A Plant Propagation Algorithm for Constrained Engineering Optimisation Problems

Abstract

Optimisation problems arising in industry are some of the hardest, often because of the tight specifications of the products involved. They are almost invariably constrained and they involve highly nonlinear, and non-convex functions both in the objective and in the constraints. It is also often the case that the solutions required must be of high quality and obtained in realistic times. Although there are already a number of well performing optimisation algorithms for such problems, here we consider the novel Plant Propagation Algorithm (PPA) which on continuous problems seems to be very competitive. It is presented in a modified form to handle a selection of problems of interest. Comparative results obtained with PPA and state-of-the-art optimisation algorithms of the Nature-inspired type are presented and discussed. On this selection of problems, PPA is found to be as good as and in some cases superior to these algorithms.

1. Introduction

Optimisation problems in design engineering are often highly nonlinear, constrained and involving continuous as well as discrete variables [1–5]. It is also often the case that some of the constraints are active at the global optimum [6]. This means that feasible approximate solutions are that much harder to find. There is a variety of algorithms for these problems, some exact, and others approximate. In the exact category, one can name Branch-and-Bound, [7], Recursive Quadratic Programming, [8], the Cutting Plane Algorithm [9], Bender’s decomposition [10]. Of the approximate variety, one can name Simulated Annealing, [11–13], the Genetic Algorithm [14–16], and the Particle Swarm Optimisation algorithm, [17, 18], to name a few. The latter category is often referred to as the metaheuristic algorithms. In general, they are characterised by two aspects of the search: exploration of the overall search space and exploitation of good areas in order to find local optima, [19–22].

A new metaheuristic, the Plant Propagation Algorithm (PPA), has recently been introduced [23]. PPA is nature inspired [19, 23, 24]; it emulates the way plants, in particular the strawberry plant, propagate. A basic PPA has been described and tested on single objective as well as multiobjective continuous optimization problems in [23]. The test problems, though standard, were of low dimension. The results showed that PPA has merits and deserves further investigation on higher dimensional problem instances as well as problems arising in practice, for these are often very challenging. PPA is attractive because, among other things, it is simple to describe and implement; it also involves only few parameters that need arbitrary setting unlike most other metaheuristics. Here, it will be tested on constrained optimisation problems arising in engineering. The paper is organised as follows. Section 2 describes PPA. Section 3 presents a modified version to handle constrained problems. Section 4 records the results obtained with PPA and a number of other heuristics. In Section 5 a conclusion and ideas for further investigation are given. The paper includes appendices that describe the problems considered.

2. The Strawberry Algorithm as PPA

The Strawberry algorithm is an exemplar PPA which can be seen as a multipath following algorithm unlike Simulated Annealing (SA) [13, 25], for instance, which is a single path following algorithm. It can, therefore, be perceived as a generalisation of SA and other path-following algorithms [26].

Exploration and exploitation are properties that effective global optimisation algorithms have [19, 24, 26]. Exploration refers to the property of covering the search space, while exploitation refers to the property of searching nearer to good solutions for local optima.

Consider what a strawberry plant and possibly any plant which propagates through runners will do to optimize its survival. If it is in a good spot of the ground, with enough water, nutrients, and light, then it is reasonable to assume that there is no pressure on it to leave that spot to guarantee its survival. So, it will send many short runners that will give new strawberry plants and occupy the neighbourhood as best they can. If, on the other hand, the mother plant is in a spot that is poor in water, nutrients, light, or any one of these necessary for a plant to survive, then it will try to find a better spot for its offspring. Therefore, it will send few runners further afield to explore distant neighbourhoods. One can also assume that it will send only a few, since sending a long runner is a big investment for a plant which is in a poor spot. We may further assume that the quality of the spot (abundance of nutrients, water, and light) is reflected in the growth of the plant. With this in mind and the following notation, PPA can be described as follows.

A plant 𝑝_𝑖 is in spot 𝑋_𝑖 in dimension 𝑛. This means 𝑋_𝑖={𝑥_𝑖,𝑗,for𝑗=1,…,𝑛}. Let 𝑁𝑃 be the number of strawberry plants to be used initially, and the PPA algorithm described in pseudo-code Algorithm 1, relies on the following strategy [23].(i)Strawberry plants which are in good spots propagate by generating many short runners.(ii)Those in poor spots propagate by generating few long runners.

