Research ArticleOn the Theoretical Analysis of the Plant Propagation Algorithms

Muhammad Sulaiman 1 Abdellah Salhi 2 Asfandyar Khan 1

Shakoor Muhammad1 andWali Khan3

1Department of Mathematics Abdul Wali Khan University Mardan Khyber Pakhtunkhwa Pakistan2Department of Mathematical Sciences University of Essex Wivenhoe Park Colchester CO4 3SQ UK3Department of Mathematics Kohat University of Science amp Technology (KUST) Khyber Pakhtunkhwa Pakistan

Correspondence should be addressed to Muhammad Sulaiman sulaiman513yahoocouk and Abdellah Salhi asessexacuk

Received 10 September 2017 Accepted 21 January 2018 Published 21 March 2018

Academic Editor Haranath Kar

Copyright copy 2018 Muhammad Sulaiman et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Plant Propagation Algorithms (PPA) are powerful and flexible solvers for optimisation problems They are nature-inspiredheuristics which can be applied to any optimisationsearch problem There is a growing body of research mainly experimentalon PPA in the literature Little however has been done on the theoretical front Given the prominence this algorithm is gainingin terms of performance on benchmark problems as well as practical ones some theoretical insight into its convergence is neededThe current paper is aimed at fulfilling this by providing a sketch for a global convergence analysis

1 Introduction

The theoretical analysis of stochastic algorithms for globaloptimisation is not new and can be found in a numberof sources such as [1ndash5] The majority of the algorithmsconsidered use random search one way or another to find theoptimum solution [6ndash13] Here we consider the algorithmicscheme of the Plant Propagation Algorithm for continuousoptimisation or PPA-C [14] and theoretically investigate itsglobal convergence to the optimum solution The optimisa-tion problems of concern are continuous and defined in finite119899-dimensional domains

The basic version of PPA [15] models the propagationof strawberry plants The scheme uses short runners forexploitation or local search refinement while long runnersare used for diversification and exploration of the searchspace Since the propagation of strawberries is due to seeds aswell as runners a Seed-based Plant Propagation Algorithm(SbPPA) has also been introduced in [16] Both PPA-C andSbPPA have been shown to be efficient on continuous uncon-strained and constrained optimisation problems statisticalconvergence analyses of PPA-C and SbPPA can be found in[9ndash11 14ndash16]

PPA-C [14 15] consists of two steps(1) Initialization a population of parent plants is gener-

ated randomly(2) Propagation a new population is created from per-

sistent parents (strawberry plants) and their children(new strawberry plants at the end of runners ie adistance away from parent plants)

Let 119878 denote the search space such that 119878 sub 119877119899 where119899 is its dimension By an iteration of PPA-C we mean a newgeneration of child plants produced by parent plants Thesechild plants are the result of either short or long runners [1416] This is the basic setup that we consider to sketch a proofof convergence to the global optimum of a given continuousoptimisation problem

The paper is organised as follows Section 2 presents theterminology used in the analysis of PPA-C Section 3 analysesa population of plants Section 31 describes the convergenceanalysis of PPA-C Section 4 is the conclusion

2 Terminology and Notation

We consider single objective minimization problems [17]119883optimal isin 119878 such that 119891(119883optimal) le 119891(119883) for all 119883 isin 119878

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6357935 8 pageshttpsdoiorg10115520186357935

2 Mathematical Problems in Engineering

where the objective function is defined as 119891 119878 sub 119877119899 rarr 119877denotes the best spot for a plant in the search space 119883 is an119899-dimensional position vector

The population at the 119892th iteration is denoted by pop119892 =1198831119892 1198832119892 119883119873119875119892 where 119873119875 is the population size Thecoordinates of runners or more precisely their endpointsare denoted by 119883119894 = (1199091119894 1199092119894 119909119899119894 )119879 where 119899 is the spacedimension of the given problem

21 Search Equations and Evaluation of New Plants Variantsof PPA can be found in [14ndash16] In this paper we analyse PPA-C as Algorithm 1 of [14]

In order to send a short or long runner 1198831015840119894 is generated[14 19ndash21] as in (1a) (1b) and (1c)

1199091015840119894119895 = 119909119894119895 + 120573119895119909119894119895 if rand119895 le 119875119898 119903 le 119873119875 (1a)

1199091015840119894119895 = 119909119894119895 + (119909119897119895 minus 119909119896119895) 120573119895 if rand119895 le 119875119898 119903 le 119873119875 (1b)

1199091015840119894119895 = 119909119894119895 + (119909119894119895 minus 119909119896119895) 120573119895if rand119895 le 119875119898 or 119868119873119895 lt 4 119903 gt 119873119875 (1c)

where 119873119875 is the population size 119903 is the Monte Carlo trialrun counter 119875119898 is the modification probability and rand119895 isin(0 1) is a randomly generated number for each 119895th entry119895 = 1 2 119899 The indices 119894 119897 119896 = 1 2 119873119875 are mutuallyexclusive that is 119894 = 119896 = 119897 Another version of PPAcalled SbPPA [16] which is inspired by propagation via seedsimplements the following search equation instead

119909lowast119894119895 = 119909119894119895 + 119871 119894 (119909119894119895 minus 120579119895) if 119875119877 le 08 120579119895 isin [119886119895119887119895] 119894 = 1 2 119873119875 119895 = 1 2 119899119909119894119895 Otherwise (2)

where 119871 119894 is a step drawn from the Levy distribution [22] and120579119895 is a random coordinate within the search space Equations(1a) (1b) (1c) and (2) perturb the current solution the resultsof which can be seen in Figures 1(a) and 1(b) respectively

22 A Case Study Let 119888 be the class of runners sent by the119894th parent plant and stored in 119865 Each runner in class 119888 isdecomposed into two vectors 120591119888 and 120596119888 where 120591119888 denotesthe vector of indices which are perturbed with respect to thecurrent position of the plant while 120596119888 represents the vectorof corresponding indices of the unperturbed coordinateswith respect to the current position of plants This can berepresented as

120591119888 cup 120596119888 = 1 2 119899 (3)

To clarify this idea let us take an example [17] of a newlygenerated runner by the 119894th plant as

1198831015840119894 = [11990910158401 11990910158402 11990910158403 11990910158404 11990910158405]119879 (4)

such that

120591119888 = 1 2 4 120596119888 = 3 5 (5)

then we can write

1198831015840119894 = (1198831015840120591119888 1198831015840120596119888) (6)

where

1198831015840120591119888 = [11990910158401 11990910158402 0 11990910158404 0] 1198831015840120596119888 = [0 0 11990910158403 0 11990910158405] (7)

The dot product of these vectors is zero which showsthat they are mutually orthogonal Mathematically this canbe written as

1198831015840120591119888 ∙ 1198831015840120596119888 = 0 (8)

Let 119881120591119888 and 119881120596119888 denote two vector spaces such that

119881120591119888 = containing vectors having dimensions as in 120591119888119881120596119888

= containing vectors having dimensions as in 120596119888(9)

119881120591119888 and119881120596119888 are subspaces of119877119899This implies that1198831015840120591119888 isin 119881120591119888and1198831015840120596119888 isin 119881120596119888

A scalar objective function 119891 defined over 1198831015840119894 can berepresented as

119891 (1198831015840119894) = 119891 (11990910158401 11990910158402 11990910158403 11990910158404 11990910158405) (10)

where 11990910158401 = 1198831205911 11990910158402 = 1198831205912 11990910158404 = 1198831205913 and 11990910158403 = 1198831205961 11990910158405 = 1198831205962 997904rArr 119891(1198831205911 1198831205912 1198831205961 1198831205913 1198831205962) = 119891 (119883120591119888 119883120596119888) (11)

where (11) represents an objective value corresponding to anew runner in position 1198831015840119894 Similarly different runners areproduced to correspond to different classes 119888 and evaluatedby the same procedureThis procedure can be generalized for119899-dimensional problems [1ndash3]

3 Graphical and Theoretical Analysis of aPopulation of Plants

Algorithm 1 states that the 119895th coordinate of an 119894th par-ent plant is perturbed with probability 119875119898 and it remains

Mathematical Problems in Engineering 3

(1) 119903 larr Counter for trial runs119873119875 larr Population size(2)(3) 119865 larr Population of runners(4)(5) for 119903 = 1 100 do(6)(7) if 119903 le 119873119875 then(8)(9) Create a random population of plants pop = 119883119894 | 119894 = 1 2 119873119875 and

gather the best solution from each run(10)(11) end if(12)(13) while 119903 gt 119873119875 do(14)(15) Use population pop119892 formed by gathering all the best solutions of previous runs

Calculate 119868119873119895 value for each column 119895 of pop119892(16)(17) end while(18)(19) Evaluate the population pop(20)(21) Assume number of runners to be 119899119903 = 3(22)(23) while (the stopping criteria is not satisfied) do(24)(25) for 119894 = 1 to119873119875 do(26)(27) for ℎ = 1 to 119899119903 do(28)(29) if 119903 le 119873119875 then(30)(31) if rand le 119875119898 then(32)(33) Generate a new solution11988310158401 according to Equation (1a)(34)(35) Evaluate it and store it in 119865(36)(37) end if(38)(39) if rand le 119875119898 then(40)(41) Generate a new solution11988310158402 according to Equation (1b)(42)(43) Evaluate it and store it in 119865(44)(45) end if(46)(47) else(48)(49) for 119895 = 1 119899 do(50)(51) if (119868119873119895 lt 4) or (rand le 119875119898) then(52)(53) Update the 119895th entry of119883119894 119894 = 1 2 119873119875 by using Equation (1c)(54)(55) end if(56)(57) Evaluate new solution11988310158403 and store it in 119865(58)(59) end for(60)

Algorithm 1 Continued

4 Mathematical Problems in Engineering

(61) end if(62)(63) end for(64)(65) end for(66)(67) Append 119865 to current population(68)(69) Sort the population in ascending order of objective values(70)(71) Update current best(72)(73) end while(74)(75) Return Updated population and global best solution(76)(77) end for(78)

Algorithm 1 PPA for constrained optimisation [14]

0 2 4 6minus15

minus10

minus5

0

5

10

15Spring design problem

Sear

ch sp

ace

times105

(a) Perturbations by (1a) (1b) and (1c)

0 5 10 15minus80

minus60

minus40

minus20

0

20

40

Sear

ch sp

ace

Spring design problem

times105

(b) Perturbations by (2)

Figure 1 Overall performance of (1a) (1b) (1c) and (2) on a design optimisation problem given in Appendix [16]

unchanged with probability 1minus119875119898Thus there are 2119899 possiblerunners to be generated for each 119894th parent plant using (1a)(1b) and (1c) where 119899 is the space dimension of the givenproblem

Let at any generation 119892 the random population berepresented as pop = 1198831119892 1198832119892 119883119873119875119892 where 119873119875denotes the population size To create a runner by using (1a)(1b) and (1c) for next generation 119892+1 PPA uses a populationof parent plants at generation 119892 for this purpose It is notrequired to know about any other runner in generation 119892+1

This shows that all runners created at generation 119892 + 1 arestatistically mutually independent Furthermore the initialpopulation is random and all parent plants do not dependon each other Thus by induction the runners at any furthergenerations are mutually independent

From (1a) (1b) and (1c) a runner 1198831015840119894 may be formedby itself or by choosing three different coordinates fromcurrent population In case of using (1a) [19] there are 119873119875possibilities to send (long or short) runners as in Figure 1On the other hand by using (1b) or (1c) 119873119875 minus 1 vectors

Mathematical Problems in Engineering 5

are used to send a new runner Thus in later cases thereare 119873119875minus11198753 possibilities to send new runners In (1a)ndash(1c)different possibilities are of the form

11988310158401119892 = 119883119894119892 + 120573 sdot 11988311989411989211988310158402119892 = 119883119894119892 + 120573 sdot 11988311989511989211988310158403119892 = 119883119894119892 + 120573 sdot 119883119896119892

(12)

where 119883119894119892 119883119895119892 and 119883119896119892 are calculated according to (1a)(1b) and (1c) in which 119892 denotes the current generation and120573 is an 119899-dimensional random vector within interval [minus1 1]The probability density functions (PDFs) [17 23] of these newvectors 119883119894119892 120573 sdot 119883119894119892 120573 sdot 119883119895119892 and 120573 sdot 119883119896119892 can be writtenas119901119883119894119892(119909119894119892)119901119883119894119892(119909119894119892120573)120573119901119883119895119892(119909119895119892120573)120573119901119883119896119892(119909119896119892120573)120573respectively

