu- ristics, by which the elements in the solution are added one by one401_{. Actually, the right choice of allocation} pattern for different HLPs needs specific knowledge of the problems.

(1) Distance-based allocation pattern by O’Kelly402

This allocation pattern, including both “nearest distance” and “second-nearest distance” allocation, was initial- ly proposed by O’Kelly, who pointed out later that when the “nearest-distance” allocation criterion is always the best for incapacitated p-median problem, it is not always true for interacting facility location problems, since it ignores the flows among facilities403_{. However, later studies on HLPs find that the “nearest distance”} allocation pattern is still an effective method, especially for p-hub median problem, under the consideration of short computational time404_.

(2) Multi –criteria allocation pattern by Klincewicz 405

A “multi-criteria” allocation pattern to the p-hub median problem was proposed by Klincewicz: initial alloca- tion of a given node is based on the sum of common traffic with certain hub and the distance to that hub. We carry out this idea by allocating non-hub nodes to “home” hub according to the two steps as follows.

(a) For the non-hub node that is covered by only one hub, it will be allocated to that hub directly;

(b) For the non-hub node

i

that can be covered by more than one hub, Index

A1

(Eq.4-8) will be used as the criterion for the allocation decision.

(

)

1

ij ji jh j N i h ih

w

w

x

A

d









(4-8)

This index considers not only the distance

d

ihbetween node

i

and hubhbut also the interchange volume

(

_{ij ji}

)

_jh

j N

w

w x







between node

i

and all nodes that have already been allocated to hub h. This index implies that if a node has heavier flow with a candidate hub and/or shorter distance from that hub, it is probably more economical to allocate this node to that hub.

As we can anticipate, part of non-hub nodes is covered by only one hub. After step one, each hub already has some subordinate demand nodes. For every not-yet-allocated demand node

i

, we calculate the Index

A1

for

401 _{See Mayer (2001), p.91.}

402 _{See O’Kelly (1987), pp.393-404.} 403 _{See O’Kelly (1992), p.303.} 404 _{See e.g. Kratica (2007), pp.15-28.} 405 _{See Klincewicz (1991), pp.25-37.}

all hubs hthat can cover it. We allocate the demand node to the hub with the highest index value. Every dou- ble- or multi-covered non-hub node is under this consideration till all find their “home” hub.

(3) Maximum flow allocation pattern by Campbell

The success of multi-criteria allocation pattern implies that traffic volume to potential “home” hub could also serve as a criterion for allocation. Travel cost by air between hubs is normally higher than that by truck, mak- ing this idea more attractive for air-ground network than multi-criteria allocation pattern. The corresponding index for the to-be allocated non-hub node

i

is as Eq.4-9.

2 =

i_h

(

_{ij ji}

)

_jh j N

A

w

w x







(4-9)

(4) Minimum cost allocation pattern by Campbell 406

Campbell proposed another allocation pattern called “minimum cost allocation”, which allocates a node to a hub so that total travel costs are minimized. It was shown that this method consistently provides a tighter bound then the above one. The corresponding index for the to-be allocated non-hub node

i

is as Eq.4-10.

3i_h ( _{ij ji}) _mj( _{ih ih km km mj mj})

j N m H

A w w x



d



d



d

 



 

   (4-10)

In Chap.5 we will test all these four allocation patterns to evaluate their performance under our instance.

4.2.5. Improvement 5: approximate of integer programming in early stage

In the basic model and Ext.1, the demand volume on each hub link can be calculated when the hub location and allocation is determined, so that the best service type can be easily chosen directly according to the cost func- tion of each service type. In other words, integer programming is not necessary. However, the air service se- lection decision in Ext.2 is determined with an integer programming problem due to the numerical constraints on the aircraft in current fleet.

If the integer programming for the service selection decision is time-costly, the total running time will grow exponentially. This is always the bottleneck to adopt hierarchical algorithm to large-scale instances. So we propose the following method, trying to make the overall algorithm more time-efficient and keep its perfor- mance at the same time. Our method is like this: if the improvement on the solution by solving the embedded sub-problem is relatively small compared with the improvement by solving the overall problem, we use an approximate result of the sub-problem in the master algorithm rather than solving the sub-problem until the master algorithm find the near-optimal region.

Generally speaking, if the algorithm for the sub-problem is time-costly and must be invoked frequently, this time-saving method can be efficient with quite small negative impact on the performance of the overall algo- rithm.

406 _{See Campbell (1996), pp.923-935.}

4.3. Summary

This chapter is contributed to the design of solution process, individual algorithms for decisions and im- provement techniques.

After we make a relatively thorough literature review on solution methods for related HLPs in Sec.4.1.1, we decide to adopt meta-heuristics under the consideration of instances scale and management requirement. In Sec.4.1.2 we briefly discuss the classification of hybrid meta-heuristics and hybrid principle. Sec.4.1.3 designs the solution process. The literature review on solution process of compound location problems demonstrates that our problem is essentially a location problem with embedded allocation and service selection problems. We divide the original problem into three hierarchical sub-problems and propose an overall solution process, connecting all subordinate algorithms with two hierarchical feedback cycles. Form Sec.4.1.4 to Sec.4.1.6 we propose specific algorithms for individual decisions. Specifically, in Sec.4.1.4 we propose the customized proce- dure of GAs for hub location decisions. In Sec. 4.1.5 we illustrate LS algorithms for allocation decisions. In order to balance the solution quality and computational time, we take up two measures, i.e. Improvement In- dex and partial LS. In Sec. 4.1.6 we use integer programming for the service selection problem with predeter- mined hub location and allocation decisions.

In Sec.4.2 we propose five improvement techniques for different procedures of SGAs. Improvement 1 is ori- ented towards initial solution generation procedure. We incorporate constructive procedure borrowed from GRASP to generate initial solutions for GAs. This method can not only yield feasible solutions but also bal- ance the diversity and intensity of the initial solution pool. Improvement 2 is oriented towards the update strategy of the solution pool. In contrast to constructive procedure for initial solution generation, injection mechanism balances the diversity and intensity of the solution pool during the whole process of GAs. Im- provement 3 tries to further improve the solutions from GAs by attaching LS after GAs for both hub location decisions and allocation decisions. Improvement 4 tries to provide a “good” initial solution for allocation LS. We list four different initialization methods. Improvement 5 is to raise the computational efficiency with no or little negative impact on the solution quality. These improvement techniques can be classified into the follow- ing categories of the hybrid heuristics according to Fig.1-4

Figure 4-12: Classification of improvement techniques

Hybrid content Hybrid level

Collaborative combination of meta with problem-

specific algorithm

Integrative combination of meta with meta

Integrative combination of meta with problem-

specific algorithm Collaborative

combination of meta with meta

Collaborative combination of meta with general tech-

niques from OR/AI

Integrative combination of meta with general techniques from OR/AI Improvement 3

In document Strategic Planning of Large-scale, Multimodal and Time-definite Networks for Overnight Express Delivery Services (Page 120-125)