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Virtual enterprise (VE) has to manage its risk effectively in order to guarantee the profit. However, restricting the risk in a VE to the acceptable level is considered difficult due to the agility and diversity of its distributed characteristics. First, in this paper, an optimization model for VE risk management based on distributed decision making model is introduced. This optimization model has two levels, namely, the top model and the base model, which describe the decision processes of the owner and the partners of the VE, respectively. In order to solve the proposed model effectively, this work then applies two powerful artificial intelligence optimization techniques known as evolutionary algorithms (EA) and swarm intelligence (SI). Experiments present comparative studies on the VE risk management problem for one EA and three state-of-the-art SI algorithms. All of the algorithms are evaluated against a test scenario, in which the VE is constructed by one owner and different partners. The simulation results show that the

A virtual enterprise (VE) [

Up to date, risk management of VE has received considerable research attentions. Various models and algorithms are developed to provide a more scientific and effective way for managing the risk of a VE. Ma and Zhang [

Nature serves as a fertile source of concepts, principles, and mechanisms for designing artificial computation systems to tackle complex computational problems. In the past few decades, many nature-inspired computational techniques were designed to deal with practical problems. Among them, the most successful are evolutionary algorithms (EA) and swarm intelligence (SI). Evolutionary algorithms are search methods that take their inspiration from natural selection and survival of the fittest in the biological world. Several different types of EA methods were developed independently. These include genetic programming (GP) [^{2}O, which extends the single population PSO to interacting multi-swarm model by constructing hierarchical interaction topologies and enhanced dynamical update equations. By incorporating the new degree of complexity, PS^{2}O can avoid premature convergence drawback of traditional SI algorithms and accommodate a considerable potential for solving more complex problems.

In this paper, we develop an optimization model for distributed decision making of risk management in VE based on the evolutionary and swarm-based methods. Here the VE risk management problem is described and formulated as a two-level DDM model, which is in order to minimize the aggregate risk level of the VE to a reasonable lower level. Then, in order to solve this complex problem effectively and efficiently, the optimization procedure based on EA and SI systems is developed. Experiments are performed on three VE risk management cases with different scales. In the experiments, a comprehensive comparative study on the performances of four well-known evolutionary and swarm-based algorithms, namely, GA, PSO, ABC, and the recently proposed PS^{2}O, is presented. Results show that the performance of the PS^{2}O is better than or similar to those of other EA and SI algorithms with the advantage of maintaining suitable diversity of the whole population in optimization process.

The paper is organized as follows. Section ^{2}O algorithms are summarized. Section

In this paper, the two-level risk management model suggested by Lu et al. [

DDM model for risk management in a VE.

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

In the base-level, the partners of VE are making their decisions according to the top-level’s instruction (i.e., the budget to partners). The base-level risk management is that the decision maker selects the optimal series of risk control actions

Genetic algorithm is a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such as inheritance, mutation, selection, and crossover. A basic GA consists of five components. These are a random number generator, a fitness evaluation unit, genetic operators for reproduction, crossover, and mutation operations. The basic algorithm is summarized in Algorithm

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At the start of the algorithm, the population initialization step randomly generates a set of number strings. Each string is a representation of a solution to the optimization problem being addressed. Continuous and discrete strings are both commonly employed. Associated with each string is a fitness value computed by the evaluation unit. The reproduction operator performs a natural selection function known as seeded selection. Individual strings are copied from one set (representing a generation of solutions) to the next according to their fitness values; the better the fitness value, the greater the probability of a string being selected for the next generation. The crossover operator chooses pairs of strings at random and produces new pairs. The simplest crossover operation is to cut the original parent strings at a randomly selected point and to exchange their tails. The number of crossover operations is governed by a crossover rate. The mutation operator, which is determined by a mutation rate, randomly mutates or reverses the values of bits in a string. A phase of the algorithm consists of applying the evaluation, reproduction, crossover, and mutation operations. A new generation of solutions is produced with each phase of the algorithm [

Artificial bee colony (ABC) algorithm is one of the most recently introduced SI algorithms. ABC simulates the intelligent foraging behavior of a honeybee swarm. In ABC model, the foraging bees are classified into three categories: employed bees, onlookers, and scout bees. The main steps of the algorithm are as shown in Algorithm

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ABC starts by associating all employed bees with randomly generated food sources (solution). In mathematical terms,

The canonical PSO is a successful SI-based technique. In PSO model, the rules that govern particles’ movements are inspired by models of fish schooling and bird flocking [

its previous best position,

best position of its neighbors.

