نوشته Fuzzy TOPSIS Software for Easy Multi-Criteria Decision Making – Excel and MATLAB اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>MATLAB is one of the most powerful programming software used for linear programming and is an ideal platform for many multi-criteria decision-making methods due to its pre-written mathematical formulas like average, maximum, minimum, Euclidean distance, etc. However, it has a high installation volume, which is its only drawback. The Data Harnsing website has made it easy to review and view post-fuzzy results with just one click, thanks to its user-friendly environment. The MATLAB code for the Fuzzy TOPSIS software is available along with a tutorial on the website. No coding knowledge is required to work with this code, as inputs are entered in the form of a matrix, and the steps are explained using simple examples. The website’s specialists are available to answer any questions users may have about the product.

Some of the key features of the Fuzzy TOPSIS software in MATLAB include:

- No coding knowledge is required
- It can be used for any desired number of criteria and alternatives
- It can handle any number of positive and negative criteria
- It has an analysis program for easy review of results
- The TOPSIS Fuzzy tutorial is available on the Data Harnsing website, divided into different sections to make learning the method easier.

Pictures of the software environment will be available soon.

نوشته Fuzzy TOPSIS Software for Easy Multi-Criteria Decision Making – Excel and MATLAB اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>نوشته DANP (DEMATEL-based analytic network process) method اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>The Fuzzy DEMATEL method is also based on expert opinions, where scores of dependency on each criterion are determined based on expert opinions. However, what it does is identify the existence of relationships among factors. This involves complex mathematical calculations that human minds may not quickly compute, especially when dealing with various criteria. Always remember that in your article or thesis, it’s better to depict internal relationships separately. This means that in addition to defining internal relationships within a network structure, it’s advisable to illustrate each of these internal relationships separately in a distinct diagram or figure. Multi-criteria decision-making techniques are flexible in such a way that if combined with other techniques, they can yield another new technique, such as the Fuzzy DANP method.

You may wonder what advantage the internal relationships determined in Fuzzy DEMATEL, based on expert opinions, have over the internal relationships specified in the classical Fuzzy ANP method. To answer this question, you need to consider the advantage of binary comparison of factors in multi-criteria decision-making. The Fuzzy DEMATEL technique examines the internal relationships between factors in a binary manner. In this regard, the accuracy of detecting dependencies between factors using this technique is higher.

The steps of the Fuzzy DANP method are as follows:

- Formation of the problem’s network structure.
- Examination of internal relationships based on the Fuzzy DEMATEL method.
- Preparation of the Fuzzy DANP questionnaire.
- Evaluation of inconsistency between pairwise comparison tables and obtaining compatible pairwise comparison tables.
- Integration of questionnaires if more than one expert is involved.
- Calculation of the weights of integrated pairwise comparison tables.
- Formation of the initial supermatrix.
- Calculation of the upper supermatrix.
- Extraction of weights.

نوشته DANP (DEMATEL-based analytic network process) method اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>نوشته Fuzzy Analytic Network Process (FANP) Method – Simple Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>Another noteworthy attribute of the FANP fuzzy method is its knack for accommodating uncertainty. In scenarios where experts grapple with ambiguity and unpredictability in their assessments, fuzzy numbers like triangular or trapezoidal distributions come to the rescue. The format (l, m, u) characterizes a triangular fuzzy number, with ‘l’ representing the lower bound and ‘u’ symbolizing the upper bound. For instance, in the context of FANP, when comparing Criterion A to Criterion B, you might assign the number (1, 3, 5), effectively attributing a range from 1 to 5 to the criterion in question. To gain a better grasp of this concept, envision students seeking admission to an elite school with a minimum GPA requirement of 17. Yet, the school principal is open to considering a GPA of 16.80. Under these circumstances, it becomes evident that GPAs exceeding 17 are also acceptable. Consequently, this GPA threshold can be represented as a triangular fuzzy number (16.80, 17, 20).

The adaptability of the Fuzzy Analytic Network Process (FANP) method extends to its synergy with other techniques, such as Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (Fuzzy TOPSIS), especially when internal relationships among criteria come into play. Typically, when these relationships are present, FANP takes the reins in calculating the weights of the criteria. Here’s a breakdown of the steps involved in the Fuzzy ANP method:

**Steps of the Fuzzy ANP Method:**

- Define the network structure for the problem.
- Create a tailored fuzzy ANP questionnaire.
- Construct the pairwise comparison matrix.
- Assess inconsistency ratios and attain a consistent pairwise comparison matrix.
- Compute the criteria weights using the Chang method (although alternative techniques are viable).
- Establish the initial supermatrix.
- Calculate the limit supermatrix and extract the final weights.

