Five Forces Model of Computational Power: A Comprehensive Measure Method

Abstract

In this paper, a model is proposed to comprehensively evaluate the computational power. The five forces model of computational power solves the problem that the measurement units of different indexes are not unified in the process of computational power evaluation. It combines the bidirectional projection method with TOPSIS method. This model is more scientific and effective in evaluating the comprehensive situation of computational power. Lastly, an example shows the validity and practicability of the model.

1. Introduction

In the era of digital economy, everything is connected to produce all kinds of big data. Data has become the fifth factor of production besides labor, capital, land and technology. Recently, data center became an important part of information infrastructure. It can be said that the takeoff of digital economy cannot be achieved without the support of computational power, and the development of computational power cannot be separated from data centers.

The industry has been constantly explored how computational power is measured and represented. In a narrow sense, computational power is a capability of a server to achieve result output by processing data. Up to now, the most widely used representation of computational power isfloating point operations per second (FLOPS), the number of floating point operations performed per second, which was first published by Frank H. McMahon [1]. Many domestic and foreign literature and server product parameters use floating point operation times to describe the computational power. For example, Y. Sun et al. [2] used FLOPS as a metric to evaluate single and double precision computational capabilities of CPU and GPU [3]. Therefore, a scientific method is urgently needed to fully express the connotation of computational power.

Multi-attribute group decision making [4-7] is an intersection of multi-attribute decision making and group decision making. It is a significant research field and has been generally used in urban planning, investment risk and other areas. At present, the problem of evaluating various metrics for computational power synthesis among different data centers could be summarized as a multi-attribute group decision problem. In order to evaluate the computational power comprehensively, this paper puts forward the "five forces model". The model integrates and compares the general computational power, intelligent computational power, computational efficiency, storage capacity and network capacity, which are highly related to the computational power of the data center. It uses the new two-way projection method [8] and TOPSIS method [9-11] to calculate the relative closeness of samples. Then the computational power of different samples is graded, which solves the difficulty of direct comparison of different dimensional variables, and improves the comprehensiveness and effectiveness of evaluation results. This paper provides a new model and method for computational power evaluation system, better guide and advise the industry to judge the development trend of the industry, and provide ideas for computational power planning and deployment.

2. Related Theories

2.1 Related definitions

In multi-attribute decision making, the scheme set is 𝐴 = {𝑎₁, 𝑎₂, ⋯ , 𝑎_𝑠}, and the attribute set is 𝐶 = {𝑐₁, 𝑐₂, ⋯ , 𝑐_𝑚}, 𝑆 = {1,2, ⋯ , 𝑠}, 𝑀 = {1,2, ⋯ , 𝑚} represents the sequence number of the scheme and the attribute, respectively. 𝑐_𝑙(𝐴), 𝑙 ∈ 𝑀 is used to represent the value of scheme 𝐴 under the attribute 𝑐_𝑙 and allow for non-comparability.

Definition 1 [12] If relation{≻, ≺, ≈, ? }satisfies the following conditions: the ordinal preference information between schemes under each attribute can be given by the decision maker; for attribute 𝑐_𝑙, 𝑙 ∈ 𝑀, any two schemes 𝑎_𝑖, 𝑎_𝑗(𝑖 ≠ 𝑗) and satisfy one of the following preference relations: 𝑎_𝑖 is superior to 𝑎_𝑗 (𝑎_𝑖 ≻ 𝑎_𝑗); 𝑎_𝑖 is as good as 𝑎_𝑗 (𝑎_𝑖 ≈ 𝑎_𝑗); 𝑎_𝑖 is worse than 𝑎_𝑗 (𝑎_𝑖 ≺ 𝑎_𝑗); The relationship between 𝑎_𝑖 and 𝑎_𝑗 is unclear (𝑎_𝑖 ? 𝑎_𝑗).

Only one of {≻, ≺, ≈, ? } relations is true between the two schemes 𝑎_𝑖 and 𝑎_𝑗 (satisfy 𝑖 ≠ 𝑗), then {≻, ≺, ≈, ? } is said to constitute a partial order preference structure. The attribute values of each scheme under different attributes are divided into N+1 levels, so that the degree of one scheme is better than another scheme is divided into N levels. Let's call it \(\begin{aligned}a_{i} \stackrel{k}{>} a_{j}\end{aligned}\), (𝑖, 𝑗 ∈ 𝑆; 0 ≤ 𝑘 ≤ 𝑛), which means 𝑎_𝑖 is superior to 𝑎_𝑗 by 𝑘 grades; \(\begin{aligned}a_{i} \stackrel{-k}{>} a_{j}\end{aligned}\), (𝑖, 𝑗 ∈ 𝑆; 0 ≤ 𝑘 ≤ 𝑛)and \(\begin{aligned}a_{i} \stackrel{k}{<} a_{j}\end{aligned}\), (𝑖, 𝑗 ∈ 𝑆; 0 ≤ 𝑘 ≤ 𝑛) all indicate that 𝑎𝑎𝑖𝑖 is inferior to 𝑎_𝑗 by 𝑘 grades.

