A Multiswarm Optimizer for Distributed Decision Making in Virtual Enterprise Risk Management

Abstract

We develop an optimization model for risk management in a virtual enterprise environment based on a novel multiswarm particle swarm optimizer called PS²O. The main idea of PS²O is to extend the single population PSO to the interacting multiswarms model by constructing hierarchical interaction topology and enhanced dynamical update equations. With the hierarchical interaction topology, a suitable diversity in the whole population can be maintained. At the same time, the enhanced dynamical update rule significantly speeds up the multiswarm to converge to the global optimum. With five mathematical benchmark functions, PS²O is proved to have considerable potential for solving complex optimization problems. PS²O is then applied to risk management in a virtual enterprise environment. Simulation results demonstrate that the PS²O algorithm is more feasible and efficient than the PSO algorithm in solving this real-world problem.

1. Introduction

Swarm Intelligence (SI) is an innovative artificial intelligence technique for solving complex optimization problems. This discipline is inspired by the collective behaviors of social animals such as fish schools, bird flocks, and ant colonies. In SI systems, there are many simple individuals who can interact locally with one another and with their environments. Although such systems are decentralized, local interactions between individuals lead to the emergence of global behavior and properties.

In recent years, many algorithmic SI methods were designed to deal with practical problems [1–5]. Among them, the most successful is Particle Swarm Optimization (PSO) that drew inspiration from the biological swarming behaviors observed in flocks of birds, schools of fish, and even human social behavior [6–8]. PSO is a population-based optimization tool, which could be implemented and applied easily to solve various function optimization problems. As a problem-solving technique, the main strength of PSO is its fast convergence, which compares favorably with Evolutionary Algorithms (EAs) and other global optimization algorithms [9–12]. However, when solving complex multimodal problems, PSO suffers from the following drawback [13]: as a population evolves, all individuals suffer premature convergence to the local optimum in the first generations. This leads to low population diversity and adaptation stagnation in successive generations. However, such loss of population diversity is not observed in natural systems. Because populations of species interact with one another in natural ecosystems, these species form biological communities which are large social systems typically consist of both heterogeneous and homogeneous aspects. The interaction between species and the complexity of their relationships in these communities exemplify what is meant by the term “symbiosis.” According to different symbiotic interrelationships, symbiotic coevolution can be classified into several categories: mutualism, commensalism, parasitism, and competition. We found that all these types are suitable to be incorporated into the standard PSO model to improve PSO’s performance on complex optimization problems. This should be a general extension of PSO with the purpose of accurately representing as many different forms of symbiotic coevolution as possible.

Thus, inspired by symbiotic cooperation (i.e., mutualism coevolution) phenomenon in nature, this paper proposed a novel multiswarm particle swarm optimizer called PS²O, which extend the single population PSO to interacting multiswarms model by constructing hierarchical interaction topologies and enhanced dynamical update equations. In PS²O, we implement a hierarchical interaction topology that consists of two levels (i.e., individual level and swarm level), in which information exchanges take place permanently. Each individual of the proposed model evolves based on the knowledge integration of itself (associate with individual’s own cognition), its swarm members (associate social interaction within each swarm), and its symbiotic partners from other swarm (associate heterogeneous cooperation between different swarms). That is, we extend the control law (i.e., the dynamic update equation) of the canonical PSO model by adding a significant ingredient, which takes into account the symbiotic coevolution (or heterogeneous cooperation) between different swarms. By incorporating this new degree of complexity, PS²O can accommodate a considerable potential for solving more complex problems. Here we provide some initial insights into this potential by evaluating PS²O on both mathematical benchmark functions and a complex real-world problem-risk management in a virtual enterprise (VE). The 5 benchmark functions used in our experiments have been widely employed by other researchers to evaluate their algorithms [14–16]. In this paper, the risk management problem in VE is modeled as a distributed decision-making (DDM) system. This novel risk management model is a complex hierarchical optimization problem with two levels, namely, the top model and the base model, which take care of the continuous decision variables and the discrete ones, respectively. The simulation results, which are compared to other methods, are reported in this paper to show the merits of the proposed algorithm.

The paper is organized as follows. Section 2 gives a review of the canonical PSO algorithm and several multi-swarm PSO variants. Section 3 describes the proposed multi-swarm coevolution algorithm. In Section 4, it will be shown that PS²O outperforms the canonical PSO and its variants on 5 benchmark test functions. Section 5 describes the risk management optimization model in VE and a detailed design algorithm of risk management by PS²O. The simulation result of risk management in a VE based on PS²O compared with canonical PSO is also presented in this section. Finally, conclusions are drawn in Section 6.

2. Review of Canonical Particle Swarm Optimization

The canonical PSO is a population-based technique, similar in some respects to evolutionary algorithms, except that potential solutions (particles) move, rather than evolve, through the search space. The rules (or particle dynamics) that govern this movement are inspired by models of swarming and flocking [7]. Each particle has a position and a velocity, and experiences linear spring-like attractions towards two attractors:(i)its previous best position,(ii)best position of its neighbors.

