Performance evaluation models for parallel computing with dual hesitant fuzzy information

Abstract

The problem of evaluating the performance of parallel computing with dual hesitant fuzzy information denotes the multiple attribute decision making issues. Afterwards, inspired by the method of Hamacher aggregation operators and the Choquet integral, in this work, we design a dual hesitant fuzzy Hamacher correlated average (DHFHCA) operator. Next, we exploit the proposed operator to implement the model to tackle the dual hesitant fuzzy multiple attribute decision making problem. In the end, an example is presented to evaluate performance of the proposed method.

Keywords

Multiple attribute decision making dual hesitant fuzzy values dual hesitant fuzzy Hamacher correlated average (DHFHCA) operator performance of parallel computing

1 Introduction

Due to the combined effect of semiconductor technology, manufacturing processes and power consumption, the high performance computing (HPC) community has seen diverse processor architectures and many kinds of parallel computers. For the green computing system, the CPU/GPU heterogeneous HPC systems have a tradeoff among the versatility, performance and effectiveness, which means that they have a very promising future. Large CPU/GPU heterogeneous systems have immense computing power, and provide a good opportunity for large-scale scientific and engineering applications. However, the complex hardware structure and specific execution scheme present a huge problem for parallel computing researchers. The parallel computing involves many research topics. We only concentrate on three of them, namely the parallel computation model, parallel programming model and parallel scalability model. Parallel computation model is an abstraction of the underlying parallel computer systems and reflects their resources and performance characteristics with several parameters. It acts as a bridge between software and hardware for parallel algorithm designers. Parallel programming model is a collection of program abstractions to provide a transparent computer software/hardware system diagram for parallel programmers. Parallel scalability model describes the scalability ofa parallel system when the system/problem size changes. The CPU/GPU heterogeneous HPC systems have specific structural characteristics and performance factors, which cannot be accurately described by the existing models. Hence, there is an urgent need to carry out parallel computing research for such HPC systems. It can provide support for the current and the future parallel application development based on these platforms. The problem of evaluating the performance of parallel computing with dual hesitant fuzzy information is the multiple attribute decision making problems [1 –23].

In this paper, we investigate the multiple attribute decision making problems with dual hesitant fuzzy information. Hence, inspired by the idea of Hamacher aggregation operators and the Choquet integral, this work provides the dual hesitant fuzzy Hamacher correlated average (DHFHCA) operator. Afterwards, this work used this operator to design the model to tackle the dual hesitant fuzzy multiple attribute decision making problems. In the end, a practical example for evaluating the performance of parallel computing is provided to test the proposed method.

2 Preliminaries

Definition 1. [24, 25] Supposing that X be a fixed set, thus, a dual hesitant fuzzy set (DHFS) based on X can be represented as follows. $D = (〈 x, h (x), g (x) 〉 | x \in X)$ (1) where h (x) and g (x) refer to two sets of some values which are ranged in [0, 1], denoting the possible membership degrees and non-membership degrees of the element x ∈ X to the set, and the following equation should be satisfied. $0 \leq γ, η \leq 1, 0 \leq γ^{+}, η^{+} \leq 1$ where γ∈ h (x) , η ∈ g (x) , γ⁺ ∈ h⁺ (x) = ∪ _γ∈h(x) max { γ }, η⁺∈ g⁺ (x) = ∪ _η∈g(x) max { η } for all x ∈ X.

Definition 2. [24, 25] Let d_i = (h_{d
_i}, g_{d
_i}) (i = 1, 2) be any two DHFEs, $s (d) = \frac{1}{# h} \sum_{γ \in h} γ - \frac{1}{# g} \sum_{η \in g} η$ the score function of d, and $p (d) = \frac{1}{# h} \sum_{γ \in h} γ + \frac{1}{# g} \sum_{η \in g} η$ the accuracy function of d, where # h and # g are the numbers of elements in set h and g.

3 Dual hesitant fuzzy Hamacher correlated aggregation operators

For the decision making problem, some degree of inter-dependent characteristics exist in different attributes. Therefore, it is of great importance to solve this problem.

Definition 3. [26] Assuming that f denotes a positive real-valued function on X, and μ refers to a fuzzy measure on X. The discrete Choquet integral of f which is corresponding to μ can be represented as follows.

$C_{μ} (f) = \sum_{i = 1}^{n} f_{σ (i)} [μ (A_{σ (i)}) - μ (A_{σ (i - 1)})]$ (2) where (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n), such that f_σ(i-1) ≥ f_σ(i) for all j = 2, …, n, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

Considering the aggregation rule for dual hesitant fuzzy sets and Choquet integral [27], next, using Hamacher operations [28 –30], we shall develop some dual hesitant fuzzy Hamacher correlated aggregation operators with dual hesitant fuzzy information.