(1) Initialization: Generate a population 𝑃={𝑋𝑖,𝑖=1,…,NP}; (2) 𝑔←1; (3) for  𝑔=1: 𝑔max  do (4)  Compute 𝑁𝑖=𝑓(𝑋𝑖),∀𝑋𝑖∈𝑃; (5)  Sort 𝑃 in ascending order of 𝑁 (for minimization); (6)  Create new population Φ; (7)  for each 𝑋𝑖,𝑖=1,…,NP  do (8)     𝛼𝑖← set of runners where both the size of the set and the distance for each runner (individually)       are proportional to 𝑁𝑖, the normalized objective value (9)     Φ← Φ∪𝛼𝑖  {append to population}; (10) end for (11) 𝑃←Φ  {new population}; (12) end for (13) Return  𝑃, the population of solutions.

Algorithm 1 

Pseudocode of PPA, [23].

It is clear that, in the above description, exploitation is implemented by sending many short runners by plants in good spots, while exploration is implemented by sending few long runners by plants in poor spots.

The parameters used in PPA are the population size 𝑁𝑃 which is the number of strawberry plants, the maximum number of generations 𝑔_max, and the maximum number of possible runners 𝑛_max per plant. 𝑔_max is effectively the stopping criterion in this initial version of PPA. The algorithm uses the objective function value at different plant positions 𝑋_𝑖, 𝑖=1,…,𝑁𝑃, in a normalised form 𝑁_𝑖, to rank them as would a fitness function in a standard genetic algorithm (note that, unlike in the GA, individuals in PPA are clones of the mother plant; they do not improve from generation to generation). The number of plant runners 𝑛^𝑖_𝛼, calculated according to (1) below, has length 𝑑𝑥^𝑖 calculated using the normalised form of the objective value at 𝑋_𝑖, each giving a 𝑑𝑥^𝑖∈(−1,1)^𝑛, as calculated with (2) below. After all individuals/plants in the population have sent out their allocated runners, new plants are evaluated and the whole increased population is sorted. To keep the population constant, individuals with lower growth are eliminated. The number of runners allocated to a given plant is proportional to its fitness as in 𝑛^𝑖_𝛼=𝑛_max𝑁_𝑖𝛼,𝛼∈(0,1). (1) Every solution 𝑋_𝑖 generates at least one runner and the length of each such runner is inversely proportional to its growth as in (2) below: 𝑑𝑥^𝑖_𝑗=21−𝑁_𝑖(𝛼−0.5),for𝑗=1,…,𝑛, (2) where 𝑛 is the problem dimension. Having calculated 𝑑𝑥^𝑖, the extent to which the runner will reach, the search equation that finds the next neighbourhood to explore is 𝑦_𝑖,𝑗=𝑥_𝑖,𝑗+𝑏_𝑗−𝑎_𝑗𝑑𝑥^𝑖_𝑗,for𝑗=1,…,𝑛. (3)

If the bounds of the search domain are violated, the point is adjusted to be within the domain [𝑎_𝑗,𝑏_𝑗], where 𝑎_𝑗 and 𝑏_𝑗 are lower and upper bounds delimiting the search space for the 𝑗th coordinate.

3. An Effective Implementation of PPA for Constrained Optimization

In this implementation of PPA, the initial population is crucial; we run the algorithm a number of times from randomly generated populations. The best individual from each run forms a member of the initial population. The number of runs to generate the initial population is 𝑁𝑃; therefore, the population size is 𝑟=𝑁𝑃. In the case of mixed integer problems, the integer variable values are fixed when they are showing a trend to converge to some values. This trend is monitored by calculating the number of times their values have not changed. When this number is greater than a certain threshold, the variables are fixed for the rest of the run. This strategy seems to work on the problems considered. Let 𝑝𝑜𝑝 be a general matrix containing the population of a given run. Its rows correspond to individuals. The following equation is used to generate a random population for each of the initial runs: 𝑥_𝑖,𝑗=𝑎_𝑗+𝑏_𝑗−𝑎_𝑗𝛼,𝑗=1,…,𝑛, (4) where 𝑥_𝑖,𝑗∈[𝑎_𝑗,𝑏_𝑗] is the 𝑗th entry of solution 𝑋_𝑖 and 𝑎_𝑗 and 𝑏_𝑗 are the 𝑗th entries of the lower and upper bounds describing the search space of the problem and 𝛼∈(0,1).