The PDFs of new runners created with (1a)ndash(1c) are givenin (15)

Definition 1 (convolution operation ⋆ [24]) Let 119891(119909) and119892(119909) be Laplace transformable piecewise continuous func-tions defined on [0 infin] The convolution product of thesetwo functions is again a function of 119909 defined as

(119891 ⋆ 119892) (119909) = int1199090119891 (120585) 119892 (119909 minus 120585) 119889120585 (13)

11990111988310158401119892 (11990910158401119892 ) = 1120573 (119901119883119894119892 (11990910158401119892 ) ⋆ 119901119883119894119892 (11990910158401119892120573 )) 11990111988310158402119892 (11990910158402119892 ) = 1120573 (119901119883119894119892 (11990910158402119892 ) ⋆ 119901119883119895119892 (11990910158402119892120573 )) 11990111988310158403119892 (11990910158403119892 ) = 1120573 (119901119883119894119892 (11990910158403119892 ) ⋆ 119901119883119896119892 (11990910158403119892120573 ))

dArr

(14)

11990111988310158401119894119892 (11990910158401119894119892)= 11205731198941119873119875 (sumsum119901119883119894119892 (11990910158401119894119892) ⋆ 119901119883119894119892 (119909101584011198941198921205731198941))

11990111988310158402119894119892 (11990910158402119894119892)= 11205731198942119873119875minus11198753 (sumsum

119894 =119895

119901119883119894119892 (11990910158402119894119892) ⋆ 119901119883119895119892 (119909101584021198941198921205731198942)) 11990111988310158403119894119892 (11990910158403119894119892)

= 11205731198943119873119875minus11198753 (sumsum119894 =119896

119901119883119894119892 (11990910158403119894119892) ⋆ 119901119883119896119892 (119909101584031198941198921205731198943))

(15)

Let 119883119892+1 be any plant in generation 119892 + 1 119883119892 a plantposition in current generation and1198831015840ℎ119892 the runner produced

by the 119894th plant 119883119894119892 at the generation 119892 Then the joint PDFof the parent plant in the next generation 119883119892+1 based on theparent119883119892 and runner1198831015840ℎ119892 where ℎ = 1 2 3 is given by

119901119883119892+1 119883119892 1198831015840ℎ119892 (119909119892+1 119909119892 1199091015840ℎ119892 )= 119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) sdot 1199011198831015840ℎ119892 |119883119892 (1199091015840ℎ119892 | 119909119892)

sdot 119901119883119892 (119909119892) sdot 1199011198831015840ℎ119892 (1199091015840ℎ119892 ) (16)

Note that a runner is selected for the next populationonly if its rank is less than or equal to 119873119875 the populationsize Its objective value is less than the maximum objectivevalue (in case of minimization problem) in the currentpopulation Note also that instead of greedy selection wesort the population and eliminate those plants whose rank ishigher than119873119875 Themodel for this selection mechanism canbe represented as follows

119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) = 120575 (119909119892+1 minus 1199091015840ℎ119892 ) forall1198831015840ℎ119892 isin 119865 forall119883119892 isin pop 119891 (119883119892) ge 119891 (1198831015840ℎ119892 ) (17)

31 ConvergenceAnalysis of PPA-C For illustration purposeswe have implemented a combined version of (1a) (1b) (1c)and (2) This version of PPA-C called H-PPA-SbPPA is ahybridisation of PPA and SbPPA [10] We have plotted theposition of plants in populations through solving the Braninand Matyas test functions (see Figures 2 3 4 and 5) Itis obvious from Figures 2 and 4 that (1a) (1b) and (1c)have generated short runners which exploit the search spacelocally On the other hand in Figures 3 and 5 (2) hasgenerated a diverse range of solutions which are spread overthe whole search space This equation helps the algorithmescape from local minima and to explore the solution spacebetter and hence the global search qualities of this algorithmMathematically this can be shown as follows

Let 119878 denote the search space containing the solution of agiven optimisation problem defined as

min 119891 (119883) | 119883 isin 119878 (18)

where 119891(119883) is the objective function Then the optimalsolution set [25] can be represented as

119878lowast= 119883optimal | 119891 (119883optimal) = min 119891 (119883) forall119883 isin 119878 (19)

where119883optimal is the optimum solution The region of attrac-tion [25] of the solution set 119878lowast is defined as

119878lowast120585 = 119883 | 119891 (119883best) minus 119891 (119883) lt 120585 (20)

where 120585 is a small positive real number and119883best is the currentbest solution

In PPA-C each parent plant produces1198831015840ℎ119892 runners (solu-tions) where ℎ = 1 2 3 The probability that at generation

6 Mathematical Problems in Engineering

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08minus1minus1

Figure 2 The exploitation capability of PPA while solving Branintest function

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08

minus1

minus1

Figure 3The exploration capability of SbPPA while solving Branintest function

119892 a subset of the temporary population 1198651015840 sub 119865 containingsolutions which are not good enough to be retained in thenext generation by the selection model as in (17) is given as

119901 1198651015840119892 cap 119878lowast120585 = le 1 minus 120585119892 (21)

where 120585119892 is a small positive real number Obviously in allprevious generations 119892 minus 1 some of the solutions died andsome succeeded to survive into the next generation Thisshows that in previous generations we have some solutions1198831015840ℎ119892 which do not belong to the region of attraction 119878lowast120585

119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (22)

At the end of each generation the temporary population119865 is appended to the main population pop119892 Then allindividuals are sorted with respect to their objective valuesThe individuals with higher rank than size of population are

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 4 A scatter plot of plants produced by PPAwhen optimisingMatyas function [15]

25

25

20

20

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10

minus15

minus15

minus20

minus20minus25

Figure 5 A scatter plot of plants produced by SbPPA whenoptimising Matyas function [16 18]

omitted Thus the probability that a generation 119892 does notcontain an optimum is given below

119901 pop119892 cap 119878lowast120585 = asymp 119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (23)

After sorting the final population at the end of eachgeneration the probability that the optimum may existin a subpopulation 1198651015840 (population of weak or dead run-nerssolutions) is less than that of the population pop119892 Thiscan be represented as

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = ge lim119892rarrinfin

119901 1198651015840119892 cap 119878lowast120585 = = 1 minus lim

119892rarrinfin119901 1198651015840119892 cap 119878lowast120585 = = 1 minus +infinprod

119892=1

(1 minus 120585119892) (24)

Mathematical Problems in Engineering 7

Following [25ndash27] the right hand side term of inequality (24)is zero if the seriessum+infin119892=1(120585119892) divergesThe convergence of PPAfollows and can be summarised in the theorem below

Theorem 2 PPA converges to the global optimumwith proba-bility 1 if it is left to run for a reasonable amount of time [1ndash3]in other words

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = = 1 (25)

Remark 3 Every population pop119892 includes some solutionswhich are in set 119878lowast120585 Remark 4 The above remark is due to the exploitationcharacteristic of PPA-C

Remark 5 Remark 3 implies that 119883best is always improvingor changing its position until the optimum is reached

Remark 6 119883best converges approximately to 119883optimal asgeneration 119892 grows

4 Conclusion

The Plant Propagation Algorithm (PPA) and its variants forcontinuous optimisation problems are getting notoriety asflexible and powerful solvers PPA is a heuristic inspiredby the way plants and in particular the strawberry plantpropagate It is also referred to as the Strawberry AlgorithmWhile there is a growing body of experimental and computa-tional works that show its good behaviour and performanceagainst well-established algorithms and heuristics there isvery little if any in terms of theoretical investigationThis gaptherefore needs to be filled The aim of course in analysingthe convergence of any algorithm (in this case PPA-C) isto give confidence to the potential users that the solutionsthat it returns are of good quality The convergence analysisput forward in this paper relies on the exploitation andexploration characteristics of the algorithm Since it doesnot get stuck in local optima and explores thoroughly thesearch space it is only a matter of time before the globaloptimum is discovered The approach is probabilistic innature and ascertains that the global optimum will be foundwith probability 1 provided the algorithm is run reasonablylong enough The argument for this to hold is that at eachiteration new and better solutions are generated whichmeansthat in the limit the global optimum is reached Questionsstill remain concerning what is considered a reasonableamount of time Bounds on the time it will take to convergeare being developed and results will be presented in a follow-up paper

Appendix

Spring Design Optimisation

Themain objective of this problem [28 29] is tominimize theweight of a tensioncompression string subject to constraintsof minimum deflection shear stress surge frequency and

limits on outside diameter and on design variables Thereare three design variables the wire diameter 1199091 the meancoil diameter 1199092 and the number of active coils 1199093 [30]The mathematical formulation of this problem where 119909119879 =(1199091 1199092 1199093) is as follows

Minimize 119891 (119909) = (1199093 + 2) 119909211990921subject to 1198921 (119909) = 1 minus 119909321199093717811990941 le 0

1198922 (119909)= 411990922 minus 1199091119909212566 (119909211990931) minus 11990941 +

1510811990921 minus 1le 01198923 (119909) = 1 minus 140451199091119909221199093 le 01198924 (119909) = 1199092 + 119909115 minus 1 le 0

(A1)

The simple limits on the design variables are 005 le 1199091 le 20025 le 1199092 le 13 and 20 le 1199093 le 150Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work has been partially sponsored by ESRC GrantESL0118591

References

[1] A Zhigljavsky and A Zilinskas Stochastic Global Optimizationvol 9 Springer Science amp Business Media 2007

[2] C A Floudas and P M Pardalos Encyclopedia of OptimizationSpringer Science amp Business Media 2008

[3] Y D Sergeyev R G Strongin and D Lera Introduction toGlobal Optimization Exploiting Space-Filling Curves SpringerScience amp Business Media 2013

[4] N Brahimi A Salhi and M Ourbih-Tari ldquoConvergence of theplant propagation algorithm for continuous global optimisa-tionrdquo RAIRO Operations Research 2018

[5] J He and L Kang ldquoOn the convergence rates of geneticalgorithmsrdquo Theoretical Computer Science vol 229 no 1-2 pp23ndash39 1999

[6] W K Mashwani A Salhi M A Jan R A Khanum and MSulaiman ldquoEnhanced version of multi-algorithm geneticallyadaptive for multiobjective optimizationrdquo Advanced ComputerScience and Applications (IJACSA) vol 12 no 6 2015

[7] W K Mashwani A Salhi O Yeniay M A Jan and R AKhanum ldquoHybrid adaptive evolutionary algorithm based ondecompositionrdquo Applied Soft Computing vol 57 pp 363ndash3782017

8 Mathematical Problems in Engineering

[8] W K Mashwani A Salhi M A Jan M Sulaiman R AKhanum and A Algarni ldquoEvolutionary algorithms based ondecomposition and indicator functions state-of-the-art surveyrdquoAdvanced Computer Science and Applications (IJACSA) vol 7no 2 2016

[9] M Sulaiman A Nature-inspired Metaheuristic The Plant Prop-agation Algorithm [PhD thesis] University of Essex 2015

[10] M Sulaiman andA Salhi ldquoA hybridisation of runner-based andseed-based plant propagation algorithmsrdquo Studies in Computa-tional Intelligence vol 637 pp 195ndash215 2016

[11] M Sulaiman A Salhi E S Fraga W K Mashwani and M MRashidi ldquoA novel plant propagation algorithm modificationsand implementationsrdquo Science International vol 28 no 1 2015

[12] B I Selamoglu A Salhi and M Sulaiman ldquoStrip algorithms asan efficient way to initialise population-based metaheuristicsrdquoin Recent Developments in Metaheuristics pp 319ndash331 Springer2018

[13] M Sulaiman A Ahmad A Khan and S MuhammadldquoHybridized Symbiotic Organism Search Algorithm for theOptimal Operation of Directional Overcurrent Relaysrdquo Com-plexity vol 2018 Article ID 4605769 11 pages 2018

[14] M Sulaiman A Salhi B I Selamoglu and O B KirikchildquoA plant propagation algorithm for constrained engineeringoptimisation problemsrdquoMathematical Problems in Engineeringvol 2014 Article ID 627416 10 pages 2014