Definition 2 In multi-attribute decision making, for convenience, \(\begin{aligned}a_{i} \stackrel{t}{>} a_{k}\end{aligned}\) is denoted as 𝑟_𝑡𝑡 ∈ {−(𝑛 − 1), ⋯ , −1,0,1, ⋯ , 𝑛 − 1}, and (1) is called hierarchical comparison set.

𝑅 = {𝑟_𝑡|𝑡 = −(𝑛 − 1), −(𝑛 − 2), ⋯ , −1,0,1, ⋯ , 𝑛 − 2, 𝑛 − 1}       (1)

The hierarchical comparison set satisfies the characteristics: (1) The set 𝑅 is ordered: if 𝛼 > 𝛽, then 𝑟_𝛼 > 𝑟_𝛽; (2) There exists a negation operator: neg(𝑟_𝛼) = 𝑟_−𝛼; (3) If 𝑟_𝛼 > 𝑟_𝛽, max{𝑟_𝛼, 𝑟_𝛽} = 𝑟_𝛼, min{𝑟_𝛼, 𝑟_𝛽} = 𝑟_𝛽.

For any degree of comparison 𝑟_𝛼, 𝑟_𝛽 ∈ 𝑅, there are three basic operators: : 𝑟_𝛼 ⊕ 𝑟_𝛽 = 𝑟_𝛼+𝛽, 𝑟_𝛼 ⊖ 𝑟_𝛽 = 𝑟_𝛼−𝛽 and 𝜆𝑟_𝛼 = 𝑟_𝜆𝛼.

The concept of generalized superior Ordinal Numbers is introduced below.

Definition 3 [13] Makes

\(\begin{aligned}a_{i j l}=\left\{\begin{array}{l}1, c_{l}\left(A_{i}\right) \stackrel{n}{>} c_{l}\left(A_{j}\right), i \neq j \\ \frac{n}{2 n-k}, c_{l}\left(A_{i}\right) \stackrel{k}{>} c_{l}\left(A_{j}\right), i \neq j \\ 0.5, c_{l}\left(A_{i}\right) \approx c_{l}\left(A_{j}\right), i \neq j \\ 0.375, c_{l}\left(A_{i}\right) ? c_{l}\left(A_{j}\right), i \neq j \\ 0, i=j \\ -\frac{n}{2 n-k}, c_{l}\left(A_{i}\right) \stackrel{-k}{>} c_{l}\left(A_{j}\right), i \neq j \\ -1, c_{l}\left(A_{i}\right) \stackrel{-n}{>} c_{l}\left(A_{j}\right), i \neq j\end{array}\right.\end{aligned}\)      (2)

Among them 𝑖, 𝑗 ∈ 𝑆, 𝑙 ∈ 𝑀. 𝑎_ijl is called the generalized superior ordinal number of scheme 𝐴_𝑖 relative to scheme 𝐴_𝑗 under the attribute 𝑐_𝑙. To make the decision result more real and reliable, five forces model combined with the concept of vector, through the transformation of vector and correlation calculation, the application of bi-directional projection method and TOPSIS method, through the size of the relative closeness to get the relevant conclusion.

Definition 4 Let \(\begin{aligned}X_{p}=\left[\underline{r}_{p j}, \bar{r}_{p j}\right]\end{aligned}\) and \(\begin{aligned}X_{q}=\left[\underline{r}_{q j}, \bar{r}_{q j}\right]\end{aligned}\) be two uncertain grade variables on set 𝑋. If we transform 𝑋_𝑝 and 𝑋_𝑞 into \(\begin{aligned}X_{p}=\left[\xi\left(\underline{r}_{p j}\right), \xi\left(\bar{r}_{p j}\right)\right]\end{aligned}\) and \(\begin{aligned}X_{q}=\left[\xi\left(\underline{r}_{q j}\right), \xi\left(\bar{r}_{q j}\right)\right]\end{aligned}\) by generalized superior ordinal number, where \(\begin{aligned}\xi\left(\underline{r}_{p j}\right)\end{aligned}\) is the value of \(\begin{aligned}\underline{r}_{p j}\end{aligned}\) transformed by generalized superior ordinal number, and other similar symbols have the same meaning, the vector formed by 𝑋_𝑝 and 𝑋_𝑞 is defined as follows:

\(\begin{aligned}X_{p} X_{q}=\left[\min \xi\left(\underline{r}_{p j}\right), \max \xi\left(\bar{r}_{p j}\right)\right]\end{aligned}\)        (3)

Among them:

\(\begin{aligned}\min \xi\left(\underline{r}_{p j}\right)=\min \left(\left|\xi\left(\underline{r}_{q j}\right)-\xi\left(\underline{r}_{p j}\right)\right|,\left|\xi\left(\bar{r}_{q j}\right)-\xi\left(\bar{r}_{p j}\right)\right|\right)\end{aligned}\).