Definition 4. Let ${\tilde{d}}_{j} (j = 1, 2, \dots, n)$ be a collection of DHFEs on X, and μ be a fuzzy measure on X, then we call

$\begin{matrix} {DHFHCA}_{μ} ({\tilde{d}}_{1}, {\tilde{d}}_{2}, \dots, {\tilde{d}}_{n}) \\ = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{d}}_{σ (j)}) \end{matrix}$ (3) the dual hesitant fuzzy Hamacher correlated aggregation (DHFHCA) operator, where (σ (1) , σ (2) , …, σ (n)) refers to a permutation of (1, 2, …, n), such that ${\tilde{d}}_{σ (j - 1)} \geq {\tilde{d}}_{σ (j)}$ for all j = 2, …, n, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

Using the operation of dual hesitant fuzzy sets, the DHFHCA operator is able to be transformed as follows:

$\begin{matrix} {DHFHCA}_{μ} ({\tilde{d}}_{1}, {\tilde{d}}_{2}, \dots, {\tilde{d}}_{n}) = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{d}}_{σ (j)}) \\ = \cup_{γ_{σ (j)} \in h_{σ (j)}, η_{σ (j)} \in g_{σ (j)}} {{\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} - \prod_{j = 1}^{n} {(1 - γ_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - γ_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}}, \\ {\frac{γ \prod_{j = 1}^{n} {(η_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - η_{σ (j)}))}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} + (γ - 1) \prod_{j = 1}^{n} {(η_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}}} \end{matrix}$ (4) of which aggregated value denotes a dual hesitant fuzzy value as well.

Especially, if μ ({x_σ(j)}) = μ (A_σ(j)) - μ (A_σ(j-1)), i = 1, 2, …, n, then DHFHCA operator reduce to DHFHWA operator. If μ (A) = ∑_{x_j∈A}μ ({ x_j }), for all A ⊆ X, where |A| is thenumber of the elements in the set A, then w_j = μ (A_σ(j)) - μ (A_σ(j-1)), i = 1, 2, …, n, where w = (w₁, w₂, …, w_n) ^T, w_j ≥ 0, i = 1, 2, …, n, and $\sum_{j = 1}^{n} w_{j} = 1$ , then, DHFHCA operator reduce to DHFHOWA operator.

4 Models for multiple attribute decision making based on dual hesitant fuzzy information

Let A ={ A₁, A₂, …, A_m } be a discrete set of alternatives, and G ={ G₁, G₂, …, G_n } be the state of nature. Suppose that the decision matrix $\tilde{D} = {({\tilde{d}}_{ij})}_{m \times n}$ is the dual hesitant fuzzy decision matrix, where ${\tilde{d}}_{ij} (i = 1, 2, \dots, m, j = 1, 2, \dots, n)$ are in the form of DHFEs. Next, we utilize DHFHCA operator to the MADM problems for evaluating the scientific and technological papers’ quality with dual hesitant fuzzy information.

Step 1. We adopt decision information which are proposed in matrix $\tilde{D}$ , and the DHFHCA operator

$\begin{matrix} {\tilde{d}}_{i} & = & {DHFHCA}_{μ} ({\tilde{d}}_{i 1}, {\tilde{d}}_{i 2}, \dots, {\tilde{d}}_{in}) = \oplus_{j = 1}^{n} ((μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)})) {\tilde{d}}_{σ (ij)}) \\ = & \cup_{γ_{σ (ij)} \in h_{σ (ij)}, η_{σ (ij)} \in g_{σ (ij)}} \\ {{\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{σ (ij)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))} - \prod_{j = 1}^{n} {(1 - γ_{σ (ij)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) γ_{σ (j)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - γ_{σ (j)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))}}}, \\ {\frac{γ \prod_{j = 1}^{n} {(η_{σ (ij)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - η_{σ (ij)}))}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))} + (γ - 1) \prod_{j = 1}^{n} {(η_{σ (ij)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))}}}} \end{matrix}$ (5) to derive the overall preference values ${\tilde{d}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i.

Step 2. Estimate the scores $S ({\tilde{d}}_{i}) (i = 1, 2, \dots, m)$ of the overall dual hesitant fuzzy preference values ${\tilde{d}}_{i} (i = 1, 2, \dots, m)$ .

Step 3. Rank all the alternatives A_i (i = 1, 2, …, m) and choose the best element considering the scores $S ({\tilde{d}}_{i}) (i = 1, 2, \dots, m)$ .

