Evolutionary Techniques in Multi-Objective Optimization Problems in Non-Standardized Production Processes

To schedule production in a Job-Shop environment means to allocate adequately the available resources. It requires to rely on efficient optimization procedures. In fact, the JobShop Scheduling Problem (JSSP) is a NP-Hard problem (Ullman, 1975), so ad-hoc algorithms have to be applied to its solution (Frutos et al., 2010). This is similar to other combinatorial programming problems (Olivera et al., 2006), (Cortés et al., 2004). Most instances of the Job-Shop Scheduling Problem involve the simultaneous optimization of two usually conflicting goals. This one, like most multi-objective problems, tends to have many solutions. The Pareto frontier reached by an optimization procedure has to contain a uniformly distributed number of solutions close to the ones in the true Pareto frontier. This feature facilitates the task of the expert who interprets the solutions (Kacem et al., 2002). In this paper we present a Genetic Algorithm linked to a Simulated Annealing procedure able to schedule the production in a Job-Shop manufacturing system (Cortés et al., 2004), (Tsai & Lin, 2003), (Wu et al., 2004), (Chao-Hsien & Han-Chiang, 2009).


Multi-objective optimization: Basic concepts
Our goal in this section is to characterize the general framework in which we will state the Job-Shop problem.We assume, without loss of generality, that there are several goals (objectives) to be minimized.Then, we seek to find a vector ** * 1 [ ,..., ] of decision variables, satisfying q inequalities ( ) 0, 1,..., as well as p equations ( ) 0, 1,...,  , a vector of k functions, each one corresponding to an objective, defined over the decision variables, attains its minimum.The class of the decision vectors satisfying the q inequalities and the p equations is denoted by Ω and each x ∈Ω  is a feasible alternative.A * x ∈Ω  is Pareto optimal if for any x ∈Ω  and every 1,..., ik = , * () That is, if there is no x  that improves some objectives without worsening the others.To simplify the notation, we say that a vector

Flexible job-shop scheduling problem
The JSSP can be described as that of organizing the execution of n jobs on m machines.We assume a finite number of tasks, 1 {} n jj J = .These tasks must be processed by a finite number of machines After allocating the operations, we obtain a finite class E of groupings of the i jk Os on the same machine.We denote each of these groupings as k E , for 1,..., km = .A key issue here is the scheduling of activities, i.e. the determination of the starting time i jk t of each i jk O .The Flexible JSSP demands a procedure to handle its two sub-problems: the allocation of the i jk Os on the different k M s and their sequencing, guided by the goals to reach.That is, to find optimal levels of Makespan (Processing Time) (see Eq. 1) and Total Operation Costs (see Eq. 2).

Hybrid genetic algorithm
Due to its many advantages, evolutionary algorithms have become very popular for solving multi-objective optimization problems (Ztzler et al., 2001), (Coello Coello et al., 2002).Among the evolutionary algorithms used, some of the most interesting are Genetic Algorithms (GA) (Goldberg, 1989).To represent the individuals, we use a variant of (Wu et al., 2004) We denote with values between 0 and (n! -1) the sequence of j J in each k M .That is, for n = 3, we may have 0→J 1 J 2 J 3 , 1→J 1 J 3 J 2 , 2→J 2 J 1 J 3 , 3→J 2 J 3 J 1 , 4→J 3 J 1 J 2 and 5→J 3 J 2 J 1 (see Table 2).
The algorithm NSGAII (Non-Dominated Sorting Genetic Algorithm II) (Deb et al., 2002), creates an initial population, be it random or otherwise.NSGAII uses an elitist strategy joint with an explicit diversity mechanism.Each individual candidate solution i is assumed to have an associated rank of non-dominance i r and a distance i d which indicates the radius of the area in the search space around i not occupied by another solution (see Eq. 3).A solution i is preferred over j if i j rr < .When i and j have the same rank, i is preferred if i j dd > .Let i Y be an ordered class of individuals with same rank as i and 1 i j f + the value for objective j for the individual after i, while 1 i j f − is the value for the individual before i. max j f is the maximal value for j among i Y while min j f is the minimal value among i Y .The distances consider all the objective functions and attach an infinite value to the extreme solutions in i Y .Since these yield the best values for one of the objective functions on the frontier, the resulting distance is the sum of the distances for the N objective functions.
MF01 / Problem 3 × 4 with 8 operations (flexible) Starting with a population t P a new population of descendants t Q obtains.These two populations mix to yield a new one, t R of size 2N (N is the original size of t P ).The individuals in t R are ranked with respect the frontier and a new population 1 t P + obtains applying a tournament selection to t R .After experimenting with several genetic operators we have chosen the uniform crossover for the crossover and two-swap for mutation (Fonseca & Fleming, 1995).After the individuals have been affected by these operators and before allowing them to become part of a new population we apply an improvement operator (Frutos & Tohmé, 2009).This operator has been designed following the guidelines of Simulated Annealing (Dowsland, 1993).This complements the genetic procedure.For the change of structure of both chromosomes we select a gene at random and change its value.This is repeated up from a cooling coefficient (α) while ω is a control parameter ensuring sufficient permutations, particularly when the temperature is high.Summarizing all this, the relevant parameters for this phase of the procedure are the initial temperature (T i ), the final one (T f ), the cooling parameter (α) and the control parameter (ω).The general layout of the whole procedure is depicted in Fig. 1.