نوشته Fuzzy Analytic Network Process (FANP) Method – Simple Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>نوشته TOPSIS in Excel with Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>Here are the TOPSIS steps. We will follow these steps in Excel, for example.

- Find weights for criteria.
- Form a decision matrix for alternatives.
- Normalize the decision matrix.
- Form a weighted normalized decision matrix.
- Find positive and negative ideals.
- Find the distance from the positive and negative ideals.
- Find the closest distance from the positive ideal and the farthest distance from the negative ideal.

To design questionnaires for criteria, you can refer to AHP tutorial part 2. In that video, we explained in detail how to tailor a questionnaire for a pairwise comparison matrix. In the third part of the AHP tutorial, we also explained how to compute the weights and inconsistency of a pairwise comparison matrix. Thus, we assume that you have watched these videos and have found the weights for the criteria. For alternatives, you can use documented data or assign a number on the Likert scale to each cell. For instance, if the convenience of an alternative is very high and your Likert scale ranges from 1 to 10, you can assign a value of 10. Indeed, for alternatives, we do not have a pairwise comparison matrix.

Download excel file for TOPSIS here

In the decision matrix columns may have different scales. In this case, you need to convert them to the same scale, usually between zero and one. We can achieve this by normalizing the numbers, either by dividing each number by the sum of numbers in the corresponding column or by dividing each number by the square root of the sum of squares of numbers in the corresponding column. We prefer to use the latter option for normalization.

So, to normalize the matrix, we first need to find the sum of squares of each column using the SUMSQ function. Simply type “=SUMSQ(” and select the column of interest, then close the parenthesis and press Enter. You can copy this formula to other columns by dragging the cell to the right. Next, divide each cell in the column by the corresponding sum of squares. To do this, enter the formula “=cell/sum_of_squares” in a new cell and press Enter. You can then copy this formula to all the cells in the column. Note that when dividing all values in a column by a fixed number, you can press F4 after typing the fixed value to automatically fix the cell reference.

To form the weighted decision matrix, you need to multiply the criteria weights by their corresponding column in the normalized decision matrix.

For positive criteria, the positive ideal corresponds to the maximum value, and the negative ideal corresponds to the minimum value. However, for negative criteria, the positive ideal is the minimum value, and the negative ideal is the maximum value. A positive criterion refers to a criterion where a larger value is preferred, while a negative criterion refers to a criterion where a smaller value is preferred. In this case, price and fuel are negative criteria, while convenience and model are positive criteria.

Finally, we need to find the distance of each row from the positive and negative ideals. To do this, you can use the following formula in Excel: =SQRT(SUMXMY2(first vector, second vector)). Here, the “first vector” refers to the row of values in the weighted normalized decision matrix, while the “second vector” refers to the values of the positive or negative ideal.

To find the final weights, we need to divide the distance of each alternative from the negative ideal (dj-) by the sum of distances from both negative and positive ideals (dj- + dj+). These weights are unnormalized. To normalize them, you can divide each weight by the sum of all weights.

نوشته TOPSIS in Excel with Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>نوشته Fuzzy Analytic Hierarchy Process (Fuzzy AHP) – With Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>In essence, Fuzzy AHP extends the AHP framework by incorporating fuzzy logic and mathematics into the decision-making process. This adaptation acknowledges that in real-world scenarios, many factors and criteria may not have clear, crisp values. Instead, they might be represented as fuzzy sets, which encompass a range of values with varying degrees of membership. This is particularly useful when dealing with subjective assessments, diverse expert opinions, or data that is inherently uncertain.