In mathematical terms, the

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Kennedy and Eberhart [

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 [

Inspired by mutualism phenomenon, in the previous works [

its own previous best position,

best position of its neighbors from its own swarm,

best position of its neighbor swarms.

Hierarchical topology of the multi-swarm.

In mathematical terms, our multi-swarm model is defined as a triplet ^{2}O algorithm is listed in Algorithm

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We should note that, for solving discrete problems, we still use (^{2}O algorithm.

The detailed design of Risk management algorithm based on EA and SI algorithms is introduced in this section. Since the risk management model described in Section

In upper-population, each individual has a dimension equal to

For example, a real-number particle (286.55, 678.33, 456.78, 701.21, 567.62) is an investment budget possible allocation of a VE consisting of 5 members. The first bit means that the owner received investment of 286.55 units. The 2 to 5 bits mean that the amounts of investment allocated to partner 1 to 4 are 678.33, 456.78, 701.21, and 567.62, respectively.

For the lower-population, in order to appropriately represent the action combination by a particle, we design an “action-to-risk-to-partner” representation for the discrete individual. Each discrete individual in each lower-population has a dimension equal to the number of

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

Then, the base-level objectives, that is, (

And the top-level objectives, that is, (

In order to easily use EA and SI algorithms to treat the risk management problem in VE, it is clear that the rewritten model, that is, (

The overall risk management process based on EA and SI algorithms can be described as follows.

The first step in top-level is to randomly initialize the EA and SI-based upper-population. Each individual

For each top-level

Compare the evaluated fitness values for all individuals in lower-population. Then update each base-level individual by its updating rules according to the selected EA and SI algorithms. 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

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// Action

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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

With the base-level reaction

Compare the evaluated fitness values for all individuals in upper-population. Then update each top-level individual by its updating rules according to the selected EA and SI algorithms. The top-level computation is repeated until the maximum number of top-level iteration is met.

The flowchart of this risk management process is illustrated in the diagram given in Figure

The risk management process based on EA and SI.

In this section, a numerical example of a VE is conducted to validate the capability of the proposed VE risk management method. Experiments were conducted with four EA and SI-based algorithms, namely, GA, PSO, ABC, and PS^{2}O, to fully evaluate the performance of the proposed optimization model.

In this section, the total investment is

Criterion of risk rating.

Value of risk probability | Risk level |
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[0.00, 0.38] | Low risk |

(0.38, 0.67] | Medium risk |

(0.67, 1.00] | High risk |

The weights of the risk factors.

Risk factor | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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0.1 | 0.15 | 0.10 | 0.05 | 0.10 | 0.10 | 0.15 | 0.10 | 0.05 | 0.10 |

The summary of parameter

1 | 2 | 3 | |

1 | 0.10 | 0.07 | 0.13 |

2 | 0.23 | 0.20 | 0.17 |

3 | 0.33 | 0.27 | 0.30 |

4 | 0.37 | 0.40 | 0.43 |

5 | 0.50 | 0.47 | 0.53 |

6 | 0.63 | 0.57 | 0.60 |

7 | 0.73 | 0.70 | 0.67 |

8 | 0.83 | 0.77 | 0.80 |

9 | 0.87 | 0.90 | 0.93 |

10 | 1.00 | 0.97 | 1.03 |

The summary of parameter

Risk factor | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |

This simulated VE environment can be constructed by one owner and different number of partners. For scalability study purpose, all involved algorithms are tested on three illustrative VE examples with 2, 4, and 9 partners (i.e.,

In applying EA and SI algorithms to this case, the continuous and binary versions of these algorithms are used in top-level and base-level of the DDM optimization model, respectively. For the top-level algorithms, the maximum generation in each execution for each algorithm is 50; the initialized population size of 10 individuals is the same for all involved algorithms, while the whole population is divided into 2 swarms (each possesses 5 individuals) for PS^{2}O in the initialization step. For the base-level algorithms, the maximum generation for each algorithm is 100; the initialized population size of 20 particles is the same for all involved algorithms, while the whole population is divided into 4 swarms (each possesses 5 individuals) for PS^{2}O in the initialization step. The experiment runs 30 times, respectively, for each algorithm. The other specific parameters of algorithms are given below.

