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3.2 Optimization objective
One of the simplest methods to combine multiple objective functions into a scalar fitness solution is the weighted sum approach. However, the monetary and nonmonetary performance measures don’t have the same physical dimension. Thus, each should be first normalized, whose method is shown as follows: (10) (11) The monetary and nonmonetary performance measures are weighted. The objective function is to minimize where are constant weights for CO, respectively and decided by experts. min 1 max min , CO CO a CO CO min 2 max min . NO NO a NO NO 1 1 2 2 ,w a w a 12 ,wwNO
4 Genetic Algorithm
4.1 Chromosome representation
In this application, each chromosome pop ( ) (the th chromosome in a population of size ) is composed of and where is the ith OP to process and the value of shows which OPs are allocated to the mth station. For example, means that the first three OPs are allocated to station 1, and then means that from the 4th to the 6th OPs are allocated to station 2, 0 means no OP. The value of shows the configuration of station m. The examples of chromosome are shown in Figs. 3(a) and 3(b). Fig. 3. Examples of chromosome n n N (,1)(, 2 ) (, 3), popnpopnpopn1(, 1) , popnO2 ,, , I OO12 (, 2),,, , M popnCCC(, 3)popn 12 , ,, , M JJJi Om C1 3 C2 6 Cm J
4.2 Initial population
For subchromosome , initial genes can be generated randomly. However, and (, 3) popn(,1) popn(popFor , the method is based on an operation precedence matrix which is detailed in Ref. [13]. For , the following steps are adopted. Step 1: Set Step 2: If is less than the number of OPs and is less than the number of stations, do the next, else END; Step 3: From to the number of OPs, select randomly a number which is allocated to station n; Step 4: Set , and go to Step 2. (,1)popn(,2)popn1,1;bn b n b p 1,1bpnn
4.3 Selection operator
Many selection procedures are currently in use, one of the simplest is Holland’s original fitness-proportionate selection, where inpiduals are selected with a probability proportional to their relative fitness. The selection operators adopted the expected value model, namely, the expected number of an inpidual in the next generation is where is the constant-size of the population. 4.3.1 Crossover operator The standard crossover operation is applied to, namely, two-point cutting. However, the crossover operation of andneeds much more efforts. For , the crossover operation may violate the precedence relationships of operations. In the paper, the crossover operator precedence preservative crossover (PPX) used by Ref. [15] is adopted for, which perfectly respects the absolute order of genes in parental chromosomes. Firstly, a vector of length of is randomly filled with the elements of the set {1, 2}, which defines the genes are drawn from parent 1 or 2. After a gene is drawn from one parent, it is deleted in the other one. The following steps are repeated until the parent chromosomes are empty and the offspring contains all genes. For, the general single-point crossover algorithm will fail, which is shown in Fig. 4. Fig. 4. Failure of crossover The chromosomes of offspring 1 and 2 are incorrect. Thus, of offspring should be adjusted after single-point crossover and the steps are as follows if the single-point is l. Step 1: Set b=the value of position l of , nl; Step 2: If b is less than the number of OPs and n is less than the number of stations, do the next, else END; Step 3: From b to the number of OPs, select randomly a number p which is allocated to station n; Step 4: Set bp1, nn1, and go to Step 2; Step 5: If n is less than the number of stations, assign 0 to station after n. 4.3.2 Mutation operator Mutation is usually required to avoid being trapped into local optima. Forthis is done by randomly selecting a gene, called gene , and changing its value to a random number from all available configurations. But for and the same problem needs further considerations. For the precedence relationship may be violated by mutation operation. Hence, an added process is introduced to preserve the feasibility of the operation sequence. The process will be explained in the following, and an example is shown in Fig. 5. Fig. 5. Mutation example Step 1: Randomly generate the mutation position m; Step 2: According to the OP sequence from OP1 to OPm and the operation precedence matrix, identify the feasible OP set at the position m. Step 3: Randomly select an element from s as the OP at the position m after mutation. Step 4: Based on similar process as generating initial population, generate the following OPs from to end. For it is evident that random single-point mutation will cause errors. Thus, it needs more steps which are shown as follows. Step 1: Randomly generate mutation position n; Step 2: If equals to 1, set else set if equals to the number of stations, set number of stations, else set Step 3: Randomly select number p betweenandwhich is the mutation value for (),()iifMNfσσN(,3)popn(,1)popn(,2)popn (,1)popn(,1)popn(,1)popn (,2)popn(,2)popn(,2)popn(,3),popnm(,1)popn (,2),popn(,1),popns 1m (,2),popnn10;s1s 1(,2);npopnn2s21(,2);nspopn 1(,2)npopn 1(,2),npopn(,2).npopn