13200 13300 13400 13500 13600 13700 13800
0 10 20 30 40 50 60
Objective function value
Time [min]
GA TS+DIV TS
Figure 2.5: Evolution of the best objective function value over time, for each algorithm
2.6 Steering strategies
An extension is now proposed, which consists in testing different steering strate-gies when, instead of havingT seconds to tackle each instance, one has to tackle the full set of instances before timeT is reached.
Having a time T to tackle a full set U = {u1, u2, . . . , uv} of truck instances points out the question of knowing when to stop the search on a certain in-stance to tackle the next one. To answer this question, different strategies are first described and then tested with exposed results. In the following, let T′ represents the current time.
The first strategy, calledfair strategy(F S), works as follows: a time Tv is given to LAG to tackle each instance, as this method is very fast and usually gets satisfying results. A second strategy, calledfair strategy with genetic (F SG), is an extension ofF S. It starts with a procedure calledLAGSORT, which consists in giving a time 13·Tv per instance toLAG(i.e., a total time of T3 is provided to test all the instances). Each time an instance ui is solved (i.e., the loading fits
in the truck), it is removed fromU and moved to a setP of solved instances.
After this procedure, during a time of T−|U|13·Tv per instance,GAis used to tackle the remaining instances (i.e., the ones that have not been previously solved by LAG), and using as initial solutions the best solution provided byLAGSORT. Renault pointed out an important goal of this problem which is trying to solve the largest number of trucks, regardless of the solution quality (i.e., as soon as f ≤13,400mm, the instance is considered as solved and keeping minimizing f is irrelevant. This stopping criterion seems the most straightforward, but for benchmarking purposes, we decided to not include it in both F S and F SG).
Thus, a third steering algorithm is then proposed, calledfit in strategy (F IS).
First, as forF S,LAGSORT is performed. After this first phase, the algorithm tackles the next instanceui∈U usingGAfor a maximum time ofTui= T−T|U|′. GA moves to the next instance ui+1 of U if the time Tui is reached or when the loading fits in the truck. Each time an instance ui is solved or the time Tui dedicated to it is reached, ui is removed fromU and moved toP. AsU is dynamically updated, so is the time dedicated to an instance. If instance ui is given a time limitTuiand is solved beforeTuiis reached, then the next instance ui+1 of U has a time Tui+1 > Tui, and Tui+1 = Tui otherwise. An extended version of this strategy, calledextended fit it strategy(EF IS), performs a sorting of the instances according to the number n of items in an increasing fashion, afterLAGSORT has been performed (thus, only the instances ofU are sorted).
Therefore,GAhas then more chance to be able to solve the first instances ofU and spare some interesting time for the next ones, which have largernvalues.
The next strategy, calledsort and perform (SP), first usesLAGSORT (but for whichLAGis stopped if the loading fits in the truck). Then for a maximum time Tui= 12·T|U|−T′ for each instanceui∈U,T S(without diversification) is used to tackle as many instances as possible (again,T Scan move to the next instance if it is able to solve the instance before the time limit). In addition, the instances that were not solved but which have a gap of more thanσ% (parameter tuned to 2, withσtested in interval [1, 30]) between fi (best encountered value of f for instanceui) andLt(length of the truck) are removed from U and added to P, as they are likely to be unsolvable. The remaining instances in U are then sorted (in a negligible computing time) in an increasing order of the remainder fi−Lt, and then theT S (without diversification) is performed on eachui∈U for a time Tui = 12 · T−T|U|′. Again, T S can move directly to the next instance
2.6. STEERING STRATEGIES 73 ui+1∈U if it has been able to solve an instanceuibefore timeTuiis reached. In SP G,SP is used butGAis used instead ofT Sfor the two last phases. Finally, inSP T G,SP is used, where the second phase is performed byT Sand the last phase by GA.
Please note that for every strategy that requires GA, N is set to 5 instead of 10, due to shorter time limits (when compared to the experimental conditions of Section 2.5).
Table 2.8: Results of the steering strategies
Strategy Number of solved instances
F S 336
F SG 338
F IS 341
EF IS 341
SP 340
SP G 338
SP T G 346
Results are provided in Table 2.8 for a time limit ofT = 1791 minutes (i.e., an average of 3 minutes for each of the 597 instances). For comparison purposes only, note that 328 instances are solved by LAG when one minute is allocated to each instance. F S solved 336 instances. Thus, eight instances are solved in the two additional minutes allocated to F S. When GA is used in F SG, two additional instances can be solved compared to F S. F IS shows the relevance of moving to the next instance when an instance is solved, which save some computing time for the remaining instances. Interestingly, EF IS does not provide any improvement over F IS. This is probably due to the fact that the prior sorting regarding the numbernof items in each instance is not relevant for the remaining instances. The last three steering strategies, namely SP, SP G and SP T G, show the relevance of the combination of two different methods.
Using together T S and GA shows the best results as it can solve up to 346 instances. Note that without steering techniques,LAGis given 10 minutes per instance to be able to solve 346 instances. Thus, LAGrequires a total time of 5970 minutes (597·10), which is almost 100 hours. In contrast,SP T Grequires 1135 minutes (597·1+(597−328)·2), which is almost 19 hours. Therefore,SP T G is more than five times faster compared toLAGwithout steering strategy. This shows the relevance of using advanced steering strategies when having a hard
deadline to tackle all the instances. Experiments show that on the full set of 597 instances, almost 250 instances are not solvable (i.e., f >13,400mm). In Section 2.5, 350 instances only are solved by GA and LAG with a time limit ofT = 60 minutes per instance (which is a total time of 597 hours). Thus 346 instances in 19 hours seems an interesting result and shows the relevance of the more refined steering strategies. Such a result is relevant from a practical standpoint: Renault has now insights to improve their approach when dealing with a complete set of instances with a hard time limit.