[15] A Salhi and E Fraga ldquoNature-inspired optimisationapproaches and the new plant propagation algorithmrdquo inProceedings of the International Conference on NumericalAnalysis and Optimization (ICeMATH rsquo11) pp K2ndash1ndashK2ndash8Yogyakarta Indonesia 2011

[16] M Sulaiman and A Salhi ldquoA seed-based plant propagationalgorithm the feeding station modelrdquo The Scientific WorldJournal vol 2015 Article ID 904364 16 pages 2015

[17] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 42 no1 pp 107ndash124 2012

[18] M Sulaiman and A Salhi ldquoThe 5th International Conferenceon Metaheuristics and Nature Inspired Computingrdquo in Pro-ceedings of the 5th International Conference on Metaheuristicsand Nature Inspired Computing Morocco UAE October 2014httpmeta2014sciencesconforg40158

[19] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[20] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[21] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2011

[23] M Taboga Lectures on Probability Theory and MathematicalStatistics CreateSpace Independent Pub 2012

[24] R N Bracewell and R Bracewell The Fourier Transform andIts Applications vol 31999 McGraw-Hill New York NY USA1986

[25] Z Hu S Xiong Q Su and X Zhang ldquoSufficient conditions forglobal convergence of differential evolution algorithmrdquo Journalof Applied Mathematics vol 2013 Article ID 193196 2013

[26] C Chen F Jin X Zhu and G Ouyan Mathematics AnalysisPress Higher Education Press 2000

[27] K Knopp Theory and Application of Infinite Series CourierDover Publications 2013

[28] J Arora Introduction to Optimum Design Academic Press2004

[29] A D Belegundu and J S Arora ldquoA study of mathematicalprogramming methods for structural optimization I TheoryrdquoInternational Journal for Numerical Methods in Engineering vol21 no 9 pp 1583ndash1599 1985

[30] L C Cagnina S C Esquivel and C A C Coello ldquoSolvingengineering optimization problemswith the simple constrainedparticle swarm optimizerrdquo Informatica (Slovenia) vol 32 pp319ndash326 2008

Mathematical Problems in Engineering 3

(1) 119903 larr Counter for trial runs119873119875 larr Population size(2)(3) 119865 larr Population of runners(4)(5) for 119903 = 1 100 do(6)(7) if 119903 le 119873119875 then(8)(9) Create a random population of plants pop = 119883119894 | 119894 = 1 2 119873119875 and

gather the best solution from each run(10)(11) end if(12)(13) while 119903 gt 119873119875 do(14)(15) Use population pop119892 formed by gathering all the best solutions of previous runs

Calculate 119868119873119895 value for each column 119895 of pop119892(16)(17) end while(18)(19) Evaluate the population pop(20)(21) Assume number of runners to be 119899119903 = 3(22)(23) while (the stopping criteria is not satisfied) do(24)(25) for 119894 = 1 to119873119875 do(26)(27) for ℎ = 1 to 119899119903 do(28)(29) if 119903 le 119873119875 then(30)(31) if rand le 119875119898 then(32)(33) Generate a new solution11988310158401 according to Equation (1a)(34)(35) Evaluate it and store it in 119865(36)(37) end if(38)(39) if rand le 119875119898 then(40)(41) Generate a new solution11988310158402 according to Equation (1b)(42)(43) Evaluate it and store it in 119865(44)(45) end if(46)(47) else(48)(49) for 119895 = 1 119899 do(50)(51) if (119868119873119895 lt 4) or (rand le 119875119898) then(52)(53) Update the 119895th entry of119883119894 119894 = 1 2 119873119875 by using Equation (1c)(54)(55) end if(56)(57) Evaluate new solution11988310158403 and store it in 119865(58)(59) end for(60)

Algorithm 1 Continued

4 Mathematical Problems in Engineering

(61) end if(62)(63) end for(64)(65) end for(66)(67) Append 119865 to current population(68)(69) Sort the population in ascending order of objective values(70)(71) Update current best(72)(73) end while(74)(75) Return Updated population and global best solution(76)(77) end for(78)

Algorithm 1 PPA for constrained optimisation [14]

0 2 4 6minus15

minus10

minus5

0

5

10

15Spring design problem

Sear

ch sp

ace

times105

(a) Perturbations by (1a) (1b) and (1c)

0 5 10 15minus80

minus60

minus40

minus20

0

20

40

Sear

ch sp

ace

Spring design problem

times105

(b) Perturbations by (2)

Figure 1 Overall performance of (1a) (1b) (1c) and (2) on a design optimisation problem given in Appendix [16]

unchanged with probability 1minus119875119898Thus there are 2119899 possiblerunners to be generated for each 119894th parent plant using (1a)(1b) and (1c) where 119899 is the space dimension of the givenproblem

Let at any generation 119892 the random population berepresented as pop = 1198831119892 1198832119892 119883119873119875119892 where 119873119875denotes the population size To create a runner by using (1a)(1b) and (1c) for next generation 119892+1 PPA uses a populationof parent plants at generation 119892 for this purpose It is notrequired to know about any other runner in generation 119892+1

This shows that all runners created at generation 119892 + 1 arestatistically mutually independent Furthermore the initialpopulation is random and all parent plants do not dependon each other Thus by induction the runners at any furthergenerations are mutually independent

From (1a) (1b) and (1c) a runner 1198831015840119894 may be formedby itself or by choosing three different coordinates fromcurrent population In case of using (1a) [19] there are 119873119875possibilities to send (long or short) runners as in Figure 1On the other hand by using (1b) or (1c) 119873119875 minus 1 vectors

Mathematical Problems in Engineering 5

are used to send a new runner Thus in later cases thereare 119873119875minus11198753 possibilities to send new runners In (1a)ndash(1c)different possibilities are of the form

11988310158401119892 = 119883119894119892 + 120573 sdot 11988311989411989211988310158402119892 = 119883119894119892 + 120573 sdot 11988311989511989211988310158403119892 = 119883119894119892 + 120573 sdot 119883119896119892

(12)

where 119883119894119892 119883119895119892 and 119883119896119892 are calculated according to (1a)(1b) and (1c) in which 119892 denotes the current generation and120573 is an 119899-dimensional random vector within interval [minus1 1]The probability density functions (PDFs) [17 23] of these newvectors 119883119894119892 120573 sdot 119883119894119892 120573 sdot 119883119895119892 and 120573 sdot 119883119896119892 can be writtenas119901119883119894119892(119909119894119892)119901119883119894119892(119909119894119892120573)120573119901119883119895119892(119909119895119892120573)120573119901119883119896119892(119909119896119892120573)120573respectively

The PDFs of new runners created with (1a)ndash(1c) are givenin (15)

Definition 1 (convolution operation ⋆ [24]) Let 119891(119909) and119892(119909) be Laplace transformable piecewise continuous func-tions defined on [0 infin] The convolution product of thesetwo functions is again a function of 119909 defined as

(119891 ⋆ 119892) (119909) = int1199090119891 (120585) 119892 (119909 minus 120585) 119889120585 (13)

11990111988310158401119892 (11990910158401119892 ) = 1120573 (119901119883119894119892 (11990910158401119892 ) ⋆ 119901119883119894119892 (11990910158401119892120573 )) 11990111988310158402119892 (11990910158402119892 ) = 1120573 (119901119883119894119892 (11990910158402119892 ) ⋆ 119901119883119895119892 (11990910158402119892120573 )) 11990111988310158403119892 (11990910158403119892 ) = 1120573 (119901119883119894119892 (11990910158403119892 ) ⋆ 119901119883119896119892 (11990910158403119892120573 ))

dArr

(14)

11990111988310158401119894119892 (11990910158401119894119892)= 11205731198941119873119875 (sumsum119901119883119894119892 (11990910158401119894119892) ⋆ 119901119883119894119892 (119909101584011198941198921205731198941))

11990111988310158402119894119892 (11990910158402119894119892)= 11205731198942119873119875minus11198753 (sumsum

119894 =119895

119901119883119894119892 (11990910158402119894119892) ⋆ 119901119883119895119892 (119909101584021198941198921205731198942)) 11990111988310158403119894119892 (11990910158403119894119892)

= 11205731198943119873119875minus11198753 (sumsum119894 =119896

119901119883119894119892 (11990910158403119894119892) ⋆ 119901119883119896119892 (119909101584031198941198921205731198943))

(15)

Let 119883119892+1 be any plant in generation 119892 + 1 119883119892 a plantposition in current generation and1198831015840ℎ119892 the runner produced

by the 119894th plant 119883119894119892 at the generation 119892 Then the joint PDFof the parent plant in the next generation 119883119892+1 based on theparent119883119892 and runner1198831015840ℎ119892 where ℎ = 1 2 3 is given by

119901119883119892+1 119883119892 1198831015840ℎ119892 (119909119892+1 119909119892 1199091015840ℎ119892 )= 119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) sdot 1199011198831015840ℎ119892 |119883119892 (1199091015840ℎ119892 | 119909119892)

sdot 119901119883119892 (119909119892) sdot 1199011198831015840ℎ119892 (1199091015840ℎ119892 ) (16)

Note that a runner is selected for the next populationonly if its rank is less than or equal to 119873119875 the populationsize Its objective value is less than the maximum objectivevalue (in case of minimization problem) in the currentpopulation Note also that instead of greedy selection wesort the population and eliminate those plants whose rank ishigher than119873119875 Themodel for this selection mechanism canbe represented as follows

119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) = 120575 (119909119892+1 minus 1199091015840ℎ119892 ) forall1198831015840ℎ119892 isin 119865 forall119883119892 isin pop 119891 (119883119892) ge 119891 (1198831015840ℎ119892 ) (17)

31 ConvergenceAnalysis of PPA-C For illustration purposeswe have implemented a combined version of (1a) (1b) (1c)and (2) This version of PPA-C called H-PPA-SbPPA is ahybridisation of PPA and SbPPA [10] We have plotted theposition of plants in populations through solving the Braninand Matyas test functions (see Figures 2 3 4 and 5) Itis obvious from Figures 2 and 4 that (1a) (1b) and (1c)have generated short runners which exploit the search spacelocally On the other hand in Figures 3 and 5 (2) hasgenerated a diverse range of solutions which are spread overthe whole search space This equation helps the algorithmescape from local minima and to explore the solution spacebetter and hence the global search qualities of this algorithmMathematically this can be shown as follows

Let 119878 denote the search space containing the solution of agiven optimisation problem defined as

min 119891 (119883) | 119883 isin 119878 (18)

where 119891(119883) is the objective function Then the optimalsolution set [25] can be represented as

119878lowast= 119883optimal | 119891 (119883optimal) = min 119891 (119883) forall119883 isin 119878 (19)

where119883optimal is the optimum solution The region of attrac-tion [25] of the solution set 119878lowast is defined as

119878lowast120585 = 119883 | 119891 (119883best) minus 119891 (119883) lt 120585 (20)

where 120585 is a small positive real number and119883best is the currentbest solution

In PPA-C each parent plant produces1198831015840ℎ119892 runners (solu-tions) where ℎ = 1 2 3 The probability that at generation

6 Mathematical Problems in Engineering

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08minus1minus1

Figure 2 The exploitation capability of PPA while solving Branintest function

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08

minus1

minus1

Figure 3The exploration capability of SbPPA while solving Branintest function

119892 a subset of the temporary population 1198651015840 sub 119865 containingsolutions which are not good enough to be retained in thenext generation by the selection model as in (17) is given as

119901 1198651015840119892 cap 119878lowast120585 = le 1 minus 120585119892 (21)

where 120585119892 is a small positive real number Obviously in allprevious generations 119892 minus 1 some of the solutions died andsome succeeded to survive into the next generation Thisshows that in previous generations we have some solutions1198831015840ℎ119892 which do not belong to the region of attraction 119878lowast120585

119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (22)

At the end of each generation the temporary population119865 is appended to the main population pop119892 Then allindividuals are sorted with respect to their objective valuesThe individuals with higher rank than size of population are

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 4 A scatter plot of plants produced by PPAwhen optimisingMatyas function [15]

25

25

20

20

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10

minus15

minus15

minus20

minus20minus25

Figure 5 A scatter plot of plants produced by SbPPA whenoptimising Matyas function [16 18]

omitted Thus the probability that a generation 119892 does notcontain an optimum is given below

119901 pop119892 cap 119878lowast120585 = asymp 119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (23)

