Computational Intelligence Based Methodologies for Modeling and Optimization

1. Computational Intelligence Based Methodologies for Modeling and Optimization Somnath Nandi Asst. Professor Dept. of Petroleum and Petrochemical Engineering MIT - Pune

2. Contents • Modeling • Artificial Neural Networks • Optimization • Genetic Algorithms • Differential Evolution • Case Study • Conclusion

3. Modeling • Engineers and scientists required to analyze the complex processes and develop mathematical models which simulate their steady-state and / or dynamic behavior. • The objective is to construct, from theoretical and empirical knowledge of the process, a mathematical description. • A mathematical model provides information on the process behavior, over important ranges of operating variables, in terms of equations, which reflects at least the major features of the underlying mechanisms.

4. Modeling (contd… ) • Phenomenological Approach: - Process behavior described in terms of the appropriate mass, momentum and energy balance equations together with the pertinent chemical engineering principles. - Mathematical formulation describing the physico-chemical phenomena underlying in the process is formulated followed by model fitting. - Regression techniques based on the least squares minimization

5. Modeling (contd… ) • Advantages : - It provides a valuable insight into the process behavior - It possesses extrapolation ability • Disadvantages: - Owing to the complex nature of many processes, the underlying physico-chemical phenomenon is seldom fully understood - Collection of the requisite phenomenological information is costly, time-consuming and tedious - Nonlinear behavior common for many processes leads to complex nonlinear models, which in most cases are not amenable to analytical solutions; thus, computationally intensive numerical methods must be utilized for obtaining solutions

6. Modeling (contd… ) • Empirical Approach: - Process behavior is modeled using appropriately chosen empirical equations, for instance, polynomial expressions. - Model can be constructed solely from the process input-output data without explicitly invoking the process phenomenology. - An appropriate functional form that possibly fits the process data is selected in advance following which the unknown model parameters are estimated using a suitable function fitting procedure.

7. Modeling (cond… ) • Artificial Intelligence based Approach: - AI is science and engineering of making “intelligent” systems, especially intelligent computer programs - Related to task of using computers to understand the “human intelligence” - Intelligence can be broadly defined as computational part of our ability to efficiently achieve goals in the world.

8. AI based Modeling Approaches • Artificial Neural Networks (ANN) • Support Vector Regression (SVR) • Genetic Programming (GP) • Fuzzy Logic (FL)

9. Artificial Neural Networks • Efforts to develop computer models of the information processing of human nervous system (Rumelhart et. al., 1986). • Simplified mathematical models describing the biological nervous system and functioning. • A highly interconnected system of simple processing elements can learn complex interrelationships between the independent and the dependent variables in a data set.

10. Artificial Neural Networks

11. Artificial Neural Networks x1 y1 x2 Output Input yK xN

12. Artificial Neural Networks • The distinct advantages of the ANN formalism are: • Can be developed solely from process input-output data. • MIMO relationships can be approximated • Possesses good generalization ability • Can tolerate noisy data or incomplete information • Can be developed even using qualitative data. • Use a generic nonlinear function for function approximation and thus there is no need to specify system-specific data fitting function as done in traditional regression.

13. Artificial Neural Networks • Principal applications of ANNs are: (i) nonlinear function approximation (i.e., process modeling), (ii) pattern recognition and classification, (iii) data reduction and compression, (iv) signal processing, (v) noise reduction.

14. What is Optimization ? • Optimization is use of specific methods to determine the most cost-effective and efficient solution to a problem or design for a process • A wide variety of problems in the design, construction, operation, and analysis of industrial processes can be resolved by optimization • The field of statistics treats various principles termed "maximum likelihood," "minimum loss," and "least squares," and business makes use of "maximum profit," "minimum cost," "maximum use of resources," "minimum effort," in its efforts to increase profits

15. What is Optimization ? • A typical engineering problem can be posed as follows: aprocess can be represented by some equations or perhaps solely by experimental data. You have a single performance criterion in mind such as minimum cost • The goal of optimization is to find the values of the variables in the process that yield the best value of the performance criterion • A trade-off usually exists between capital and operating costs. The described factors-process or model and the performance criterion-constitute the optimization "problem."

