The seasonal production of fruit and seeds is akin to opening a feeding station, such as a restaurant. Agents coming to feed on the fruit are like customers attending the restaurant; they arrive at a certain rate and get served at a certain rate following some appropriate processes. The same applies to birds and animals visiting and feeding on ripe fruit produced by plants such as the strawberry plant. This phenomenon underpins the seed dispersion of the plants. Modelling it as a queuing process results in a seedbased search/optimisation algorithm. This variant of the Plant Propagation Algorithm is described, analysed, tested on nontrivial problems, and compared with well established algorithms. The results are included.
Plants have evolved a variety of ways to propagate. Propagation with seeds is perhaps the most common of them all and one which takes advantage of all sorts of agents ranging from wind to water, birds, and animals. In [
Plants rely heavily on the dispersion of their seeds to colonise new territories and to improve their survival [
Strawberry plant propagation: through seed dispersion [
Strawberry fruit with seeds
Strawberry flower
A strawberry eaten by bird(s)
A bird eating strawberries
Strawberry plants showing runners and seeded fruit
There are many biologically inspired optimization algorithms in the literature [
The Plant Propagation Algorithm (PPA) also known as the strawberry algorithm was inspired by the way plants and specifically the strawberry plants propagate using runners [
SbPPA is tested on both unconstrained and constrained benchmark problems also used in [
Some animals and plants depend on each other to conserve their species [
Seed dispersal by different agents is also called “seed shadow” [
the quantity of fruit or seeds available to agents which implies the rate at which the agents will visit the plants,
a probability density function that tells us about the service rate with which the agents are served by the plants.
The model estimates the quantity of seeds that is spread locally compared to that dispersed globally [
Agents arrive at plants in a random process. Assume that at most one agent arrives to the plants in any unit of time (orderliness condition). It is further supposed that the probability of arrivals of agents to the plants remains the same for a particular period of time. This period corresponds to when the plants are in fruit and during which time the number of visitors is stable (stationarity condition). Furthermore, it is assumed that the arrival of one agent does not affect the rest of arrivals (independence).
With these assumptions in mind, the arrival of agents to plants follows a Poisson process [
We assume that the system is in steady state. Let
Since the plant needs to maximise dispersion, this is equivalent to having a large
Frugivores may travel far away from the plants and hence will disperse the seeds far and wide. This feeding behaviour typically follows a Lévy distribution [
The Lévy distribution is a probability density distribution for random variables. Here the random variables represent the directions of flights of arbitrary birds. This function ranges over real numbers in the domain represented by the problem search space.
The flight lengths of the agents served by the plants follow a heavy tailed power law distribution [
We assume that the arrival of different agents (birds and animals) to the plants to feed is according to the Poisson distribution [
Distribution of agents arriving at strawberry plants to eat fruit and disperse seeds.
Overall performance of SbPPA on Spring Design Problem.
Perturbations by (
Perturbations by (
As already stated, it is essential in this algorithm to balance exploration and exploitation. To this end, we choose a threshold value of the Poisson probability that dictates how much exploration and exploitation are done during the search. The probability
On the other hand, if
As mentioned in Algorithm
(1) NP
(2)
(3)
(4) Create a random population of seeds
using (
(5) Evaluate the population
(6)
(7)
(8) Use updated population
(9)
(10)
(11)
(12)
(13)
(14)
(15) Update the current entry according to (
(16)
(17)
(18)
(19)
(20)
(21) Update the current entry according to (
(22)
(23)
(24)
(25)
(26) Update current best
(27)
(28)
(29)
Performance of SbPPA on unconstrained global optimization problems.
Performance of SbPPA on constrained global optimization problems (see Appendices).
The seedbased propagation process of SP can be represented in the following steps.
The dispersal of seeds in the neighbourhood of the SP, as shown in Figure
Seeds are spread globally through agents, as shown in Figures
The probabilities,
For implementation purposes, we assume that each SP produces one fruit, and each fruit is assumed to have one seed; by a solution
In our experiments we tested SbPPA against some recently developed algorithms and some well established and standard ones. Our set of test problems includes benchmark constrained and unconstrained optimization problems [
In Tables
(+) when SbPPA is better;
(
(−) when SbPPA is worse.
The parameter settings are given in Tables
Parameters used for each algorithm for solving unconstrained global optimization problems
PSO [ 
ABC [ 
HPA [ 
SbPPA [ 


SN = 100  Agents = 100  NP = 10 

MCN = 
Iteration number = 
Iteration number = 

MR = 0.8 

PR = 0.8 

Limit = 



—  Limit = 
— 
—  — 

— 
Experimental setup used for each algorithm for solving unconstrained global optimization problems
CEP [ 
FEP [ 
SbPPA 

Population size 
Population size 

Tournament size 
Tournament size 
PR = 0.8 



Parameters used for each algorithm for solving constrained optimization problems. All experiments are repeated 30 times.
PSO [ 
ABC [ 
FF [ 
SSOC [ 
SbPPA [ 


SN = 40  Fireflies = 25 

NP = 10 

MCN = 6000  Iteration number = 2000  Iteration number = 500  Iteration number = 800 

