The relationship between all the improvement techniques proposed in this section and customized SGAs in- troduced in Section 4.1.4 are illustrated in Fig.4-8. Improvement techniques 1 and 4 are integrative combina- tion with meta-heuristics, while Improvement techniques 3 belongs to cooperative combination with meta- heuristics (see Fig.4-1).
Figure 4-8: Relationship between SGAs and improvement techniques
4.2.1. Improvement 1: constructive procedure for initial solution generation
The choice of starting point is very crucial for the performance of iterative improvement heuristics such as GAs. In this section, we try to improve the performance of GAs with a “good” initial population.
It is common and easy for SGAs to randomly generate initial population for HLPs, if the hub number p is ex- ogenously fixed. However, for HLPs with endogenous hub number, such as hub set covering problem or hub location problem with fixed cost, it is quite unreasonable and arbitrary to choose random number of hubs at random place from the candidate hub set H. For the hub set covering problem, the primary problem is the feasibility of the solutions. Randomly generated solution has low possibility that all demand nodes can be cov-
Yes Initial solutions Feasibility adjustment Crossover Mutation Stop criterion? Feasibility adjustment No 4.1.5 Allocation
4.2.1 Improvement 1
Stop
4.2.2 Improvement 2
4.2.3 Improvement 3
4.2.4 Improvement 4
4.1.4 SGAs
Initialize solution pool
Update solution pool
ered by the selected hubs, so that the program spends a lot of time to judge the feasibility of the new generated solution and to repair the unfeasible solutions. For hub location problem with fixed cost, the primary problem is too large search space, so that the program is likely to run randomly and yields pool solution even after many generations. Moreover, randomly generated solutions are also not applicable for network with unevenly spread demand nodes, which is exactly the situation in our case.
In the light of these problems, we replace the random procedure for generating initial solutions in SGAs with “constructive procedure” borrowed from GRASP, by which feasible solutions can be generated. We name GAs incorporated with “constructive procedure” Constructive GAs (CGAs) in the following. This “constructive procedure” is iterative, greedy, random and adaptive. It is iterative because every initial solution is constructed by choosing one element at a time until the solution is feasible. It is greedy because the addition of each ele- ment is guided by the myopic criterion or greedy function. It is random because every element is chosen ran- domly from a Restricted Candidate List (RCL). The RCL technique allows the procedure to be repeated but to generate different initial solutions every time. Finally, it is adaptive because the RCL is updated after each choice.
An appropriate composition of RCL signifies a good balance between the diversity and the intensity of initial population. The RCL is constructed by repeatedly calculating the greedy function
f k( )
(see Eq.4-7) and ranking the results in decreasing order. Actually the functionf k( )
denotes the distance-weighted flow of those still uncovered nodes that hub k can cover. This greedy function is based on the assumption that the busier a hub is and the nearer a hub to demand nodes is, the more value it will bring to the network and the more travel cost it will save.( )
(
)
/
crasp ij ji ik ik i N j Nf k
w
w
y
d
forkH
craspand
1
0
ik ikd
D
y
otherwise
(4-7)In the beginning, the to-be-assigned hub set
H
crasp contains all potential hubs inH
; the to-be-allocated de- mand node setN
craspcontains all elements inN . The to-be-constructed location solution is
. We calculatethe objective value of
f k( )
for everyk k(
H
crasp)
.Then we assignf
max andf
min with the largest and the smallest values off k( )
respectively. We also define a threshold parameter (
0,1 )
. The RCL is com- posed of all elements inH
crasp , whose objective values off k( )
are superior to the threshold, i.e.min max min max
(
),
f
f
f
f
. After one node is randomly and uniformly selected from the RCL as hub node, it is deleted fromH
crasp and added in the location solution. At the same time all demand nodes inN
crasp that can be covered by this selected hub are eliminated fromN
crasp. SoH
crasp,N
crasp,f
max,f
minand the loca- tion solution are updated after each selection. This procedure goes on until all demand nodes are covered, i.e.crasp
N
is
(see Fig.4-9). It will be repeatedN
pop times to generate enough initial solutions for the solution pool.As we can see, this constructive procedure can not only yield feasible solutions, but also control the random- ness of the initial population. Borrowed from GRASP, it can be regarded as a repetitive sampling technique. Each reiteration produces a sample from a distribution, whose mean and variance are dependent on the param-
eter
. The parameter
controls the greediness and randomness or, in other words, intensity and diversity of the population. The case
1corresponds to a pure greedy algorithm, while the case
0corresponds to a pure random algorithm.Figure 4-9: Constructive procedure for initial solution generation
4.2.2. Improvement 2: injection mechanism
The injection mechanism is to introduce new chromosomes into the population at every generation or at regu- lar generation intervals. Contrary to the constructive procedure that controls the diversity of the initial popu- lation, the injection mechanism controls the diversity of the population during the GAs process. Some new generated individuals are injected into the solution pool to increase the diversity of the population throughout generations and to prevent premature of the algorithms. This idea is borrowed from the natural phenomenon, simulating the immigration of people between countries or regions. This scheme has been proved effective when embedded in GAs to solve location problems385.
The parameterPnew is applied to control the diversity of the population during the GAs process. In order to keep the population size constant,
(1P
new)
N
pop best individuals are selected from the former generation with “Plus update strategy”, together with the new generated individuals, to compose the next generation (see Fig.4-10).385 See Salhi/ Gamal (2003), p214.
No No
Yes
Yes Initialize RCL, n=0
n=n+1
Randomly select one element from RCL Update RCL, Ngrasp, Hgrasp, solution
N>Npop?
Restore the solution in solution pool Ngrasp is null?
Figure 4-10: Plus update strategy with the injection mechanism
4.2.3. Improvement 3: local search after GAs
As a population-based algorithm, SGAs is good at quickly locating the regions with high performance in vast and complex search space. Once those regions are located, it may not be able to explore the complex space sufficiently386. That is to say, it is relatively weak in exploiting the regions that have been found. It is believed that GAs can be more efficient (i.e. need less time) and more effective (i.e. find better solutions) for most com- binatorial optimization problems when embedded with LS techniques387. One possible reason given by Jaszkiewicz388 is that in many cases local optimum constitutes a relatively small part of the search space and thus can be achieved in an efficient way. LS provides the potential to cover the weakness of GAs in searching local areas after GAs efficiently find the vicinity of the optimal solution from a wide range.
GAs, which use certain kind of interaction with local searchers, are named Memetic Algorithms (MAs)389. The name MAs is inspired by Richard Dawkin’s concept of a meme390, which in GAs refers to the strategies (e.g.
386 This may be one of the reasons why the performance of SGAs deteriorates significantly as the size of the problem increases. See Lim et al (2000),
p.249.
387 See e.g. Merz /Freisleben (1997), pp.159-164; Galinier /Hao (1999), pp.379-397; Jaszkiewicz (2002), pp.50-71; Goldberg/ Voessner (1999), pp.220-
228; Galinier/ Hao (1999), pp.379–397.
388 See Jaszkiewicz (2002), pp. 50–71.
389 It was so named firstly by Moscato (1989), p.2003; 390 See Dawkins (2006), p.189.
Incumbent generation
Without selection Individuals with the best fitness Compare Add AddAll individuals in incumbent generation
Offspring from crossover operation
Offspring from mutation operation
Fitness of all individuals