# Goal Programming Models with Linear and Exponential Fuzzy Preference Relations

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## Abstract

**:**

## 1. Introduction

## 2. Goal Programming Formulation

#### 2.1. Representation of Fuzzy Preference Relations as Membership Functions

#### 2.2. Proposed Approach

**Model****1:**- This model is derived from Arenas-Parra et al. [35] and considers a linear membership function for fuzzy preference relations.
**Model****2:**- This model is an extension of Model 1 with an integration of the proposed formulated exponential membership functions for linguistic preferences among the objectives. This model uses nonlinearities in preferences among the objectives.
**Model****3:**- It is another extension of Model 1, with a new objective function considering a linear membership function for linguistic preferences between the objectives.
**Model****4:**- It is an extension of Model 2, incorporating a new objective function and an exponential membership function for linguistic preferences.

**Model**

**1:**This model was initially proposed by Arenas-Parra et al. [35] and later utilized by Bilbao-Terol et al. [33]. Bilbao-Terol et al. [33] applied the extended model in forest planning. Here, Model 1 is the same model proposed by Arenas-Parra et al. [35] with a linear membership function for linguistic preferences among the objectives as explained. The objective functions are defined as:

**Model**

**2:**This model is the extension of Model 1. It utilizes an exponential membership function for the fuzzy hierarchies between the goals. The formulation is presented below:

**Model**

**3:**It is derived from a hybridization of the Torabi and Hassini [47] approach with the goal programming model of Arenas-Parra et al. [35]. Objective 1 in Model 1 only maximizes the sum of satisfaction levels of normalized values; it does not considers a minimum satisfaction of all the objectives. By integrating the Torabi and Hassini [47] approach, we get more flexibility as it considers both the minimum satisfaction and the sum of satisfaction of all the objectives.

**Model**

**4:**This model is similar to Model 3, the only difference being in the linguistic preferences, which are represented here by an exponential membership function. Hence, Model 4 is formulated as follows:

#### 2.3. Comparison with Other Approaches

- Comparison with respect to the objective function

- 2.
- Comparison with respect to Fuzzy preference membership function

- 3.
- Comparison with respect to GP models

## 3. Solution Approach

## 4. Experimental Study

- Goal 1 is significantly more important than Goal 2.
- Goal 2 is significantly more important than Goal 4.
- Goal 2 is significantly more important than Goal 5.
- Goal 3 is fully more important than Goal 2.

**Step:1**Start

**Step:2**The optimization model is formulated as single objective problem and result values are obtained for all the objectives. It will provide us the best and worst values for the objectives.

**Step:3**Formulate the normalized function and membership grades for linear and exponential preference relations.

**Step:4**Now, formulate the problem as Models 1–4, as detailed in Section 2.2. The formulation of Model 1 becomes:

**Step:5**As the formulation is done, then all the models are modeled in AMPL language (Fourer et al. [51]) and solved by the CONOPT solver (Drud [55]) using the NEOS server online facility provided by Wisconsin Institutes for Discovery at the University of Wisconsin in Madison for solving Optimization problems (see Gropp, W. Moré [52], Czyzyk et al. [53], Dolan [54], and Server [50]). The solution obtained is feasible and optimal.

**Step:6**The method is stopped and the solution is presented to the DM.

#### 4.1. Results and Discussion

- Goal 1 is significantly more important than Goal 2.
- Goal 2 is significantly more important than Goal 4.
- Goal 2 is significantly more important than Goal 5.
- Goal 3 is fully more important than Goal 2.

- Goal 1 is significantly more important than Goal 2.
- Goal 2 is significantly more important than Goal 4.
- Goal 2 is significantly more important than Goal 5.
- Goal 3 is significantly more important than Goal 2.

- Goal 1 is significantly more important than Goal 2.
- Goal 2 is partially equal to Goal 4.
- Goal 2 is significantly more important than Goal 5.
- Goal 3 is fully more important than Goal 2.

- Goal 1 is significantly more important than Goal 2.
- Goal 2 is completely more important than Goal 4.
- Goal 2 is moderately more important than Goal 5.
- Goal 3 is fully more important than Goal 2.

- Goal 1 is significantly more important than Goal 2.
- Goal 2 is significantly more important than Goal 4.
- Goal 2 is significantly more important than Goal 5.
- Goal 3 is partially more important than Goal 2.

