Master TOPSIS to operationalise multi-criteria trade‑offs using distance‑to‑ideal reasoning and transparent rankings.

1.TOPSIS is best described as

1. A single-objective optimization method for continuous design variables
2. A compensatory MCDM method for ranking finite alternatives across multiple criteria
3. A simulation technique for stochastic hydrologic models
4. A clustering algorithm for grouping similar alternatives

2.In TOPSIS, the Positive Ideal Solution (PIS) represents

1. The alternative with the smallest Euclidean norm in the decision space
2. A hypothetical alternative with the best performance on every criterion
3. The real alternative that appears first in the decision matrix
4. The average of all alternatives over all criteria

3.The Negative Ideal Solution (NIS) in TOPSIS is

1. The worst-performing actual alternative in the dataset
2. A hypothetical alternative with the worst value of each criterion
3. The alternative with the largest index in the matrix
4. The alternative that violates the most constraints

4.For benefit-type criteria (to be maximized), the PIS component is taken as

1. The minimum observed value of that criterion
2. The maximum observed value of that criterion
3. The mean observed value of that criterion
4. Zero after normalization

5.The main purpose of vector normalization in TOPSIS is to

1. Change the ranking by amplifying large values
2. Remove units and rescale each criterion to a dimensionless form
3. Force all criteria to have the same variance
4. Reduce the number of alternatives to a smaller subset

6.In the weighted normalized decision matrix VV, each entry vijvij is obtained by

1. Dividing xijxij by the maximum value in column jj
2. Multiplying the normalized value rijrij by the criterion weight wjwj
3. Adding the weight wjwj directly to xijxij
4. Subtracting the minimum value of column jj from xijxij

7.The Euclidean distance Di+Di+ in TOPSIS measures

1. The distance from alternative ii to the origin in the raw decision space
2. How far alternative ii is from the Negative Ideal Solution
3. How far alternative ii is from the Positive Ideal Solution in weighted-normalized space
4. The difference between best and worst criteria values for alternative ii

8.The closeness coefficient CiCi in TOPSIS is defined as

1. Ci=Di+/(Di++Di−)Ci=Di+/(Di++Di−)
2. Ci=Di−/(Di++Di−)Ci=Di−/(Di++Di−)
3. Ci=Di+−Di−Ci=Di+−Di−
4. Ci=Di+/Di−Ci=Di+/Di−

9.Regarding the interpretation of the closeness coefficient CiCi, TOPSIS assumes that

1. CiCi can take any real value between −∞−∞ and +∞+∞
2. Larger CiCi indicates an alternative closer to NIS and farther from PIS
3. CiCi lies in [0,1][0,1] and larger values are more preferred
4. CiCi is only used for normalization and not for ranking

10.A key motivation for using TOPSIS in MCDM is that it

1. Ignores trade-offs and focuses on a single aggregated cost function
2. Hides conflicting criteria by forcing them into one unweighted index
3. Provides a transparent geometric ranking by closeness to best and separation from worst performance
4. Eliminates the need to define benefit or cost type for criteria

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