MODE Algorithm

This section outlines classic DE steps, namely, initialization, mutation, crossover and selection, followed by its adaptation for MOO. In the initialization step of DE, a population of NP individuals is randomly initialized inside the bounds on decision variables.

xj = xjL + rand(0,1) (xjU – xjL) j = 1, 2, …, n (6.5) Here, rand (0, 1) is a random number from uniform distribution between 0 and 1, and xjL and xjU are the lower and upper bounds on j^th decision variable. Equation 6.5 has to be applied on each decision variable, for producing a single individual (i.e., NP×n times for producing the initial population).

There are several strategies for performing mutation operation (Price et al., 2005), and selection of a strategy depends on the type of

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optimization problem. The general convention used in the below strategies (Equations 6.6 to 6.10) is DE/p/q. Here, p stands for perturbation vector (first term on the right hand side) and q is the number of difference vectors (remaining terms on the right hand side). xr0, xr1, xr2

and xr3 are randomly selected individuals from the parent population, and xbest and xi are respectively the best and target individuals from the parent population.

DE/rand/1 vi = xr0 + F(xr1 – xr2) (6.6) DE/best/1 vi = xbest + F(xr1 – xr2) (6.7) DE/rand/2 vi = xr0 + F(xr1 – xr2) + F(xr3 – xr4) (6.8) DE/best/2 vi = xbest + F(xr1 – xr2) + F(xr3 – xr4) (6.9) DE/rand-to-best/1 vi = xi + F(xbest – xi) + F(xr3 – xr4) (6.10) Of these, the first one is generally good and used in classic DE.

There are two commonly used crossover operations in DE:

exponential and binomial. In the exponential crossover, one randomly selected element of mutant individual is copied into the trial individual, and it ensures that trial individual will be different from the target individual. After that, a random number is generated between 0 and 1 using uniform distribution. If this random number is lower than Cr, then one randomly selected element of mutant individual is copied into trial individual. This process is repeated until the generated random number is greater than Cr; then, current and remaining elements of the target individual are copied into the trial individual. In the binomial crossover, elements of mutant (v) and target (x) individuals compete with each other, with crossover probability, Cr to generate trial individual (u). The following equation ensures that at least one element of x is copied to u.

u_, = x_, if rand(0,1) ≤ Cr or j = j

v_, otherwise (6.11) After crossover, the trial individual is checked for decision variable violation. If a trial individual violates lower or upper bound for any decision variable, then that particular decision variable is randomly re-initialized within its bounds on that decision variable. Finally, DE performs selection between the trial and target individuals based on the objective function value. This selection is performed as follows; here, x^G+1 is the selected individual for the next generation.

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= if f ≤ f

otherwise (6.12) MODE employed in R program is based on the algorithm in Sharma and Rangaiah (2013b). A flowchart of this program is shown in Fig. 6.1.

An initial population of NP individuals is randomly generated inside the bounds on decision variables (using Equation 6.5), and values of objectives and constraints are calculated for all individuals. In the generation loop, a mutant individual is created for each target individual, by mutation using individuals from the initial/current/parent population and DE/rand/1 strategy in Equation 6.6. The target and mutant individuals produce a trial individual by binomial crossover (Equation 6.11). In many process optimization problems, evaluation of objectives and constraints for each trial individual requires simulation of the entire process, and so it is often computationally expensive. Use of tabu list (TL) avoids the revisit of search space by keeping a record of recently visited points (Srinivas and Rangaiah, 2007). In MODE-R program, if the trial individual is near to any individual in the TL by a specified distance (in normalized decision variable space), then it is rejected without calculating objectives and constraints. Accepted trial individual is stored in the child population, and also added to the TL. Similarly, NP trial individuals (child population) are generated, and child population is evaluated for values of objectives and constraints.

The child population is mixed with the parent population. The combined population undergoes non-dominated sorting and crowding distance calculations to select the population for the next generation. If the MOO problem has no constraints, then NP individuals are selected from the combined population (maximum 2NP individuals depending upon the rejected individuals by tabu check), as follows.

a. One individual is dominating another individual, if it is better than the other in all objectives. Two individuals, i1 and i2 are non-dominated to each other if i1 is better than i2 in at least one objective, and also i2 is better than i1 in at least one other objective.

b. The number of individuals dominating each individual (nd) is calculated for each and every individual in the combined population. First rank (best Pareto front) is assigned to the non-dominated individuals with nd = 0.

c. Individuals with first rank are excluded from the combined population. Then, non-dominated individuals from the remaining

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population are found and assigned second rank. This procedure is repeated until all individuals are ranked.

d. The first/best NP individuals are selected as the population for the next generation. For this, individuals are first selected based on the Pareto rank given in steps (b) and (c). If all the individuals of a Pareto front cannot be selected for the subsequent generation, then least crowded individuals (based on the crowding distance measure) are selected to complete the population size. The crowding distance measures distribution of non-dominated solutions on the Pareto-optimal front by calculating distance using neighboring non-dominated solutions (Deb, 2001).

If the optimization problem has inequality constraints, MODE-R handles them by constrained dominance, also known as feasibility approach (Deb et al., 2002). Following definition of constrained dominance is applied between two individuals, i1 and i2 (in step a above).

If any of the following conditions is true, then individual i1 is dominating individual i2.

a. Both the individuals are feasible, and individual i1 is better than i2 in all objectives.

b. Individual i1 is feasible and i2 is infeasible.

c. Both the individuals are infeasible, and individual i1 has lesser total constraint violation compared to individual i2.

As shown in Fig. 6.1, the above search procedure is repeated for the specified maximum number of generations (MNG). Note that self-adaptation of DE parameters, described in Chapter 5 of this book, is also implemented in MODE-R program.

MODE-R takes two sets of input data from the user before performing the optimization. One set of inputs is MODE algorithm parameters, namely, crossover probability, mutation factor, tabu radius, tabu list size, population size and maximum number of generations. The second set of inputs consists of data specifically for HEN retrofitting.

These inputs include details about the existing HEN, namely, number of hot and cold streams present in the network, product of mass flow rate and heat capacity of each of the streams, either the overall heat transfer coefficient of the heat exchanger or the film heat transfer coefficients of each of the streams. These data are essential for HEN model calculations.

Cost details like the cost of additional area in the existing heat exchanger, capital cost of a new heat exchanger, re-piping/re-locating costs, and

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cost, supply and return temperatures of each of the utilities, are necessary for economic analysis of HEN retrofitting. If there are constraints on the approach temperature, then value of the minimum approach temperature (MAT) needs to be specified.

The output file of MODE-R is a text file, which stores values of

decision variables, objective functions and constraints at the user-specified intermediate generations and also at the final generation.

MODE-R program files with inputs and output file for case study 2 are available from the website of this book maintained by World Scientific.