# Equivalent Exchange Method for Decision-Making in Case of Alternatives with Incomparable Attributes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Description

- $q=1.\phantom{\rule{0.166667em}{0ex}}.Q$ is the iteration index, where Q is the total number of the iterations, which is written as an upper index in variables.
- $\mathbf{X}=\left({X}_{1},{X}_{2},\dots ,{X}_{i},\dots ,{X}_{I}\right),i=1.\phantom{\rule{0.166667em}{0ex}}.I$ is the initial set of alternatives, or decision options.
- I denotes the total number of alternatives to be compared.
- ${\mathbf{X}}^{q}\subseteq \mathbf{X}$ is the set of alternatives at the qth iteration. During the run of the algorithm, the number of alternatives in subsequent iterations is getting reduced, and the alternatives are renumbered.
- ${I}^{q}$ marks the number of alternatives at the qth iteration.
- ${L}^{q}$ designates the number of attributes at the qth iteration. In subsequent iterations, the number of attributes cannot increase and tends to go down.
- l denotes the indices of the attributes against which the decision options are compared, $l=1.\phantom{\rule{0.166667em}{0ex}}.{L}^{q}$.
- ${\mathbf{F}}^{q}$ is the preference matrix at the qth iteration whose elements ${F}_{il}^{q},i=1.\phantom{\rule{0.166667em}{0ex}}.{I}^{q},l=1.\phantom{\rule{0.166667em}{0ex}}.{L}^{q}$ are estimates of the ith alternative by the lth attribute. Since the number of alternatives and attributes tend to reduce as the iterations go on, the size of this matrix will also go down.
- ${Y}_{ikl}^{q},l=1.\phantom{\rule{0.166667em}{0ex}}.{L}^{q}$ stands for the elements of comparison matrices ${\mathbf{Y}}^{q}$ for alternative pairs in the qth iteration. Each pair of alternatives being compared corresponds to a unique row of the matrix.
- ${Y}_{ik}^{q}$ designates the sum of matrix elements ${Y}_{ikl}^{q}$ over all the attributes, i.e., the sum of the elements in the row indexed by $(i,k)$.

#### 2.2. Basic Concepts

- ${X}_{i}\u2ab0{X}_{k}$, i.e., ${X}_{i}$ is weakly dominating over the alternative denoted by ${X}_{k}$,AND
- there is at least one attribute $\tilde{l}$ such that the ith alternative is better than the kth alternative measured with it.

#### 2.3. The Iteration of the EEM Algorithm

- The set of alternatives at the qth iteration: $\left\{{X}_{i}^{q}\right\},i=1\dots {I}^{q}$;
- The set of attributes at the qth iteration $l=1\dots {L}^{q}$;
- The preference matrix at the qth iteration: ${\mathbf{F}}^{q}=\parallel {F}_{il}^{q}\parallel ,i=1\dots {I}^{q},l=1\dots {L}^{q}$ is made of the value of preference scalars against the lth attribute for alternative ${X}_{i}$.

#### 2.3.1. Step 1: Computing the Comparison Matrix

#### 2.3.2. Step 2: Search for Insignificant Attributes

#### 2.3.3. Step 3: Search for Dominated Alternatives

**Case A**happens if ${Y}_{ik}^{q}={L}^{q}$, i.e., the sum of elements in the row of matrix ${\mathbf{Y}}^{q}$ corresponding to the pair of alternatives $({X}_{i},{X}_{k})$ in the qth iteration is equal to the number of attributes ${L}^{q}$ in the same iteration. This is the case where the weak dominance takes place but not the strict dominance. The DM should make a decision about which of the alternatives is to be kept, and which is declared dominated. The DM may make such a decision straightforwardly or with taking into consideration additional factors, which have not been introduced into the task yet.**Case B**happens if ${Y}_{ik}^{q}={L}^{q}-1,{L}^{q}1$, i.e., the sum of elements in the row of matrix ${\mathbf{Y}}^{q}$ corresponding to the pair of alternatives $({X}_{i},{X}_{k})$ in the qth iteration is less than the number of attributes ${L}^{q}$ by 1. Consequently, the estimate of alternatives $({X}_{i},{X}_{k})$ is equal among all attributes except only one attribute denoted with $\tilde{l}$. In formal words, $\exists !\tilde{l}:{Y}_{ik\tilde{l}}^{q}=0$. The DM must make a decision, which of the estimates by attribute $\tilde{l}$ is preferable: ${F}_{i\tilde{l}}^{q}$ or ${F}_{k\tilde{l}}^{q}$. If estimate ${F}_{i\tilde{l}}^{q}$ is considered preferable, i.e., ${F}_{i\tilde{l}}\succ {F}_{k\tilde{l}}$, then alternative ${X}_{i}$ should be declared as strictly dominating over alternative ${X}_{k}$, ${X}_{i}\succ {X}_{k}$. Otherwise, alternative ${X}_{k}$ would be declared as strictly dominating over alternative ${X}_{i}$.**Case C**is the situation where neither Case A nor Case B take place. Then, there are no dominated alternatives which have been identified in the step.