In mathematical terms, the 𝑖th particle is represented as 𝑥_𝑖=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝐷) in the D-dimensional space, where 𝑥_𝑖𝑑∈[𝑙_𝑑,𝑢_𝑑], 𝑑∈[1,𝐷], 𝑙_𝑑, 𝑢_𝑑 are the lower and upper bounds for the 𝑑th dimension, respectively. The rate of velocity for particle 𝑖 is represented as 𝜈_𝑖=(𝜈_𝑖1,𝜈_𝑖2,…,𝜈_𝑖𝐷) is clamped to a maximum velocity 𝑉_max which is specified by the user. In each time step 𝑡, the particles are manipulated according to the following equations:𝜈_𝑖𝑑(𝑡)=𝜒𝜈_𝑖𝑑(𝑡−1)+𝑅₁𝑐₁𝑝_𝑖𝑑−𝑥_𝑖𝑑(𝑡−1)+𝑅₂𝑐₂𝑝_𝑔𝑑−𝑥_𝑖𝑑(𝑡−1), (2.1)𝑥_𝑖𝑑(𝑡)=𝑥_𝑖𝑑(𝑡−1)+𝜈_𝑖𝑑(𝑡), (2.2) where 𝑅₁ and 𝑅₂ are random values between 0 and 1, 𝑐₁ and 𝑐₂ are learning rates, which control how far a particle will move in a single iteration, 𝑝_𝑖𝑑 is the best position found so far of the 𝑖th particle, 𝑝_𝑔𝑑 is the best position of any particles in its neighborhood, and 𝜒 is called constriction factor [17], given by2𝜒=2−𝜑−𝜑²−4𝜑, (2.3) where 𝜑=𝑐₁+𝑐₂,𝜑>4.

Kennedy and Eberhart [18] proposed a binary PSO in which a particle moves in a state space restricted to zero and one on each dimension, in terms of the changes in probabilities that a bit will be in one state or the other. The velocity formula (2.1) remains unchanged except that 𝑥_𝑖𝑑, 𝑝_𝑖𝑑, and 𝑝_𝑔𝑑 are integers in {0,1}, and 𝜈_𝑖𝑑 must be constrained to the interval [0.0,1.0]. This can be accomplished by introducing a sigmoid function 𝑆(𝜈), and the new particle position is calculated using the following rule:ifrand<𝑆𝜈_𝑖𝑑,then𝑥_𝑖𝑑=1;else𝑥_𝑖𝑑=0, (2.4) where rand is a random number selected from a uniform distribution in [0.0, 1.0] and the function 𝑆(𝜈) is a sigmoid limiting transformation as follows:𝑆(𝜈)=11+𝑒^−𝜈. (2.5)

3. PS²O Algorithm

Straight PSO uses the analogy of a single-species population and the suitable definition of the particle dynamics and the particle information network (interaction topology) to reflect the social evolution in the population. However, the situation in nature is much more complex than what this simple metaphor seems to suggest. Indeed, in biological populations there is a continuous interplay between individuals of the same species, and also encounters and interactions of various kinds with other species [19]. The points at issue can be clearly seen when one observes such ecological systems as symbiosis, host-parasite systems, and prey-predator systems, in which two organisms mutually support each other, one exploits the other, or they fight against each other. For instance, mutualistic relations between plants and fungi are very common. The fungus invades and lives among the cortex cells of the secondary roots and, in turn, helps the host plant absorb minerals from the soil. Another well-known example is the “association” between the Nile crocodile and the Egyptian plover, a bird that feeds on any leeches attached to the crocodile’s gums, thus keeping them clean. This kind of “cleaning symbiosis” is also common in fish.

Inspired by mutualism phenomenon, we extend the single population PSO to the interacting multi-swarms model by constructing hierarchical information networks and enhanced particle dynamics. In our multi-swarms approach, the interaction occurs not only between the particles within each swarm but also between different swarms. That is, the information exchanges on a hierarchical topology of two levels (i.e., the individual level and the swarm level). Many patterns of connection can be used in different levels of our model. The most common ones are rings, two-dimensional and three-dimensional lattices, stars, and hypercubes. Two example hierarchical topologies are illustrated in Figure 1. In Figure 1(a), four swarms at the upper level are connected by a ring, while each swarm (possesses four individual particles at the lower level) is structured as a star. While in Figure 1(b), both levels are structured as rings. Then, we suggest in the proposed model that each individual moving through the solution space should be influenced by three attractors:(i)its own previous best position,(ii)best position of its neighbors from its own swarm,(iii)best position of its neighbor swarms.

(a)

(b)

Figure 1 

Hierarchical topologies of the multiswarm.