After sorting the final population at the end of eachgeneration the probability that the optimum may existin a subpopulation 1198651015840 (population of weak or dead run-nerssolutions) is less than that of the population pop119892 Thiscan be represented as

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = ge lim119892rarrinfin

119901 1198651015840119892 cap 119878lowast120585 = = 1 minus lim

119892rarrinfin119901 1198651015840119892 cap 119878lowast120585 = = 1 minus +infinprod

119892=1

(1 minus 120585119892) (24)

Mathematical Problems in Engineering 7

Following [25ndash27] the right hand side term of inequality (24)is zero if the seriessum+infin119892=1(120585119892) divergesThe convergence of PPAfollows and can be summarised in the theorem below

Theorem 2 PPA converges to the global optimumwith proba-bility 1 if it is left to run for a reasonable amount of time [1ndash3]in other words

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = = 1 (25)

Remark 3 Every population pop119892 includes some solutionswhich are in set 119878lowast120585 Remark 4 The above remark is due to the exploitationcharacteristic of PPA-C

Remark 5 Remark 3 implies that 119883best is always improvingor changing its position until the optimum is reached

Remark 6 119883best converges approximately to 119883optimal asgeneration 119892 grows

4 Conclusion

The Plant Propagation Algorithm (PPA) and its variants forcontinuous optimisation problems are getting notoriety asflexible and powerful solvers PPA is a heuristic inspiredby the way plants and in particular the strawberry plantpropagate It is also referred to as the Strawberry AlgorithmWhile there is a growing body of experimental and computa-tional works that show its good behaviour and performanceagainst well-established algorithms and heuristics there isvery little if any in terms of theoretical investigationThis gaptherefore needs to be filled The aim of course in analysingthe convergence of any algorithm (in this case PPA-C) isto give confidence to the potential users that the solutionsthat it returns are of good quality The convergence analysisput forward in this paper relies on the exploitation andexploration characteristics of the algorithm Since it doesnot get stuck in local optima and explores thoroughly thesearch space it is only a matter of time before the globaloptimum is discovered The approach is probabilistic innature and ascertains that the global optimum will be foundwith probability 1 provided the algorithm is run reasonablylong enough The argument for this to hold is that at eachiteration new and better solutions are generated whichmeansthat in the limit the global optimum is reached Questionsstill remain concerning what is considered a reasonableamount of time Bounds on the time it will take to convergeare being developed and results will be presented in a follow-up paper

Appendix

Spring Design Optimisation

Themain objective of this problem [28 29] is tominimize theweight of a tensioncompression string subject to constraintsof minimum deflection shear stress surge frequency and

limits on outside diameter and on design variables Thereare three design variables the wire diameter 1199091 the meancoil diameter 1199092 and the number of active coils 1199093 [30]The mathematical formulation of this problem where 119909119879 =(1199091 1199092 1199093) is as follows

Minimize 119891 (119909) = (1199093 + 2) 119909211990921subject to 1198921 (119909) = 1 minus 119909321199093717811990941 le 0

1198922 (119909)= 411990922 minus 1199091119909212566 (119909211990931) minus 11990941 +

1510811990921 minus 1le 01198923 (119909) = 1 minus 140451199091119909221199093 le 01198924 (119909) = 1199092 + 119909115 minus 1 le 0

(A1)

The simple limits on the design variables are 005 le 1199091 le 20025 le 1199092 le 13 and 20 le 1199093 le 150Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work has been partially sponsored by ESRC GrantESL0118591

References

[1] A Zhigljavsky and A Zilinskas Stochastic Global Optimizationvol 9 Springer Science amp Business Media 2007

[2] C A Floudas and P M Pardalos Encyclopedia of OptimizationSpringer Science amp Business Media 2008

[3] Y D Sergeyev R G Strongin and D Lera Introduction toGlobal Optimization Exploiting Space-Filling Curves SpringerScience amp Business Media 2013

[4] N Brahimi A Salhi and M Ourbih-Tari ldquoConvergence of theplant propagation algorithm for continuous global optimisa-tionrdquo RAIRO Operations Research 2018

[5] J He and L Kang ldquoOn the convergence rates of geneticalgorithmsrdquo Theoretical Computer Science vol 229 no 1-2 pp23ndash39 1999

[6] W K Mashwani A Salhi M A Jan R A Khanum and MSulaiman ldquoEnhanced version of multi-algorithm geneticallyadaptive for multiobjective optimizationrdquo Advanced ComputerScience and Applications (IJACSA) vol 12 no 6 2015

[7] W K Mashwani A Salhi O Yeniay M A Jan and R AKhanum ldquoHybrid adaptive evolutionary algorithm based ondecompositionrdquo Applied Soft Computing vol 57 pp 363ndash3782017

8 Mathematical Problems in Engineering

[8] W K Mashwani A Salhi M A Jan M Sulaiman R AKhanum and A Algarni ldquoEvolutionary algorithms based ondecomposition and indicator functions state-of-the-art surveyrdquoAdvanced Computer Science and Applications (IJACSA) vol 7no 2 2016

[9] M Sulaiman A Nature-inspired Metaheuristic The Plant Prop-agation Algorithm [PhD thesis] University of Essex 2015

[10] M Sulaiman andA Salhi ldquoA hybridisation of runner-based andseed-based plant propagation algorithmsrdquo Studies in Computa-tional Intelligence vol 637 pp 195ndash215 2016

[11] M Sulaiman A Salhi E S Fraga W K Mashwani and M MRashidi ldquoA novel plant propagation algorithm modificationsand implementationsrdquo Science International vol 28 no 1 2015

[12] B I Selamoglu A Salhi and M Sulaiman ldquoStrip algorithms asan efficient way to initialise population-based metaheuristicsrdquoin Recent Developments in Metaheuristics pp 319ndash331 Springer2018

[13] M Sulaiman A Ahmad A Khan and S MuhammadldquoHybridized Symbiotic Organism Search Algorithm for theOptimal Operation of Directional Overcurrent Relaysrdquo Com-plexity vol 2018 Article ID 4605769 11 pages 2018

[14] M Sulaiman A Salhi B I Selamoglu and O B KirikchildquoA plant propagation algorithm for constrained engineeringoptimisation problemsrdquoMathematical Problems in Engineeringvol 2014 Article ID 627416 10 pages 2014

[15] A Salhi and E Fraga ldquoNature-inspired optimisationapproaches and the new plant propagation algorithmrdquo inProceedings of the International Conference on NumericalAnalysis and Optimization (ICeMATH rsquo11) pp K2ndash1ndashK2ndash8Yogyakarta Indonesia 2011

[16] M Sulaiman and A Salhi ldquoA seed-based plant propagationalgorithm the feeding station modelrdquo The Scientific WorldJournal vol 2015 Article ID 904364 16 pages 2015

[17] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 42 no1 pp 107ndash124 2012

[18] M Sulaiman and A Salhi ldquoThe 5th International Conferenceon Metaheuristics and Nature Inspired Computingrdquo in Pro-ceedings of the 5th International Conference on Metaheuristicsand Nature Inspired Computing Morocco UAE October 2014httpmeta2014sciencesconforg40158

[19] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[20] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[21] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2011

[23] M Taboga Lectures on Probability Theory and MathematicalStatistics CreateSpace Independent Pub 2012

[24] R N Bracewell and R Bracewell The Fourier Transform andIts Applications vol 31999 McGraw-Hill New York NY USA1986

[25] Z Hu S Xiong Q Su and X Zhang ldquoSufficient conditions forglobal convergence of differential evolution algorithmrdquo Journalof Applied Mathematics vol 2013 Article ID 193196 2013

[26] C Chen F Jin X Zhu and G Ouyan Mathematics AnalysisPress Higher Education Press 2000

[27] K Knopp Theory and Application of Infinite Series CourierDover Publications 2013

[28] J Arora Introduction to Optimum Design Academic Press2004

[29] A D Belegundu and J S Arora ldquoA study of mathematicalprogramming methods for structural optimization I TheoryrdquoInternational Journal for Numerical Methods in Engineering vol21 no 9 pp 1583ndash1599 1985

[30] L C Cagnina S C Esquivel and C A C Coello ldquoSolvingengineering optimization problemswith the simple constrainedparticle swarm optimizerrdquo Informatica (Slovenia) vol 32 pp319ndash326 2008

4 Mathematical Problems in Engineering

(61) end if(62)(63) end for(64)(65) end for(66)(67) Append 119865 to current population(68)(69) Sort the population in ascending order of objective values(70)(71) Update current best(72)(73) end while(74)(75) Return Updated population and global best solution(76)(77) end for(78)

Algorithm 1 PPA for constrained optimisation [14]

0 2 4 6minus15

minus10

minus5

0

5

10

15Spring design problem

Sear

ch sp

ace

times105

(a) Perturbations by (1a) (1b) and (1c)

0 5 10 15minus80

minus60

minus40

minus20

0

20

40

Sear

ch sp

ace

Spring design problem

times105

(b) Perturbations by (2)

Figure 1 Overall performance of (1a) (1b) (1c) and (2) on a design optimisation problem given in Appendix [16]

unchanged with probability 1minus119875119898Thus there are 2119899 possiblerunners to be generated for each 119894th parent plant using (1a)(1b) and (1c) where 119899 is the space dimension of the givenproblem

Let at any generation 119892 the random population berepresented as pop = 1198831119892 1198832119892 119883119873119875119892 where 119873119875denotes the population size To create a runner by using (1a)(1b) and (1c) for next generation 119892+1 PPA uses a populationof parent plants at generation 119892 for this purpose It is notrequired to know about any other runner in generation 119892+1

This shows that all runners created at generation 119892 + 1 arestatistically mutually independent Furthermore the initialpopulation is random and all parent plants do not dependon each other Thus by induction the runners at any furthergenerations are mutually independent

From (1a) (1b) and (1c) a runner 1198831015840119894 may be formedby itself or by choosing three different coordinates fromcurrent population In case of using (1a) [19] there are 119873119875possibilities to send (long or short) runners as in Figure 1On the other hand by using (1b) or (1c) 119873119875 minus 1 vectors

Mathematical Problems in Engineering 5

are used to send a new runner Thus in later cases thereare 119873119875minus11198753 possibilities to send new runners In (1a)ndash(1c)different possibilities are of the form

11988310158401119892 = 119883119894119892 + 120573 sdot 11988311989411989211988310158402119892 = 119883119894119892 + 120573 sdot 11988311989511989211988310158403119892 = 119883119894119892 + 120573 sdot 119883119896119892

(12)

where 119883119894119892 119883119895119892 and 119883119896119892 are calculated according to (1a)(1b) and (1c) in which 119892 denotes the current generation and120573 is an 119899-dimensional random vector within interval [minus1 1]The probability density functions (PDFs) [17 23] of these newvectors 119883119894119892 120573 sdot 119883119894119892 120573 sdot 119883119895119892 and 120573 sdot 119883119896119892 can be writtenas119901119883119894119892(119909119894119892)119901119883119894119892(119909119894119892120573)120573119901119883119895119892(119909119895119892120573)120573119901119883119896119892(119909119896119892120573)120573respectively

The PDFs of new runners created with (1a)ndash(1c) are givenin (15)

Definition 1 (convolution operation ⋆ [24]) Let 119891(119909) and119892(119909) be Laplace transformable piecewise continuous func-tions defined on [0 infin] The convolution product of thesetwo functions is again a function of 119909 defined as

(119891 ⋆ 119892) (119909) = int1199090119891 (120585) 119892 (119909 minus 120585) 119889120585 (13)

11990111988310158401119892 (11990910158401119892 ) = 1120573 (119901119883119894119892 (11990910158401119892 ) ⋆ 119901119883119894119892 (11990910158401119892120573 )) 11990111988310158402119892 (11990910158402119892 ) = 1120573 (119901119883119894119892 (11990910158402119892 ) ⋆ 119901119883119895119892 (11990910158402119892120573 )) 11990111988310158403119892 (11990910158403119892 ) = 1120573 (119901119883119894119892 (11990910158403119892 ) ⋆ 119901119883119896119892 (11990910158403119892120573 ))

dArr

(14)

11990111988310158401119894119892 (11990910158401119894119892)= 11205731198941119873119875 (sumsum119901119883119894119892 (11990910158401119894119892) ⋆ 119901119883119894119892 (119909101584011198941198921205731198941))

11990111988310158402119894119892 (11990910158402119894119892)= 11205731198942119873119875minus11198753 (sumsum