#### 2.3.4. Step 4: Shrinkage of the Preference Matrix

- The dominated alternatives, which are related to both cases A and B in the previous step, have to be excluded from vector ${X}^{q}$. The list of remaining alternatives are renumbered to restore the consequent indexing of the alternatives;
- The rows corresponding to the dominated alternatives and the columns corresponding to insignificant attributes are deleted from matrix ${\mathbf{F}}^{q}$;
- The rows in the matrix corresponding to the pairs of alternatives where at least one of the alternatives is being dominated have to be excluded from matrix ${\mathbf{Y}}^{q}$. The columns corresponding to insignificant attributes are also excluded;
- Variables ${I}^{q}$ and ${L}^{q}$ take on the new values equal to the number of, respectively, alternatives and attributes remaining for further consideration.

#### 2.3.5. Step 5: The Single Alternative Found

#### 2.3.6. Step 6: The Single Attribute Remains

#### 2.3.7. Step 7: Search for the Alternative Suitable for Equivalent Exchange

#### 2.3.8. Step 8: Choice of an Exchangeable Attribute

- Attribute ${l}^{*}$ cannot be the same as attribute $\widehat{l}$, selected in Step 7;
- The element of the comparison matrix that corresponds to this attribute and the pair $({X}_{\widehat{i}},{X}_{\widehat{k}})$ selected in step 7 must be zero: ${Y}_{\widehat{i}\widehat{k}{l}^{*}}^{q}=0$.

#### 2.3.9. Step 9: Performing the Equivalent Exchange

“Let us suppose that the cost of implementation changes from 220 to 200 thousand euros. By how many days should the implementation period be increased from the initial 20 weeks, so that both changes would be equivalent from the DM’s own perspective?”

“To reduce the cost of implementation by 20 thousand euros, we can give 3 extra weeks for system implementation”.

#### 2.4. The EDSS Description

## 3. Results

#### 3.1. Iteration 1

**Time**attributes for the exchange has been chosen automatically. This selection can be easily explained since there is a unity in the first column of Table 3 while all other columns contained zeros only. This choice will head for a faster elimination of the column as soon as its attribute has been found to be insignificant.

**Equipment**attribute for the equivalent exchange with

**Time**. Then, in Step 9, the dialog window, which is shown in Figure 3, asks the DM for entering the new value of

**Equipment**attribute. The estimate of ${X}_{2}$ by

**Time**attribute is going to decrease from 28 to 25, whiich will be thought of as a positive effect. The DM considers that equivalent decrease in

**Equipment**should be its change from 80 to 78. Although this decision may well look arbitrary, it is the simulation of the DM’s personal decision here. In other words, the DM is ready to sacrifice just a small piece of office equipment for a three-minute decrease in time that most of the staff will spend to obtain an office.

#### 3.2. Iteration 2

**Time**is an insignificant attribute and has to be excluded. This exclusion has led to the reduction in the number of columns constituting matrix ${\mathbf{F}}^{2}$. The new matrices ${\mathbf{F}}^{2}$ and ${\mathbf{Y}}^{2}$ are shown in Table 6 and Table 7 correspondingly. The number of attributes has decreased, and the rest of the attributes have been renumbered. It is important to emphasize the fact that the insignificance of

**Time**and the deletion of its column have become possible because of the equivalent exchange done at the previous iteration—Iteration 1.

**Location**, while the attribute for exchange has to be chosen by the DM. Suppose they have picked up

**Costs**and decided that the increase in

**Location**from 70 to 80 can reasonably be achieved with an increase in

**Costs**from 1500 to 1600. The result of the second iteration is the equivalent exchange made between ${X}_{1}$ and ${X}_{2}$ alternatives. The resultant matrix is shown in Table 8.