In the main body of the algorithm, before updating the population we create a temporary population Φ to hold new solutions generated from each individual in the population. Then we implement three rules with fixed modification parameter 𝑃_𝑚, chosen here, as 𝑃_𝑚=0.8. The first two rules are implemented if the population is initialized randomly. Rule 01 uses the following equation to update the population: 𝑥^∗_𝑖,𝑗=𝑥_𝑖,𝑗1+𝛽,𝑗=1,…,𝑛, (5) where 𝛽∈[−1,1] and 𝑥^∗_𝑖,𝑗∈[𝑎_𝑗,𝑏_𝑗].

The generated individual 𝑋^∗_𝑖 is evaluated according to the objective function and is stored in Φ. In rule 02 another individual is created with the same modification parameter 𝑃_𝑚=0.8 as in the following equation: 𝑥^∗_𝑖,𝑗=𝑥_𝑖,𝑗+𝑥_𝑙,𝑗−𝑥_𝑘,𝑗𝛽,𝑗=1,…,𝑛, (6) where 𝛽∈[−1,1], 𝑥^∗_𝑖,𝑗∈[𝑎_𝑗,𝑏_𝑗]. 𝑙, 𝑘 are mutually exclusive indices and are different from 𝑖.

The generated individual 𝑋^∗_𝑖 is evaluated according to the objective function and is stored in Φ. The first two rules are applicable for 𝑟≤𝑁𝑃 the number of runs. For 𝑟>𝑁𝑃 the algorithm also tries to recognise entries which are settling to their final values through a counter 𝐼𝑁. If the 𝑗th entry in current population has a low 𝐼𝑁 value, then it is modified by implementing (7); otherwise it is left as it is. The value (for 𝐼𝑁) that is suggested by experimentation over a number of problems is 4. The following equation is used when modification is necessary: 𝑥^∗_𝑖,𝑗=𝑥_𝑖,𝑗+𝑥_𝑖,𝑗−𝑥_𝑘,𝑗𝛽,𝑗=1,…,𝑛, (7) where 𝛽∈[−1,1], 𝑥^∗_𝑖,𝑗∈[𝑎_𝑗,𝑏_𝑗], and 𝑘 is different from 𝑖. To keep the size of the population constant, the extra plants at the bottom of the sorted population are eliminated.

4. Examples of Structural Engineering Optimization Problems

PPA as explained in the pseudo-code Algorithm 1 is extended to cater for constrained optimisation problems to be found in the appendices [6, 27]. This extended version of PPA is fully explained in the pseudo-code Algorithm 2. Note that the penalty function approach is used to handle the constraints [19, 22]. Equations are first transformed into inequality constraints before they are taken into consideration. Table 1 records the parameter values used in the implementation. Column 4 shows the value of 𝑃_𝑚 used throughout the experiments. This value has been found through experimentation on the problem described in Appendix B. The different runs are represented in Figure 1 where for 40 trials corresponding to the 30000 function evaluations threshold, 𝑃_𝑚=0.8 seems to be the optimal value for this parameter. Other aspects of the extended algorithm such as exploration and exploitation are investigated in Figures 2 and 3. These figures, as one expects, show that the magnitudes of the steps/perturbations of the plant positions, that is the lengths of the runners, get shorter and shorter as the search progresses. Figure 4 is a representation of the convergence of the objective values of the problems described in Appendices A and C. In both cases, these values fluctuate wildly before they settle down to very good approximate values. The numerical results of the experiments on all problems described in Appendices A, through G are compiled in Tables 2, 3, 4, 5, 6, 7, and 8.

Table 1 

Parameters used in PPA for seven engineering problems.

Table 2 

Welded beam design optimisation.

Table 3 

Pressure vessel design optimisation.

Table 4 

Minimization of the weight of a compression spring.

Table 5 

Speed reducer design optimisation.

Table 6 

Constrained optimization problem I.

Table 7 

Constrained optimization problem II.

Table 8 

Optimal solutions for a variation on Himmelblau's function.

