Using hyperheuristics to improve the determination of the kinetic - - PowerPoint PPT Presentation

Using hyperheuristics to improve the determination of the kinetic constants of a chemical reaction in heterogeneous phase

Jos´ e Mat´ ıas Cutillas Lozano and Domingo Gim´ enez

Departamento de Inform´ atica y Sistemas, University of Murcia

CCA, June 10-12, 2014

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 1 / 21

Contents

1

Motivation

2

Determination of kinetic constants

3

Scheme for metaheuristics and hyperheuristics

4

Experimental results

5

Conclusions

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 2 / 21

Motivation

Kinetic constants of a chemical reaction

Kinetic parameters of a chemical reaction are determined with metaheuristic methods. The processes occurring in the human stomach when neutralizing the acid with an antacid tablet are simulated. It is a reaction combined with mass transfer of carbonate ions present in the solid phase upon contact with an acid solution. Solving the problem requires the calculation of the whole chemical system using the Euler numerical method.

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 3 / 21

Motivation

Hyperheuristics based on parameterized metaheuristics

Selecting the appropriate values of parameters to apply a satisfactory metaheuristic to a particular problem can be difficult and is computationally demanding. Hyperheuristics based on a metaheuristic scheme (HMS) are used to select these values. The hyperheuristics are metaheuristics searching in the space of metaheuristics. Previous results obtained applying metaheuristics are improved (CCA2013).

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 4 / 21

Motivation

Parallel-parameterized metaheuristics and hyperheuristics

The application of hyperheuristics is computationally demanding, and parallel versions are used. The same parallelization techniques used for metaheuristics are applicable to hyperheuristics based on the same metaheuristic scheme. A parallel metaheuristic is obtained by selecting the values of metaheuristic and parallelism parameters. Although parallelism can be applied in the hyperheuristic and in the metaheuristics, it is usually applied only in the hyperheuristic.

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 5 / 21

Determination of kinetic constants

The problem of determination of kinetic constants (PKICO)

Our search for the kinetic parameters of a chemical reaction that

• ccurs in heterogeneous phase involves the simulation of the

processes occurring in the human stomach. Depending on the pH, there are three main ways in which the dissolution of calcium carbonate occurs:

By reaction with acetic acid. CaCO3 + H3O+ ↔ Ca2+ + HCO−

3 + H2O

By reaction with carbonic acid. CaCO3 + H2CO3 ↔ Ca2+ + 2 · HCO−

3

And by the hydrolysis reaction. CaCO3 + H2O ↔ Ca2+ + HCO−

3 + OH−

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 6 / 21

Determination of kinetic constants

When the reaction occurs in several independent paths, the overall rate is simply the sum of all individual rates. So, the kinetic of dissolution of calcium carbonate is a function of the concentration of carbonic acid in the solution, the pH and the mass transfer area: 1 V dNCa2+ dt = −k1an1 H3O+n2 − k2an3 [H2CO3]n4 − k3

k1, k2 and k3 are the combined reaction rate constants. n1, n2, n3 and n4 are the reaction orders. a is the area of the tablet.

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 7 / 21

Determination of kinetic constants

Metaheuristics for PKICO

An individual is represented by a real vector of size seven that is the set of kinetic constants. The ranges of values for the constants are set following empirical criteria. Every time the fitness of an individual is calculated, the whole chemical system is solved: for i = 0 → N do Calculate at instant i:

• Ca2+

, a, [H3O+] , [HCO−] , [H2CO3] , pHcal, ∆

• Ca2+

, [CH3COOH] , [CH3COO−] Fitness = Fitness + (pHexp,i − pHcal,i)2 end for

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 8 / 21

Scheme for metaheuristics and hyperheuristics

Shared-memory-parameterized scheme

Initialize(S,ParamIni,ThreadsIni) while (not EndCondition(S,ParamEnd,ThreadsEnd)) SS = Select(S,ParamSel,ThreadsSel) SS1 = Combine(SS,ParamCom,ThreadsCom) SS2 = Improve(SS1,ParamImp,ThreadsImp) S = Include(SS2,ParamInc,ThreadsInc) Independent parallelization of the functions, with parallelism parameters (number of threads) for each function. The optimum value of the parallelism parameters depends on the values of the metaheuristic parameters (the metaheuristic or the combination of metaheuristics). Hyperheuristics are implemented with the same scheme, and they search for satisfactory metaheuristics implemented with this scheme (satisfactory values of the metaheuristic parameters).