In mathematical terms, our multi-swarm model is defined as a triplet⟨𝑃,𝑇,𝐶⟩, where 𝑃={𝑆₁,𝑆₂,…,𝑆_𝑀} is a collection of 𝑀 swarms, and each swarm possesses a members set 𝑆_𝑘={𝑋^𝑘₁,𝑋^𝑘₂,…,𝑋^𝑘_𝑁}of 𝑁 individuals. 𝑇 is the hierarchical topology of the multi-swarm. 𝐶 is the enhanced control low of the particle dynamics, which can be formulated as𝜈^𝑘_𝑖𝑑(𝑡)=𝜒𝜈^𝑘_𝑖𝑑(𝑡−1)+𝑅₁𝑐₁𝑝^𝑘_𝑖𝑑−𝑥^𝑘_𝑖𝑑(𝑡−1)+𝑅₂𝑐₂𝑝^𝑘_𝑔𝑑−𝑥^𝑘_𝑖𝑑(𝑡−1)+𝑅₃𝑐₃𝑝^𝜃_𝑔𝑑−𝑥^𝑘_𝑖𝑑(),𝑥𝑡−1_𝑘𝑖𝑑(𝑡)=𝑥^𝑘_𝑖𝑑(𝑡−1)+𝜈^𝑘_𝑖𝑑(𝑡), (3.1)(3.2) where 𝑥^𝑘_𝑖𝑑 represents the position of the 𝑖th particle of the 𝑘th swarm, 𝑝^𝑘_𝑖𝑑 is the personal best position found so far by 𝑥^𝑘_𝑖𝑑, 𝑝^𝑘_𝑔𝑑 is the best position found so far by this particle’s neighbors within swarm 𝑘, 𝑝^𝜃_𝑔𝑑 is the best position found so far by the other swarms in the neighborhood of swarm 𝑘 (here 𝜃 is the index of the swarm which the best position belongs to), 𝑐₁ are the individual learning rates, 𝑐₂ are the social learning rate between particles within each swarm, 𝑐₃ are the social learning rate between different swarms, and 𝑅₁,𝑅₂,𝑅₃∈ℜ^𝑑 are random vectors uniformly distributed in [0,1]. Here, the term 𝑅₁𝑐₁(𝑝^𝑘_𝑖𝑑−𝑥^𝑘_𝑖𝑑) is associated with cognition since it takes into account the individual’s own experiences; the term 𝑅₂𝑐₂(𝑝^𝑘_𝑔𝑑−𝑥^𝑘_𝑖𝑑) represents the social interaction within swarm 𝑘; the term 𝑅₃𝑐₃(𝑝^𝜃_𝑔𝑑−𝑥^𝑘_𝑖𝑑) takes into account the symbiotic coevolution between dissimilar swarms.

When constriction factor is implemented as in the canonical PSO above, 𝜒 is calculated from the values of the acceleration coefficients (i.e., the learning rate) 𝑐₁ and 𝑐₂, importantly, it is the sum 𝜑 of these two coefficients that determines what 𝜒 to use [17]. This fact implies that the particle’s velocity can be adjusted by any number of terms, as long as the acceleration coefficients sum to an appropriate value. Thus, the constriction factor 𝜒 in velocity formula of PS²O can be calculated by2𝜒=2−𝜙−𝜙²−4𝜙, (3.3) where 𝜙=𝑐₁+𝑐₂+𝑐₃, 𝜙>4. Then the algorithm will behave properly, at least as far as its convergence and explosion characteristics, whether all of 𝜙 is allocated to one term, or it is divided into thirds, fourths, and so forth.

We should note that, for solving discrete problems, we still use (2.4) and (2.5) to discrete the position vectors in PS²O algorithm. The pseudocode for the PS²O algorithm is listed in Table 1. The flowchart of the PS²O algorithm is presented in Figure 2, and according variables used in PS²O are summarized in Table 2.

Table 1 

Pseudocode for the PS²O algorithm.

Table 2 

List of variables used in PS²O.

Figure 2 

The flowchart of the PS2O algorithm.

4. Benchmark Test

4.1. Test Function

A set of 5 benchmark functions, which are commonly used in evolutionary computation literature [16, 20] to show solution quality and convergence rate, was employed to evaluate the PS²O algorithm in comparison to others. The first problem is the unimodal Sphere function that is easy to solve. The second problem is the Rosenbrock function, which has a narrow valley from the perceived local optima to the global optimum and can be treat as a multimodal problem. The remaining three functions are multimodal problem. Griewank’s function has a ∏^𝐷_𝑖=1cos(𝑥_𝑖/√𝑖) component causing linkages among variables, thereby making it difficult to reach the global optimum. The Weierstrass function is continuous but differentiable only on a set of points. The composition functions are a set of novel challenging problems, which are constructed using some basic benchmark function with a random located global optimum and several randomly located deep local optima. The Gaussian function is used to combine the basic functions and blur the function structures. CF1 is constructed using 10 sphere functions which is an asymmetrical multimodal function with 1 global optimum and 9 local optima (the landscape of CF1 is illustrated in Figure 3). The variables of the CF1 formulation can be referred to [20]. The formulas of these functions are presented below.(1)Sphere function: 𝑓₁(𝑥)=^𝐷_𝑖=1𝑥²_𝑖. (4.1)(2)Rosenbrock function: 𝑓₂(𝑥)=^𝐷_𝑖=1100×𝑥_𝑖+1−𝑥²_𝑖²+1−𝑥_𝑖². (4.2)(3)Griewank function: 𝑓₃(𝑥)=14000^𝐷_𝑖=1𝑥²_𝑖−^𝐷_𝑖=1𝑥cos^𝑖√𝑖+1. (4.3)(4)Weierstrass function: 𝑓₄(𝑥)=^𝐷_𝑖=1^𝑘max_𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘𝑥_𝑖+0.5−𝑛^𝑘max_𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘•0.5, (4.4) where 𝑎=0.5, 𝑏=3, 𝑘max=20.(5)Composition function 1: 𝑓₅(𝑥)=^𝑛_𝑖=1𝑤_𝑖∗𝑓_𝑖𝑥−𝑜_𝑖+𝑜_𝑖old𝜆_𝑖∗𝑀_𝑖+bias_𝑖+𝑓₋bias, (4.5)