(1) Initialization: 𝑔max← Maximum number of generations; NP ← population size; 𝑟← trial run (2) if  𝑟≤NP  then (3)  Create a random population of plants 𝑝𝑜𝑝={𝑋𝑖∣𝑖=1,2,…,NP}, using (4) and gather the best solutions. (4) end if (5) while  𝑟>NP  do (6)  Use population 𝑝𝑜𝑝𝑔 formed by gathering all best solutions from previous runs.    Calculate 𝐼𝑁𝑗 value for each column 𝑗 of 𝑝𝑜𝑝𝑔 (see Section 3). (7) end while (8) Evaluate the population. In case of 𝑝𝑜𝑝𝑔 the algorithm does not need to evaluate the population, (9) Set number of runners, 𝑛𝑟=3, 𝑛𝑔𝑒𝑛=1, (10) while (𝑛𝑔𝑒𝑛<𝑔max) or (𝑛_𝑒𝑣𝑎𝑙<𝑚𝑎𝑥_𝑒𝑣𝑎𝑙) do (11)  Create Φ: (12)  for  𝑖=1 to NP  do (13)   for  𝑘=1 to 𝑛𝑟  do (14)    if  𝑟≤NP  then (15)     if  rand≤𝑃𝑚  then (16)      Generate a new solution 𝑋∗ according to (5); (17)       Evaluate it and store it in Φ; (18)     end if (19)     if  rand≤𝑃𝑚  then (20)      Generate a new solution 𝑋∗ according to (6); (21)       Evaluate it and store it in Φ; (22)     end if (23)    else (24)     for  𝑗=1: 𝑛  do (25)      if (𝐼𝑁𝑗<4) or (rand≤𝑃𝑚) then (26)       update the 𝑗th entry of 𝑋𝑖, 𝑖=1,2,…,NP, according to (7); (27)      end if (28)      Evaluate new solution 𝑋∗ and store it in Φ; (29)     end for (30)    end if (31)   end for (32)  end for (33)  Add Φ to current population; (34)  Sort the population in ascending order of the objective values; (35)  Update current best; (36) end while (37) Return: Updated population.

Algorithm 2 

Pseudocode of PPA for constrained optimisation.

Figure 1 

Impact of the particular probability 𝑃𝑚 on the performance of PPA.

Figure 2 

Exploitation characteristic of PPA using rules 2-3.

Figure 3 

Exploration characteristic of PPA using rule 1.

(a)

(b)

Figure 4 

Convergence plot of welded beam design and tension compression spring design optimization problems over 100 trial runs.

5. Conclusion

We have implemented PPA to solve seven well known difficult constrained optimization problems arising in engineering design with continuous domains. PPA found either near best known solutions or optimal ones to all of them. The results are compared to those obtained with other algorithms found in the literature, namely GA (and variants of it, here denoted EC and EP), PSO, HSA (and variants of it namely PHS and IPHS), IP, MP, and BDA. Note that some of the problems have not been solved by all algorithms. For instance, the pressure vessel design problem has not been solved with BDA or MP as far as we know. Similarly the spring compression problem has not been solved with IP and HSA. These gaps in the computational results found in the literature on the considered problems are not substantial to hinder our experimental work and conclusions. Indeed, the recorded evidence in the large majority of cases points to the overwhelming superiority of PPA. Having said that, it must be added that further improvements to PPA and testing are being carried out on a more extensive collection of test problems including discrete ones.