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 9 / 21

Scheme for metaheuristics and hyperheuristics

Metaheuristics, functions and fitness calculation

Four pure metaheuristics, GRASP (GR), Genetic algorithm (GA), Scatter search (SS), Tabu Search (TS), and some combinations of the type GR+GA+SS+TS are considered for meta and hyperheuristics. The basic functions are similar for meta and hyperheuristics, with smaller sizes for hyperheuristic sets and parameters due to their higher computational cost. Fitness computation in hyperheuristics was made as FitSP1E (Fitness with several problem inputs in one execution) which is a way of reducing the dependence on the input and the increase of the execution time. Other possibilities give worse results. A Common Indicator calculated as the inverse of the product of fitness and execution time (CI = 1

ft ) is used to evaluate the global

quality of the solutions. High values are desirable.

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 10 / 21

Experimental results

Hyperheuristic parameters

Values of the hyperheuristic parameters used for the selection of metaheuristics: Reduced Hybrid Hyperheuristic (Hre) Genetic Algorithm based Hyperheuristic (Hge)

INEIni FNEIni PEIIni IIEIni STMIni NBESel NWESel NBBCom NBWCom Hre 5 5 50 3 2 3 2 2 3 Hge 20 20 20 10 NWWCom PEIImp IIEImp SMIImp PEDImp IDEImp SMDImp NBEInc LTMInc Hre 2 50 3 2 10 5 2 3 5 Hge 10 5 20 Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 11 / 21

Experimental results

Metaheuristic parameters intervals

Lower and upper limits of the metaheuristic parameters. The hyperheuristics search for metaheuristic parameters in these intervals.

INEIni FNEIni PEIIni IIEIni STMIni NBESel NWESel NBBCom NBWCom Lower 5 5 1 2 2 5 5 Upper 200 100 100 20 15 100 100 100 100 NWWCom PEIImp IIEImp SMIImp PEDImp IDEImp SMDImp NBEInc LTMInc Lower 5 1 1 2 Upper 100 100 20 15 100 10 15 100 15 Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 12 / 21

Experimental results

Metaheuristic parameters for direct application of metaheuristics

The results obtained with the application of hyperheuristics are compared with those with basic metaheuristics (GRASP, GA, SS and TS) and the best hybrid metaheuristic for the problem (Mhy, CCA2013). Values of the parameters for the four pure metaheuristics and the hybrid metaheuristic:

INEIni FNEIni PEIIni IIEIni STMIni NBESel NWESel NBBCom NBWCom GR 200 1 100 50 GA 100 100 100 50 SS 100 20 100 50 10 10 90 100 TS 200 1 100 10 5 1 Mhy 50 15 100 15 2 8 7 15 20 NWWCom PEIImp IIEImp SMIImp PEDImp IDEImp SMDImp NBEInc LTMInc GR GA 10 5 100 SS 90 100 5 10 TS 100 5 5 1 20 Mhy 15 100 5 2 20 5 2 8 3 Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 13 / 21

Experimental results

Problem variables

Values of the problem variables considered in the experiments:

Series S1 S2 S3 S4 S5 S6 N 70 50 34 77 81 69 MT 1.33 1.33 1.31 1.32 1.32 1.30 [HAc]0 · 102 3.98 3.31 2.39 2.18 3.01 1.99 V 250 250 250 100 150 200 N: number of experimental points of time and pH to be integrated. MT (g): mass of the tablet. [HAc]0 (mol/L): initial concentration of acetic acid in each series. V (mL): volume of the solution. Other variables common to all series of experiments: MA (g) = 0.68 is the active mass of calcium carbonate in the tablet. a0 (cm2) = 7.11 is the initial area of the tablet.

• H2CO∗

3

• (mol/L) = 10−5 is the total concentration of carbonate species in the solution.