The Fuzzy AHP process typically involves the following key steps:

**Hierarchical Structure**: Establish a hierarchical structure that breaks down the decision problem into a series of levels and criteria. This structure helps organize the decision-making process.**Pairwise Comparisons**: In the traditional AHP, pairwise comparisons are made to assess the relative importance or preference between criteria and alternatives. In Fuzzy AHP, these comparisons account for fuzziness and uncertainty. Decision-makers use linguistic variables, fuzzy numbers, or other fuzzy sets to express their judgments.**Fuzzy Number Aggregation**: Aggregating the fuzzy numbers derived from pairwise comparisons to determine the overall preferences and weights of criteria and alternatives. This step involves fuzzy arithmetic operations, such as fuzzy addition and multiplication.**Consistency Assessment**: Checking the consistency of the derived fuzzy judgments to ensure that they are logically sound and do not contain contradictions. Inconsistent judgments can be refined through discussions or adjustments.**Defuzzification**: Converting the fuzzy preferences and weights into crisp values if needed for further analysis or decision-making.**Ranking and Decision-Making**: Using the aggregated preferences and weights to rank alternatives or make informed decisions.

Fuzzy AHP is particularly valuable in complex decision scenarios, such as project selection, supplier evaluation, or product prioritization, where imprecise data and diverse perspectives need to be considered. It allows decision-makers to weigh the impact of different factors and arrive at well-informed decisions, even in situations where precise numerical values are elusive.

In summary, Fuzzy AHP is an extended version of the Analytic Hierarchy Process that accommodates fuzziness and uncertainty in decision-making, making it a versatile and powerful tool for tackling real-world problems.

نوشته Fuzzy Analytic Hierarchy Process (Fuzzy AHP) – With Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>نوشته Fuzzy TOPSIS Method with a Simple Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>In Fuzzy TOPSIS, the decision-making process involves several steps:

**Identification of Criteria:**The first step is to identify the criteria that will be used to evaluate the alternatives. These criteria should be relevant to the decision problem and reflect the decision maker’s objectives. For example, in selecting a supplier for a manufacturing company, criteria might include cost, quality, delivery time, and reliability.**Fuzzy Evaluation:**The decision maker assesses the performance of each alternative with respect to each criterion. Instead of providing precise values, they use linguistic variables (e.g., high, medium, low) or fuzzy numbers to express the degree of satisfaction or preference. For instance, if evaluating the cost criterion, the decision maker might assign the label “medium” to one alternative’s cost.**Normalization:**To make the evaluations comparable, fuzzy values are normalized to a common scale ranging from 0 to 1, where 1 represents the best performance for a particular criterion. Normalization is crucial because different criteria may have different units or measurement scales.**Weight Assignment:**The decision maker assigns weights to each criterion to reflect their relative importance. These weights are typically determined through discussions, expert opinions, or analytical methods like the Analytic Hierarchy Process (AHP).**Constructing the Fuzzy Decision Matrix:**Using the normalized fuzzy evaluations and weights, a fuzzy decision matrix is constructed, where each row represents an alternative, and each column represents a criterion. This matrix captures the information needed for decision-making.**Determining the Positive and Negative Ideal Solutions:**In Fuzzy TOPSIS, the positive ideal solution represents the best values for each criterion, while the negative ideal solution represents the worst values. These solutions are determined based on the maximization or minimization of criteria and their weights.**Calculating Proximity to Ideal Solutions:**Proximity measures (closeness coefficients) are calculated for each alternative relative to the positive and negative ideal solutions. These measures quantify how well each alternative aligns with the desired outcomes.**Ranking Alternatives:**Alternatives are ranked based on their proximity values. The alternative with the highest proximity to the positive ideal solution is considered the most preferred or best choice.

Fuzzy TOPSIS provides a systematic framework for decision-makers to consider both the quantitative and qualitative aspects of decision problems. It acknowledges that real-world decisions often involve imprecise information and subjective judgments. By incorporating fuzzy logic and linguistic variables, it allows decision-makers to express their preferences in a more flexible and realistic manner.

This method has been applied in various domains. For example, in environmental management, Fuzzy TOPSIS can be used to select the best location for a waste disposal site by considering criteria such as distance to populated areas, environmental impact, and cost. In financial portfolio management, it can help investors choose the optimal investment portfolio by evaluating assets based on criteria like return, risk, and liquidity.

In summary, Fuzzy TOPSIS is a valuable tool for decision-makers facing complex, multi-criteria problems with uncertain or vague data. It provides a structured approach to handle fuzziness in decision-making, making it a powerful technique in the field of operations research and decision analysis.