Appendices

A. Welded Beam Design Optimisation

The welded beam design is a standard test problem for constrained design optimisation [6, 22]. There are four design variables: the width 𝑤 and length 𝐿 of the welded area, the depth 𝑑 and thickness ℎ of the main beam. The objective is to minimise the overall fabrication cost, under the appropriate constraints of shear stress 𝜏, bending stress 𝜎, buckling load 𝑃, and maximum end deflection 𝛿. The optimization model is summarized as follows, where 𝑥^𝑇=(𝑤,𝐿,𝑑,ℎ): Minimise𝑓(𝑥)=1.10471𝑤²𝐿+0.04811𝑑ℎ(14.0+𝐿),subjectto𝑔₁(𝑥)𝑔=𝑤−ℎ≤0,₂(𝑥)(𝑥)𝑔=𝛿−0.25≤0,₃(𝑥)(𝑥)𝑔=𝜏−13,600≤0,₄(𝑥)(𝑥)𝑔=𝜎−30,000≤0,₅(𝑥)=1.10471𝑤²()𝑔+0.04811𝑑ℎ14.0+𝐿−5.0≤0,₆(𝑥)𝑔=0.125−𝑤≤0,₇(𝑥)=6000−𝑃(𝑥)≤0, (A.1) where 𝜎(𝑥)=504,000ℎ𝑑²1,𝐷=2𝐿²+(𝑤+𝑑)²,𝐿𝑄=600014+2,𝛿=65,85630,000ℎ𝑑³,𝐽=√2𝑤𝐿𝐿²6+(𝑤+𝑑)²2,𝛼=6000√,2𝑤𝐿𝛽=𝑄𝐷𝐽,𝑃=0.61423×10⁶𝑑ℎ³61−^𝑑30/48,𝜏(𝑥)=28𝛼²+𝛼𝛽𝐿𝐷+𝛽². (A.2)

The simple limit or bounds are 0.1≤𝐿, 𝑑≤10 and 0.1≤𝑤, ℎ≤2.0.

B. Pressure Vessel Design Optimisation

Pressure vessels are widely used in our daily life, such as champagne bottles and gas tanks [6, 39]. For a given volume and working pressure, the basic aim of designing a cylindrical vessel is to minimize the total cost. Typically, the design variables are the thickness 𝑑₁ of the head, the thickness 𝑑₂ of the body, the inner radius 𝑟, and the length 𝐿 of the cylindrical section [6]. This is a well-known test problem for optimization, where 𝑥^𝑇=(𝑑₁,𝑑₂,𝑟,𝐿), and it can be written as Minimise𝑓(𝑥)=0.6224𝑑₁𝑟𝐿+1.7781𝑑₂𝑟²+3.1661𝑑²₁𝐿+19.84𝑑²₁𝑟,subjectto𝑔₁(𝑥)=−𝑑₁𝑔+0.0193𝑟≤0,₂(𝑥)=−𝑑₂𝑔+0.00954𝑟≤0,₃(𝑥)=−𝜋𝑟²𝐿−4𝜋3𝑟³𝑔+1296000≤0,₄(𝑥)=−𝐿−240≤0. (B.1)

The simple limits on the design variables are 0.0625≤𝑑₁,𝑑₂≤99×0.0625,10.0≤𝑟,𝐿≤200. (B.2)

C. Spring Design Optimisation

The main objective of this problem [33, 34] is to minimize the weight of a tension/compression spring, subject to constraints of minimum deflection, shear stress, surge frequency, and limits on outside diameter and on design variables. There are three design variables: the wire diameter 𝑥₁, the mean coil diameter 𝑥₂, and the number of active coils 𝑥₃ [6]. The mathematical formulation of this problem, where 𝑥^𝑇=(𝑥₁,𝑥₂,𝑥₃), is as follows: Minimize𝑓(𝑥)=𝑥₃+2𝑥₂𝑥²₁,subjectto𝑔₁(𝑥)𝑥=1−³²𝑥₃7,178𝑥⁴₁𝑔≤0,₂(𝑥)=4𝑥²₂−𝑥₁𝑥₂12,566𝑥₂𝑥³₁−𝑥⁴₁+15,108𝑥²₁𝑔−1≤0,₃(𝑥)=1−140.45𝑥₁𝑥²₂𝑥₃𝑔≤0,₄(𝑥)=𝑥₂+𝑥₁1.5−1≤0. (C.1)

The simple limits on the design variables are 0.05≤𝑥₁≤2.0, 0.25≤𝑥₂≤1.3, and 2.0≤𝑥₃≤15.0.