\(\begin{aligned}\max \xi\left(\bar{r}_{p j}\right)=\max \left(\left|\xi\left(\underline{r}_{q j}\right)-\xi\left(\underline{r}_{p j}\right)\right|,\left|\xi\left(\bar{r}_{q j}\right)-\xi\left(\bar{r}_{p j}\right)\right|\right)\end{aligned}\).

Let 𝐴 = {𝑎₁, 𝑎₂, ⋯ , 𝑎_𝑚} be the scheme set, 𝐶 = {𝑐₁, 𝑐₂, ⋯ , 𝑐_𝑛} be the attribute set, and attribute 𝑐_𝑙(1 ≤ 𝑙 ≤ 𝑛) be divided into 5 grades. There is a grade comparison set 𝑅 = {𝑟_𝑖|𝑖 = −4, ⋯ ,0, ⋯ ,4, ?}, where?represents the case where the merits and demerits of the two schemes are unknown, and 𝑡 = 4. We know that the two rank variables are 𝑋_𝑝 = [𝑟₂, 𝑟₃] and 𝑋_𝑞 = [𝑟₃, 𝑟₄].

𝑋_𝑝 = [𝑟₂, 𝑟₃] and 𝑋_𝑞 = [𝑟₃, 𝑟₄] can be transformed into \(\begin{aligned}X_{p}=\left[\frac{2}{3}, \frac{4}{5}\right]\end{aligned}\) and \(\begin{aligned}X_{q}=\left[\frac{4}{5}, 1\right]\end{aligned}\) by means of generalized superior ordinal number. Then, through definition 4, the vector formed by 𝑋_𝑝 and 𝑋_𝑞 is calculated as \(\begin{aligned}X_{p} X_{q}=\left[\frac{2}{15}, \frac{3}{15}\right]\end{aligned}\).

2.2 Bidirectional Projection Model

Projection method [14] is a appropriate method to dispose of decision problems. However, in some cases, the traditional projection method will be ineffective. Bidirectional projection [8] can handle problems that traditional projection method cannot handle. It has the characteristics of scientific rationality, simplicity and high distinction. This section combines the bidirectional projection method and TOPSIS method [15] to give a bidirectional projection model.

Assume that \(\begin{aligned}X_{i}=\left[\xi\left(\underline{r}_{i j}^{t}\right), \xi\left(\bar{r}_{i j}^{t}\right)\right]\end{aligned}\) is the information of the grade transformation of alternative 𝑋_𝑖 under the j-th attribute 𝑐_𝑗.

Firstly, under attribute 𝑗, positive and negative ideal schemes are expressed as \(\begin{aligned}X^+=\left[\max _{1 \leq i \leq n} \xi\left(r_{i j}\right), \max _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)\right]\end{aligned}\), \(\begin{aligned}X^{-}=\left[\min _{1 \leq i \leq n} \xi\left(\underline{r}_{i j}\right), \min _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)\right]\end{aligned}\), where 𝑛 is the number of overall alternatives. According to Definition 4, the vector formed by positive and negative ideal schemes could be calculated and presented as \(\begin{aligned}X^{-} X^{+}=\left[\xi\left(\underline{r}_{i j}^{t}\right), \xi\left(\bar{r}_{i j}^{t}\right)\right]\end{aligned}\), where:

\(\begin{aligned}\begin{array}{l}\xi\left(\underline{r}_{i j}^{t}\right)=\min \left(\left(\max _{1 \leq i \leq n} \xi\left(\underline{r}_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\underline{r}_{i j}\right)\right),\left(\max _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)\right)\right) \\ \xi\left(\bar{r}_{i j}^{t}\right)=\max \left(\left(\max _{1 \leq i \leq n} \xi\left(\underline{r}_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\underline{r}_{i j}\right)\right),\left(\max _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)\right)\right)\end{array}\end{aligned}\)

Also according to definition 4, under attribute 𝑗, the vector formed by 𝑋_𝑖 and positive and negative ideal schemes 𝑋⁺, 𝑋⁻ is expressed as \(\begin{aligned}X^{-} X_{i}=\left[\xi\left(\underline{r}_{i j}^{-}\right), \xi\left(\bar{r}_{i j}^{-}\right)\right]\end{aligned}\), \(\begin{aligned}X_iX^+=\left[\xi\left(\underline{r}_{i j}^{t}\right), \xi\left(\bar{r}_{i j}^{t}\right)\right]\end{aligned}\), where,