5 Numerical example

Entering the era of big data, the parallel machine architectures, scalable ability of computing resources and industry application mode are changed greatly and rapidly, which bring both opportunities and challenges for parallel computing. As a bridge between hardware and software, parallel computational model is an important technology to promote the research and development of big data. Recently, industry has proposed several big data programming models which are widely used on terabyte and petabyte data processing, while academics are studying big data computation models which can reflect the parallel machine architectures, reveal the principle of computation, communication and I/O behavior of big data jobs, analyzes the popular big data systems theoretically, and provide optimization guidelines for big data applications. Inspired by traditional parallel computational model, popular big data programming model and current big data computational model, we conclude that there are three theoretical problems needed to be solved. They are three elements (parameters, behaviors, and cost functions) problem, scalability and fault-tolerance problem and performance optimization problem. Focusing on these problems, on the one hand, we study the big data computational model and its optimization methods theoretically; on the other hand, we apply these methods in real big data case studies. Therefore, in this part, we illustrate a numerical example for evaluating the performance of parallel computing with dual hesitant fuzzy information to describe our approach. Particularly, we give a panel using five parallel computing models A_i (i = 1, 2, …, 5) to evaluate. The education institution should be given using the given 4 attributes, that is, ding172G₁ refers to the CPU computing performance; ding173G₂ means the memory bandwidth and capacity; ding174G₃ is the network bandwidth; ding175G₄ is the network latancy. To alleviate the effect between each other, the decision makers should be used to estimate the five parallel computing models A_i (i = 1, 2, 3, 4, 5) with the given attributes in anonymity and the decision matrix $\tilde{D} = {({\tilde{d}}_{ij})}_{5 \times 4}$ is given in Table 1, where ${\tilde{d}}_{ij} (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4)$ conform to the definition of DHFEs.

Attribute weights is allocated using the condition: ω = (0.30, 0.10, 0.40, 0.20).

Next, we exploit the method developed to test performance of parallel computing with dual hesitant fuzzy information.

Step 1. Assume that the fuzzy mode of attribute of G_j (j = 1, 2, …, n) and attribute sets of G as follows: $\begin{matrix} μ (G_{1}) = 0.20, μ (G_{2}) = 0.30, μ (G_{3}) = 0.25, μ (G_{4}) = 0.15 \\ μ (G_{1}, G_{2}) = 0.65, μ (G_{1}, G_{3}) = 0.50, μ (G_{1}, G_{4}) = 0.55, μ (G_{2}, G_{3}) = 0.45 \\ μ (G_{2}, G_{4}) = 0.40, μ (G_{3}, G_{4}) = 0.45, μ (G_{1}, G_{2}, G_{3}) = 0.80, μ (G_{1}, G_{2}, G_{4}) = 0.85 \\ μ (G_{1}, G_{3}, G_{4}) = 0.70, μ (G_{2}, G_{3}, G_{4}) = 0.65, μ (G_{1}, G_{2}, G_{3}, G_{4}) = 1.00 \end{matrix}$

Step 2. We use the decision information which are defined in matrix $\tilde{D}$ , and the DHFHCA operator to achieve the overall preference values ${\tilde{d}}_{i}$ of the five parallel computing models A_i (i = 1, 2, 3, 4, 5) and calculate the scores $s ({\tilde{d}}_{i}) (i = 1, 2, 3, 4, 5)$ of the five parallel computing models A_i: $\begin{matrix} s ({\tilde{d}}_{1}) & = & - 0.2245, s ({\tilde{d}}_{2}) = - 0.1786 \\ s ({\tilde{d}}_{3}) & = & - 0.1128, s ({\tilde{d}}_{4}) = - 0.2354 . \\ s ({\tilde{d}}_{5}) & = & 0.2013 \end{matrix}$

Step 3. Rank all the five parallel computing models according to the scores $s ({\tilde{d}}_{i}) (i = 1, 2, 3, 4, 5)$ of the overall dual hesitant fuzzy values ${\tilde{d}}_{i} (i = 1, 2, \dots, 5)$ : A₅ ≻ A₃ ≻ A₂ ≻ A₁ ≻ A₄, and thus the most desirable parallel computing model is A₅.

6 Conclusion

The high performance computing (HPC) has been widely applied in the field of military simulation. Along with the improvements of both simulation scale and simulation resolution, the demand for communication and computing resources continuously increased. As a result, the HLA simulation based on PC network cannot meet the demand. The high performance computing environment can provide powerful computing and communication capabilities for large-scale simulation applications. However most of current RTI software was based on distributed network environments. The advantages of the high performance computing environment cannot be fully exploited on the communication structure, message processing, data consistency maintenance aspects. Therefore, the research on the techniques of HLA/RTI based on high performance computing environments, give full play to the platform for efficient communication and computing power to improve the operating efficiency of the simulation system, promote the development of large-scale simulation applications and so has a very important significance. The problem of evaluating the performance of parallel computing using dual hesitant fuzzy information are regarded as the multiple attribute decision making problem. Afterwards, inspired by the method of Hamacher aggregation operators and the Choquet integral, we have designed the dual hesitant fuzzy Hamacher correlated average (DHFHCA) operator. Next, we have used this operator to design the model to tackle the dual hesitant fuzzy multiple attribute decision making problems. In the end, we propose an example to test performance of our proposed method.

Footnotes

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2011B09414) and the Natural Science Foundation of Tibet Autonomous Region.

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