D. Speed Reducer Design Optimization

The problem of designing a speed reducer [40] is a standard test problem. It consists of the design variables as face width 𝑥₁, module of teeth 𝑥₂, number of teeth on pinion 𝑥₃, length of the first shaft between bearings 𝑥₄, length of the second shaft between bearings 𝑥₅, diameter of the first shaft 𝑥₆, and diameter of the first shaft 𝑥₇ (all variables continuous except 𝑥₃ that is integer). The weight of the speed reducer is to be minimized subject to constraints on bending stress of the gear teeth, surface stress, transverse deflections of the shafts, and stresses in the shaft [6]. The mathematical formulation of the problem, where 𝑥^𝑇=(𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₇), is as follows: Minimise𝑓(𝑥)=0.7854𝑥₁𝑥²₂3.3333𝑥²₃+14.9334𝑥₃43.0934−1.508𝑥₁𝑥²₆+𝑥³₇+7.4777𝑥³₆+𝑥³₇+0.7854𝑥₄𝑥²₆+𝑥₅𝑥²₇,subjectto𝑔₁(𝑥)=27𝑥₁𝑥²₂𝑥₃𝑔−1≤0,₂(𝑥)=397.5𝑥₁𝑥²₂𝑥²₃𝑔−1≤0,₃(𝑥)=1.93𝑥³₄𝑥₂𝑥₃𝑥⁴₆𝑔−1≤0,₄(𝑥)=1.93𝑥³₅𝑥₂𝑥₃𝑥⁴₇𝑔−1≤0,₅(𝑥)=1.0110𝑥³₆745.0𝑥₄𝑥₂𝑥₃²+16.9×10⁶𝑔−1≤0,₆(𝑥)=1.085𝑥³₇745.0𝑥₅𝑥₂𝑥₃²+157.5×10⁶𝑔−1≤0,₇(𝑥)=𝑥₂𝑥₃𝑔40−1≤0,₈(𝑥)=5𝑥₂𝑥₁𝑔−1≤0,₉(𝑥)=𝑥₁12𝑥₂𝑔−1≤0,₁₀(𝑥)=1.5𝑥₆+1.9𝑥₄𝑔−1≤0,₁₁(𝑥)=1.1𝑥₇+1.9𝑥₅−1≤0. (D.1)

The simple limits on the design variables are 2.6≤𝑥₁≤3.6,0.7≤𝑥₂≤0.8,17≤𝑥₃≤28,7.3≤𝑥₄≤8.3,7.8≤𝑥₅≤8.3,2.9≤𝑥₆≤3.9,5.0≤𝑥₇≤5.5. (D.2)

E. Constrained Optimization Problem 1

Minimize𝑓(𝑥)=𝑥₁−2²+𝑥₂−1²,subjectto𝑔₁(𝑥)=𝑥₁−2𝑥₂𝑔+1=0,₂(𝑥)=−𝑥₁4−𝑥²₂+1≥0,−10≤𝑥₁,𝑥₂≤10. (E.1)

F. Constrained Optimization Problem 2

Minimize𝑓(𝑥)=𝑥²₁+𝑥₂−11²+𝑥₁+𝑥²₂−7²,subjectto𝑔₁(𝑥)=4.84−𝑥₁−0.05²−𝑥₂−2.5²𝑔≥0,₂(𝑥)=𝑥²₁+𝑥₂−2.5²−4.84≥0,0≤𝑥₁,𝑥₂≤6. (F.1)

G. Himmelblau’s Optimization Problem

Minimize𝑓(𝑥)=5.357847𝑥²₃+0.8356891𝑥₁𝑥₅+37.293239𝑥₁−40792.141,subjectto𝑔₁(𝑥)=85.334407+0.0056858𝑥₂𝑥₅+0.00026𝑥₁𝑥₄−0.0022053𝑥₃𝑥₅,𝑔₂(𝑥)=80.51249+0.0071317𝑥₂𝑥₅+0.0029955𝑥₁𝑥₂+0.0021813𝑥²₃,𝑔₃(𝑥)=9.300961+0.0047026𝑥₃𝑥₅+0.0012547𝑥₁𝑥₃+0.0019085𝑥₃𝑥₄,0≤𝑔₁(𝑥)≤92,90≤𝑔₂(𝑥)≤110,20≤𝑔₃(𝑥)≤25,78≤𝑥₁≤102,33≤𝑥₂≤45,27≤𝑥_𝑖≤45,𝑖=3,4,5. (G.1)

Conflict of Interests

The authors have no conflict of interests.

Acknowledgments

This work is supported by Abdul Wali Khan University, Mardan, Pakistan, Grant no. F.16-5/P& D/AWKUM/238.