Figure 3 

The landscape maps of CF1 function.

where𝑛 is the number of basic function, 𝑤_𝑖 is weight value for each 𝑓_𝑖(𝑥),𝑓_𝑖(𝑥) is 𝑖th basic function used to construct the composition function (here 𝑓₁-𝑓₁₀: Sphere Function), 𝑜_𝑖 is new shifted optimum position for each 𝑓_𝑖(𝑥),𝑜_𝑖old is old optimum position for each 𝑓_𝑖(𝑥), 𝜆_𝑖is used to stretch or compress the function, 𝑀_𝑖 is orthogonal rotation matrix for each 𝑓_𝑖(𝑥), and bias_𝑖 is defines which optimum is global optimum.

4.2. Experimental Setting

Experiments were conducted with PS²O compared with four successful variants of PSO:(i)local version of PSO with constriction factor (CPSO) [21];(ii)fully informed particle swarm (FIPS) [22];(iii)unified particle swarm (UPSO) [23];(iv)fitness-distance-ratio-based PSO (FDR-PSO) [24].

Among these variations, UPSO combined the global version and local version PSO together to construct a unified particle swarm optimizer; FIPS used all the neighbors’ knowledge of the particle to update the velocity; the FDR-PSO selects one other particle, which has a higher fitness value and is nearer to the particle being updated, to update each velocity dimension.

The number of swarms 𝑀 needs be tuned. Three 10D functions, namely Sphere, Rosenbrock, and Griewank, are used to investigate the impact of this parameter. Experiments were executed by changing the number of swarms and fixing each swarm size at 10. The average test results obtained form 30 runs are plotted in Figure 4. From Figure 4, we can observe that the performance of PS²O is influenced by 𝑀. When 𝑀 increases, we obtained faster convergence velocity and better results.

(a)

(b)

(c)

Figure 4 

PS2O’s results on 3 test functions with different 𝑀.

For fair comparison, the population size of all algorithms used in our experiments was set at 100 (all the swarms of PS²O include the same particle numbers of 10). The maximum velocity of all PSO variants was set to be 5% of the search space for unimodal functions and 50% for multimodal functions. For canonical PSO and UPSO, the learning rates 𝑐₁ and 𝑐₂ were both 2.05 and the constriction factor 𝜒=0.729. For FIPS, the constriction factor 𝜒 equals 0.729 and the U-ring topology that achieved highest success rate is used. For FDR-PSO, the inertia weight 𝜔 started at 0.9 and ended at 0.5 and a setting of 𝑐₁=𝑐₂=2.0 was adopted. For PS²O, the interaction topology illustrated in Figure 1(b) is used; the constriction factor in PS²O is also used with 𝜒=0.729 according to Clerc’s method; correspondingly, the 𝜙 coefficient must sum to 4.1 and then the learning rates 𝑐₁=𝑐₂=𝑐₃=𝜙/3≈1.3667.

4.3. Simulation Results

The experiment runs 50 times, respectively, for each algorithm on each benchmark function of 30 dimensions. The numbers of generations were set to be 10000. The representative results obtained are presented in Table 4, including the best, worst, mean, and standard deviation of the function values found in 50 runs. Figures 5, 6, 7, 8, and 9 presents the evolution process for all algorithms according to the reported results in Table 3.

Table 3 

Performance of all algorithms. In bold are the best results.

Table 4 

Criterion of risk rating.

Figure 5 

The median convergence results of Sphere function.

Figure 6 

The median convergence results of Rosenbrock function.

Figure 7 

The median convergence results of Griewank function.

Figure 8 

The median convergence results of Weierstrass function.

Figure 9 

The median convergence results Composition function 1.

From the results, we can observe that the PS²O algorithm obtains an obviously remarkable performance. We can see it clearly that PS²O converged with greatly faster speed to significantly better results than the other PSO variants for both unimodal and multimodal cases. It should be mentioned that the PS²O were the only ones able to consistently find the minimum of the Sphere function, Griewank’s function, Weierstrass function, and Composition function 1, while the other algorithms generated poorer results on them. The result on Rosenbrock obtained by PS²O is also very good. Since a result within 40.0 on 30 D Rosenbrock reported in other EA and SI works is considered well, the PS²O algorithm’s performance on Rosenbrock function is remarkable good.