The experiment employed a binary coded standard GA having random selection, crossover, mutation, and elite units. Stochastic uniform sampling technique was the chosen selection method. Single-point crossover operation with the rate of 0.8 was employed. Mutation operation restores genetic diversity lost during the application of reproduction and crossover. Mutation rate in the experiment was set to be 0.01.

For continuous PSO, the learning rates

The basic ABC is used in the study. Since there are no literatures using ABC for discrete optimization so far, this experiment just used crossover operation to update individuals in ABC population. That is, the ABC position update (

For continuous PS^{2}O, the parameters were set to the values ^{2}O, the parameters were set to the values

All algorithms are tested on the risk management problems with 3, 5, and 10 VE members. The representative results obtained are presented in Table ^{2}O algorithm can consistently converge to better results than the other three algorithms for all test cases. Also, PS^{2}O is the most fast one for finding good results within relatively few generations.

Results of all algorithms. In bold are the best.

Scale of VE | PSO | PS^{2}O | GA | ABC | |
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3 members | Best | 0.2167 | 0.2065 | 0.2240 | |

Worst | 0.3034 | 0.5420 | 0.2650 | ||

Mean | 0.2514 | 0.2356 | 0.2436 | ||

Std | 0.0202 | 0.0626 | 0.0112 | ||

5 members | Best | 0.3396 | 0.3218 | 0.3166 | |

Worst | 0.4804 | 3.2243 | 0.3891 | ||

Mean | 0.3739 | 0.5139 | 0.3593 | ||

Std | 0.0331 | 0.6110 | 0.0148 | ||

10 members | Best | 0.3320 | 0.2641 | 0.3940 | |

Worst | 4.7246 | 2.7449 | 4.2746 | ||

Mean | 0.7786 | 0.8987 | 0.7578 | ||

Std | 0.9049 | 0.6972 | 0.9817 |

The iteration courses of all algorithms on different VE scales. (a) 3 members. (b) 4 members. (c) 10 members.

In this experiment, the analysis of variance (ANOVA) test was also carried out to validate the efficacy of four tested EA and SI methods. The graphical statistics analyses are done through box plot. A box plot is a graphical tool, which provides an excellent visual summary of many important aspects of a distribution. The box stretches from the lower hinge (defined as the 25th percentile) to the upper hinge (the 75th percentile) and therefore contains the middle half of the scores in the distribution. The median is shown as a line across the box. Therefore, one-fourth of the distribution is between this line and the top of the box and one-fourth of the distribution is between this line and the bottom of the box.

First, the box plots for the results presented in Table ^{2}O provides better results for all the test cases than those of the other three algorithms.

ANOVA test for all candidate algorithms on different VE scales of (a) 3 members, (b) 4 members, and (c) 10 members. (Here 1, 2, 3, and 4 are the algorithm index of PSO, PS^{2}O, GA, ABC, resp.).

Second, to compare the robustness of the involved algorithms on the risk manage problem, the experiment can be statistically considered as one-factor experiment, in which the optimization result was the response variable and the scales of the VE were the factor, which had 3 levels: 3, 5, and 10. The results of the ANOVA for the VE risk management in different scales using these algorithms were presented in Figure ^{2}O algorithm. Therefore, against the scales variation of the testing VE cases, the robustness of the PS^{2}O is much better than those of the PSO, GA, and ABC.

ANOVA test for the risk management results with different VE scales optimized by (a) PSO, (b) PS^{2}O, (c) GA, and (d) ABC.

In this paper, we develop an optimization model for minimizing the risks of the virtual enterprise based on evolutionary and swarm intelligence methods. First, a two-level risk management model 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 management problem in VE, is elaborately developed. We should note that the proposed optimization model is genetic and extendible: the model does not depend on the optimization algorithm used and other evolutionary and swarm intelligence techniques could be equally well adopted, which enable a comparison of various algorithms for the same application scenario.

Experiments show comparative studies on the VE risk management problem for the GA, PSO, ABC, and PS^{2}O. The simulation results show that the PS^{2}O algorithm obtains superior solutions on three testing cases than the other algorithms in terms of optimization accuracy and computation robustness. That is, in PS^{2}O, 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.

This work is supported by the National 863 plans projects of China under Grants nos. 2006AA04A117 and 08H2010201. The first author would like to thank Dr. Fuqing Lu for helpful discussions and constructive comments.