#### 3.3. Iteration 3

**Location**attribute, the EDSS suggests this attribute as $\widehat{l}$ again, while the DM picks up

**Size**for the exchange. The decrease in

**Location**from 85 to 80 can be tolerated if the equivalent increase in

**Location**takes place from 800 to 850.

#### 3.4. Iteration 4

**Location**has become an insignificant attribute and has to be excluded. This leads to the reduction in the number of columns constituting matrix ${\mathbf{F}}^{4}$ since the corresponding column has to be deleted. The new matrices ${\mathbf{F}}^{4}$ and ${\mathbf{Y}}^{4}$ are shown in Table 12 and Table 13 correspondingly.

**Equipment**from 78 to 60 and the decrease in

**Costs**from 1600 to 1200. The resultant preference matrix is shown in Table 14.

#### 3.5. Iteration 5

**Size**from 500 to 700 is equivalent to the increase in

**Costs**from 1200 to 1500. The modified matrix after the fifth iteration is in Table 16.

#### 3.6. Iteration 6

**Cost**, while the estimates for two other attributes are equal for the alternatives in the pair $({X}_{1},{X}_{2})$. The shrunk matrix with renumbered alternatives is shown in Table 18. The matrix containing the result of pairwise comparison is depicted in Table 19.

**Equipment**from 50 to 60 is treated as equivalent to the increase in

**Costs**from 1900 to 2000. It provides the modified preference matrix shown in Table 20.

#### 3.7. Iteration 7

**Equipment**takes the same value over both alternatives, so it appears to be insignificant and must be excluded. The shrunk preference matrix is shown in Table 22, and the matrix of pair-wise comparisons is presented in Table 23. Then, the equivalent exchange is being conducted where the decrease in

**Size**from 850 to 700 is assumed to be equivalent to the decrease in

**Costs**from 2000 to 1650. This exchange leads to the preference matrix depicted in Table 24.

#### 3.8. Iteration 8

**Size**becomes insignificant and has to be put out of the consideration. The resultant matrix containing the rest column is shown in Table 26.

#### 3.9. The Final Decision

**Costs**. Thus, ‘Office B’ is chosen as a unique optimal solution, since this choice implies a lower amount of money spent on renting the office.

## 4. Discussion

#### 4.1. The Description of the Attributes

#### 4.2. The Foundation of EEM

#### 4.3. The Main Strategy

- The first strategy consists of conducting such a sequence of exchanges which results in the equal values for the same attribute which the estimates take on in all the alternatives.
- The second strategy consists of the equalization of two alternatives by all parameters except one; the only parameter with different values will determine the dominating alternative.

#### 4.4. The Multiple Algorithm Outputs

#### 4.5. The Convergence of the Algorithm

#### 4.6. The Multiple Attribute Dominance

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AIM | Aspiration-level Interactive Method |

DM | Decision Maker |

DSS | Decision Support System |

EDSS | Expert Decision Support System |

EEM | Equivalent Exchange Method |

ELECTRE | ÉLimination Et Choix Traduisant la REalité |

(Elimination and Choice Translating Reality) | |

GUEST | Acronym for the methodology: Go, Uniform, Evaluate, Solve, Test |

PROMETHEE | Preference Ranking Organization METHod for Enrichment of Evaluations |

SMAA | Stochastic Multiobjective Acceptability Analysis |

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Pairs of the Alternatives | ||
---|---|---|

i | k | ${\mathit{Y}}_{\mathit{ik}}^{\mathit{q}}$ |

1 | 2 | 2 |

1 | 3 | 2 |

1 | 4 | 2 |

2 | 3 | 1 |

2 | 4 | 1 |

3 | 4 | 4 |

1: Time | 2: Location | 3: Equipment | 4: Size | 5: Costs | |
---|---|---|---|---|---|

${X}_{1}$: Office A | 25.0 | 80.0 | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 28.0 | 70.0 | 80.0 | 500.0 | 1500.0 |

${X}_{3}$: Office C | 25.0 | 85.0 | 50.0 | 800.0 | 1900.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | 4 | 5 | ${\mathit{Y}}_{\mathit{i}\mathit{k}}^{1}(\mathbf{\Sigma})$ |
---|---|---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 0 | 0 | 0 | 0 | 0 | 0 |

