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Difference between TOPSIS and VIKOR

Different normalisation procedure and aggregating function.

VIKOR limitations

Difficult to select a value for v. It is recommend to use 0.5 but this value heavily influences the result; More research is required to create guidelines for setting v effectively; Rank reversal

How to normalise units for WSM?

Divide the value that needs to be normalised by the sum of the all the values of that criteria, I.e. calculate the proportion of the value in relation to other values for the same criterion.

Fuzzy sets

Expresses a membership with a value between 0 and 1.

What is the essence of WSM

It allows to define criteria weights (importance of that criterion); The score calculated for each alternative is equal to the sum of the product of the weights and decision variables

Weighted goal programming

Like simple goal programming, but you simply assign penalty weights to each of the deviations. Then you minimise the weighted sum of the deviations.

Procedure for WSM

Step 1: Maximisation. If some of the criteria are minimising, they need to be inverted to be maximising—use 1/p formula. Step 2: Normalisation. If different criteria are measured using different scales (units), they need to be normalised—use x/(x1 + x2 + ... + xn) formula. Step 3: Score Calculation. Now that we have normalised values for each criteria, those values need to be multiplied by their weight and added together—use SUM(weight x normalised value) formula. Step 4: Score Comparison. Once you have scores for each alternative, the one with the highest score is the best.

TOPSIS stands for

Technique for Order Preference by Similarity to Ideal Solutions

Distance measures

The Euclidean distance is the distance between two points that you would measure with a ruler. Manhattan (Squares); Chebyshev (a perpendicular line)

Weighed Sum Methods (WSM) is also known as

The Simple Additive Weight Method (SAW) and the Simple Multi Attribute Rating Technique (SMART)

Analytic Network Process (ANP)

A decision making method for choosing between alternatives similar to AHP. However, compared to AHP, it Uses network rather hierarchical structure, but requires a much greater amount of input.

What does Linear optimisation require?

A linear function to be maximised/minimised - objective function; Input variables; Problem constraints

consistency ratio

A numerical measure of the degree of consistency in a series of pairwise comparisons. Values less than or equal to 0.10 are considered reasonable.

Variable

A value that varies; location capable of storing temporary data.

AHP limitations

AHP scale (not equal dispersion across the scale); rank reversal when an alternative is added or removed (ANP is a solution); time consumption if more than 3 criteria x 3 alternatives; pairwise inconsistency (CR is a solution)

discrete variable

Counted

Limitations of TOPSIS

Rank reversal

Weighted Sum Method (WSM)

a decision making method for choosing between alternatives. It allows the decision-maker to define criteria weights. Each weight signifies the importance of that criterion. The score calculated for each alternative is equal to the sum of the product of the weights and decision variables.

VIKOR

a decision making method to choose between alternatives, which assess alternatives on the basis of their separation from a positive ideal point. The optimal alternative has the shortest distance from the positive ideal solution.

TOPSIS

a decision making method to choose between alternatives, which assess alternatives on the basis of their separation from an negative and positive Ideal point. The optimal alternative has the shortest distance from the positive ideal solution the furthest distance from the negative ideal solution.

optimum

an ideal point between a number of trade-offs; Edgeworth-Pareto optimal/Pareto optimal/Pareto efficient solution

Analytical Hierarchy Process (AHP)

A decision making method for choosing between alternatives that uses a hierarchical structure of criteria and alternatives, and pairwise comparisons (scale 1-9) to input qualitative information. In pairwise comparisons, each alternative is compared head-to-head (one-to-one) with each of the other alternatives. Similarly, criteria weights can also be evaluated using pairwise comparisons.

numerical variable

Between any two continuous values there may be an infinite number of other values; measured

Lexicographic goal programming procedure

Goals / deviations are listed in the order of their importance. You focus on finding a solution that satisfies the most important goal. Next you do the same for the second most important goal (as is possible without changing the first goal). You then repeat this procedure until you have worked though all of the priority levels.

Lexicographic goal programming

Like simple goal programming, but you simply assign priority levels to the goals / deviations. The most important goals / deviations will be at the highest level of priority, while the less important ones will be at the lower priority levels.