With the hierarchical interaction topology, a suitable diversity in the whole population can be maintained. At the same time, the enhanced dynamical update rule significantly speeds up the multi-swarm to converge to the global optimum. Because of this, the PS²O performs considerably better than many PSO variants.

5. Virtual Enterprise Risk Management Based on PS²O

A virtual enterprise (VE) [25] is a temporary consortium of autonomous, diverse, and possibly geographically dispersed organizations that pool their resource to meet short-term objectives and exploit fast changing market trends. A VE is a dynamic alliance of member companies (owner and partners), which join to take advantage of a market opportunity. Each member company will provide its own core competencies in areas such as marketing, engineering, and manufacturing to the VE. When the market opportunity has passed, the VE is dissolved. In a VE environment, there are various sources of risks that may threaten the security of VE, such as market risk, credit risk, operational risk, and others [26]. Recently, risk management of a VE has attracted much research attention.

5.1. The Two-Level Optimization Model for Risk Management in a Virtual Enterprise

In this paper, the two-level risk manage model suggested by Huang and Lu [27] is employed to evaluate the performance of the proposed PS²O algorithm. This model can be described as a two-level Distributed Decision Making (DDM) system that is depicted in Figure 10(a).

Figure 10 

The exchange of information between (a) the owner and partners, (b) the upper-swarms, and lower-swarms.

In the top level, the decision maker is the owner who allocates the budget (i.e., the risk cost investment) to each member of VE. The decision variables are therefore given by 𝐼=(𝐼₀,𝐼₁,…,𝐼_𝑛). Here 𝐼₀ denotes the budget to owner and 𝐼_𝑖 (𝑖=1,2,…,𝑛) represents the budget to Partner 𝑖. That is, there are 𝑛+1 members in a VE. Then the top-level objective of risk management in a VE is to best allocate the budget of each member so as to minimize the total risk level of the VE. The top-level model can be formulated as a continuous optimization problem that is given in what follows:min_𝐼𝐹_𝑇(𝐼)=^𝑛_𝑖=0𝑤_𝑖𝑅_𝑖𝐼_𝑖,s.t.^𝑛_𝑖=0𝐼_𝑖≤𝐼_max,𝑅_𝑖𝐼_𝑖≤𝑅_max, (5.1) where 𝑅_𝑖(𝐼_𝑖)is the risk level of 𝑖th member under risk cost investment 𝐼_𝑖, 𝑤_𝑖 represents the weight of member 𝑖, 𝐼_max is the maximum total investment budget, and 𝑅_max stands for the maximum risk level for each member in the VE.

In the base level, the partners of VE are making their decisions in view of the top-level’s instruction (i.e., the budget to partners). The base-level risk management is that the decision makers select the optimal series of risk control actions 𝐴_𝑖=(𝑎^𝑖₁,𝑎^𝑖₂,…,𝑎^𝑖_𝑚) for each partner 𝑖(𝑖=1,2,…,𝑛) to minimize the risk level with respect to the allocated budget 𝐼_𝑖. Here 𝑚 is the number of risk factors that affect each partner’s security. Then the base-level model can be formulated as a discrete optimization problem that is given in what follows:min_𝐴𝐹_𝐵(𝐴)=^𝑛_𝑖=1𝑤_𝑖𝑅_𝑖𝐴_𝑖𝐼_𝑖,s.t.^𝑚_𝑗=1𝐶^𝑖_𝑗𝑎^𝑖_𝑗≤𝐼_𝑖,𝑎^𝑖_𝑗∈{0,1,2,…,𝑊}, (5.2) where 𝑅_𝑖(𝐴_𝑖|𝐼_𝑖) is the risk level of 𝑖th partner under risk control action 𝐴_𝑖 with respect to the top-level investment budget 𝐼_𝑖, 𝐶^𝑖_𝑗(𝑎^𝑖_𝑗) represents the cost of partner 𝑖 under the risk control action 𝑎^𝑖_𝑗 for the risk factor 𝑗, and 𝑊 stands for the number of available actions for each risk factor of each partner.

5.2. Risk Management in VE Based on PS²O

The detailed design of risk management algorithm based on PS²O is introduced in this section. Since the optimizing problem has a two-level hierarchical structure, this risk management algorithm is composed of two types of swarms that search in different levels, respectively, namely, the upper swarm and the lower swarm. The algorithm design reflects a two-phase searching process as Figure 10(b) illustrates. In the top-level searching process, the upper swarms that are designed based on the continuous PS²O, search a continuous space for the investment budget allocation for all VE members. While the lower swarms, which are designed based on the discrete PS²O, receive information from upper swarms, and must search the discrete space for a best action combination for risk management of all partners. The overall searching process can be described as follows.