#### 4.2. Efficiency Analysis

## 5. Conclusions, Limitations, and Future Directions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Linguistic Term | Fuzzy Relation | Membership Values | Transform Function |
---|---|---|---|

${\tilde{\mathit{R}}}_{\mathit{q}},\mathit{q}=\mathbf{1},\mathbf{\dots},\mathbf{10}$ | ${\mathit{\mu}}_{{\tilde{\mathit{R}}}_{\mathit{q}}}$ | ||

Exactly equal | ${\tilde{R}}_{1}$ | ${\mu}_{{\tilde{R}}_{1}}$ | |

Partially equal | ${\tilde{R}}_{2}$ | ${\mu}_{{\tilde{R}}_{2}}$ | |

Partially more important than | ${\tilde{R}}_{3}$ | ${\mu}_{{\tilde{R}}_{3}}$ | |

Slightly more important than | ${\tilde{R}}_{4}$ | ${\mu}_{{\tilde{R}}_{4}}$ | |

Moderately more important than | ${\tilde{R}}_{5}$ | ${\mu}_{{\tilde{R}}_{5}}$ | ${\mu}_{k}(X)-{\mu}_{l}(X)$ |

Significantly more important than | ${\tilde{R}}_{6}$ | ${\mu}_{{\tilde{R}}_{6}}$ | $\forall k,l\in 1,\dots ,K$ |

Completely more important than | ${\tilde{R}}_{7}$ | ${\mu}_{{\tilde{R}}_{7}}$ | |

Fully more important than | ${\tilde{R}}_{8}$ | ${\mu}_{{\tilde{R}}_{8}}$ | |

Extremely more important than | ${\tilde{R}}_{9}$ | ${\mu}_{{\tilde{R}}_{9}}$ | |

Incomparable | ${\tilde{R}}_{10}$ | ${\mu}_{{\tilde{R}}_{10}}$ |

Variables | Model 1 (Linear) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |

Z | 2.7085 | 2.696 | 2.6834 | 2.67086 | 2.8468 | 3.1662 | 3.486 | 3.8059 | 4.1257 | 4.4456 | 4.7655 |

${f}_{1}$ | 12 | 12 | 12 | 12 | 45.9139 | 46.3816 | 46.3816 | 46.3816 | 46.3816 | 46.3816 | 46.3816 |

${f}_{2}$ | 24 | 24 | 24 | 24 | 100 | 100.526 | 100.526 | 100.526 | 100.526 | 100.526 | 100.526 |

${f}_{3}$ | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 | 120 |

${f}_{4}$ | 24 | 24 | 24 | 24 | 57.0168 | 57.1053 | 57.1053 | 57.1053 | 57.1053 | 57.1053 | 57.1053 |

${f}_{5}$ | 0 | 0 | 0 | 0 | 39.6639 | 40 | 40 | 40 | 40 | 40 | 40 |

${x}_{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${x}_{2}$ | 0 | 0 | 0 | 0 | 8.2563 | 8.2894 | 8.2894 | 8.2894 | 8.2894 | 8.2894 | 8.2894 |

${x}_{3}$ | 0 | 0 | 0 | 0 | 1.6596 | 1.7105 | 1.7105 | 1.7105 | 1.7105 | 1.7105 | 1.7105 |

${x}_{4}$ | 12 | 12 | 12 | 12 | 16.1239 | 16.1184 | 16.1184 | 16.1184 | 16.1184 | 16.1184 | 16.1184 |

${n}_{1}$ | 1 | 1 | 1 | 1 | 0.9517 | 0.9497 | 0.9497 | 0.9497 | 0.9497 | 0.9497 | 0.9497 |

${n}_{2}$ | 0.24 | 0.24 | 0.24 | 0.24 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${n}_{3}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${n}_{4}$ | 0.3428 | 0.3428 | 0.3428 | 0.3428 | 0.8145 | 0.8157 | 0.8157 | 0.8157 | 0.8157 | 0.8157 | 0.8157 |

${n}_{5}$ | 0 | 0 | 0 | 0 | 0.9915 | 1 | 1 | 1 | 1 | 1 | 1 |

${\mu}_{12}$ | 0.88 | 0.88 | 0.88 | 0.88 | 0.4758 | 0.4748 | 0.4748 | 0.4748 | 0.4748 | 0.4748 | 0.4748 |

${\mu}_{24}$ | 0.4485 | 0.4485 | 0.4485 | 0.4485 | 0.5927 | 0.5921 | 0.5921 | 0.5921 | 0.592105 | 0.5921 | 0.5921 |