119894 =119895

119901119883119894119892 (11990910158402119894119892) ⋆ 119901119883119895119892 (119909101584021198941198921205731198942)) 11990111988310158403119894119892 (11990910158403119894119892)

= 11205731198943119873119875minus11198753 (sumsum119894 =119896

119901119883119894119892 (11990910158403119894119892) ⋆ 119901119883119896119892 (119909101584031198941198921205731198943))

(15)

Let 119883119892+1 be any plant in generation 119892 + 1 119883119892 a plantposition in current generation and1198831015840ℎ119892 the runner produced

by the 119894th plant 119883119894119892 at the generation 119892 Then the joint PDFof the parent plant in the next generation 119883119892+1 based on theparent119883119892 and runner1198831015840ℎ119892 where ℎ = 1 2 3 is given by

119901119883119892+1 119883119892 1198831015840ℎ119892 (119909119892+1 119909119892 1199091015840ℎ119892 )= 119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) sdot 1199011198831015840ℎ119892 |119883119892 (1199091015840ℎ119892 | 119909119892)

sdot 119901119883119892 (119909119892) sdot 1199011198831015840ℎ119892 (1199091015840ℎ119892 ) (16)

Note that a runner is selected for the next populationonly if its rank is less than or equal to 119873119875 the populationsize Its objective value is less than the maximum objectivevalue (in case of minimization problem) in the currentpopulation Note also that instead of greedy selection wesort the population and eliminate those plants whose rank ishigher than119873119875 Themodel for this selection mechanism canbe represented as follows

119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) = 120575 (119909119892+1 minus 1199091015840ℎ119892 ) forall1198831015840ℎ119892 isin 119865 forall119883119892 isin pop 119891 (119883119892) ge 119891 (1198831015840ℎ119892 ) (17)

31 ConvergenceAnalysis of PPA-C For illustration purposeswe have implemented a combined version of (1a) (1b) (1c)and (2) This version of PPA-C called H-PPA-SbPPA is ahybridisation of PPA and SbPPA [10] We have plotted theposition of plants in populations through solving the Braninand Matyas test functions (see Figures 2 3 4 and 5) Itis obvious from Figures 2 and 4 that (1a) (1b) and (1c)have generated short runners which exploit the search spacelocally On the other hand in Figures 3 and 5 (2) hasgenerated a diverse range of solutions which are spread overthe whole search space This equation helps the algorithmescape from local minima and to explore the solution spacebetter and hence the global search qualities of this algorithmMathematically this can be shown as follows

Let 119878 denote the search space containing the solution of agiven optimisation problem defined as

min 119891 (119883) | 119883 isin 119878 (18)

where 119891(119883) is the objective function Then the optimalsolution set [25] can be represented as

119878lowast= 119883optimal | 119891 (119883optimal) = min 119891 (119883) forall119883 isin 119878 (19)

where119883optimal is the optimum solution The region of attrac-tion [25] of the solution set 119878lowast is defined as

119878lowast120585 = 119883 | 119891 (119883best) minus 119891 (119883) lt 120585 (20)

where 120585 is a small positive real number and119883best is the currentbest solution

In PPA-C each parent plant produces1198831015840ℎ119892 runners (solu-tions) where ℎ = 1 2 3 The probability that at generation

6 Mathematical Problems in Engineering

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08minus1minus1

Figure 2 The exploitation capability of PPA while solving Branintest function

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08

minus1

minus1

Figure 3The exploration capability of SbPPA while solving Branintest function

119892 a subset of the temporary population 1198651015840 sub 119865 containingsolutions which are not good enough to be retained in thenext generation by the selection model as in (17) is given as

119901 1198651015840119892 cap 119878lowast120585 = le 1 minus 120585119892 (21)

where 120585119892 is a small positive real number Obviously in allprevious generations 119892 minus 1 some of the solutions died andsome succeeded to survive into the next generation Thisshows that in previous generations we have some solutions1198831015840ℎ119892 which do not belong to the region of attraction 119878lowast120585

119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (22)

At the end of each generation the temporary population119865 is appended to the main population pop119892 Then allindividuals are sorted with respect to their objective valuesThe individuals with higher rank than size of population are

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 4 A scatter plot of plants produced by PPAwhen optimisingMatyas function [15]

25

25

20

20

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10

minus15

minus15

minus20

minus20minus25

Figure 5 A scatter plot of plants produced by SbPPA whenoptimising Matyas function [16 18]

omitted Thus the probability that a generation 119892 does notcontain an optimum is given below

119901 pop119892 cap 119878lowast120585 = asymp 119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (23)

After sorting the final population at the end of eachgeneration the probability that the optimum may existin a subpopulation 1198651015840 (population of weak or dead run-nerssolutions) is less than that of the population pop119892 Thiscan be represented as

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = ge lim119892rarrinfin

119901 1198651015840119892 cap 119878lowast120585 = = 1 minus lim

119892rarrinfin119901 1198651015840119892 cap 119878lowast120585 = = 1 minus +infinprod

119892=1

(1 minus 120585119892) (24)

Mathematical Problems in Engineering 7

Following [25ndash27] the right hand side term of inequality (24)is zero if the seriessum+infin119892=1(120585119892) divergesThe convergence of PPAfollows and can be summarised in the theorem below

Theorem 2 PPA converges to the global optimumwith proba-bility 1 if it is left to run for a reasonable amount of time [1ndash3]in other words

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = = 1 (25)

Remark 3 Every population pop119892 includes some solutionswhich are in set 119878lowast120585 Remark 4 The above remark is due to the exploitationcharacteristic of PPA-C

Remark 5 Remark 3 implies that 119883best is always improvingor changing its position until the optimum is reached

Remark 6 119883best converges approximately to 119883optimal asgeneration 119892 grows

4 Conclusion

The Plant Propagation Algorithm (PPA) and its variants forcontinuous optimisation problems are getting notoriety asflexible and powerful solvers PPA is a heuristic inspiredby the way plants and in particular the strawberry plantpropagate It is also referred to as the Strawberry AlgorithmWhile there is a growing body of experimental and computa-tional works that show its good behaviour and performanceagainst well-established algorithms and heuristics there isvery little if any in terms of theoretical investigationThis gaptherefore needs to be filled The aim of course in analysingthe convergence of any algorithm (in this case PPA-C) isto give confidence to the potential users that the solutionsthat it returns are of good quality The convergence analysisput forward in this paper relies on the exploitation andexploration characteristics of the algorithm Since it doesnot get stuck in local optima and explores thoroughly thesearch space it is only a matter of time before the globaloptimum is discovered The approach is probabilistic innature and ascertains that the global optimum will be foundwith probability 1 provided the algorithm is run reasonablylong enough The argument for this to hold is that at eachiteration new and better solutions are generated whichmeansthat in the limit the global optimum is reached Questionsstill remain concerning what is considered a reasonableamount of time Bounds on the time it will take to convergeare being developed and results will be presented in a follow-up paper

Appendix

Spring Design Optimisation

Themain objective of this problem [28 29] is tominimize theweight of a tensioncompression string subject to constraintsof minimum deflection shear stress surge frequency and

limits on outside diameter and on design variables Thereare three design variables the wire diameter 1199091 the meancoil diameter 1199092 and the number of active coils 1199093 [30]The mathematical formulation of this problem where 119909119879 =(1199091 1199092 1199093) is as follows

Minimize 119891 (119909) = (1199093 + 2) 119909211990921subject to 1198921 (119909) = 1 minus 119909321199093717811990941 le 0

1198922 (119909)= 411990922 minus 1199091119909212566 (119909211990931) minus 11990941 +

1510811990921 minus 1le 01198923 (119909) = 1 minus 140451199091119909221199093 le 01198924 (119909) = 1199092 + 119909115 minus 1 le 0

(A1)

The simple limits on the design variables are 005 le 1199091 le 20025 le 1199092 le 13 and 20 le 1199093 le 150Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work has been partially sponsored by ESRC GrantESL0118591

References

[1] A Zhigljavsky and A Zilinskas Stochastic Global Optimizationvol 9 Springer Science amp Business Media 2007

[2] C A Floudas and P M Pardalos Encyclopedia of OptimizationSpringer Science amp Business Media 2008

[3] Y D Sergeyev R G Strongin and D Lera Introduction toGlobal Optimization Exploiting Space-Filling Curves SpringerScience amp Business Media 2013

[4] N Brahimi A Salhi and M Ourbih-Tari ldquoConvergence of theplant propagation algorithm for continuous global optimisa-tionrdquo RAIRO Operations Research 2018

[5] J He and L Kang ldquoOn the convergence rates of geneticalgorithmsrdquo Theoretical Computer Science vol 229 no 1-2 pp23ndash39 1999

[6] W K Mashwani A Salhi M A Jan R A Khanum and MSulaiman ldquoEnhanced version of multi-algorithm geneticallyadaptive for multiobjective optimizationrdquo Advanced ComputerScience and Applications (IJACSA) vol 12 no 6 2015

[7] W K Mashwani A Salhi O Yeniay M A Jan and R AKhanum ldquoHybrid adaptive evolutionary algorithm based ondecompositionrdquo Applied Soft Computing vol 57 pp 363ndash3782017

8 Mathematical Problems in Engineering

[8] W K Mashwani A Salhi M A Jan M Sulaiman R AKhanum and A Algarni ldquoEvolutionary algorithms based ondecomposition and indicator functions state-of-the-art surveyrdquoAdvanced Computer Science and Applications (IJACSA) vol 7no 2 2016

[9] M Sulaiman A Nature-inspired Metaheuristic The Plant Prop-agation Algorithm [PhD thesis] University of Essex 2015

[10] M Sulaiman andA Salhi ldquoA hybridisation of runner-based andseed-based plant propagation algorithmsrdquo Studies in Computa-tional Intelligence vol 637 pp 195ndash215 2016

[11] M Sulaiman A Salhi E S Fraga W K Mashwani and M MRashidi ldquoA novel plant propagation algorithm modificationsand implementationsrdquo Science International vol 28 no 1 2015

[12] B I Selamoglu A Salhi and M Sulaiman ldquoStrip algorithms asan efficient way to initialise population-based metaheuristicsrdquoin Recent Developments in Metaheuristics pp 319ndash331 Springer2018

[13] M Sulaiman A Ahmad A Khan and S MuhammadldquoHybridized Symbiotic Organism Search Algorithm for theOptimal Operation of Directional Overcurrent Relaysrdquo Com-plexity vol 2018 Article ID 4605769 11 pages 2018

[14] M Sulaiman A Salhi B I Selamoglu and O B KirikchildquoA plant propagation algorithm for constrained engineeringoptimisation problemsrdquoMathematical Problems in Engineeringvol 2014 Article ID 627416 10 pages 2014

[15] A Salhi and E Fraga ldquoNature-inspired optimisationapproaches and the new plant propagation algorithmrdquo inProceedings of the International Conference on NumericalAnalysis and Optimization (ICeMATH rsquo11) pp K2ndash1ndashK2ndash8Yogyakarta Indonesia 2011

[16] M Sulaiman and A Salhi ldquoA seed-based plant propagationalgorithm the feeding station modelrdquo The Scientific WorldJournal vol 2015 Article ID 904364 16 pages 2015

[17] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 42 no1 pp 107ndash124 2012

[18] M Sulaiman and A Salhi ldquoThe 5th International Conferenceon Metaheuristics and Nature Inspired Computingrdquo in Pro-ceedings of the 5th International Conference on Metaheuristicsand Nature Inspired Computing Morocco UAE October 2014httpmeta2014sciencesconforg40158

[19] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[20] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[21] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2011

[23] M Taboga Lectures on Probability Theory and MathematicalStatistics CreateSpace Independent Pub 2012

[24] R N Bracewell and R Bracewell The Fourier Transform andIts Applications vol 31999 McGraw-Hill New York NY USA1986

[25] Z Hu S Xiong Q Su and X Zhang ldquoSufficient conditions forglobal convergence of differential evolution algorithmrdquo Journalof Applied Mathematics vol 2013 Article ID 193196 2013

[26] C Chen F Jin X Zhu and G Ouyan Mathematics AnalysisPress Higher Education Press 2000

[27] K Knopp Theory and Application of Infinite Series CourierDover Publications 2013