\(\begin{aligned}\begin{array}{l}\xi\left(\underline{r}_{i j}^{-}\right)=\min \left(\left(\xi\left(\underline{r}_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\underline{r}_{i j}\right)\right),\left(\xi\left(\bar{r}_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)\right)\right) \\ \xi\left(\bar{r}_{i j}^{-}\right)=\max \left(\left(\xi\left(r_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\underline{r}_{i j}\right)\right),\left(\xi\left(\bar{r}_{i j}\right)-\min _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)\right)\right) \\ \xi\left(\underline{r}_{\underline{i}}^{+}\right)=\min \left(\left(\max _{1 \leq i \leq n} \xi\left(r_{i j}\right)-\xi\left(r_{i j}\right)\right),\left(\max _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)-\xi\left(\bar{r}_{i j}\right)\right)\right) \\ \xi\left(\bar{r}_{i j}^{+}\right)=\max \left(\left(\max _{1 \leq i \leq n} \xi\left(r_{i j}\right)-\xi\left(\underline{r}_{i j}\right)\right),\left(\max _{1 \leq i \leq n} \xi\left(\bar{r}_{i j}\right)-\xi\left(\bar{r}_{i j}\right)\right)\right)\end{array}\end{aligned}\)

The modulus of the corresponding vector can be calculated by the following formula:

\(\begin{aligned}\left|X^{-} X^{+}\right| & =\sqrt{\sum_{j=1}^{m}\left(\left(\xi\left(\underline{r}_{i j}^{t}\right)\right)^{2}+\left(\xi\left(\bar{r}_{i j}^{t}\right)\right)^{2}\right)} \\ \left|X^{-} X_{i}\right| & =\sqrt{\sum_{j=1}^{m}\left(\left(\xi\left(\underline{r}_{i j}^{-}\right)\right)^{2}+\left(\xi\left(\bar{r}_{i j}^{-}\right)\right)^{2}\right)} \\ \left|X_{i} X^{+}\right| & =\sqrt{\sum_{j=1}^{m}\left(\left(\xi\left(\underline{r}_{i j}^{+}\right)\right)^{2}+\left(\xi\left(\bar{r}_{i j}^{+}\right)\right)^{2}\right)}\end{aligned}\)

Then, the cosine values cos(𝑋⁻𝑋_𝑖, 𝑋⁻𝑋⁺) and cos(𝑋⁻𝑋_𝑖, 𝑋⁻𝑋⁺) can be calculated as:

\(\begin{aligned}\cos \left(X^{-} X_{i}, X^{-} X^{+}\right)=\frac{\sum_{j=1}^{m}\left(\xi\left(\underline{r}_{i j}^{t}\right) \cdot \xi\left(\underline{r}_{i j}^{-}\right)+\xi\left(\bar{r}_{i j}^{t}\right) \cdot \xi\left(\bar{r}_{i j}^{-}\right)\right)}{\left|X^{-} X_{i}\right| \cdot\left|X^{-} X^{+}\right|}\end{aligned}\)       (4)

\(\begin{aligned}\cos \left(X^{-} X_{i}, X^{-} X^{+}\right)=\frac{\sum_{j=1}^{m}\left(\xi\left(r_{-i j}^{t}\right) \cdot \xi\left({ }_{-i j}^{+}\right)+\xi\left(r_{i j}^{-t}\right) \cdot \xi\left(\bar{r}_{i j}^{+}\right)\right)}{\left|X_{i} X^{+}\right| \cdot\left|X^{-} X^{+}\right|}\end{aligned}\)        (5)

Then, the projection values of vectors 𝑋⁻𝑋_𝑖 to 𝑋⁻𝑋⁺ and 𝑋⁻𝑋⁺ to 𝑋_𝑖𝑋⁺ are:

\(\begin{aligned}\begin{array}{r}\operatorname{prj}_{X^{-} X^{+}}\left(X^{-} X_{i}\right)=\left|X^{-} X_{i}\right| \cdot \cos \left(X^{-} X_{i}, X^{-} X^{+}\right) \\ =\frac{\sum_{j=1}^{m}\left(\xi\left(r_{i j}^{t}\right) \cdot \xi\left(\underline{\underline{r}}_{i j}^{-}\right)+\xi\left(\bar{r}_{i j}^{t}\right) \cdot \xi\left(\bar{r}_{i j}^{-}\right)\right)}{\left|X^{-} X^{+}\right|}\end{array}\end{aligned}\)       (6)