نوشته Fuzzy TOPSIS Method with a Simple Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>نوشته ANP Method or Analytical Network Process with a Simple Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>The ANP method, introduced by Thomas L. Saaty, aims to select appropriate options based on multiple criteria. It is also employed for weighting criteria and sub-criteria. In ANP’s network analysis process, pairwise comparisons are used to determine criterion weights.

- Introduction to Multi-Criteria Decision Making and the Superiority of the ANP Method over Other Techniques.
- Teaching the Preparation of ANP Questionnaires.
- Calculating the Weights of Pairwise Comparison Matrices.
- Calculating the Inconsistency of Pairwise Comparison Matrices.
- Integrating Questionnaires When Multiple Experts’ Opinions Are Utilized.
- Manual Implementation of the ANP Method.
- Implementing the ANP Method in Expert Choice Software.

The ANP method, proposed by Saaty and Zaras in 1986, extends the AHP method. In situations where lower-level elements impact higher-level ones, or where elements on the same level are not independent, the hierarchical approach becomes less applicable. The ANP technique presents complex relationships among different decision levels in a network format, accounting for interactions and feedback among criteria and alternatives.

According to the model above, the four criteria C1 to C4 represent the primary decision-making criteria. These internal relationships indicate which criterion is more dependent on another, for example, how the first criterion relates to the second. The box with lines drawn around the criteria signifies the existence of internal relationships among all of them with each other, as mentioned in the previous sentence.

The Analytical Network Process (ANP) allows decision-makers to construct a network, enabling the examination of internal relationships among elements. The nodes within this network correspond to criteria or alternatives, and the branches connecting these nodes represent the degree of their interdependence. Determining the relationships within the network structure or assessing the mutual dependencies among criteria and alternatives is the most crucial aspect of the ANP method.

Relationships and dependencies can manifest as connections between different levels of the network, either external or internal. The relative importance of each member of the set – at its relevant level – is assessed through a series of pairwise comparisons, similar to the Analytic Hierarchy Process (AHP). Further instruction on ANP, along with a numerical example, will be provided in the following sections. For more detailed guidance on this technique, please download the ANP method package.

In this section, a two-level ANP example is presented, and for higher-level structures, you can refer to the ANP method package. A two-level scenario is one where we have criteria and alternatives, and if sub-criteria are also considered, it transforms into a three-level structure. Imagine you are selecting the best “Data Scientist” for your company from three candidates, and you also want to rank these three candidates. Below, the network structure for this problem is outlined.

To determine internal relationships using the ANP method, we answer the question of which factor’s change influences the change in another factor. If a change in the first factor leads to a change in the second factor, an arrow is drawn from the second factor to the first. In the example above, a high GPA and experience indicate high competency. Additionally, a high level of experience signifies older age.

If an arrow is drawn from one factor to more than one other factor, then a pairwise comparison table is formed among the factors that have the arrowhead pointing towards them relative to the factor at the arrow’s end. The questionnaire format for internal relationships and other tables differs. Below are tables showing inconsistency values and weights. It’s worth noting that the scores are determined based on the following scale (for more details, refer to the ANP method package).

نوشته ANP Method or Analytical Network Process with a Simple Example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>نوشته Analytical Hierarchy Process (AHP): Step-by-step example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>**Introduction to Analytic Hierarchy Process (AHP)**

Multi-criteria decision making (MCDM) is a decision-making approach that allows experts to take into account multiple criteria when making decisions. Analytic Hierarchy Process (AHP) is a popular MCDM technique that helps decision-makers to evaluate and prioritize alternatives based on a set of criteria.

MCDM can be categorized into two main groups, namely multi-objective decision making (MODM) and multi-attribute decision making (MADM). MODM is a category of MCDM that aims to minimize or maximize multiple objectives or criteria simultaneously. It is characterized by the presence of an objective function for each criterion. The criteria can be either in the form of maximization for positive criteria or minimization for negative criteria. However, MODM can be computationally expensive and often requires heuristic algorithms to solve. Generally, MODM applies to quantitative variables, but qualitative variables can be transformed into quantitative variables and still be used in MODM.

On the other hand, MADM methods like AHP, ANP, TOPSIS, etc. aim to address the limitations of MODM. MADM is a category of MCDM that deals with the evaluation of alternatives based on a set of attributes or criteria. The attributes or criteria can be at different levels, such as criteria, alternatives, or any finer level. Sub-criteria can also be included, although the model does not necessarily require sub-criteria for each criterion.