[28] J Arora Introduction to Optimum Design Academic Press2004

[29] A D Belegundu and J S Arora ldquoA study of mathematicalprogramming methods for structural optimization I TheoryrdquoInternational Journal for Numerical Methods in Engineering vol21 no 9 pp 1583ndash1599 1985

[30] L C Cagnina S C Esquivel and C A C Coello ldquoSolvingengineering optimization problemswith the simple constrainedparticle swarm optimizerrdquo Informatica (Slovenia) vol 32 pp319ndash326 2008

Mathematical Problems in Engineering 5

are used to send a new runner Thus in later cases thereare 119873119875minus11198753 possibilities to send new runners In (1a)ndash(1c)different possibilities are of the form

11988310158401119892 = 119883119894119892 + 120573 sdot 11988311989411989211988310158402119892 = 119883119894119892 + 120573 sdot 11988311989511989211988310158403119892 = 119883119894119892 + 120573 sdot 119883119896119892

(12)

where 119883119894119892 119883119895119892 and 119883119896119892 are calculated according to (1a)(1b) and (1c) in which 119892 denotes the current generation and120573 is an 119899-dimensional random vector within interval [minus1 1]The probability density functions (PDFs) [17 23] of these newvectors 119883119894119892 120573 sdot 119883119894119892 120573 sdot 119883119895119892 and 120573 sdot 119883119896119892 can be writtenas119901119883119894119892(119909119894119892)119901119883119894119892(119909119894119892120573)120573119901119883119895119892(119909119895119892120573)120573119901119883119896119892(119909119896119892120573)120573respectively

The PDFs of new runners created with (1a)ndash(1c) are givenin (15)

Definition 1 (convolution operation ⋆ [24]) Let 119891(119909) and119892(119909) be Laplace transformable piecewise continuous func-tions defined on [0 infin] The convolution product of thesetwo functions is again a function of 119909 defined as

(119891 ⋆ 119892) (119909) = int1199090119891 (120585) 119892 (119909 minus 120585) 119889120585 (13)

11990111988310158401119892 (11990910158401119892 ) = 1120573 (119901119883119894119892 (11990910158401119892 ) ⋆ 119901119883119894119892 (11990910158401119892120573 )) 11990111988310158402119892 (11990910158402119892 ) = 1120573 (119901119883119894119892 (11990910158402119892 ) ⋆ 119901119883119895119892 (11990910158402119892120573 )) 11990111988310158403119892 (11990910158403119892 ) = 1120573 (119901119883119894119892 (11990910158403119892 ) ⋆ 119901119883119896119892 (11990910158403119892120573 ))

dArr

(14)

11990111988310158401119894119892 (11990910158401119894119892)= 11205731198941119873119875 (sumsum119901119883119894119892 (11990910158401119894119892) ⋆ 119901119883119894119892 (119909101584011198941198921205731198941))

11990111988310158402119894119892 (11990910158402119894119892)= 11205731198942119873119875minus11198753 (sumsum

119894 =119895

119901119883119894119892 (11990910158402119894119892) ⋆ 119901119883119895119892 (119909101584021198941198921205731198942)) 11990111988310158403119894119892 (11990910158403119894119892)

= 11205731198943119873119875minus11198753 (sumsum119894 =119896

119901119883119894119892 (11990910158403119894119892) ⋆ 119901119883119896119892 (119909101584031198941198921205731198943))

(15)

Let 119883119892+1 be any plant in generation 119892 + 1 119883119892 a plantposition in current generation and1198831015840ℎ119892 the runner produced

by the 119894th plant 119883119894119892 at the generation 119892 Then the joint PDFof the parent plant in the next generation 119883119892+1 based on theparent119883119892 and runner1198831015840ℎ119892 where ℎ = 1 2 3 is given by

119901119883119892+1 119883119892 1198831015840ℎ119892 (119909119892+1 119909119892 1199091015840ℎ119892 )= 119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) sdot 1199011198831015840ℎ119892 |119883119892 (1199091015840ℎ119892 | 119909119892)

sdot 119901119883119892 (119909119892) sdot 1199011198831015840ℎ119892 (1199091015840ℎ119892 ) (16)

Note that a runner is selected for the next populationonly if its rank is less than or equal to 119873119875 the populationsize Its objective value is less than the maximum objectivevalue (in case of minimization problem) in the currentpopulation Note also that instead of greedy selection wesort the population and eliminate those plants whose rank ishigher than119873119875 Themodel for this selection mechanism canbe represented as follows

119901119883119892+1|119883119892 1198831015840ℎ119892 (119909119892+1 | 119909119892 1199091015840ℎ119892 ) = 120575 (119909119892+1 minus 1199091015840ℎ119892 ) forall1198831015840ℎ119892 isin 119865 forall119883119892 isin pop 119891 (119883119892) ge 119891 (1198831015840ℎ119892 ) (17)

31 ConvergenceAnalysis of PPA-C For illustration purposeswe have implemented a combined version of (1a) (1b) (1c)and (2) This version of PPA-C called H-PPA-SbPPA is ahybridisation of PPA and SbPPA [10] We have plotted theposition of plants in populations through solving the Braninand Matyas test functions (see Figures 2 3 4 and 5) Itis obvious from Figures 2 and 4 that (1a) (1b) and (1c)have generated short runners which exploit the search spacelocally On the other hand in Figures 3 and 5 (2) hasgenerated a diverse range of solutions which are spread overthe whole search space This equation helps the algorithmescape from local minima and to explore the solution spacebetter and hence the global search qualities of this algorithmMathematically this can be shown as follows

Let 119878 denote the search space containing the solution of agiven optimisation problem defined as

min 119891 (119883) | 119883 isin 119878 (18)

where 119891(119883) is the objective function Then the optimalsolution set [25] can be represented as

119878lowast= 119883optimal | 119891 (119883optimal) = min 119891 (119883) forall119883 isin 119878 (19)

where119883optimal is the optimum solution The region of attrac-tion [25] of the solution set 119878lowast is defined as

119878lowast120585 = 119883 | 119891 (119883best) minus 119891 (119883) lt 120585 (20)

where 120585 is a small positive real number and119883best is the currentbest solution

In PPA-C each parent plant produces1198831015840ℎ119892 runners (solu-tions) where ℎ = 1 2 3 The probability that at generation

6 Mathematical Problems in Engineering

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08minus1minus1

Figure 2 The exploitation capability of PPA while solving Branintest function

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08

minus1

minus1

Figure 3The exploration capability of SbPPA while solving Branintest function

119892 a subset of the temporary population 1198651015840 sub 119865 containingsolutions which are not good enough to be retained in thenext generation by the selection model as in (17) is given as

119901 1198651015840119892 cap 119878lowast120585 = le 1 minus 120585119892 (21)

where 120585119892 is a small positive real number Obviously in allprevious generations 119892 minus 1 some of the solutions died andsome succeeded to survive into the next generation Thisshows that in previous generations we have some solutions1198831015840ℎ119892 which do not belong to the region of attraction 119878lowast120585

119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (22)

At the end of each generation the temporary population119865 is appended to the main population pop119892 Then allindividuals are sorted with respect to their objective valuesThe individuals with higher rank than size of population are

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 4 A scatter plot of plants produced by PPAwhen optimisingMatyas function [15]

25

25

20

20

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10

minus15

minus15

minus20

minus20minus25

Figure 5 A scatter plot of plants produced by SbPPA whenoptimising Matyas function [16 18]

omitted Thus the probability that a generation 119892 does notcontain an optimum is given below

119901 pop119892 cap 119878lowast120585 = asymp 119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (23)

After sorting the final population at the end of eachgeneration the probability that the optimum may existin a subpopulation 1198651015840 (population of weak or dead run-nerssolutions) is less than that of the population pop119892 Thiscan be represented as

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = ge lim119892rarrinfin

119901 1198651015840119892 cap 119878lowast120585 = = 1 minus lim

119892rarrinfin119901 1198651015840119892 cap 119878lowast120585 = = 1 minus +infinprod

119892=1

(1 minus 120585119892) (24)

Mathematical Problems in Engineering 7

Following [25ndash27] the right hand side term of inequality (24)is zero if the seriessum+infin119892=1(120585119892) divergesThe convergence of PPAfollows and can be summarised in the theorem below

Theorem 2 PPA converges to the global optimumwith proba-bility 1 if it is left to run for a reasonable amount of time [1ndash3]in other words

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = = 1 (25)

Remark 3 Every population pop119892 includes some solutionswhich are in set 119878lowast120585 Remark 4 The above remark is due to the exploitationcharacteristic of PPA-C

Remark 5 Remark 3 implies that 119883best is always improvingor changing its position until the optimum is reached

Remark 6 119883best converges approximately to 119883optimal asgeneration 119892 grows

4 Conclusion

The Plant Propagation Algorithm (PPA) and its variants forcontinuous optimisation problems are getting notoriety asflexible and powerful solvers PPA is a heuristic inspiredby the way plants and in particular the strawberry plantpropagate It is also referred to as the Strawberry AlgorithmWhile there is a growing body of experimental and computa-tional works that show its good behaviour and performanceagainst well-established algorithms and heuristics there isvery little if any in terms of theoretical investigationThis gaptherefore needs to be filled The aim of course in analysingthe convergence of any algorithm (in this case PPA-C) isto give confidence to the potential users that the solutionsthat it returns are of good quality The convergence analysisput forward in this paper relies on the exploitation andexploration characteristics of the algorithm Since it doesnot get stuck in local optima and explores thoroughly thesearch space it is only a matter of time before the globaloptimum is discovered The approach is probabilistic innature and ascertains that the global optimum will be foundwith probability 1 provided the algorithm is run reasonablylong enough The argument for this to hold is that at eachiteration new and better solutions are generated whichmeansthat in the limit the global optimum is reached Questionsstill remain concerning what is considered a reasonableamount of time Bounds on the time it will take to convergeare being developed and results will be presented in a follow-up paper

Appendix

Spring Design Optimisation

Themain objective of this problem [28 29] is tominimize theweight of a tensioncompression string subject to constraintsof minimum deflection shear stress surge frequency and

limits on outside diameter and on design variables Thereare three design variables the wire diameter 1199091 the meancoil diameter 1199092 and the number of active coils 1199093 [30]The mathematical formulation of this problem where 119909119879 =(1199091 1199092 1199093) is as follows

Minimize 119891 (119909) = (1199093 + 2) 119909211990921subject to 1198921 (119909) = 1 minus 119909321199093717811990941 le 0

1198922 (119909)= 411990922 minus 1199091119909212566 (119909211990931) minus 11990941 +

1510811990921 minus 1le 01198923 (119909) = 1 minus 140451199091119909221199093 le 01198924 (119909) = 1199092 + 119909115 minus 1 le 0

(A1)

The simple limits on the design variables are 005 le 1199091 le 20025 le 1199092 le 13 and 20 le 1199093 le 150Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work has been partially sponsored by ESRC GrantESL0118591

References

[1] A Zhigljavsky and A Zilinskas Stochastic Global Optimizationvol 9 Springer Science amp Business Media 2007

[2] C A Floudas and P M Pardalos Encyclopedia of OptimizationSpringer Science amp Business Media 2008

[3] Y D Sergeyev R G Strongin and D Lera Introduction toGlobal Optimization Exploiting Space-Filling Curves SpringerScience amp Business Media 2013

[4] N Brahimi A Salhi and M Ourbih-Tari ldquoConvergence of theplant propagation algorithm for continuous global optimisa-tionrdquo RAIRO Operations Research 2018

[5] J He and L Kang ldquoOn the convergence rates of geneticalgorithmsrdquo Theoretical Computer Science vol 229 no 1-2 pp23ndash39 1999

[6] W K Mashwani A Salhi M A Jan R A Khanum and MSulaiman ldquoEnhanced version of multi-algorithm geneticallyadaptive for multiobjective optimizationrdquo Advanced ComputerScience and Applications (IJACSA) vol 12 no 6 2015

[7] W K Mashwani A Salhi O Yeniay M A Jan and R AKhanum ldquoHybrid adaptive evolutionary algorithm based ondecompositionrdquo Applied Soft Computing vol 57 pp 363ndash3782017

8 Mathematical Problems in Engineering

[8] W K Mashwani A Salhi M A Jan M Sulaiman R AKhanum and A Algarni ldquoEvolutionary algorithms based ondecomposition and indicator functions state-of-the-art surveyrdquoAdvanced Computer Science and Applications (IJACSA) vol 7no 2 2016