\(\begin{aligned} \operatorname{prj}_{X_{i} X^{+}}\left(X^{-} X^{+}\right) & =\left|X^{-} X^{+}\right| \cdot \cos \left(X_{i} X^{+}, X^{-} X^{+}\right) \\ & =\frac{\sum_{j=1}^{m}\left(\xi\left(\underline{r}_{i j}^{t}\right) \cdot \xi\left(\underline{r}_{i j}^{+}\right)+\xi\left(\bar{r}_{i j}^{t}\right) \cdot \xi\left(\bar{r}_{i j}^{+}\right)\right)}{\left|X_{i} X^{+}\right|}\end{aligned}\)       (7)

Note 1: In the Fig. 1, the higher value of prj_{𝑋⁻𝑋⁺}(𝑋⁻𝑋_𝑖) is, the closer 𝑋_𝑖 is to 𝑋⁺ ; On the contrary, the larger prj_{𝑋𝑖𝑋⁺}(𝑋⁻𝑋⁺) is, the closer 𝑋_𝑖 is to 𝑋⁻. Accordingly, the pros and cons of each alternative scheme can be judged according to the calculation of their projection values.

Fig. 1. the projection values of vectors 𝑋⁻𝑋_𝑖 to 𝑋⁻𝑋⁺ and 𝑋⁻𝑋⁺ to 𝑋_𝑖𝑋⁺

Through calculation, the projection formula describes the similarity and considers the distance (value) and direction between the alternative plan and the positive and negative ideal plan. In order to make the result more accurate and reasonable, alternative plan 𝑋_𝑖𝑋_𝑖 needs to be combined with the projection of positive and negative rational plan 𝑋⁺𝑋⁺ and 𝑋⁻, which is represented by relative closeness degree 𝐶(𝑋_𝑖) and calculated by the following formula:

\(\begin{aligned}C\left(X_{i}\right)=\frac{\operatorname{prj}_{X^{-} X^{+}}\left(X^{-} X_{i}\right)}{\operatorname{prj}_{X^{-} X^{+}}\left(X^{-} X_{i}\right)+\operatorname{prj}_{X_{i} X^{+}}\left(X^{-} X^{+}\right)}\end{aligned}\)       (8)

where pr𝑗_{𝑋⁻𝑋⁺}(𝑋⁻𝑋_𝑖) and prj_{𝑋𝑖𝑋⁺}(𝑋⁻𝑋⁺) represent the projection values of vectors 𝑋⁻𝑋_𝑖 to 𝑋⁻𝑋⁺ and 𝑋⁻𝑋⁺ to 𝑋_𝑖𝑋⁺ severally.

On these basis , the specific steps of hierarchical bidirectional projection model are as shown below :

Step 2.2.1: Construct decision moment 𝐷 with rank variables and normalize it;

Step 2.2.2: Correlation transformation of grade variables is carried out through the above generalized superior ordinal number operation;

Step 2.2.3: Determine positive ideal scheme 𝑋⁺ = {𝑋₁⁺, 𝑋₂⁺, ⋯ , 𝑋_𝑚⁺} and negative ideal scheme 𝑋⁻ = {𝑋₁⁻, 𝑋₂⁻, ⋯ , 𝑋_𝑚⁻};

Step 2.2.4: Calculate vectors 𝑋⁻𝑋⁺ and 𝑋_𝑖 formed by 𝑋⁺ 𝑋⁻ and vectors 𝑋⁻𝑋_𝑖 and 𝑋_𝑖𝑋⁺ formed by 𝑋⁺ 𝑋⁻ respectively;

Step 2.2.5: According to (6) and (7), calculate the projection values prj_{𝑋⁻𝑋⁺}(𝑋⁻𝑋_𝑖) and prj_{𝑋𝑖𝑋⁺}(𝑋⁻𝑋⁺) from vector 𝑋⁻𝑋_𝑖, vector 𝑋⁻𝑋⁺ and vector 𝑋⁻𝑋⁺ to vector 𝑋_𝑖𝑋⁺ respectively;

Step 2.2.6: Calculate the relative closeness between scheme 𝑋_𝑖 and the ideal scheme according to (8);

Step 2.2.7: Sort relative closeness degree (from large to small), and select the scheme with the maximum closeness degree as the optimal scheme.

3. Five Forces Model of Computational Power

3.1 Selection of Computational Power Measurement Indicators

Computational power is the ability of servers in a data center to output results after data processing. It is a comprehensive indicator to measure the computational power of a data center. A higher value indicates a stronger comprehensive computational power [16]. On a server main-board, data is transferred from the CPU [17], memory, hard disk, and NIC in sequence. If graphics processing is needed, GPU [18] is also required. In a broad sense, computational power is a comprehensive concept including computing, storage, transmission (network) and many other connotations. Computational power is a comprehensive index to measure the computational power of data centers [19].