16. What is Optimization ? • Optimization is minimization or maximization of an objective function (also called a performance index or goal function) that may be subject to certain constraints min f (x) Goal function subject to, g (x) = 0 Equality constraints h (x) < 0 Inequality constraints

17. Need for Optimization • Typical problems in engineering process design or plant operation have many (possibly an infinite number) solutions • Optimization is concerned with selecting the best among the entire set by efficient quantitative methods • Computers and associated software make the necessary computations feasible and cost effective • To obtain useful information using computers, however, requires (1) critical analysis of the process or design, (2) insight about what appropriate performance objectives are (what is to be accomplished), (3)use of past experience, sometimes called engineering judgment.

18. Applications of Optimization • Determining the best sites for plant location • Routing tankers for the distribution of crude and refined products • Sizing and layout of a pipeline • Designing equipment and an entire plant • Scheduling maintenance and equipment replacement • Operating equipment, such as tubular reactors, columns, and absorbers • Evaluating plant data to construct a model of a process • Minimizing inventory charges • Allocating resources or services among several processes • Planning and scheduling construction

19. 1-Dimensional Search

20. 2-Dimensional Search

21. 2-Dimensional Search

22. 2-Dimensional Search

23. Unimodal Optimization

24. Multi-modal Optimization A function exhibiting different types of stationary points. a-inflection point (scalar equivalent to a saddle point); b-global maximum; c-local minimum; d-local maximum

25. Global Methods of Optimization

26. Performance of Classical Techniques

27. Multiobjective Optimization • A MOO problem will have two or more objectives involving many decision variables and constraints • Consider an MOO problem with two objectives: f1(x) and f2(x), and several decision variables (x) Minimize f1(x) (1) Minimize f2(x) (2) With respect to x Subject to xL ≤ x ≤ xU (3) h (x) = 0 (4) g (x) ≤ 0 (5)

28. Multiobjective Optimization

29. Different Evolutionary Techniques • Genetic Algorithms (GA) • Simulated Annealing (SA) • Ant Colony Optimization (ACO) • Tabu Search (TS) • Particle Swarm Optimization (PSO) • Differential Evolution (DE) • Memetic Algorithm (MA) • Simultaneous Perturbation Stochastic Approximation (SPSA)

30. What is GA ? • GAs are computer based search and optimization algorithms based on mechanics of natural genetics and natural selection • A population of initial solution is generated within feasible region • The main idea is - Survival of the fittest - Evolution of species with time • Only best solution will survive till end

31. What is GA ? • Genetic Algorithms (GAs) were invented by John Holland and developed by him and his students and colleagues. This lead to Holland's book "Adaptation in Natural and Artificial Systems" published in 1975. • All living organisms consist of cells. In each cell there is the same set of chromosomes • A chromosome consists of genes, blocks of DNA. Each gene encodes a particular protein • Complete set of genetic material (all chromosomes) is called genome. • Particular set of genes in genome is called genotype.

32. Working Principle Let us consider the maximization problem: • Coding: - Variable xi are first coded into binary strings - Length of string is determined based on desired accuracy of solution

33. Working Principle • Fitness function - GA are based on survival-of-the-fittest - Naturally suitable for solving maximization problems - Minimization are transformed to suitable maximization ones - Fitness function is a measure of goodness of the string - Our target is to keep on increasing the overall fitness functions of all the strings - Genetic operators perform duty to manipulate binary strings so that fitness function is keep on increasing on successive iterations

34. Working Principle • GA Operators - Reproduction / Selection • Selects good strings of a population • Forms a mating pool • Above – average stings are picked from current population • Multiple copies of selected strings are placed in mating pool in a probabilistic manner • No new strings are formed in this phase • Roulette – Wheel or Stochastic Remainder Selection methods

