Extensive experiments are conducted to investigate the strengths and weaknesses of the algorithms. Thanks to the University of Turin for making my PhD studies possible there with the major scholarships.
INTRODUCTION
Li and \Vu [150] conSi(lere(l) class of Columnwise Pair-wise (CP) algorithms in the context of the construction of optimal oversaturated plans. We present mathematical models of packing problems, which will be considered in the thesis.
OVERVIEW OF HEURISTIC APPROACFIES
ILS approach
In practice, much of the potential complexity of ILS is hidden in the history dependency. Local search applied to the initial solution therefore gives the starting point s of the step in the set S.
MBH approach
The core idea of this approach is the measurement of the difficulty of the problems through the concept of funnel (see again Figure 2.1(b)). Such mechanisms include some device to enforce diversity among members of the population, to prevent all or most of them from converging to the same solutions.
MAXIMIN LHD
Proposed ILS heuristic for maxirnin LHD
Cyclic Order Exchange (COE): Our first disruption movement procedure is Cyclic Order Exchange (COE). Note that we need j - i > 2, because otherwise the perturbation would be a special case of the local displacement used in the local search procedure.
Experiments on Local Search procedure
On the other hand, the MS search space is a relatively small subset. First, we present experiments on the influence of three variants of COE perturbation displacement procedures.
Impact of Stopping Rule
From the figure we notice that the trends of the smoothed curves are almost linear for the k values considered. We define the AAO MaxNonlrnp value as the MaxNonlmp value at which a result is obtained within a given small percentage of the best (obviously achieved for MaxNonlmp=10000). As expected, the computational cost of the algorithm with MaxNonlrnp=MNI-EF is slightly higher in relation to that with MNI-1000 (the cost for the former is at most twice the latter for k = 10), while the computational .
Considering that the gain in quality with MNI-10000 with respect to MNI-EF is not very significant, we can claim from these experiments that the MaxNonlmp value produced by formula (4.2) is quite a good compromise between the quality of the result and computational cost.
Comparison of ILS with the existing literature
Regarding the number of starts for each LHD, we follow the settings given in Table 4.12. Although the quality of the results obtained with ILS appears to be quite good, it is important to determine whether such results were obtained only by brute force, ie. at the cost of a very large computational effort. Since computational cost is one of the important issues for heuristic approaches, we report the computational cost of the above experiments in Table 4.14.
And we have already compared the PC perturbation based ILS approach with SA approach in Table 4.13.
Experiments about the complexity analysis
We observe that the number of critical points is most often 1 or 2, and only occasionally is greater than 6. Then we would like to find out the effect of N as well as k on the number of perturbations under each. From the experiments (see Figure 4.25) we note that there is a significant impact of N on the number of perturbations invoked for all k considered.
It appears that the number of called perturbations is somewhat logarithmic with respect to N (see (tear curve in Figure 4.25).
COMPUTATIONAL EXPERIMENTS AND DISCUSSION ABOUT ILS WITFI OPT()
Impact of the neighborhood structure and the optirnality criterion
MJO,L, I neighborhood structure with Opt(Do , q5) optional criterion and (c) neighborhood structure with Opt(Di ,) optimal criterion. the additional light cost of keeping track of the best values observed during a run). Moreover, it is guaranteed that the results with Opt(Do , ) are always at least as good as those with Opt(). In Figure 5.1 we notice that for small N values the quality of the average Mm values of the two approaches is basically the same, while the quality with Opt(Di , gets slightly better) as N increases.
The significantly longer times with Opt(Di,) and the local shift of £MRD1 relative to those with Opt(Di, J1) were somehow not expected.
Further experiments
However, in what follows we will not look for the best compromise between the quality of the results and the calculation time, but we will only perform experiments with MaxNonlmp= 100, which will turn out to be already quite significant. Searches performed by ILS() are definitely more expensive, but, according to the results, such higher costs are well paid in terms of the quality of the results. For k > 6, except for a small number of low N values, all solutions returned by ILS() are improvements of the best known results.
Such higher costs are clearly rewarded in terms of the quality of the results, but we may wonder about the quality of the results if we impose time constraints on ILS.
Experiments about the complexity analysis of ILS()
But the number of swap moves to be attempted at each iteration is now different. To derive the overall time complexity of ILS(q), we still need to derive a forniula for the number of perturbations (ie, the number of local searches) performed during each ILS run. From the experiments (see figure 5.15) we notice that there is a marked influence of N on the number of disturbances.