MR = 0.8 

PF = 0.7  PR = 0.8 

— 

— 

Weight factors = 0.9 to 0.4  —  —  —  — 
Results obtained by SbPPA, HPA, PSO, and ABC. All problems in this table are unconstrained.
Fun.  Dim  Algorithm  Best  Worst  Mean  SD 


4  ABC 




PSO 





HPA 





SbPPA 








2  ABC 




PSO 





HPA 





SbPPA 








2  ABC 




PSO 





HPA 





SbPPA 








2  ABC 




PSO 





HPA 





SbPPA  −1.031628  −1.031628  −1.031628 





6  ABC 




PSO 





HPA 





SbPPA 








10  ABC 




PSO 





HPA 





SbPPA 








30  ABC 




PSO 





HPA 





SbPPA 








30  ABC 




PSO 





HPA 





SbPPA 



 



30  ABC 




PSO 





HPA 





SbPPA 








30  ABC 




PSO 





HPA 





SbPPA 




Results obtained by SbPPA, CEP, and FEP. All problems in this table are unconstrained [
Function number  Algorithm  Maximum generations  Mean  SD 


CEP  2000 


FEP 



SbPPA 






CEP  5000 


FEP 



SbPPA 






CEP  20000 


FEP 



SbPPA 






CEP  1500 


FEP 



SbPPA 






CEP  3000 


FEP 



SbPPA 






CEP  9000 


FEP 



SbPPA 






CEP  5000 


FEP 



SbPPA 






CEP  100 


FEP 



SbPPA 


Results obtained by SbPPA, CEP, and FEP. All problems in this table are unconstrained [
Function number  Algorithm  Maximum generations  Mean  SD 


CEP  100 


FEP 



SbPPA 






CEP  100 


FEP 



SbPPA 






CEP  200 


FEP 



SbPPA 






CEP  100 


FEP 



SbPPA 






CEP  100 


FEP 



SbPPA 






CEP  100 


FEP 



SbPPA 






CEP  100 


FEP 



SbPPA 






CEP  4000 


FEP 



SbPPA 


Results obtained by SbPPA, PSO, ABC, FF, and SSOC. All problems in this table are standard constrained optimization problems.
Fun. name  Optimal  Algorithm  Best  Mean  Worst  SD 

CP1  −15  PSO 




ABC 





FF 





SSOC 





SbPPA 







CP2 

PSO 




ABC 





FF 





SSOC 





SbPPA 







CP3 

PSO 




ABC 





FF 





SSOC 





SbPPA 







CP4 

PSO 




ABC 





FF 





SSOC 





SbPPA 







CP5 

PSO 




ABC 





FF 





SSOC 





SbPPA 







Spring Design Problem  Not known  PSO 




ABC 





FF 





SSOC 





SbPPA 







Welded beam design problem  Not known  PSO 




ABC 





FF 





SSOC 





SbPPA 







Speed reducer design optimization  Not known  PSO 




ABC 





FF 





SSOC 





SbPPA 




In this paper, a new metaheuristic referred to as the SeedBased Plant Propagation Algorithm (SbPPA) [
To capture the dispersal process, we adopt a queuing approach which, given the extent of fruit produced, indicates the extent of seeds dispersed and hence the effectiveness of the search/optimization algorithm based on this process. Looking at the random process of agents using the plants (feeding station) it is reasonable to assume that it is of the Poisson type. On the other hand, the time taken by agents in successfully eating fruit and leaving to disperse its seeds, in other words the service time for agents, is expressed by a random variable which follows the exponential probability distribution. To this end, we choose a threshold value of the Poisson probability that dictates how much exploration and exploitation are done during the search. An alternative strategy has been adopted here. This strategy consists in making sure that the initial population is as good as the user can afford it to be by using best solutions found so far. The effects of this strategy on convergence are shown through convergence plots of Figures
See Tables
Unconstrained global optimization problems (Set1) used in our experiments.
Fun.  Fun. name 


Range  Min  Formulation 


Colville  4  UN 

0 






Matyas  2  UN 

0 




Schaffer  2  MN 

0 




Six Hump Camel Back  2  MN 






Trid6  6  UN 






Trid10  10  UN 






Sphere  30  US 






SumSquares  30  US 






Griewank  30  MN 






Ackley  30  MN 



Unconstrained global optimization problems (Set2) used in our experiments [
Fun. number  Range 

Function  Formulation 





30  Schwefel's Problem 2.22 

0 




30  Schwefel's Problem 2.21 

0 




30  Rosenbrock 

0 




30  Step 

0 




30  Quartic (noise) 

0 




30  Schwefel 






30  Rastrigin 






2  Branin 






2  GoldsteinPrice 






4  Hartman's Family ( 






6  Hartman's Family ( 






4  Shekel's Family ( 






4  Shekel's Family ( 






4  Shekel's Family ( 






2  Shekel's Foxholes 






4  Kowalik 


Consider the following:
Consider the following:
Consider the following:
Consider the following:
Consider the following:
The welded beam design is a standard test problem for constrained design optimisation [
The problem of designing a speed reducer [
The main objective of this problem [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to anonymous reviewers for their valuable reviews and constructive criticism on earlier version of this paper. This work is supported by Abdul Wali Khan University, Mardan, Pakistan, Grant no. F.165/P& D/AWKUM/238.