35. Working Principle • Crossover - New strings are created - It exchanges information among strings of mating pool - 2 strings are picked at random - Point of crossover is probabilistically chosen Children Strings Parent Strings

36. Working Principle • Mutation - It changes 1 to 0 and vice versa - Small probability pm generally < 0.1 - Need is to create a point in the neighborhood of the current point - Performs local search around current solution - It maintains diversity of population Mutation

37. Diversification • Generate initial population covering entire range • Visit new places • Extract characteristics of each region • Cover as much as possible • Performing Global Search • All are done by Crossover operator

38. Intensification • Should be started once search space is well scanned • Visit zones adjacent / nearby to already visited • Check the performance • Perform local search • This is done by Mutation operator

39. Algorithm • Step 1: Do coding, choose selection operator, crossover and mutation probability (pc and pm). Choose population size (n), string length (l), max. no. of iterations (Nmax) • Step 2: Evaluate each string of population • Step 3: Perform Reproduction on population • Step 4: Perform crossover on random pairs of strings • Step 5: Perform mutation on each string • Step 6: Evaluate strings of new population • Step 7: Set N = N + 1 and go to step 3 • Terminate if N > Nmax or no further improvement on string performance

40. Advanced GA • Multi Point Crossover : • Real Coded GA : - Real variables are directly used - Optimal point of any desired accuracy obtained • Non – dominated Sorting : - To keep versatility of population - Give more chance to a poor performer to enhance its skills • Pareto GA : - Population in a GA simulation is adaptively divided into separate subpopulation, corresponding to each optimum point by use of sharing functions - Can get all the solutions of Pareto Optimal front in one shot

41. GA - Applications • Reactor Design – Ammonia Synthesis • Process Optimization – Cumene Synthesis - Phenol Production • Scheduling – Refinery Operations • Multiphase – Trickle Bed Reactor • Polymerization Processes – MMA Synthesis - Polyethylene Plant - Nylon Manufacture • Water Distribution

42. Differential Evolution • Introduced by Storn and Price in 1996 • Algorithm works with a population of size N • Algorithm iterates as follows: - Generate new vector by adding weighted difference of two vectors to third - Mix new vector with target vector to yield trial vector - Replace target vector with trial vector if latter is strictly superior

43. Differential Evolution

45. Differential Evolution • F and CR are DE control parameters • F is a real-valued factor in the range (0.0,1.0+] • Upper limit on F has been empirically determined. • CR is a real-valued crossover factor in range [0.0,1.0] • CR controls the probability that a trial vector parameter will come from the randomly chosen noise vector

46. Importance of Parameters • Optimal values are dependent both on objective function characteristics and on the population size, NP • Practical advice on how to select control parameters NP, F and CR can be found in the literature

47. Crossover in DE

48. DE - Applications • Multiprocessor synthesis • Power minimisation • Neural network learning. • Crystallographic characterization • Design of Shell-and-Tube Heat Exchangers • Heat transfer parameter estimation in a trickle bed reactor • Gas Transmission Network • Water Pumping and Distribution Systems • Optimization of Ammonia Synthesis Reactor • Design and Operation of Thermal Cracker

49. DE - Advantages • Powerful algorithm- multidimensional functions • Easy applicable to various problems. • Widely used • Literature and other materials available • Generally good accuracy for real world problems • Easy to implement as same parameter settings work fine for a wide range of problems • Drawback : Somewhat slow during initial iterations

50. Cumene Synthesis • Main reaction : Benzene + Isopropyl Alcohol  Cumene + Water (benzene alkylation) • Secondary reactions : Cumene + Isopropyl Alcohol  p-Di-isopropyl Benzene + Water (cumene alkylation) p-Di-isopropyl Benzene  m-Di-isopropyl Benzene (isomerization) 2 Isopropyl alcohol  Di-isopropyl ether + Water (alcohol dehydration)