(1) Particle Representation
(a) Definition of Continuous Particle
In each upper-swarm, each particle has a dimension equal to 𝑛+1(i.e., the number of VE members). Each particle has a real number representation and is a possible allocation of investment budget for all members. The 𝑖th particle of the 𝑘th upper swarm is defined as follows: 𝑋^𝑇_𝑖𝑘=𝑥^𝑇_𝑖1𝑘,𝑥^𝑇_𝑖2𝑘,…,𝑥^𝑇_{𝑖()𝑘𝑛+1},𝑥^𝑇_𝑖𝑗𝑘∈ℜ. (5.3)
For example, a real-number particle (286.55, 678.33, 456.78, 701.21, 567.62) is a possible allocation of investment budget of 5 VE members. The first bit means that the owner received investment of 286.55 units. The 2 to 5 bits mean that the amount of investment allocated to partner 1 to 4 is 678.33, 456.78, 701.21, and 567.62 respectively.(b) Definition of Discrete Particle
For the lower swarms, in order to appropriately represent the action combination by a particle, we design an “action-to-risk-to-partner” representation for the discrete particle. Each discrete particle in each lower swarm has a dimension equal to the number of 𝑛×𝑚×𝑊, here 𝑊 is the number of available actions for each risk factor, 𝑚 is the number of risk factors of each partner, and 𝑛 is the number of VE partners. The 𝑖th particle of the 𝑘th lower swarm is defined as follows: 𝑋^𝐿_𝑖𝑘=𝑥^𝐿_{𝑖()𝑘111},𝑥^𝐿_{𝑖()𝑘112},…,𝑥^𝐿_{𝑖()𝑘𝑛×𝑚×𝑊},𝑥^𝐿_{𝑖𝑘𝛼𝛽𝛾}∈{0,1}, (5.4) where 𝑥^𝐿_{𝑖(𝛼𝛽𝛾)𝑘} equals 1 if the risk factor 𝛽 of VE partner 𝛼 is solved by the 𝛾th action and 0 otherwise. For example, set 𝑛=2,𝑚=4,𝑊=4, suppose the action combination of two partners is (2314, 2401), here 0 stands for no action is selected for the third risk factor of the second partner in VE. By our definition, we have 𝑥^𝐿_{𝑖(112)𝑘}=𝑥^𝐿_{𝑖(123)𝑘}=𝑥^𝐿_{𝑖(131)𝑘}=𝑥^𝐿_{𝑖(144)𝑘}=𝑥^𝐿_{𝑖(212)𝑘}=𝑥^𝐿_{𝑖(224)𝑘}=𝑥^𝐿_{𝑖(241)𝑘}=1 and all other 𝑥^𝐿_{𝑖(𝛼𝛽𝛾)𝑘}=0 (see Figure 11).

(a)

(b)

Figure 11 

Definition of a discrete particle (2314, 2401) for the action combination of two partners.

