---
title: "AHP-TOPSIS-2N Example"
author: "Raquel Dourado Coutinho, Marcos dos Santos"
date: '`r Sys.Date()`'
output: 
  rmarkdown::html_vignette:
    toc: yes
    toc_depth: 2
vignette: >
  %\VignetteIndexEntry{AHP-TOPSIS-2N Example}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Introduction
##### Multicriteria Decision Analysis (MCDA) and the AHP-TOPSIS-2N method.
The objective of MCDA is to assist in decisions with multiple criteria. Multicriteria methods can be useful in several problems in business and even in personal life. AHP-TOPSIS-2N is a hybrid method build from the AHP (Analytic Hierarchy Process) and TOPSIS-2N (Technique for Order of Preference by Similarity to Ideal Solution - with two normalizations).

AHP-TOPSIS-2N uses the AHP to calculate the criteria weights and uses TOPSIS  twice to generate rankings, each time with a different kind of normalization. This can allow the comparison of results and the analysis of the robustness. A consistency ratio is calculated, and when it is higher than 10%, it is required to check judgments on criteria comparison.  


## Example
As an example, this vignette uses a case of row material supplier evaluation.
The goal is to choose among A, B, and C suppliers based on the product cost, product quality (1 to 5), and the lead time.
Below we have the decision matrix (alternatives x criteria).

| Alternatives | Cost | Quality | Lead Time|
|:------------:|-----:|--------:|---------:|
|       A1     |  1100|    5    |    25    |
|       A2     |   850|    3.5  |    10    |
|       A3     |   950|    4    |    30    |


After defining the decision matrix, it's time to define a matrix with a pairwise comparison of the criteria, using the Saaty scale (1-9). 


|             | Cost | Quality | Lead Time|
|:-----------:|-----:|--------:|---------:|
|   Cost      |     1|    1    |    3     |
|   Quality   |     1|    1    |    5     |
|   Lead Time |   1/3|    1/5  |    1     |


The criteria comparison matrix can be read like this: "Cost is so important as Quality, Cost has moderate importance over Lead Time, Quality has strong importance over Lead Time."



```{r setup}
library(ahptopsis2n)

# define the decision matrix
decision<-matrix(c(1100, 5, 25,
                   850, 3.5, 10,
                   950, 4, 30), ncol=3, byrow=TRUE)

rownames(decision)<- c("A1", "A2", "A3")

#define criteria matrix with pairwise comparison
criteria<-matrix(c(1, 1, 3,
                   1, 1, 5,
                   1/3, 1/5, 1), ncol=3, byrow=TRUE)

# define each criterion objective
minmax<-c("min", "max", "min")

# associate the objects to the function arguments and run the function
ahptopsis2n(decision=decision, criteria=criteria, minmax=minmax)

```
As we can see, the result is a list with a consistency ratio and two data frames with priority sorting of the alternatives.

### References


De Souza, L. P., Gomes, C. F. S. and De Barros, A. P. (2018). Implementation of New Hybrid AHP–TOPSIS-2N Method in Sorting and Prioritizing of an it CAPEX Project Portfolio. International Journal of Information Technology & Decision Making. DOI: 10.1142/S0219622018500207.

Saaty, T.L. The Analytic Hierarchy Process. McGraw-Hill, New York. (1980)