[9] M Sulaiman A Nature-inspired Metaheuristic The Plant Prop-agation Algorithm [PhD thesis] University of Essex 2015

[10] M Sulaiman andA Salhi ldquoA hybridisation of runner-based andseed-based plant propagation algorithmsrdquo Studies in Computa-tional Intelligence vol 637 pp 195ndash215 2016

[11] M Sulaiman A Salhi E S Fraga W K Mashwani and M MRashidi ldquoA novel plant propagation algorithm modificationsand implementationsrdquo Science International vol 28 no 1 2015

[12] B I Selamoglu A Salhi and M Sulaiman ldquoStrip algorithms asan efficient way to initialise population-based metaheuristicsrdquoin Recent Developments in Metaheuristics pp 319ndash331 Springer2018

[13] M Sulaiman A Ahmad A Khan and S MuhammadldquoHybridized Symbiotic Organism Search Algorithm for theOptimal Operation of Directional Overcurrent Relaysrdquo Com-plexity vol 2018 Article ID 4605769 11 pages 2018

[14] M Sulaiman A Salhi B I Selamoglu and O B KirikchildquoA plant propagation algorithm for constrained engineeringoptimisation problemsrdquoMathematical Problems in Engineeringvol 2014 Article ID 627416 10 pages 2014

[15] A Salhi and E Fraga ldquoNature-inspired optimisationapproaches and the new plant propagation algorithmrdquo inProceedings of the International Conference on NumericalAnalysis and Optimization (ICeMATH rsquo11) pp K2ndash1ndashK2ndash8Yogyakarta Indonesia 2011

[16] M Sulaiman and A Salhi ldquoA seed-based plant propagationalgorithm the feeding station modelrdquo The Scientific WorldJournal vol 2015 Article ID 904364 16 pages 2015

[17] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 42 no1 pp 107ndash124 2012

[18] M Sulaiman and A Salhi ldquoThe 5th International Conferenceon Metaheuristics and Nature Inspired Computingrdquo in Pro-ceedings of the 5th International Conference on Metaheuristicsand Nature Inspired Computing Morocco UAE October 2014httpmeta2014sciencesconforg40158

[19] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[20] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[21] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2011

[23] M Taboga Lectures on Probability Theory and MathematicalStatistics CreateSpace Independent Pub 2012

[24] R N Bracewell and R Bracewell The Fourier Transform andIts Applications vol 31999 McGraw-Hill New York NY USA1986

[25] Z Hu S Xiong Q Su and X Zhang ldquoSufficient conditions forglobal convergence of differential evolution algorithmrdquo Journalof Applied Mathematics vol 2013 Article ID 193196 2013

[26] C Chen F Jin X Zhu and G Ouyan Mathematics AnalysisPress Higher Education Press 2000

[27] K Knopp Theory and Application of Infinite Series CourierDover Publications 2013

[28] J Arora Introduction to Optimum Design Academic Press2004

[29] A D Belegundu and J S Arora ldquoA study of mathematicalprogramming methods for structural optimization I TheoryrdquoInternational Journal for Numerical Methods in Engineering vol21 no 9 pp 1583ndash1599 1985

[30] L C Cagnina S C Esquivel and C A C Coello ldquoSolvingengineering optimization problemswith the simple constrainedparticle swarm optimizerrdquo Informatica (Slovenia) vol 32 pp319ndash326 2008

6 Mathematical Problems in Engineering

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08minus1minus1

Figure 2 The exploitation capability of PPA while solving Branintest function

1

1

08

08

06

06

04

04

02

02

0

0

minus02

minus02

minus04

minus04

minus06

minus06

minus08

minus08

minus1

minus1

Figure 3The exploration capability of SbPPA while solving Branintest function

119892 a subset of the temporary population 1198651015840 sub 119865 containingsolutions which are not good enough to be retained in thenext generation by the selection model as in (17) is given as

119901 1198651015840119892 cap 119878lowast120585 = le 1 minus 120585119892 (21)

where 120585119892 is a small positive real number Obviously in allprevious generations 119892 minus 1 some of the solutions died andsome succeeded to survive into the next generation Thisshows that in previous generations we have some solutions1198831015840ℎ119892 which do not belong to the region of attraction 119878lowast120585

119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (22)

At the end of each generation the temporary population119865 is appended to the main population pop119892 Then allindividuals are sorted with respect to their objective valuesThe individuals with higher rank than size of population are

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10minus15

Figure 4 A scatter plot of plants produced by PPAwhen optimisingMatyas function [15]

25

25

20

20

15

15

10

10

5

5

0

0

minus5

minus5

minus10

minus10

minus15

minus15

minus20

minus20minus25

Figure 5 A scatter plot of plants produced by SbPPA whenoptimising Matyas function [16 18]

omitted Thus the probability that a generation 119892 does notcontain an optimum is given below

119901 pop119892 cap 119878lowast120585 = asymp 119892minus1prod119894=1

119901 1198651015840119894 cap 119878lowast120585 = le 119892minus1prod119895=1

(1 minus 120585119895) (23)

After sorting the final population at the end of eachgeneration the probability that the optimum may existin a subpopulation 1198651015840 (population of weak or dead run-nerssolutions) is less than that of the population pop119892 Thiscan be represented as

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = ge lim119892rarrinfin

119901 1198651015840119892 cap 119878lowast120585 = = 1 minus lim

119892rarrinfin119901 1198651015840119892 cap 119878lowast120585 = = 1 minus +infinprod

119892=1

(1 minus 120585119892) (24)

Mathematical Problems in Engineering 7

Following [25ndash27] the right hand side term of inequality (24)is zero if the seriessum+infin119892=1(120585119892) divergesThe convergence of PPAfollows and can be summarised in the theorem below

Theorem 2 PPA converges to the global optimumwith proba-bility 1 if it is left to run for a reasonable amount of time [1ndash3]in other words

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = = 1 (25)

Remark 3 Every population pop119892 includes some solutionswhich are in set 119878lowast120585 Remark 4 The above remark is due to the exploitationcharacteristic of PPA-C

Remark 5 Remark 3 implies that 119883best is always improvingor changing its position until the optimum is reached

Remark 6 119883best converges approximately to 119883optimal asgeneration 119892 grows

4 Conclusion

The Plant Propagation Algorithm (PPA) and its variants forcontinuous optimisation problems are getting notoriety asflexible and powerful solvers PPA is a heuristic inspiredby the way plants and in particular the strawberry plantpropagate It is also referred to as the Strawberry AlgorithmWhile there is a growing body of experimental and computa-tional works that show its good behaviour and performanceagainst well-established algorithms and heuristics there isvery little if any in terms of theoretical investigationThis gaptherefore needs to be filled The aim of course in analysingthe convergence of any algorithm (in this case PPA-C) isto give confidence to the potential users that the solutionsthat it returns are of good quality The convergence analysisput forward in this paper relies on the exploitation andexploration characteristics of the algorithm Since it doesnot get stuck in local optima and explores thoroughly thesearch space it is only a matter of time before the globaloptimum is discovered The approach is probabilistic innature and ascertains that the global optimum will be foundwith probability 1 provided the algorithm is run reasonablylong enough The argument for this to hold is that at eachiteration new and better solutions are generated whichmeansthat in the limit the global optimum is reached Questionsstill remain concerning what is considered a reasonableamount of time Bounds on the time it will take to convergeare being developed and results will be presented in a follow-up paper

Appendix

Spring Design Optimisation

Themain objective of this problem [28 29] is tominimize theweight of a tensioncompression string subject to constraintsof minimum deflection shear stress surge frequency and

limits on outside diameter and on design variables Thereare three design variables the wire diameter 1199091 the meancoil diameter 1199092 and the number of active coils 1199093 [30]The mathematical formulation of this problem where 119909119879 =(1199091 1199092 1199093) is as follows

Minimize 119891 (119909) = (1199093 + 2) 119909211990921subject to 1198921 (119909) = 1 minus 119909321199093717811990941 le 0

1198922 (119909)= 411990922 minus 1199091119909212566 (119909211990931) minus 11990941 +

1510811990921 minus 1le 01198923 (119909) = 1 minus 140451199091119909221199093 le 01198924 (119909) = 1199092 + 119909115 minus 1 le 0

(A1)

The simple limits on the design variables are 005 le 1199091 le 20025 le 1199092 le 13 and 20 le 1199093 le 150Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work has been partially sponsored by ESRC GrantESL0118591

References

[1] A Zhigljavsky and A Zilinskas Stochastic Global Optimizationvol 9 Springer Science amp Business Media 2007

[2] C A Floudas and P M Pardalos Encyclopedia of OptimizationSpringer Science amp Business Media 2008

[3] Y D Sergeyev R G Strongin and D Lera Introduction toGlobal Optimization Exploiting Space-Filling Curves SpringerScience amp Business Media 2013

[4] N Brahimi A Salhi and M Ourbih-Tari ldquoConvergence of theplant propagation algorithm for continuous global optimisa-tionrdquo RAIRO Operations Research 2018

[5] J He and L Kang ldquoOn the convergence rates of geneticalgorithmsrdquo Theoretical Computer Science vol 229 no 1-2 pp23ndash39 1999

[6] W K Mashwani A Salhi M A Jan R A Khanum and MSulaiman ldquoEnhanced version of multi-algorithm geneticallyadaptive for multiobjective optimizationrdquo Advanced ComputerScience and Applications (IJACSA) vol 12 no 6 2015

[7] W K Mashwani A Salhi O Yeniay M A Jan and R AKhanum ldquoHybrid adaptive evolutionary algorithm based ondecompositionrdquo Applied Soft Computing vol 57 pp 363ndash3782017

8 Mathematical Problems in Engineering

[8] W K Mashwani A Salhi M A Jan M Sulaiman R AKhanum and A Algarni ldquoEvolutionary algorithms based ondecomposition and indicator functions state-of-the-art surveyrdquoAdvanced Computer Science and Applications (IJACSA) vol 7no 2 2016

[9] M Sulaiman A Nature-inspired Metaheuristic The Plant Prop-agation Algorithm [PhD thesis] University of Essex 2015

[10] M Sulaiman andA Salhi ldquoA hybridisation of runner-based andseed-based plant propagation algorithmsrdquo Studies in Computa-tional Intelligence vol 637 pp 195ndash215 2016

[11] M Sulaiman A Salhi E S Fraga W K Mashwani and M MRashidi ldquoA novel plant propagation algorithm modificationsand implementationsrdquo Science International vol 28 no 1 2015

[12] B I Selamoglu A Salhi and M Sulaiman ldquoStrip algorithms asan efficient way to initialise population-based metaheuristicsrdquoin Recent Developments in Metaheuristics pp 319ndash331 Springer2018

[13] M Sulaiman A Ahmad A Khan and S MuhammadldquoHybridized Symbiotic Organism Search Algorithm for theOptimal Operation of Directional Overcurrent Relaysrdquo Com-plexity vol 2018 Article ID 4605769 11 pages 2018

[14] M Sulaiman A Salhi B I Selamoglu and O B KirikchildquoA plant propagation algorithm for constrained engineeringoptimisation problemsrdquoMathematical Problems in Engineeringvol 2014 Article ID 627416 10 pages 2014

[15] A Salhi and E Fraga ldquoNature-inspired optimisationapproaches and the new plant propagation algorithmrdquo inProceedings of the International Conference on NumericalAnalysis and Optimization (ICeMATH rsquo11) pp K2ndash1ndashK2ndash8Yogyakarta Indonesia 2011

[16] M Sulaiman and A Salhi ldquoA seed-based plant propagationalgorithm the feeding station modelrdquo The Scientific WorldJournal vol 2015 Article ID 904364 16 pages 2015

[17] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 42 no1 pp 107ndash124 2012

[18] M Sulaiman and A Salhi ldquoThe 5th International Conferenceon Metaheuristics and Nature Inspired Computingrdquo in Pro-ceedings of the 5th International Conference on Metaheuristicsand Nature Inspired Computing Morocco UAE October 2014httpmeta2014sciencesconforg40158

[19] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[20] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[21] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2011

[23] M Taboga Lectures on Probability Theory and MathematicalStatistics CreateSpace Independent Pub 2012

[24] R N Bracewell and R Bracewell The Fourier Transform andIts Applications vol 31999 McGraw-Hill New York NY USA1986