${\mu}_{32}$ | 0.76 | 0.76 | 0.76 | 0.76 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mu}_{25}$ | 0.62 | 0.62 | 0.62 | 0.62 | 0.5042 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

$\sum {n}_{k}$ | 2.5828 | 2.5828 | 2.5828 | 2.5828 | 4.7579 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.7085 | 2.7085 | 2.7085 | 2.7085 | 1.5728 | 1.5669 | 1.5669 | 1.5669 | 1.5669 | 1.5669 | 0 |

Variables | Model 2 (Exponential) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\alpha $ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |

Z | 3.0707 | 3.0219 | 2.9731 | 2.9474 | 3.0969 | 3.3464 | 3.6302 | 3.914 | 4.1978 | 4.4816 | 4.7655 |

${f}_{1}$ | 12 | 12 | 12 | 35 | 35 | 46.3816 | 46.3816 | 46.3816 | 46.3816 | 46.3816 | 46.3816 |

${f}_{2}$ | 24 | 24 | 24 | 45.7463 | 84.8031 | 100.526 | 100.526 | 100.526 | 100.526 | 100.526 | 100.526 |

${f}_{3}$ | 120 | 120 | 120 | 134.416 | 120 | 120 | 120 | 120 | 120 | 120 | 120 |

${f}_{4}$ | 24 | 24 | 24 | 27.1662 | 54.1798 | 57.1053 | 57.1053 | 57.1053 | 57.1053 | 57.1053 | 57.1053 |

${f}_{5}$ | 0 | 0 | 0 | 13.3127 | 31.0515 | 40 | 40 | 40 | 40 | 40 | 40 |

${x}_{1}$ | 0 | 0 | 0 | 0 | 0.2852 | 0 | 0 | 0 | 0 | 0 | 0 |

${x}_{2}$ | 0 | 0 | 0 | 0.6944 | 6.9784 | 8.2894 | 8.2894 | 8.2894 | 8.2894 | 8.2894 | 8.2894 |

${x}_{3}$ | 0 | 0 | 0 | 2.6337 | 0.4991 | 1.7105 | 1.7105 | 1.7105 | 1.7105 | 1.7105 | 1.7105 |

${x}_{4}$ | 12 | 12 | 12 | 12.5414 | 15.909 | 16.1184 | 16.1184 | 16.1184 | 16.1184 | 16.1184 | 16.1184 |

${n}_{1}$ | 1 | 1 | 1 | 1 | 1 | 0.9497 | 0.9497 | 0.9497 | 0.9497 | 0.9497 | 0.9497 |

${n}_{2}$ | 0.24 | 0.24 | 0.24 | 0.4574 | 0.848 | 1 | 1 | 1 | 1 | 1 | 1 |

${n}_{3}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${n}_{4}$ | 0.3428 | 0.3428 | 0.3428 | 0.3880 | 0.7739 | 0.8157 | 0.8157 | 0.8157 | 0.8157 | 0.8157 | 0.8157 |

${n}_{5}$ | 0 | 0 | 0 | 0.3328 | 0.7762 | 1 | 1 | 1 | 1 | 1 | 1 |

${\mu}_{12}$ | 0.9258 | 0.9258 | 0.9258 | 0.8504 | 0.6926 | 0.598 | 0.5980 | 0.5980 | 0.5980 | 0.5980 | 0 |

${\mu}_{24}$ | 0.5718 | 0.5718 | 0.5718 | 0.6551 | 0.6573 | 0.7068 | 0.7068 | 0.7068 | 0.7068 | 0.7068 | 0 |

${\mu}_{32}$ | 0.8421 | 0.8421 | 0.8421 | 0.6624 | 0.223 | 0 | 0 | 0 | 0 | 0 | 0 |

${\mu}_{25}$ | 0.7309 | 0.7309 | 0.7309 | 0.6804 | 0.6562 | 0.6224 | 0.6224 | 0.6224 | 0.6224 | 0.6224 | 0 |

$\sum {n}_{k}$ | 2.5828 | 2.5828 | 2.5828 | 3.1783 | 4.3983 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0707 | 3.0707 | 3.0707 | 2.8484 | 2.2293 | 1.9273 | 1.9273 | 1.9273 | 1.9273 | 1.9273 | 0 |

Variables | Model 3 (Linear) | |||||
---|---|---|---|---|---|---|

(${\mathit{\gamma}}_{\mathbf{1}},{\mathit{\gamma}}_{\mathbf{2}},{\mathit{\gamma}}_{\mathbf{3}}$) | c1 | c2 | c3 | c4 | c5 | c6 |