How to maximise minimising criteria for WSM?

Use 1/p, where p - the minimising criterion value

Optimal set

a set of optimal solutions

Positive and negative ideal solutions

artificial alternatives which are hypothesised by the decision maker, based on the ideal solution for all criteria and the worst solution which possesses the most inferior decision variables.

How does TOPSIS calculate results?

by comparing Euclidean distances between the actual alternatives and the hypothesised ones.

Pareto Frontier

optimal set drawn in 2 or 3 dimensions

VIKOR stands for

Multi-criteria Optimisation and Compromise Solution.

Steps for WSM

1. Maximise criteria (if needed) 2. Normalise parameters 3. Calculate scores The one with the highest score is the best

Why WSM requires normalisation?

Most problems have criteria that use different scales (unjts) and therefore the direct comparison is not available and more universal (relative) way of comparison is needed.

Procedure for TOPSIS

Step 1: Create a Decision Matrix. The matrix must include all the alternatives, criteria and criteria weights. Step 2: Normalise the Decision Matrix. In TOPSIS, vector normalisation is used for criteria values. Use r(ij) = x(ij)/(sqrt(sum(I)x^2(ij) *Criteria weights are normalised using simple normalisation x/SUM. Step 3: Create a Weighted Normalised Matrix. Multiply the normalised values by the weights. Step 4: Determine Positive Ideal Solutions (PIS). Positive ideal solution is an artificial alternative with the best values on each criterion: the highest weighted normalised matrix value for maximising criteria or the lowest for minimising criteria. Step 5: Determine Negative Ideal Solutions (NIS). Negative ideal solution is an artificial alternative with the worst values on each criterion: the lowest weighted normalised matrix value for maximising criteria or the highest for minimising criteria. Step 6: Determine Separation from PIS. Subtract the PIS value on each criterion from each weighted normalised matrix value and square the difference. Then, for each alternative, add up all of these squared differences and take a square root of the sum (as in SD). Step 7: Determine Separation from NIS. Subtract the NIS value on each criterion from each weighted normalised matrix value and square the difference. Then, for each alternative, add up all of these squared differences and take a square root of the sum (as in SD). Step 8: Calculate Relative Closeness to the Ideal Solution. For each alternative, take its separation from NIS and divide it by the sum of separation from NIS added to the separation from PIS. Use Si-/(Si+ + Si-). Step 9: Choose the best alternative. The best alternative is the farthest from the NIS, thus the one with the highest value for relative closeness to the ideal solution.

Procedure for AHP

Step 1: Reciprocal Matrix. Enter pairwise comparisons into a matrix. Score = Row/Column Step 2: Score Elicitation. There are a number of different ways to elicit scores from a reciprocal matrix. The most common ways are: A. Principle eigenvectors of the matrix. B. Geometric mean. * A: 1. Calculate the (pairwise matrix)^2. Multiple the values for the rows by the values for the columns of each cell. Use MMULT + ctrl+shift+enter. 2. Calculate the sum of each row. Sum the squared pairwise matrix values for each row. Use SUM. 3. Normalise these values. Divide each value by the sum of all the values. 4. Repeat until the differences between the sums in the two consecutive calculations are smaller than a prescribed value. E.g. no differences until 5 decimal points. 5. Calculate the scores for each alternative. Multiply the criteria values for each alternative by the criteria weights. * B: 1. Calculate the geometric mean. Use GEOMEAN for all the rows. 2. Normalise these values. Divide each value by the sum of all the values. 3. Calculate the scores for each alternative. Multiply the criteria values for each alternative by the criteria weights. Step 3: Score Comparisons. The alternative with the highest scores is the best.

Linear optimisation

a decision making method used to find the single most optimal solution that meets all the stated criteria. Often used in management for maximising profits and minimising costs.

Goal programming

a decision making method which can be used to solve multi-objective problems. In goal programming you need to set a specific numerical goal for each objective. Deviational variables are then minimised in an objective function to try to find a solution nearest to the goals.