${X}_{1}:{X}_{3}$ | 1 | 0 | 0 | 0 | 0 | 1 |

${X}_{2}:{X}_{3}$ | 0 | 0 | 0 | 0 | 0 | 0 |

1: Time | 2: Location | 3: Equipment | 4: Size | 5: Costs | |
---|---|---|---|---|---|

${X}_{1}$: Office A | 25.0 | 80.0 | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 25.0 | 70.0 | 78.0 | 500.0 | 1500.0 |

${X}_{3}$: Office C | 25.0 | 85.0 | 50.0 | 800.0 | 1900.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | 4 | 5 | ${\mathit{Y}}_{\mathit{ik}}^{2}(\mathbf{\Sigma})$ |
---|---|---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 1 | 0 | 0 | 0 | 0 | 1 |

${X}_{1}:{X}_{3}$ | 1 | 0 | 0 | 0 | 0 | 1 |

${X}_{2}:{X}_{3}$ | 1 | 0 | 0 | 0 | 0 | 1 |

1: Location | 2: Equipment | 3: Size | 4: Costs | |
---|---|---|---|---|

${X}_{1}$: Office A | 80.0 | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 70.0 | 78.0 | 500.0 | 1500.0 |

${X}_{3}$: Office C | 85.0 | 50.0 | 800.0 | 1900.0 |

**Table 7.**The pair-wise comparison matrix at Iteration 2 after

**Time**has been excluded—${\mathbf{Y}}^{2}$.

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | 4 | ${\mathit{Y}}_{\mathit{ik}}^{2}(\mathbf{\Sigma})$ |
---|---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 0 | 0 | 0 | 0 | 0 |

${X}_{1}:{X}_{3}$ | 0 | 0 | 0 | 0 | 0 |

${X}_{2}:{X}_{3}$ | 0 | 0 | 0 | 0 | 0 |

1: Location | 2: Equipment | 3: Size | 4: Costs | |
---|---|---|---|---|

${X}_{1}$: Office A | 80.0 | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 80.0 | 78.0 | 500.0 | 1600.0 |

${X}_{3}$: Office C | 85.0 | 50.0 | 800.0 | 1900.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | 4 | ${\mathit{Y}}_{\mathit{ik}}^{3}(\mathbf{\Sigma})$ |
---|---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 1 | 0 | 0 | 0 | 1 |

${X}_{1}:{X}_{3}$ | 0 | 0 | 0 | 0 | 0 |

${X}_{2}:{X}_{3}$ | 0 | 0 | 0 | 0 | 0 |

1: Location | 2: Equipment | 3: Size | 4: Costs | |
---|---|---|---|---|

${X}_{1}$: Office A | 80.0 | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 80.0 | 78.0 | 500.0 | 1600.0 |

${X}_{3}$: Office C | 80.0 | 50.0 | 850.0 | 1900.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | 4 | ${\mathit{Y}}_{\mathit{ik}}^{4}(\mathbf{\Sigma})$ |
---|---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 1 | 0 | 0 | 0 | 1 |

${X}_{1}:{X}_{3}$ | 1 | 0 | 0 | 0 | 1 |

${X}_{2}:{X}_{3}$ | 1 | 0 | 0 | 0 | 1 |

1: Equipment | 2: Size | 3: Costs | |
---|---|---|---|

${X}_{1}$: Office A | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 78.0 | 500.0 | 1600.0 |

${X}_{3}$: Office C | 50.0 | 850.0 | 1900.0 |

**Table 13.**The pair-wise comparison matrix at Iteration 4 after

**Location**has been excluded—${\mathbf{Y}}^{4}$.

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | ${\mathit{Y}}_{\mathit{ik}}^{4}(\mathbf{\Sigma})$ |
---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 0 | 0 | 0 | 0 |

${X}_{1}:{X}_{3}$ | 0 | 0 | 0 | 0 |

${X}_{2}:{X}_{3}$ | 0 | 0 | 0 | 0 |

1: Equipment | 2: Size | 3: Costs | |
---|---|---|---|

${X}_{1}$: Office A | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 60.0 | 500.0 | 1200.0 |

