Ajuste del cuestionario Servqual (versión preliminar)

Crossover (also termed reproduction) is a GA operation which creates a new solution (child) by combining two or more selected solutions (parents). A standard crossover operator combines two parents by various techniques (e.g. single point, double point, and uniform) to form two children (Ting 2005). However, in PEPSO, a customized crossover technique has used that results in just one child. In this technique, at first, a pump schedule as the main parent will be selected. The more desirable solutions have a higher chance to be selected as the parent. Then one or multiple time blocks of the parent pump schedule that are not desirable will be selected to be replaced with potentially better time blocks of other solutions. When a time block of the parent pump schedule is selected for replacement, a better time block needed to be found to replace it. Therefore, the same time blocks of all available solutions will be ranked and by using the roulette wheel method a time block will be selected for replacement. Time blocks with a higher rank have a greater chance of selection to replace the undesirable time block of the parent pump schedule. By this method, we can expect that promising solutions will be selected and their undesirable time blocks will be replaced with better time blocks to form a more acceptable solution.

The selection of parent solutions requires the ranking of all solutions according to the non-domination rank and crowding distance in each Pareto frontier (as explained in section 2.2.2.6). The rank of solutions is used to calculate the PI value of each solution. Solutions with the highest rank (on the edge of the first Pareto frontier) are the most important solutions, and solutions with the lowest rank (in a crowded section of the last Pareto frontier) are the least important solutions. PI values of solutions will eventually be used to calculate the cumulative PI vector that will be used for selecting parents using the

roulette wheel sampling method. However, before calculating the cumulative PI vector, PI of solutions should be adjusted based on 1) number of Negative Pressure Warnings (NPW) and 2) final tank level status of each solution.

For adjusting PI value of the solutions, at first, PI of the solutions that have NPW will be reduced by dividing it by a number that is calculated based on the number of NPW of the solution. A higher number of NPW increase the size of the denominator and reduces the importance value. By default, the function that calculates denominator value, adjusted in a way that if all solutions in a population except one has maximum number of NPW, probability of selecting the solution without NPW is 20% of probability of selecting one solution from the group of all other solutions with maximum possible number of NPW (see Equation 8). When there is no NPW associated with the solution, the minimum value of the denominator is 1 and when the solution has the maximum number of NPW the denominator value is equal to 1+(20% of the size of the population).

NPW Probability Reducer Denominator = 1 + (Population Size × 0.2 × (No. of NPW of solution /

Maximum No. of NPW)) Equation 8

After reducing PI of the solution based on the number of its NPWs, the status of final tank level of solution will be investigated. If final tank level is equal to or greater than the initial tank level, it is a desirable solution. However, if the final tank level is smaller than the initial solution, the amount of the tank level deficit will be calculated. Based on the calculated tank level deficit, PI of the solution will be reduced again. The formula and logic of calculating the tank level deficiency is similar to the calculation of the NPW Probability Reducer Denominator and appears in Equation 9.

Tank Level Deficiency Probability Reducer Denominator = 1 + (Population Size × 0.2 × (Tank

Accordingly, the effect of PI reduction of both NPW and tank level deficiency can be imposed on initial importance value of the solution by Equation 10.

Adjusted Importance of the solution = Initial Importance of Solution / (NPW Probability Reducer ×

Tank Level Deficiency Probability Reducer) Equation 10

Larger values of the adjusted PI indicate that the solution has a higher rank in the non-dominated sorted population and has fewer NPWs and a less significant tank level deficiency. It means that in comparison to all solutions in the population, a solution with higher adjusted PI value is closer to the optimum solution and is feasible and desirable from the operation perspective. It makes the solution a good candidate to be parent and generator of the next generation of better solutions. The adjusted PI of solutions can be used to create the cumulative PI vector that will be used for selecting parents by roulette wheel technique with replacement. This means that a solution can be selected as a parent multiple times.

The TTSUI will be used for selecting some candidate undesirable time blocks that need to be replaced to create a better child solution. This means that, at first, the TTSUI of all time steps of each solution is calculated. Then TTSUIs are used as PI values to create the cumulative PI vector. This new cumulative PI vector will be used in the roulette wheel method (without replacement) to select time blocks with high TTSUI that are good candidates for replacement. Sampling with the roulette wheel method without replacement prevents a time block of the selected pump schedule to be selected multiple times for the replacement.

Finally, after selecting the parent and selecting those time blocks that are undesirable, it is necessary to select better time step from other solutions to replace the undesirable time blocks. For doing this, we need to compare the same time block of all

solutions and rank solutions for each time step separately. PEPSO ranks time steps by using a combined factor that includes the rank of the solution in population and value of TTSUI of the same time step of each solution. The reciprocal of the rank of a solution will be added to the reciprocal of the TTSUI multiplied with a factor (by default, 15) to calculate PI of each solution for the time step (see Equation 11). By this method, the calculated PI includes the effect of both the desirability of the time step (reciprocal of TTSUI) and the rank of the solution. This means that to consider a time step of a pump schedule as a promising time step, we need to make sure that 1) the desirability of the time step is high, and 2) it comes from a high-rank solution (that means the time step can lead to a good solution when combined with other time steps of a pump schedule). It should be noted that the multiplier of 15 for the desirability part of formula puts the main emphasis on the desirability instead of the solution rank. PI of each time step of each solution will be added to the PI of the same time block of other solutions to create the cumulative PI vector of the time block. This cumulative PI vector will be used for selecting the solution that has the most promising time step by using the roulette wheel technique with replacement. The time step of the selected promising solution will replace the undesirable time step of a parent to form a better child.

PI of the Time Step of the Solution = (1 / Rank of the solution) + 15 × (1 / TTSUI) Equation 11

By using this customized crossover technique, each parent generates one child that is mainly created from the one parent but may have some time blocks from other solutions. The focus of this crossover technique is on improving the solution condition by changing some time blocks of the pump schedule (columns), and it will not affect a row (the whole operation plan of a single pump) or a cell of pump schedule individually.

The number of parents involved in each iteration and number of time steps that need to be replaced can be defined by users. The first value that users define is the crossover percentage, which defines the percentage of the solution in the population which should be selected as parents. The second crossover parameter that users define is the crossover rate, which represents the percentage of the number of time steps of a solution that should be replaced with the similar time steps of other solutions. Both of these parameters can vary between 0 and 100%. By default, they are both set to 50%. This means that during each optimization iteration, by default, 50% of solutions will be selected as parents and 50% of time blocks of each selected parent will be replaced with promising time block of other solutions. It should be noted that if users want to input this numbers via the user interface they can use percentage values. However, inside the PEPSO, these percentages will be changed to a number between 0 to 1, and if users want to change them by editing the project file manually, they should convert percentages to a number between 0 to 1.