(2) Risk Management Procedure
The processing performed by this algorithm is best illustrated in the diagram given in Figure 12.Step 1. The first step in top level is to randomly initialize 𝑀^𝑇 upper swarms each possesses 𝑁^𝑇 particles (totally 𝑀^𝑇×𝑁^𝑇 individuals). Each particle 𝑋^𝑇_𝑖𝑘 in the top level is an instruction and is communicated to the base level to drive a base level search process (Steps 2–4).Step 2. For each top-level instruction 𝑋^𝑇_𝑖𝑘, the base-level randomly initialize 𝑀^𝐵 lower swarms each possesses 𝑁^𝐵 particles (totally 𝑀^𝐵×𝑁^𝐵 individuals). At each iteration in base level, for each particle 𝑋^𝐿_𝑖𝑘 (i.e., the 𝑖th particle of the 𝑘th lower-swarm), evaluate its fitness using the base-level optimization function as follows: 𝐹_𝐵𝑋^𝐿_𝑖𝑘=^𝑛_𝛼=1𝑤_𝛼𝑅_𝛼𝑋^𝐿_𝑖𝛼𝑘∣𝑋^𝑇_𝑖𝛼𝑘=^𝑛𝑚_𝛼=1^𝑙_𝛽=1𝜆=1𝑤_𝛼𝑢_𝛽𝑓_𝛽𝜆𝑥^𝐿_{𝑖𝛼𝛽𝑘}𝑑_𝜆,+𝜑^𝑛_𝛼=1^𝑚_𝛽=1𝐶^𝛼_𝛽𝑥^𝐿_{𝑖𝛼𝛽𝑘}−𝑥^𝑇_𝑖𝛼𝑘⁺, (5.5) where 𝑢_𝛽 is the weight of the risk factor 𝛽, 𝑑_𝜆is the value corresponding to the risk rating 𝜆, l is the number of risk ratings, and 𝜑 is the punishment coefficient. 𝑥^𝐿_{𝑖𝛼𝛽𝑘}=(𝑥^𝐿_{𝑖(𝛼𝛽1)𝑘},𝑥^𝐿_{𝑖(𝛼𝛽2)𝑘},…,𝑥^𝐿_{𝑖(𝛼𝛽𝑊)𝑘}) and |𝑥^𝐿_{𝑖𝛼𝛽𝑘}| is defined as the position index of 1 in 𝑥^𝐿_{𝑖𝛼𝛽𝑘}. For example, if 𝑥^𝐿_{𝑖𝛼𝛽𝑘}=(0010), the value of |𝑥^𝐿_{𝑖𝛼𝛽𝑘}| is 3. 𝑓_𝛽𝜆(|𝑥^𝐿_{𝑖𝛼𝛽𝑘}|) is a convex decreasing function, which is approximated by 𝑓_𝛽𝜆𝑥^𝐿_{𝑖𝛼𝛽𝑘}=exp−𝜃_𝛽𝜆𝑥^𝐿_{𝑖𝛼𝛽𝑘} (5.6) to assesses the probability of risk occurrence at risk rating 𝜆 under action |𝑥^𝐿_{𝑖𝛼𝛽𝑘}|. Here the parameter 𝜃_𝛽𝜆 is used to describe the effects of different risk factors under different risk ratings. The cost of the action 𝐶^𝛼_𝛽(|𝑥^𝐿_{𝑖𝛼𝛽𝑘}|) is assumed to be a concave increasing function of the corresponding action, which is approximated by 𝐶^𝛼_𝛽𝑥^𝐿_{𝑖𝛼𝛽𝑘}=1001−exp−𝜏^𝛼_𝛽𝑥^𝐿_{𝑖𝛼𝛽𝑘} (5.7) and the parameter 𝜏^𝛼_𝛽 describes the effects of different risk factors of different partners. The notation (⋅)⁺ is defined as follows: (𝑥)⁺=𝑥if𝑥>0,0else. (5.8)Step 3. Compare the evaluated fitness values and select pbest, sbest, and cbest for each lower swarm. Then update the velocity 𝑣^𝐿_{𝑖(𝛼𝛽𝛾)𝑘} of each base level particle according to (3.1). For our problem, each partner can only select one action for each risk factor or do nothing with this factor. In order to take care of this problem, for each particle, action 𝛾 is selected for risk factor 𝛽 of partner 𝛼 according to following probability: 𝑝_{𝑖𝑘𝛼𝛽𝛾}=𝑠𝜈^𝐿_{𝑖𝑘𝛼𝛽𝛾}∑^𝑊_𝛾=1𝑠𝜈^𝐿_{𝑖𝑘𝛼𝛽𝛾}. (5.9) Then the position of each base-level particle is updated by Algorithm 1.Step 4. The base-level search process is repeated until the maximum number of base-level iteration is met. Then send the last best base-level decision variable 𝑋^𝐿∗_𝑖𝑘 to the top-level for the fitness computation of the top-level particle 𝑋^𝑇_𝑖𝑘.Step 5. With the base-level reaction 𝑋^𝐿∗_𝑖𝑘, each top-level particle 𝑋^𝑇_𝑖𝑘 is evaluated by the following top-level fitness function: 𝐹_𝑇𝑋^𝑇_𝑖𝑘=^𝑛_𝛼=0𝑤_𝛼𝑅_𝛼𝑋^𝑇_{𝑖()𝑘𝛼+1}=𝑤₀𝑅₀𝑋^𝑇_𝑖1𝑘+𝐹_𝐵𝑋^𝐿∗_𝑖𝑘+𝜙^𝑛+1_𝛼=1𝑥^𝑇_𝑖𝛼𝑘−𝐼_max⁺+𝜂^𝑛_𝛼=1^𝑚𝑙_𝛽=1𝜆=1𝑤_𝛼𝑢_𝛽𝑓_𝛽𝜆𝑥^𝐿_{𝑖𝛼𝛽𝑘}𝑑_𝜆−𝑅_max⁺, (5.10) where 𝜙 and 𝜂 is the punishment coefficient and the risk level of the owner 𝑅₀(𝑋^𝑇_𝑖1𝑘) is approximated by a convex decreasing function as follows: 𝑅₀𝑋^𝑇_𝑖1𝑘=exp−0.001𝑋^𝑇_𝑖1𝑘. (5.11)Step 6. Compare the evaluated fitness values and select pbest, sbest, and cbest for each upper swarm. Then update the velocity and position of each top-level particle according to (3.1) and (3.2). The top-level computation is repeated until the maximum number of top-level iteration is met.

Begin Let 𝑋temp be a zero vector that has a dimension equal to 𝑛×𝑚×𝑊. For   (𝛼=1𝑡𝑜𝑛)  For   (𝛽=1𝑡𝑜𝑚)   For   (𝛾=1𝑡𝑜𝑊)    If   (rand≤𝑝𝑖(𝛼𝛽𝛾)𝑘)     //Action𝛾 is selected for risk𝛽 of partner𝛼 𝑋temp𝑡𝛼𝛽𝛾=1;     Break;    End if   End for  End for End for 𝑋𝐿𝑖𝑘=𝑋temp END

Algorithm 1 

Figure 12 

The risk management algorithm based on PS2O.

5.3. An Illustrative Example

In this section, a numerical example of a VE is conducted to validate the capability of VE risk management based on the proposed PS²O. In order to show the superiority of PS²O, the risk management algorithm designed by canonical PSO is also applied to the same case.