To illustrate the above concepts, consider an example of each category. For MODM, consider minimizing a two-objective function, and for MADM, consider a model with three levels of criteria, sub-criteria, and alternatives.

In this second part of the AHP tutorial, the focus is on designing a questionnaire to evaluate the pairwise comparisons of criteria and alternatives. A hierarchical structure is used to form the pairwise comparison matrices for both the criteria and alternatives. Since all the criteria are connected to the goal, a pairwise comparison matrix for it is created. Similarly, pairwise comparison matrices are created for the alternatives corresponding to each criterion. A sample of the pairwise comparison matrices is presented, and each cell is filled based on a linguistic scale, such as the Likert scale, which ranges from 1 to 5 or 1 to 10. Experts are asked to assign the appropriate value to each question based on their preference, with the numerical value matched to a linguistic variable. For criteria, experts are asked to compare the preference between two options and indicate the degree of relative preference. For alternatives, the experts are asked to indicate which option is preferred more in terms of a criterion and the degree of relative preference. Once the questionnaire is completed and the expert’s answers are obtained, the pairwise matrices are filled in. It is important to ensure that the answers meet the consistency rate before calculating the weights of the matrices. Aggregating the questionnaires into one is required to solve the AHP problem. The geometric mean is the most popular method used for aggregating the questionnaires, where the nth root is taken after multiplying the values.

In this third part of the AHP tutorial, Ishmael from dataharnessing.com explains how to calculate the weights and inconsistency rate for a single matrix. These calculations can be repeated for each matrix in the analysis or for aggregated matrices from multiple experts. To calculate the weights, one needs to find \lambda_{max}, which is the maximum eigenvalue obtained by setting the determinant of det(A-\lambda*I)=0. This value is then used to solve for the weights by plugging it into the previous equation and multiplying it by the vector W, (A-\lambda_{max}*I)W=0. To convert the weights to a contracted scale between 0 and 1, normalization can be done by dividing each weight by the sum of all weights.

Calculating the inconsistency rate involves determining the inconsistency index, which depends on the number of attributes being compared. For example, in the matrix provided, there are 4 attributes. The inconsistency index is divided by the inconsistency ratio, which is a predetermined value for a given number of attributes, to determine the inconsistency rate. In this case, the inconsistency ratio is 0.9, which can be found in a table provided.

function [W, IR] = WtIR(A) %% Find weights Lambda_max = max(eig(A)); W = null(A- Lambda_max * eye(size(A))); W_normalized = W./sum(W); %% inconsistency rate RI = [0.00, 0.00, 0.58, 0.90, 1.12, 1.24, 1.32, 1.41, 1.45, 1.49, 1.51, 1.54, 1.56, 1.58, 1.59, 1.61]; II = (Lambda_max-size(A, 1))/(size(A, 1) - 1); IR = II/RI(size(A, 1)); if IR < 0.1 disp('The imported pairwise comparison matrix is consistent') else disp('The imported pairwise comparison matrix is inconsistent, and questions should answer again.') end end

Click here to download the AHP codes in matlab

The calculations for weights and inconsistency rate can be performed using MATLAB software. A function has been created that combines the calculation of both inconsistency rate and weights in a single code. However, the word “function” should be excluded from the first line of the code when you run it in MATLAB to yield both results.

Once the weights of the pairwise comparison matrices have been calculated, they are organized into decision matrices for each level. In our previous example, the decision matrix for alternatives is a 3 by 4 matrix where each column represents the weights of the alternatives for a specific criterion. It is important to note that the decision matrix for criteria is a 4 by 1 vector because there is only one pairwise comparison matrix at this level.

To determine the final aggregated weights of alternatives in the Analytic Hierarchy Process, the decision matrices need to be multiplied from the bottom to the top of the hierarchy. This requires multiplying the decision matrix of alternatives by the criteria vector, which yields a 3×1 vector. It is necessary to normalize these weights if they have not already been normalized.

In the next video we introduce software and online resources that you can easily run your AHP projects benefiting from machine learning techniques.

نوشته Analytical Hierarchy Process (AHP): Step-by-step example اولین بار در Data Harnessing | Unlock the Power of Machine Learning. پدیدار شد.

]]>