Z | 2.4251 | 2.4785 | 4.0506 | 2.1444 | 2.9408 | 2.0899 |

${f}_{1}$ | 12 | 35 | 46.3816 | 46.3816 | 46.3816 | 65.1907 |

${f}_{2}$ | 24 | 87.7187 | 100.526 | 100.526 | 100.526 | 121.692 |

${f}_{3}$ | 120 | 120 | 120 | 120 | 120 | 120 |

${f}_{4}$ | 24 | 54.9527 | 57.1053 | 57.1053 | 57.1053 | 60.6625 |

${f}_{5}$ | 0 | 31.8203 | 40 | 40 | 40 | 53.5177 |

${x}_{1}$ | 0 | 0 | 0 | 0 | 0 | 0 |

${x}_{2}$ | 0 | 7.4822 | 8.2894 | 8.2894 | 8.2894 | 9.62345 |

${x}_{3}$ | 0 | 0.4728 | 1.7105 | 1.7105 | 1.7105 | 3.7559 |

${x}_{4}$ | 12 | 16.253 | 16.1184 | 16.1184 | 16.1184 | 15.8961 |

$\lambda $ | 0 | 0.7850 | 0.8157 | 0.8157 | 0.8157 | 0.8666 |

${n}_{1}$ | 1 | 1 | 0.9497 | 0.9497 | 0.9497 | 0.8666 |

${n}_{2}$ | 0.24 | 0.8771 | 1 | 1 | 1 | 1 |

${n}_{3}$ | 1 | 1 | 1 | 1 | 1 | 1 |

${n}_{4}$ | 0.3428 | 0.7850 | 0.8157 | 0.8157 | 0.8157 | 0.8666 |

${n}_{5}$ | 0 | 0.7955 | 1 | 1 | 1 | 1 |

${\mu}_{12}$ | 0.88 | 0.5614 | 0.4748 | 0.4748 | 0.4748 | 0.4333 |

${\mu}_{24}$ | 0.4485 | 0.5460 | 0.5921 | 0.5921 | 0.5921 | 0.5666 |

${\mu}_{32}$ | 0.76 | 0.1228 | 0 | 0 | 0 | 0 |

${\mu}_{25}$ | 0.62 | 0.5408 | 0.5 | 0.5 | 0.5 | 0.5 |

$\sum {n}_{k}$ | 2.5828 | 4.4577 | 4.7655 | 4.7655 | 4.7655 | 4.7332 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.7085 | 1.7711 | 1.5669 | 1.5669 | 1.5669 | 1.5 |

Variables | Model 4 (Exponential) | |||||
---|---|---|---|---|---|---|

(${\mathit{\gamma}}_{\mathbf{1}},{\mathit{\gamma}}_{\mathbf{2}},{\mathit{\gamma}}_{\mathbf{3}}$) | c1 | c2 | c3 | c4 | c5 | c6 |