${X}_{3}$: Office C | 50.0 | 850.0 | 1900.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | ${\mathit{Y}}_{\mathit{ik}}^{5}(\mathbf{\Sigma})$ |
---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 1 | 0 | 0 | 1 |

${X}_{1}:{X}_{3}$ | 0 | 0 | 0 | 0 |

${X}_{2}:{X}_{3}$ | 0 | 0 | 0 | 0 |

1: Equipment | 2: Size | 3: Costs | |
---|---|---|---|

${X}_{1}$: Office A | 60.0 | 700.0 | 1700.0 |

${X}_{2}$: Office B | 60.0 | 700.0 | 1500.0 |

${X}_{3}$: Office C | 50.0 | 850.0 | 1900.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | ${\mathit{Y}}_{\mathit{ik}}^{6}(\mathbf{\Sigma})$ |
---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 1 | 1 | 0 | 2 |

${X}_{1}:{X}_{3}$ | 0 | 0 | 0 | 0 |

${X}_{2}:{X}_{3}$ | 0 | 0 | 0 | 0 |

**Table 18.**The preference matrix at Iteration 6 after ’Office A’ has been excluded—${\mathbf{F}}^{6}$.

1: Equipment | 2: Size | 3: Costs | |
---|---|---|---|

${X}_{1}$: Office B | 60.0 | 700.0 | 1500.0 |

${X}_{2}$: Office C | 50.0 | 850.0 | 1900.0 |

**Table 19.**The pair-wise comparison matrix at Iteration 6 after ’Office A’ has been excluded—${\mathbf{Y}}^{6}$.

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | ${\mathit{Y}}_{\mathit{ik}}^{6}(\mathbf{\Sigma})$ |
---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 0 | 0 | 0 | 0 |

1: Equipment | 2: Size | 3: Costs | |
---|---|---|---|

${X}_{1}$: Office B | 60.0 | 700.0 | 1500.0 |

${X}_{2}$: Office C | 60.0 | 850.0 | 2000.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | 3 | ${\mathit{Y}}_{\mathit{ik}}^{7}(\mathbf{\Sigma})$ |
---|---|---|---|---|

${X}_{1}:{X}_{2}$ | 1 | 0 | 0 | 1 |

**Table 22.**The preference matrix at Iteration 7 after ’Equipment’ has been excluded—${\mathbf{F}}^{7}$.

1: Size | 2: Costs | |
---|---|---|

${X}_{1}$: Office B | 700.0 | 1500.0 |

${X}_{2}$: Office C | 850.0 | 2000.0 |

**Table 23.**The pair-wise comparison matrix at Iteration 7 after ’Equipment’ has been excluded—${\mathbf{Y}}^{7}$.

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | ${\mathit{Y}}_{\mathit{ik}}^{7}(\mathbf{\Sigma})$ |
---|---|---|---|

${X}_{1}:{X}_{2}$ | 0 | 0 | 0 |

1: Size | 2: Costs | |
---|---|---|

${X}_{1}$: Office B | 700.0 | 1500.0 |

${X}_{2}$: Office C | 700.0 | 1650.0 |

$({\mathit{X}}_{\mathit{i}}:{\mathit{X}}_{\mathit{k}})/\mathit{l}$ | 1 | 2 | ${\mathit{Y}}_{\mathit{ik}}^{8}(\mathbf{\Sigma})$ |
---|---|---|---|

${X}_{1}:{X}_{2}$ | 1 | 0 | 1 |

1: Costs | |
---|---|

${X}_{1}$: Office B | 1500.0 |

${X}_{2}$: Office C | 1650.0 |

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## Share and Cite

**MDPI and ACS Style**

Kravchenko, T.; Shevgunov, T.
Equivalent Exchange Method for Decision-Making in Case of Alternatives with Incomparable Attributes. *Inventions* **2023**, *8*, 12.
https://doi.org/10.3390/inventions8010012

**AMA Style**

Kravchenko T, Shevgunov T.
Equivalent Exchange Method for Decision-Making in Case of Alternatives with Incomparable Attributes. *Inventions*. 2023; 8(1):12.
https://doi.org/10.3390/inventions8010012

**Chicago/Turabian Style**

Kravchenko, Tatiana, and Timofey Shevgunov.
2023. "Equivalent Exchange Method for Decision-Making in Case of Alternatives with Incomparable Attributes" *Inventions* 8, no. 1: 12.
https://doi.org/10.3390/inventions8010012