From the experiments, we note that despite a peak at k = 6, there is no significant influence of k on the number of perturbations invoked during a run (see the bar chart in Figure 5.16).
PACKING PROBLEMS: DEFINITIONS AND MATHEMATICAL MODELS
The above model can be niodified in such a way that we can get rid of the square roots. For example, if the optimal solution has free circles (see bound 3), then we can move them around and thus obtain an infinite set of solutions, all with the same optimal radius of the container. Problem E-1: Find the value of the maximum circle radius such that n identical non-overlapping circles can be placed in a unit circular container.
Problem E-3 : Instead of fixing the radius of the circular container and searching for the maximum radius of the circles in the packing, one can equivalently search for the minimum ratio of the radius of the container to the radius of the circles in the packaging without fixing them.
BASIN FlOPPING ALGORITHMS FOR TFIE ICPCC PROBLEM
MBII approach for ICPCC problem
During this search, members of the population cooperate with each other to guarantee. If we denote by S the space of the solutions that we are interested in (in ICPCC basically the local minimizers), the inequality measure can be defined as the following function. Basically, at each iteration: a set Y of new candidates is generated through the application of the perturbation move to each member of the population.
The local search procedure and perturbation techniques of the PBH approach are the same as those for the MBH approach.
COMPUTATIONAL EXPERIMENTS AND DISCUSSION ABOUT ICPCC PACKING PROBLEMS
- Number of local minimizers
- Choice of the stopping parameter MaxNonImp)
- The impact of the parameter A with different perturbation moves
- Comparison among different perturbation strategies
In Column Nrsuccess, we report for each MaxNonlmp value the number of times the best solution reported in [274] was reached (or improved). As a further test, we decided to increase the number of MBH runs from 5 to 50 for the 9 cases where a failure occurred with MaxNonlmp= 100. For a fair comparison, we allowed a number of local searches in MS (a) slightly larger than the largest number of local searches required by MBH with the three perturbation moves (we will denote these Multistart runs by MS(L) in what follows) (b) equal to twice the same number of local searches (we will Multistart runs denoted by MS(D) in what follows).
Although the number of local searches in MS(L) and MBH(FJ) is almost equivalent, the execution time of MS(L) fluctuates from twice to eight times that of MBH(FJ) for each n.
PACKING PROBLEMS: NON-IDENTICAL CIRCLES IN A SMALLEST CIRCULAR CONTAINER
New perturbation moves
As already pointed out, when dealing with unequal circles we can add new perturbation moves to the minor variant of the perturbation moves used for even circles. In particular, here we propose two further perturbation motions, namely (1) the random jump (RJ) perturbation motion and (ii) the radius-based random swap (RBRS) perturbation motion. In Section 8.4, we developed the Jerk Perturbation (JP) movement technique in which circles' centers are perturbed within a neighboring space.
In the example, the RJ perturbation randomly selects a single circle (the red one in the figure).
Experiments and discussion
As for SIB-MBH, in Table 9.2 we report for each test instance the total number of rounds for the instance and the reduced number of rounds after removing the "small" rounds. The last three columns report the number of successes per case for each of the three perturbation shifts tested (FJ, RJ, RBRS) with SIB-MBH, i.e. we first consider the operation of various disturbance movements within the standard MBH approach, i.e.
Things definitely get better when we consider the performance of the various perturbations of the SIB-MBH approach.
CONCLUSION AND FUTURE RESEARCH
Eglais, New Approach to Scheduling Out of Experiments, Dynamics and Force Problems, Vol. in Russian). Wynn, Experimental design and observation for large systems, Journal of the Royal Statistical Society: Series B, vol. Baxter J., Avoiding local optima in warehouse location, Journal of the Operational Research Society, vol.
Ylvisaker, Bayesian prediction of deterministic functions, with applications to the design and analysis of computational experiments, Journal of the American Statistical Association, vol.
APPENDIX
OVERALL IMPROVED VALUES
- Overall improved radii in ICPCC
- Overall improved radii in NICPCC
Because n our proposed algorithms are able to reach all the best-known values in the literature (available in [274]) and to obtain 21 improved configurations, the radii of which are reported in Table A.1. For these cases, our algorithms were able to achieve all the best-known results and improve some of them, the radii of which are reported in Table A.2.
SOME IMPROVED SOLUTIONS