Z | 2.7148 | 2.7444 | 4.0867 | 2.2526 | 3.0129 | 2.1262 |

${f}_{1}$ | 12 | 35 | 46.3816 | 46.3816 | 46.3816 | 65.1907 |

${f}_{2}$ | 24 | 70.4306 | 100.526 | 100.526 | 100.526 | 121.692 |

${f}_{3}$ | 120 | 120 | 120 | 120 | 120 | 120 |

${f}_{4}$ | 24 | 50.3698 | 57.1053 | 57.1053 | 57.1053 | 60.6625 |

${f}_{5}$ | 0 | 27.2614 | 40 | 40 | 40 | 53.5177 |

${x}_{1}$ | 0 | 1.6916 | 0 | 0 | 0 | 0 |

${x}_{2}$ | 0 | 4.4949 | 8.2894 | 8.2894 | 8.2894 | 9.6234 |

${x}_{3}$ | 0 | 0.6287 | 1.7105 | 1.7105 | 1.7105 | 3.7559 |

${x}_{4}$ | 12 | 14.2134 | 16.1184 | 16.1184 | 16.1184 | 15.8961 |

$\lambda $ | 0 | 0.6815 | 0.8157 | 0.8157 | 0.8157 | 0.8666 |

${n}_{1}$ | 1 | 1 | 0.9497 | 0.9497 | 0.9497 | 0.8666 |

${n}_{2}$ | 0.24 | 0.7043 | 1 | 1 | 1 | 1 |

${n}_{3}$ | 1 | 1 | 1 | 1 | 1 | 1 |

${n}_{4}$ | 0.3428 | 0.7195 | 0.8157 | 0.8157 | 0.8157 | 0.8666 |

${n}_{5}$ | 0 | 0.6815 | 1 | 1 | 1 | 1 |

${\mu}_{12}$ | 0.9258 | 0.7543 | 0.5980 | 0.5980 | 0.5980 | 0.5562 |

${\mu}_{24}$ | 0.5718 | 0.6151 | 0.7068 | 0.7068 | 0.7068 | 0.6843 |

${\mu}_{32}$ | 0.8421 | 0.4049 | 0 | 0 | 5.38E-17 | 0 |

${\mu}_{25}$ | 0.7309 | 0.6333 | 0.6224 | 0.6224 | 0.6224 | 0.6224 |

$\sum {n}_{k}$ | 2.5828 | 4.1054 | 4.7655 | 4.7655 | 4.7655 | 4.7332 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0707 | 2.4077 | 1.9273 | 1.9273 | 1.9273 | 1.8631 |

PR Type | Model 1 (Linear) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |

PRT1 | $\sum {n}_{k}$ | 2.5828 | 2.5828 | 2.5828 | 2.5828 | 4.7579 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.7085 | 2.7085 | 2.7085 | 2.70857 | 1.5728 | 1.5669 | 1.56696 | 1.56696 | 1.56696 | 1.56696 | 0 | |

PRT2 | $\sum {n}_{k}$ | 2.5828 | 4.4577 | 4.4577 | 4.4577 | 4.7579 | 4.7655 | 4.7655 | 4.7655 | 4.786 | 4.786 | 4.786 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.8285 | 2.2097 | 2.2097 | 2.2097 | 2.0728 | 2.0669 | 2.0669 | 2.0669 | 2.012 | 2.012 | 0 | |

PRT3 | $\sum {n}_{k}$ | 2.8 | 2.8 | 2.8 | 2.8 | 4.1554 | 4.1554 | 4.6364 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.1 | 3.1 | 3.1 | 3.1 | 2.4222 | 2.4222 | 1.8249 | 1.6064 | 1.6064 | 1.6064 | 0 | |

PRT4 | $\sum {n}_{k}$ | 2.5828 | 2.5828 | 2.5828 | 2.5828 | 4.7579 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.7314 | 2.7314 | 2.7314 | 2.7314 | 1.6051 | 1.5976 | 1.5976 | 1.5976 | 1.5976 | 1.5976 | 0 | |

PRT5 | $\sum {n}_{k}$ | 1.4949 | 2.85 | 4.7616 | 4.7616 | 4.7616 | 4.786 | 4.786 | 4.786 | 4.786 | 4.786 | 4.786 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3 | 2.925 | 2.5759 | 2.5759 | 2.5759 | 2.5576 | 2.5576 | 2.5576 | 2.5576 | 2.5576 | 0 |

PR Type | Model 2 (Exponential) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |

PRT1 | $\sum {n}_{k}$ | 2.5828 | 2.5828 | 2.5828 | 3.17837 | 4.3983 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0707 | 3.0707 | 3.0707 | 2.84845 | 2.2293 | 1.9273 | 1.92737 | 1.92737 | 1.92737 | 1.92737 | 0 | |

PRT2 | $\sum {n}_{k}$ | 2.5828 | 2.5828 | 3.2 | 4.5716 | 4.7579 | 4.7655 | 4.7655 | 4.7655 | 4.786 | 4.786 | 4.786 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.1543 | 3.1543 | 3.0372 | 2.6378 | 2.5554 | 2.5498 | 2.5498 | 2.5498 | 2.4987 | 2.4987 | 0 | |

PRT3 | $\sum {n}_{k}$ | 2.7526 | 2.8 | 2.8 | 3.1477 | 4.1554 | 4.1554 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.3765 | 3.3757 | 3.3757 | 3.2372 | 2.757 | 2.757 | 1.9612 | 1.9612 | 1.9612 | 1.9612 | 0 | |

PRT4 | $\sum {n}_{k}$ | 2.5828 | 2.5828 | 2.8261 | 3.1876 | 4.4577 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0257 | 3.0257 | 2.9734 | 2.8471 | 2.1912 | 1.9472 | 1.9472 | 1.9472 | 1.9472 | 1.9472 | 0 | |

PRT5 | $\sum {n}_{k}$ | 1.4949 | 3.2 | 4.7616 | 4.7616 | 4.7616 | 4.786 | 4.786 | 4.786 | 4.786 | 4.786 | 4.786 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.2449 | 3.1955 | 2.9382 | 2.9382 | 2.9382 | 2.921 | 2.921 | 2.921 | 2.921 | 2.921 | 0 |