Computational power is determined by three metrics: data processing capacity, data storage capacity, and data circulation capacity. The computational power of a data center is determined by data processing capability, data storage capability and data circulation capability. In the process of responding to the industrial trend of the new generation of digital technology represented by big data and AI [20], data processing capability can be divided into general computational power represented by CPU and intelligent computational power represented by GPU and AI chip. The former is mainly used to perform general tasks, while the latter mainly undertakes computationally intensive tasks such as graphics display, big data analysis, signal processing, artificial intelligence and physical simulation. In summary, this paper selects five metrics for computational power evaluation, namely: general computational power, intelligent computational power, computational efficiency, network capacity, and storage capacity.

General computational power: General computational power refers to the general computational power represented by CPU. In this paper, the average floating point operations per second (FLOPS) of single rack is used to evaluate general computational power output by CPU in the data center, and the unit is TFLOPS (FP32). CPU chips are divided into a variety of architectures, including x86, ARM and so on.

CP_Gernal = ∑(𝑎_𝑖 × CPU_𝑖)       (9)

CP_Gernal stands for general computational power of the data center;

𝑎_𝑖 indicates the number of CPU servers of a certain type;

CPU_𝑖 indicates the computational power of this type of CPU server.

Intelligent computational power: The intelligent computational power mainly refers to the intelligent computational power represented by GPU and AI chip. In this paper, the average floating point operations per second (FLOPS) of single rack is used to evaluate intelligent computational power output by GPU and AI chips in the data center, and the unit is TFLOPS (FP32).

CP_Intelligent = ∑(𝑏_𝑖 × 𝐺PU_𝑖 + 𝑐_𝑖 × FPGA_𝑖 + 𝑑_𝑖 × ASIC_𝑖)       (10)

CP_Intelligent stands for intelligent computational power of the data center;

𝑏_𝑖 indicates the number of GPU servers of a certain type;

𝑐_𝑖 indicates the number of FPGA servers of a certain type;

𝑑_𝑖 indicates the number of ASIC servers of a certain type;

GPU_𝑖 indicates the computational power of this type of GPU server;

FPGA_𝑖 indicates the computational power of this type of FPGA server;

ASIC_𝑖 indicates the computational power of this type of ASIC server.

Computational efficiency: Computational efficiency (CE) refers to the ratio of computational power to the power consumption of all IT devices. IT is an efficiency that considers both computational performance and power of the data center. The computational efficiency unit in this paper is GFLOPS/W (FP32).

\(\begin{aligned}C E=\frac{C P}{\sum I T \text { equipment power }}\end{aligned}\)       (11)

CP stands for computational power, which is the sum of general computational power and intelligent computational power.∑ IT equipment power indicates the sum of the power of all IT devices in the data center. In (11), the calculation efficiency represents the computational power generated by the power consumption per watt of the IT equipment in the data center.

Network capability: There are many indicators to measure network capability. In this paper, network bandwidth speed is used to measure network performance. The unit is Mbit/s (Megabits per second), that is, the number of bits transmitted per second.

Storage capability: Storage capability is determined by storage capacity, storage performance, and storage security. This section uses input/output operations per second (IOPS) to measure storage performance, which is the read/write times per second.

3.2 Model Construction

In this section, the measurement information of data center is combined with the bidirectional projection model, and a decision model is given to process the information of different indexes of computational power in different data centers by using bidirectional projection method, and construct “Five Forces Model of Computational Power”. An example is analyzed to show that the decision model has good effect. The specific steps of the model are as shown below:

Step 3.2.1: The indicators are graded according to the data. Maximize and minimize existing data. In the middle of the maximum value and minimum value, according to the number of grades required isometric division. In general, the more desirable the value, the higher the grade value.

Step 3.2.2: Grade the data and convert it into a real number (between 0 and 1) of superior order. According to the ranking number table divided in Step 3.2.1, The corresponding grade number is determined according to the data under each index. In each index, the grades of different data centers are compared in pairs, and the grades of one data center are better or worse than another data center in one index. Through the calculation formula of the superior ordinal number, the number of each grade is converted to 0-1, which is convenient for subsequent processing.

Step 3.2.3: Determine 𝑋⁺ and 𝑋⁻ and determine the corresponding vectors by calculation according to Definition 4. It is necessary to take the maximum and minimum value of the corresponding optimal ordinal number under each index to obtain the 𝑋⁺ and 𝑋⁻ under this index. Calculate the corresponding vector by (3).