Equilibrium constants: Ka = 10−4.76, K

′

a = 10−6.35, KH = 10−1.5. Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 14 / 21

Experimental results

Metaheuristic parameters obtained by the hyperheuristics

Values of the metaheuristic parameters obtained when applying the reduced hybrid (Hre) and the genetic hyperheuristics (Hge) to three training instances of the problem varying the instance for each iteration in the same execution (FitSP1E)

INEIni FNEIni PEIIni IIEIni STMIni NBESel NWESel NBBCom NBWCom HreFitSP1E 24 17 45 2 12 6 9 99 1 HgeFitSP1E 145 32 66 11 8 5 21 71 77 NWWCom PEIImp IIEImp SMIImp PEDImp IDEImp SMDImp NBEInc LTMInc HreFitSP1E 8 86 10 11 86 4 9 16 9 HgeFitSP1E 62 72 14 8 74 8 12 27 3 Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 15 / 21

Experimental results

Fitness for different metaheuristics and problem sizes

The best metaheuristics were obtained from the validation set (S1, S2 and S3) in one execution of each hyperheuristic. On average, metaheuristics obtained with FitSP1E appreciably improve the results achieved by applying the pure metaheuristics, and they provide fitness values that are close to the best obtained by applying the hyperheuristic to the validation problems directly (Hge and Hre)

1 2 3 4 5 6 7 s4 s5 s6 average

fitness/fitnessbest

TS GR SS GA HreFitSP1E HgeFitSP1E Hre Hge Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 16 / 21

Experimental results

Fitness, time and CI comparison

Indicators for hyperheuristics: the Hre hyperheuristic is faster than Hge, so it has a higher value of CI with similar fitness

Hre Hge FitSP1E (f) 2.1444 2.0348 (t) 59.93 783.17 (CI) 77.81 6.28

Hyperheuristics - Best metaheuristic comparison: both Hre and Hge produce better values of fitness than the best metaheuristic combination

Hre Hge Mhy S4 0.8937 0.8931 1.7600 S5 3.8209 3.8311 4.0178 S6 1.3023 1.3022 1.7953 Mean 2.0055 2.0088 2.5243

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 17 / 21

Experimental results

Statistical summary of the results

B−M H M−H P−M 1 2 3 4 5 6 7

metaheuristics fitness

(a) PKICO S4

B−M H M−H P−M 3.8 4.0 4.2 4.4 4.6 4.8

metaheuristics fitness

(b) PKICO S5

B−M H M−H P−M 1.5 2.0 2.5

metaheuristics fitness

(c) PKICO S6 Figure: Statistical summary of the fitness means obtained by applying different

metaheuristics to several sizes of the problem PKICO (S4, S5 and S6). Four sets of algorithms are considered: the direct application of the two hyperheuristic configurations (H), the metaheuristics obtained from hyperheuristics (M-H) with fitness computation as FitSP1E, the application of the best metaheuristic not selected automatically (B-M), and the four pure metaheuristics (P-M).

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 18 / 21

Experimental results

Statistical summary of the results

The Kruskal-Wallis test revealed statistical differences in the means for the sets of instances in the three problem sizes considered. The best algorithm in all problem sizes was the hyperheuristic (H) applied directly to each instance, with the other algorithms being worse in all cases. The metaheuristics obtained with the hyperheuristics (M-H) improve the best results obtained with the best metaheuristic (B-M) obtained in a previous work.

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 19 / 21

Conclusions

Conclusions

Previous fitness results in the determination of kinetic constants are improved by using hyperheuristics based on parameterized schemes for metaheuristics. Hyperheuristics significantly improve the results of applying pure metaheuristics and also improve the fitness resulting from the application of the best known hybrid metaheuristic. This leads us to think that hyperheuristics over parameterized metaheuristics are a good tool for selection of optimal metaheuristics. Similar results were obtained regarding fitness with the two hyperheuristics considered. However, it would be advisable to use the reduced hyperheuristic because it significantly reduces the execution time.

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 20 / 21

Conclusions

Future research

Future work could include the application of the same methodology to other optimization problems of chemical parameters and processes. For problems with a high computational cost it is convenient to develop unified parameterized schemes in parallel systems of different architectures: shared memory, message passing, GPU and hybrid-heterogeneous clusters. Hyperheuristics for bi-objective problems could be applied for the time-fitness optimization of metaheuristics.

Cutillas-Lozano, Gim´ enez (UMU) Hyperheuristics for kinetic constants CCA, June 10-12, 2014 21 / 21