Decision Model

1 Introduction

In our project, we develop a mathematical method to decide which pathway we could use because we have several candidates and need to conduct a scientific and rational method to find out the most suitable one. Therefore, an Analytic Hierarchy Process (AHP) is performed. By doing so, we can combine our subjective and objective factors into consideration and finally get the one from candidates.

2 Hierarchy model Construction

To use the AHP model, we need to list our candidates first. Combining with our project design and references we have read, we finally select 3 candidate pathways, which are Astaxanthin pathway, Ginsenosides pathway1, and Cannabinoid pathway2 respective- ly. In addition, during our discussion and interviews with different expertise, we select 5 most critical criteria we need to consider. Thanks for our human practice working, we can finally determine the criteria we need to use in our AHP model.

By using 3 candidates and 5 criteria, the hierarchy model is established. (Figure 1)

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Figure 1: Hierarchy Model Structure

3 Comparison matrix construction

After having the hierarchy structure, we further construct comparison matrix. When comparing the relative importance of the ith element to the jth element to the factor above, we use the quantified relative weight $a_{ij}$ to describe. Suppose there are a total of n elements to participate in the comparison, then matrix $ A={\{a_{ij}\}}_{n\times n}$ called comparison matrix. The value of $a_{ij}$ in the comparison matrix is assigned according to the following scale.

$a_{ij}=1$ which implies that element $i$ and element $j$ have the same importance to the previous level factor.
$a_{ij}=3$ which implies that element $i$ is slightly more important than element $j$.
$a_{ij}=5$ which implies that element $i$ is more important than element $j$.
$a_{ij}=7$ which implies that element $i$ is much more important than element $j$.
$a_{ij}=9$ which implies that element $i$ is extremely more important than element$j$.
$a_{ij}=2n, n=1,2,3,4$ which implies that the importance of the elements $i$ and$j$ is between $a_{ij}=2n-1$ and $a_{ij}=2n+1$.
$a_{ij}=\frac{1}{n}, n=1,2,\cdots,9,$ if and only if $a_{ij}=n$.

By using the scale above, we construct our comparison matrix as follows.

\begin{equation} Goal\:\:Matrix:\quad \left( \begin{array}{ccccc} 1 & 2 & 7 & 5 & 5 \\ 1/2 & 1 & 4 & 3 & 3 \\ 1/7 & 1/4 & 1 & 1/2 & 1/3 \\ 1/5 & 1/3 & 2 & 1 & 1 \\ 1/5 & 1/3 & 3 & 1 & 1 \end{array} \right) \end{equation}

Criteria Matrix

\begin{equation} B1:\quad \left( \begin{array}{ccc} 1 & 1/8 & 1/3 \\ 3 & 1 & 1/3 \\ 8 & 3 & 1 \end{array} \right) \end{equation} \begin{equation} B2:\quad \left( \begin{array}{ccc} 1 & 2 & 5 \\ 1/2 & 1 & 2 \\ 1/5 & 1/2 & 1 \end{array} \right) \end{equation} \begin{equation} B3:\quad \left( \begin{array}{ccc} 1 & 1 & 3 \\ 1 & 1 & 3 \\ 1/3 & 1/3 & 1 \end{array} \right) \end{equation} \begin{equation} B4:\quad \left( \begin{array}{ccc} 1 & 3 & 4 \\ 1/3 & 1 & 1 \\ 1/4 & 1 & 1 \end{array} \right) \end{equation} \begin{equation} B5:\quad \left( \begin{array}{ccc} 1 & 4 & 1/4 \\ 1 & 1 & 1/4 \\ 4 & 1 & 1 \end{array} \right) \end{equation}

Where, the Goal Matrix represents different degrees of impor- tance of criteria to the final goal(pathway selection), and Criteria Matrix B1 to B5 represent different degrees of capablity of alter- native pathway to each criteria.

4 Consistency test

Obviously, we can find out the matrix above has a characteristic:

$$ a_{ij}>0 $$ $$ a_{ji}=\frac{1}{a_{ij}} $$

which means our matrix is a Reciprocal matrix: Further analysis finds that if $A $ is a completely consistent reciprocal matrix, we have a relationship $a_{ij}a_{jk}=a_{ik},\;\; \forall i,j,k=1,2,\cdots,n$ . In fact, however, it is impossible to satisfy the equations above when constructing a comparison matrix. As a result, we use CR, which called Random Consistency Ratio to measure the degree of inconsistancy of a matrix and evaluate whether we could accept such an inconsistancy. The algorithm is listed as follows.

Step 1

Calculate an indicator CI (Consistency Index) that measures the degree of inconsistency in a comparison matrix A ($n>1$ matrix)

\begin{equation}\nonumber CI=\frac{\lambda_{max}-n}{n-1} \end{equation}

$\lambda_{max}$ means the maximum eigenvalue, and $n$ is the order of the matrix

Step 2

Calculate an indicator $RI$ (Random consistency Index), a comparison matrix consistency test standards, which only determined by the order of the matrix.

Instead of using theRI presented by Saaty (1980) direct- ly,we conduct a new method to calculateRI by ourselves. By doing so, our model can be more precise. The improved algorithm is presented below.

i For a certain matrix order 5, we randomly extract $10$ values from 1, 2, ..., 9, 1/2, 1/3, ..., 1/9 as the upper triangular element, the main diagonal element takes 1 and the lower triangular element goes to the reciprocal of the triangular element, thereby obtaining the matrix R.
ii Calculate CI value of the matrix R.
iii Iterate the previous step.
iv Calculate the average of all CI values.

After 10 iterations, we get the value of RI when matrix order is 5. And the value of RI is 1.1108.

Step 3

Calculate the $CR$ (Consistency Ratio) of the comparison matrix $A$ according to the following formula,

\begin{equation} CR=\frac{CI}{RI} \end{equation}

Step 4

When $CR < 0.1$, we can determine that the comparison matrix $A$ has satisfactory consistency, or the degree of inconsistency is acceptable; otherwise, the comparison matrix $A$ is adjusted until satisfactory consistency is achieved.

5 Weight value and hierarchical total order calculation

After getting the relatively consistant matrix, we use MATLAB to calculate eigenvector of each matrix. And then use Z-score standardization to ensure all values in eigenvector are within the range of 0-1. The vector which is standardized is called weight vector.

By calculating matrixes above, we get 6 weight vectors. Where, $\omega_{U}$ represents the weight vector of Goal Matrix (1)

$$\omega_{U}=(0.475,0.263,0.051,0.103,0.126)^{T}$$

and $\omega_{B_{i}},\;\;i=1,2,3,4$ represents the weight values of Criteria Matrix B1 to B5

$$\omega_{B1}=(0.082,0.236,0.682)^{T}$$ $$\omega_{B2}=(0.606,0.265,0.129)^{T}$$ $$\omega_{B3}=(0.429,0.429,0.143)^{T}$$ $$\omega_{B4}=(0.636,0.185,0.179)^{T}$$ $$\omega_{B5}=(0.167,0.167,0.667)^{T}$$

And also, they could be regarded as Our lab's Capability score, Degree of Innovation score, Time Consuming score, Market Potential score and Experiment Cost score of each candidate.

Finally, according to the weight vectors above, we can get a total score of each candidate.

\begin{equation} Total\;\;Score=\sum_{n=1}^{5}\omega_{Ui}\times\omega_{Bnj}\quad\quad (i=1,2,3,4,5;\;j=1,2,3) \end{equation}

Figure 2: Total score of each candidate

According to the total score of each candidate, we choose the one with highest score, which is Astaxanthin pathway From now on, our project starts!

Team:CAU China/Decision Model