In this case, the VE is constructed by one owner and four partners (i.e., 𝑛=4) and the total investment is 𝐵_max=3500; 10 risk factors are considered for each partner and 4 actions can be selected for each risk factor (i.e., 𝑚=10 and 𝑊=4); the number of risk ratings 𝑙=3 and the value of each rating is 𝑑₁=0.165, 𝑑₂=0.335, and 𝑑₃=0.500, respectively (according to the values of ratings, the criterion of risk rating is shown in Table 4); the maximum risk level 𝑅_max=0.67, which means that the risk level of each member must be below the medium level; the weight of risk level of each VE member is 𝑤₀=𝑤₁=𝑤₂=𝑤₃=𝑤₄ and the weights 𝑢_𝛽 of each risk factors for each partner are listed in Table 5; the values of the parameter 𝜃_𝛽𝜆 and 𝜏^𝛼_𝛽 are presented in Tables 6 and 7, respectively; the punishment coefficient 𝜙, 𝜂, and 𝜑 are given as 1.5, 28 and 0.2.

Table 5 

The weights of the risk factors.

Table 6 

The summary of parameter 𝜃_𝛽𝜆.

Table 7 

The summary of parameter 𝜏^𝛼_𝛽.

In applying PS²O and PSO to this case, the continuous and binary algorithms are used in top level and base level of the optimization model respectively. For the top-level algorithms, the maximum generation in each execution for each algorithm is 100; the initialized population size of 10 particles is the same for PS²O and PSO, while the whole population is divided into 2 swarms (each possesses 5 individuals) for PS²O in the initialization step; the interaction topology illustrated in Figure 1(a) is used for continuous PS²O; the other parameters of continuous PS²O and PSO were set to the same values as in Section 4. For the base-level algorithms, the maximum generation for each algorithm is 100; the initialized population size of 20 particles is the same for PS²O and PSO, while the whole population is divided into 4 swarms (each possesses 5 individuals) for PS²O in the initialization step; the interaction topology illustrated in Figure 1(b) is used for binary PS²O; the other parameters of binary PS²O and PSO were set to the same values as in Section 4. The experiment runs 30 times, respectively, for each algorithm.

The top-level and base-level search progresses of the averaged best-so-far fitness values over 20 runs are shown in Figures 13 and 14, respectively. It should be noted that the total iteration of base-level searching is 100 (base-level maximum generation) ×10 (top-level population size) ×10 (top-level maximum generation) = 10⁴. That is, the base-level algorithms will be restarted after every 100 iterations. From the figures, we can see that PS²O converges with a higher speed compared to PSO and obtains better results in both levels searching progresses.

Figure 13 

The top-level search process based on PS2O and PSO.

Figure 14 

The base-level search process based on PS2O and PSO.

The average solutions over 30 runs obtained by PS²O and PSO are summarized in Table 8. Before proceeding with the risk management procedure, the risk levels are one for both the VE and the partner, which is a high risk level. Table 8 shows that the resulting risk levels of the VE and the owner are in the low risk level, while all the partners are in the medium risk level. Therefore the budget and the actions selected by the owner and the partner are very effective to reduce the risk of the VE.

Table 8 

Simulation results of both algorithms.

To fully demonstrate the risk management performance using the PS²O algorithm, risk investment budget, and risk level controlling processes of each VE member based on PS²O is shown in Figure 15. Generally, an effective actions sequence corresponds to a higher cost and a lower risk level. From the figures, it can be concluded that the additional cost of selecting effect actions can not result in a remarkable decrease in the risk level.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 15 

The base-level search process based on PS2O and PSO.

6. Conclusions

In this paper, we develop an optimization model for minimizing the risks of the virtual enterprise based on a novel multi swarm optimizer PS²O. In PS²O, the hierarchical interaction topology consists of two levels (i.e., the individual level and the swarm level), in which information exchanges take place not only between the particles within each swarm but also between different swarms. The dynamical update equations of our multi-swarm approach are enhanced by a significant ingredient, which takes into account the symbiotic coevolution (or heterogeneous cooperation) between different swarms. Because of this, each individual of the proposed model evolves based on the knowledge integration of itself (associate with individual’s own cognition), its swarm members (associate social interaction within each swarm), and its symbiotic partners from other swarm (associate heterogeneous cooperation between different swarms). With five mathematical benchmark functions, PS²O is proved to have significantly better performance than four successful variants of PSO.

In the proposed risk management model of VE, a two-level optimization scheme was introduced to describe the decision processes of the owner and the partners. This DDM model considers the situation that the owner allocates the budget to each member of the VE in order to minimize the risk level of the VE. Accordingly, a transfer optimization model, which can easily use EA and SI algorithms to treat the risk manage problem in VE, is elaborately developed. PS²O is then employed to solve the real-world VE risk management problem. The simulation studies, which compared to Canonical PSO algorithm, show that the PS²O obtains superior risk management solutions than PSO methods in terms of optimization accuracy and convergence speed.

Acknowledgments

This work is supported by the Natural Science Foundation of Liaoning Province of China under Grant 20082006, the Support Program for the Outstanding Technological Person in Liaoning Province of China under Grant lr2011035, and the National Natural Science Foundation of China under Grants 61105067 and 61174164.