Practical experiences
The parameters and characteristics of the computing equipment used during these experiments were as follows: size of the population: 200, number of generations: 500, type of crossover: uniform, probability of crossover: 0.90, type of mutation: two-swap, probability of mutation: 0.01, type of local search: simulated annealing (T i : 850, T f : 0.01, α: 0.95, ω: 10), probability of local search: 0.01, CPU: 3.00 GHZ and RAM: 1.00 GB.We worked with the PISA tool (A Platform and Programming Language Independent Interface for Search Algorithms) (Bleuler et al., 2003).The results obtained by means of HGA were compared to those yield by Greedy Randomized Adaptive Search Procedures (GRASP) (Binato et al., 2001), Taboo Search (TS) (Armentano & Scrich, 2000) and Ant Colony Optimization (ACO) (Heinonen & Pettersson, 2007).For the problems MF01, MF02, MF03, MF04 and MF05 (Frutos et al., 2010), we show the results for the multi-objective analysis based on Makespan (f 1 , (1)) and Total Operation Costs (f 2 , (2)).They were obtained by running each algorithm 10 times.
. From each superpopulation a class of undominated solutions was extracted, constituting the Pareto frontier for each algorithm.To obtain an approximation to the true Pareto front (Approximate Pareto Frontier), we take the fronts of each algorithm, from which all the dominated solutions are eliminated.These are detailed in Table 3 (MF01), Table 4 (MF02), Table 5 (MF03), Table 6 (MF04) and Table 7 (

Conclusions
We presented a Hybrid Genetic Algorithm (HGA) intended to solve the Flexible Job-Shop Scheduling Problem (Flexible JSSP).The application of HGA required the calibration of parameters, in order to yield valid values for the problem at hand, which constitute also a reference for similar problems.We have shown that this HGA yields more solutions in the Approximate Pareto Frontier than other algorithms.As said above, PISA has been used here as a guide for the implementation of our HGA.Nevertheless, PISA itself has features that we tried to overcome, making the understanding and extension of its outcomes a little bit hard.JMetal (Meta-heuristic Algorithms in Java) (Durillo et al., 2006) is already an alternative to PISA implemented on JAVA.We are currently experimenting with other techniques of local search in order to achieve a more aggressive exploration.We are also interested in evaluating the performance of the procedure over other kinds of problems to see whether it saves resources without sacrificing precision in convergence.
of the Pareto frontier is the main goal of Multi-Objective Optimization.
T corresponds to the actual temperature determined www.intechopen.comEvolutionary Techniques in Multi-Objective Optimization Problems in Non-Standardized Production Processes 113

Table 2 .
Chromosome encoding process

Table 8 (
MF01), Table 9 (MF02), Table 10 (MF03), Table 11 (MF04) and Table 12 (MF05)).The outcomes are summarized in Table 1.None of the results for MF01, MF02, MF03, MF04 and MF05 is statistically significant at an overall significance level α=0.05.This indicates that no algorithm generate approximation sets that are significantly better.Next, we considered unary quality indicators using normalized approximation sets.Then, we applied the unary indicators (unary hypervolume indicator I H , unary epsilon indicatior I e 1 and R indicator I R2 1 ) on the normalized approximation sets as well as on the reference set generated by PISA (I H , I e 1 and I R2 1 , Table 8 (MF01), Table 9 (MF02), Table 10 (MF03), Table 11 (MF04) and Table 12 (MF05)).Again, no significant differences were found at the 0.05 level.