[25] Z Hu S Xiong Q Su and X Zhang ldquoSufficient conditions forglobal convergence of differential evolution algorithmrdquo Journalof Applied Mathematics vol 2013 Article ID 193196 2013

[26] C Chen F Jin X Zhu and G Ouyan Mathematics AnalysisPress Higher Education Press 2000

[27] K Knopp Theory and Application of Infinite Series CourierDover Publications 2013

[28] J Arora Introduction to Optimum Design Academic Press2004

[29] A D Belegundu and J S Arora ldquoA study of mathematicalprogramming methods for structural optimization I TheoryrdquoInternational Journal for Numerical Methods in Engineering vol21 no 9 pp 1583ndash1599 1985

[30] L C Cagnina S C Esquivel and C A C Coello ldquoSolvingengineering optimization problemswith the simple constrainedparticle swarm optimizerrdquo Informatica (Slovenia) vol 32 pp319ndash326 2008

Mathematical Problems in Engineering 7

Following [25ndash27] the right hand side term of inequality (24)is zero if the seriessum+infin119892=1(120585119892) divergesThe convergence of PPAfollows and can be summarised in the theorem below

Theorem 2 PPA converges to the global optimumwith proba-bility 1 if it is left to run for a reasonable amount of time [1ndash3]in other words

lim119892rarrinfin

119901 pop119892 cap 119878lowast120585 = = 1 (25)

Remark 3 Every population pop119892 includes some solutionswhich are in set 119878lowast120585 Remark 4 The above remark is due to the exploitationcharacteristic of PPA-C

Remark 5 Remark 3 implies that 119883best is always improvingor changing its position until the optimum is reached

Remark 6 119883best converges approximately to 119883optimal asgeneration 119892 grows

4 Conclusion

The Plant Propagation Algorithm (PPA) and its variants forcontinuous optimisation problems are getting notoriety asflexible and powerful solvers PPA is a heuristic inspiredby the way plants and in particular the strawberry plantpropagate It is also referred to as the Strawberry AlgorithmWhile there is a growing body of experimental and computa-tional works that show its good behaviour and performanceagainst well-established algorithms and heuristics there isvery little if any in terms of theoretical investigationThis gaptherefore needs to be filled The aim of course in analysingthe convergence of any algorithm (in this case PPA-C) isto give confidence to the potential users that the solutionsthat it returns are of good quality The convergence analysisput forward in this paper relies on the exploitation andexploration characteristics of the algorithm Since it doesnot get stuck in local optima and explores thoroughly thesearch space it is only a matter of time before the globaloptimum is discovered The approach is probabilistic innature and ascertains that the global optimum will be foundwith probability 1 provided the algorithm is run reasonablylong enough The argument for this to hold is that at eachiteration new and better solutions are generated whichmeansthat in the limit the global optimum is reached Questionsstill remain concerning what is considered a reasonableamount of time Bounds on the time it will take to convergeare being developed and results will be presented in a follow-up paper

Appendix

Spring Design Optimisation

Themain objective of this problem [28 29] is tominimize theweight of a tensioncompression string subject to constraintsof minimum deflection shear stress surge frequency and

limits on outside diameter and on design variables Thereare three design variables the wire diameter 1199091 the meancoil diameter 1199092 and the number of active coils 1199093 [30]The mathematical formulation of this problem where 119909119879 =(1199091 1199092 1199093) is as follows

Minimize 119891 (119909) = (1199093 + 2) 119909211990921subject to 1198921 (119909) = 1 minus 119909321199093717811990941 le 0

1198922 (119909)= 411990922 minus 1199091119909212566 (119909211990931) minus 11990941 +

1510811990921 minus 1le 01198923 (119909) = 1 minus 140451199091119909221199093 le 01198924 (119909) = 1199092 + 119909115 minus 1 le 0

(A1)

The simple limits on the design variables are 005 le 1199091 le 20025 le 1199092 le 13 and 20 le 1199093 le 150Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work has been partially sponsored by ESRC GrantESL0118591

References

[1] A Zhigljavsky and A Zilinskas Stochastic Global Optimizationvol 9 Springer Science amp Business Media 2007

[2] C A Floudas and P M Pardalos Encyclopedia of OptimizationSpringer Science amp Business Media 2008

[3] Y D Sergeyev R G Strongin and D Lera Introduction toGlobal Optimization Exploiting Space-Filling Curves SpringerScience amp Business Media 2013

[4] N Brahimi A Salhi and M Ourbih-Tari ldquoConvergence of theplant propagation algorithm for continuous global optimisa-tionrdquo RAIRO Operations Research 2018

[5] J He and L Kang ldquoOn the convergence rates of geneticalgorithmsrdquo Theoretical Computer Science vol 229 no 1-2 pp23ndash39 1999

[6] W K Mashwani A Salhi M A Jan R A Khanum and MSulaiman ldquoEnhanced version of multi-algorithm geneticallyadaptive for multiobjective optimizationrdquo Advanced ComputerScience and Applications (IJACSA) vol 12 no 6 2015

[7] W K Mashwani A Salhi O Yeniay M A Jan and R AKhanum ldquoHybrid adaptive evolutionary algorithm based ondecompositionrdquo Applied Soft Computing vol 57 pp 363ndash3782017

8 Mathematical Problems in Engineering

[8] W K Mashwani A Salhi M A Jan M Sulaiman R AKhanum and A Algarni ldquoEvolutionary algorithms based ondecomposition and indicator functions state-of-the-art surveyrdquoAdvanced Computer Science and Applications (IJACSA) vol 7no 2 2016

[9] M Sulaiman A Nature-inspired Metaheuristic The Plant Prop-agation Algorithm [PhD thesis] University of Essex 2015

[10] M Sulaiman andA Salhi ldquoA hybridisation of runner-based andseed-based plant propagation algorithmsrdquo Studies in Computa-tional Intelligence vol 637 pp 195ndash215 2016

[11] M Sulaiman A Salhi E S Fraga W K Mashwani and M MRashidi ldquoA novel plant propagation algorithm modificationsand implementationsrdquo Science International vol 28 no 1 2015

[12] B I Selamoglu A Salhi and M Sulaiman ldquoStrip algorithms asan efficient way to initialise population-based metaheuristicsrdquoin Recent Developments in Metaheuristics pp 319ndash331 Springer2018

[13] M Sulaiman A Ahmad A Khan and S MuhammadldquoHybridized Symbiotic Organism Search Algorithm for theOptimal Operation of Directional Overcurrent Relaysrdquo Com-plexity vol 2018 Article ID 4605769 11 pages 2018

[14] M Sulaiman A Salhi B I Selamoglu and O B KirikchildquoA plant propagation algorithm for constrained engineeringoptimisation problemsrdquoMathematical Problems in Engineeringvol 2014 Article ID 627416 10 pages 2014

[15] A Salhi and E Fraga ldquoNature-inspired optimisationapproaches and the new plant propagation algorithmrdquo inProceedings of the International Conference on NumericalAnalysis and Optimization (ICeMATH rsquo11) pp K2ndash1ndashK2ndash8Yogyakarta Indonesia 2011

[16] M Sulaiman and A Salhi ldquoA seed-based plant propagationalgorithm the feeding station modelrdquo The Scientific WorldJournal vol 2015 Article ID 904364 16 pages 2015

[17] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 42 no1 pp 107ndash124 2012

[18] M Sulaiman and A Salhi ldquoThe 5th International Conferenceon Metaheuristics and Nature Inspired Computingrdquo in Pro-ceedings of the 5th International Conference on Metaheuristicsand Nature Inspired Computing Morocco UAE October 2014httpmeta2014sciencesconforg40158

[19] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[20] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[21] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2011

[23] M Taboga Lectures on Probability Theory and MathematicalStatistics CreateSpace Independent Pub 2012

[24] R N Bracewell and R Bracewell The Fourier Transform andIts Applications vol 31999 McGraw-Hill New York NY USA1986

[25] Z Hu S Xiong Q Su and X Zhang ldquoSufficient conditions forglobal convergence of differential evolution algorithmrdquo Journalof Applied Mathematics vol 2013 Article ID 193196 2013

[26] C Chen F Jin X Zhu and G Ouyan Mathematics AnalysisPress Higher Education Press 2000

[27] K Knopp Theory and Application of Infinite Series CourierDover Publications 2013

[28] J Arora Introduction to Optimum Design Academic Press2004

[29] A D Belegundu and J S Arora ldquoA study of mathematicalprogramming methods for structural optimization I TheoryrdquoInternational Journal for Numerical Methods in Engineering vol21 no 9 pp 1583ndash1599 1985

[30] L C Cagnina S C Esquivel and C A C Coello ldquoSolvingengineering optimization problemswith the simple constrainedparticle swarm optimizerrdquo Informatica (Slovenia) vol 32 pp319ndash326 2008

8 Mathematical Problems in Engineering

[8] W K Mashwani A Salhi M A Jan M Sulaiman R AKhanum and A Algarni ldquoEvolutionary algorithms based ondecomposition and indicator functions state-of-the-art surveyrdquoAdvanced Computer Science and Applications (IJACSA) vol 7no 2 2016

[9] M Sulaiman A Nature-inspired Metaheuristic The Plant Prop-agation Algorithm [PhD thesis] University of Essex 2015

[10] M Sulaiman andA Salhi ldquoA hybridisation of runner-based andseed-based plant propagation algorithmsrdquo Studies in Computa-tional Intelligence vol 637 pp 195ndash215 2016

[11] M Sulaiman A Salhi E S Fraga W K Mashwani and M MRashidi ldquoA novel plant propagation algorithm modificationsand implementationsrdquo Science International vol 28 no 1 2015

[12] B I Selamoglu A Salhi and M Sulaiman ldquoStrip algorithms asan efficient way to initialise population-based metaheuristicsrdquoin Recent Developments in Metaheuristics pp 319ndash331 Springer2018

[13] M Sulaiman A Ahmad A Khan and S MuhammadldquoHybridized Symbiotic Organism Search Algorithm for theOptimal Operation of Directional Overcurrent Relaysrdquo Com-plexity vol 2018 Article ID 4605769 11 pages 2018

[14] M Sulaiman A Salhi B I Selamoglu and O B KirikchildquoA plant propagation algorithm for constrained engineeringoptimisation problemsrdquoMathematical Problems in Engineeringvol 2014 Article ID 627416 10 pages 2014

[15] A Salhi and E Fraga ldquoNature-inspired optimisationapproaches and the new plant propagation algorithmrdquo inProceedings of the International Conference on NumericalAnalysis and Optimization (ICeMATH rsquo11) pp K2ndash1ndashK2ndash8Yogyakarta Indonesia 2011

[16] M Sulaiman and A Salhi ldquoA seed-based plant propagationalgorithm the feeding station modelrdquo The Scientific WorldJournal vol 2015 Article ID 904364 16 pages 2015

[17] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 42 no1 pp 107ndash124 2012

[18] M Sulaiman and A Salhi ldquoThe 5th International Conferenceon Metaheuristics and Nature Inspired Computingrdquo in Pro-ceedings of the 5th International Conference on Metaheuristicsand Nature Inspired Computing Morocco UAE October 2014httpmeta2014sciencesconforg40158

[19] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[20] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008

[21] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2011

[23] M Taboga Lectures on Probability Theory and MathematicalStatistics CreateSpace Independent Pub 2012

[24] R N Bracewell and R Bracewell The Fourier Transform andIts Applications vol 31999 McGraw-Hill New York NY USA1986

[25] Z Hu S Xiong Q Su and X Zhang ldquoSufficient conditions forglobal convergence of differential evolution algorithmrdquo Journalof Applied Mathematics vol 2013 Article ID 193196 2013

[26] C Chen F Jin X Zhu and G Ouyan Mathematics AnalysisPress Higher Education Press 2000

[27] K Knopp Theory and Application of Infinite Series CourierDover Publications 2013

[28] J Arora Introduction to Optimum Design Academic Press2004

[29] A D Belegundu and J S Arora ldquoA study of mathematicalprogramming methods for structural optimization I TheoryrdquoInternational Journal for Numerical Methods in Engineering vol21 no 9 pp 1583ndash1599 1985

[30] L C Cagnina S C Esquivel and C A C Coello ldquoSolvingengineering optimization problemswith the simple constrainedparticle swarm optimizerrdquo Informatica (Slovenia) vol 32 pp319ndash326 2008

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