Step 3.2.4: Each vector is obtained by computing model, we get cos(𝑋⁻𝑋_𝑖, 𝑋⁻𝑋⁺), cos(𝑋⁻𝑋_𝑖, 𝑋⁻𝑋⁺) and other data. According to (6) and (7), the corresponding vector 𝑋⁻𝑋_𝑖 to vector 𝑋⁻𝑋⁺ , and the projection value of vector 𝑋⁻𝑋⁺ to vector 𝑋_𝑖𝑋⁺ are calculated respectively for each data center.

Step 3.2.5: calculating the relative closeness of each data center from (8).

Step 3.2.6: Sort the size according to the value of relative closeness. Star rating is performed for each data center according to the relative proximity value and the star rating table. Finally draw a conclusion.

4. Case Study

Table 1 shows the five indicators (general computational power, intelligent computational power, computational efficiency, network capability and storage capability) corresponding to the six data centers.

Table 1. Sample data

Based on the experience of experts and data distribution, the evaluation steps are as follows: Step 4.1.1: Classify the five indicators according to the data, as shown below;

Step 4.1.2: Grade the data in Table 1 according to the data center computational power index classification table in Table 2. Under each index, compare the grades of each data center in pairs to obtain the relative grades and convert them into superior numbers, as shown in Table 3, Table 4 and Table 5.

Table 2. Classification of five forces index of data center computational power

Table 3. Data center computational power index grading

Table 4. Pairwise comparison of different indicators in different data centers

Table 5. Transformation table of superior Ordinal Numbers for comparison of data center levels

Step 4.1.3: Determine 𝑋⁺ and 𝑋⁻, and calculate and determine the corresponding vectors;

𝑋⁺ = {[0.57,0,0.67,0,0.8],[0,0,1,0.57,0,1],[0,0.57,0.67,0,0,0.8], , [0.57,0,0.57,0.57,0.8,0.8],[0.57,0,0.67,0,0.67,0.67]};

𝑋⁻ = {[−0.67, −0.8, −0.57, −0.57, −0.8,0],[−1, −1,0, −0.8, −1,0], [0.8,0.67, −0.57,0.8,0.8,0],[−0.67, −0.8, −0.67, −0.67,0,0], [−0.57, −0.67,0, −0.67,0,0]};

𝑋⁻𝑋⁺ = {[0.8,1.24],[1,1.37],[0.8,1.24],[0.8,1.24],[0.67,1.14]};

𝑋⁻𝑋₁ = {[0.23,1.14],[1,1.37],[0.8,1.24],[0.23,0.67],[0.10,0.57]};

𝑋₁𝑋⁺ = {[0.10,0.57],[0,0],[0,0],[0.13,0.57],[0.10,0.57]};

𝑋⁻𝑋₂ = {[0.8,1.24],[1,1.37],[0.23,1.14],[0.8,1.24],[0.67,1.14]};

𝑋₂𝑋⁺ = {[0,0],[0,0],[0.10,0.57],[0,0],[0,0]};

𝑋⁻𝑋₃ = {[0.10,0.57],[0,0],[0.10,0.57],[0.23,0.67],[0,0]};

𝑋₃𝑋⁺ = {[0.23,1.14],[1,1.37],[0.23,1.14],[0.13,0.57],[0.67,1.14]};

𝑋⁻𝑋₄ = {[0.10,0.57],[0.43,0.8],[0.8,1.24],[0.23,0.67],[0.67,1.14]};

𝑋₄𝑋⁺ = {[0.23,1.14],[0.2,0.57],[0,0],[0.13,0.57],[0,0]};

𝑋⁻𝑋₅ = {[0.8,1.24],[1,1.37],[0.8,1.24],[0,0],[0,0]};

𝑋₅𝑋⁺ = {[0,0],[0,0],[0,0],[0.8,1.24],[0.67,1.14]};

𝑋⁻𝑋₆ = {[0,0],[0,0],[0,0],[0,0],[0,0]};

𝑋₆𝑋⁺ = {[0.8,1.24],[1,1.37],[0.8,1.24],[0.8,1.24],[0.67,1.14]}.

In Fig. 2, the red coil represents the distance between each indicator of each data center and 𝑋⁺. If the red coil is smaller, it indicates that the data center is closer to 𝑋⁺ of each indicator, and the data center is better, otherwise the opposite. The blue coil represents the distance of each metric for each data center relative to 𝑋⁻. The larger the blue coil is, the farther the index of the data center is from 𝑋⁻, the better the data center is, otherwise the opposite. In Fig. 2, we can intuitively see that DC.6 is the worst overall, while DC.1 and DC.2 are relatively better.

Fig. 2. DC.1-DC.6 Distance between each index and positive and negative ideal schemes

Step 4.1.4: Calculate the corresponding projection value;

Corresponding modules are obtained as follows:

|𝑋⁻𝑋⁺| ≈ 3.3392; |𝑋⁻𝑋₁| ≈ 2.6915; |𝑋₁𝑋⁺| ≈ 1.0077; |𝑋⁻𝑋₂| ≈ 3.2149;

|𝑋₂𝑋⁺| ≈ 0.5793; |𝑋⁻𝑋₃| ≈ 1.0807; |𝑋₃𝑋⁺| ≈ 2.7735; |𝑋⁻𝑋₄| ≈ 2.3621;

|𝑋₄𝑋⁺| ≈ 1.4385; |𝑋⁻𝑋₅| ≈ 2.6882; |𝑋₅𝑋⁺| ≈ 1.9808; |𝑋⁻𝑋₆| ≈ 0;

|𝑋₆𝑋⁺| ≈ 3.3392.

Here, the modulus obtained by each data center (overall distance from positive and negative ideal schemes) is drawn into Fig. 3 In the figure, the closer the data center corresponding to the red coil is to the central position, the closer the overall distance of the data center is to 𝑋⁺, and the better the overall situation is, and vice versa. The closer the data center in the blue circle is to the extents of the diagram, the farther the data center is from 𝑋⁻, and the better the overall situation is, and vice versa. It can be seen intuitively that DC.1, DC.2, DC.4, DC.5 are relatively good, while DC.3, DC.6 are relatively poor, which coincides with the final conclusion.

Fig. 3. Distance between sample data and positive and negative ideal schemes

Then the projection value is obtained as follows:

Pr 𝑗_{𝑋⁻𝑋⁺} (𝑋⁻𝑋₁) ≈ 2.5085; Prj_𝑋1𝑋⁺(𝑋⁻𝑋⁺) ≈ 2.2966;

Pr 𝑗_{𝑋⁻𝑋⁺} (𝑋⁻𝑋₂) ≈ 3.1669; Prj_𝑋2𝑋⁺(𝑋⁻𝑋⁺) ≈ 1.3528;

Pr 𝑗_{𝑋⁻𝑋⁺} (𝑋⁻𝑋₃) ≈ 0.7713; Prj_𝑋3𝑋⁺(𝑋⁻𝑋⁺) ≈ 3.1156;

Pr 𝑗_{𝑋⁻𝑋⁺} (𝑋⁻𝑋₄) ≈ 2.1685; Prj_𝑋4𝑋⁺(𝑋⁻𝑋⁺) ≈ 2.3606;

Pr 𝑗_{𝑋⁻𝑋⁺} (𝑋⁻𝑋₅) ≈ 2.1642; Prj_𝑋5𝑋⁺(𝑋⁻𝑋⁺) ≈ 1.9808;

Pr 𝑗_{𝑋⁻𝑋⁺} (𝑋⁻𝑋₆) = 0; Prj_𝑋6𝑋⁺(𝑋⁻𝑋⁺) ≈ 3.3391.

Step 4.1.5: Calculate the relative closeness degree of each data center;

𝐶(DC. 1) ≈ 0.5220; 𝐶(DC. 2) ≈ 0.7007; 𝐶(DC. 3) ≈ 0.1984;

𝐶(DC. 4) ≈ 0.4788; 𝐶(DC. 5) ≈ 0.5221; 𝐶(DC. 6) = 0.

Step 4.1.6: Sort the size according to the value of relative closeness, determine the star rating of each data center according to Table 6, and finally draw a conclusion.

Based on the calculated relative closeness with Table 7, the corresponding star rating for each data center is derived as follows.

Table 6. Corresponding table of relative closeness degree and star rating

Table 7. Data Center Class Comparison Table

5. Conclusion

As an important capability of data center, computational power supports the development of application scenarios such as artificial intelligence, IoT and AR/VR. Meanwhile, the rapid popularization of these application scenarios also needs higher level of computational power in data center. Computational power is affected by computational, network, storage and power consumption. If it cannot be comprehensively measured, it cannot be improved. Consequently, it is particularly significant to thoroughly evaluate the computational power of data center.

Based on the research on computational power of data center, the "five forces model" is proposed. Firstly, the data center is divided into numerical grades in general computational power, intelligent computational power, computational efficiency, network capability and storage capability. Secondly, the bidirectional projection method and TOPSIS method are used to calculate each data center. Finally, star rating is made for different data centers by comparing the relative proximity value. The model makes the results more differentiated and reflects the advantages and disadvantages of different data centers more truly. However, the method proposed in this paper does not consider such factors as data measurement error and whether there is a fixed proportion between different indicators to make the overall efficiency better, which will be explored in future studies.

National Key R&D Program of China (2021ZD0113003)