Variables | Model 3 (Linear) | ||||||
---|---|---|---|---|---|---|---|

PR Type | (${\mathit{\gamma}}_{\mathbf{1}},{\mathit{\gamma}}_{\mathbf{2}},{\mathit{\gamma}}_{\mathbf{3}}$) | c1 | c2 | c3 | c4 | c5 | c6 |

PRT1 | $\sum {n}_{k}$ | 2.5828 | 4.4577 | 4.7655 | 4.7655 | 4.7655 | 4.7332 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.7085 | 1.77113 | 1.5669 | 1.5669 | 1.5669 | 1.5 | |

PRT2 | $\sum {n}_{k}$ | 2.5828 | 4.7579 | 4.786 | 4.786 | 4.7851 | 4.7556 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.8285 | 2.0728 | 2.012 | 2.012 | 1.9965 | 1.9592 | |

PRT3 | $\sum {n}_{k}$ | 2.8 | 4.1554 | 4.7655 | 4.4987 | 4.7332 | 4.7332 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.1 | 2.4222 | 1.6064 | 2.0085 | 1.6665 | 1.6665 | |

PRT4 | $\sum {n}_{k}$ | 2.5828 | 4.4577 | 4.7655 | 4.7655 | 4.7655 | 4.7655 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.7314 | 1.8001 | 1.5976 | 1.5976 | 1.5976 | 1.5976 | |

PRT5 | $\sum {n}_{k}$ | 3.2 | 4.7616 | 4.786 | 4.7851 | 4.7851 | 4.7851 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.8812 | 2.5759 | 2.5576 | 2.5485 | 2.5485 | 2.5485 |

Variables | Model 4 (Exponential) | ||||||
---|---|---|---|---|---|---|---|

PR Type | (${\mathit{\gamma}}_{\mathbf{1}},{\mathit{\gamma}}_{\mathbf{2}},{\mathit{\gamma}}_{\mathbf{3}}$) | c1 | c2 | c3 | c4 | c5 | c6 |

PRT1 | $\sum {n}_{k}$ | 2.5828 | 4.1054 | 4.7655 | 4.7655 | 4.7655 | 4.7332 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0707 | 2.4077 | 1.9273 | 1.9273 | 1.9273 | 1.8631 | |

PRT2 | $\sum {n}_{k}$ | 2.99 | 4.7579 | 4.786 | 4.786 | 4.7851 | 4.7556 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0787 | 2.5554 | 2.4987 | 2.4987 | 2.4839 | 2.4483 | |

PRT3 | $\sum {n}_{k}$ | 4.7154 | 4.7174 | 4.7655 | 4.7332 | 4.7332 | 4.7332 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 2.0184 | 2.0165 | 1.9612 | 2.0008 | 2.0008 | 2.0008 | |

PRT4 | $\sum {n}_{k}$ | 2.7121 | 4.097 | 4.7655 | 4.7655 | 4.7655 | 4.7332 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0001 | 2.4024 | 1.9472 | 1.9472 | 1.9472 | 1.8709 | |

PRT5 | $\sum {n}_{k}$ | 4.4126 | 4.7616 | 4.786 | 4.7851 | 4.7851 | 4.7835 |

$\sum {\mu}_{{R}_{q}}(k,l)$ | 3.0136 | 2.9382 | 2.921 | 2.9126 | 2.9126 | 2.91 |

PR Type | Distance Measure for Model 1 and 2 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |

PRT1 | Model 1 | 1.5905 | 1.5905 | 1.5905 | 1.5905 | 1.3126 | 1.3147 | 1.3147 | 1.3147 | 1.3147 | 1.3147 | 2.009 |

Model 2 | 1.5151 | 1.5151 | 1.5151 | 1.2129 | 1.0285 | 1.1943 | 1.1943 | 1.1943 | 1.1943 | 1.1943 | 2.009 | |

PRT2 | Model 1 | 1.5769 | 1.5769 | 1.5769 | 0.9511 | 0.9864 | 0.9892 | 0.9892 | 0.9892 | 1.0079 | 1.0079 | 2.0056 |

Model 2 | 1.5087 | 1.5087 | 1.1816 | 0.7274 | 0.7518 | 0.7543 | 0.7543 | 0.7543 | 0.7687 | 0.7687 | 2.0056 | |

PRT3 | Model 1 | 1.3909 | 1.3909 | 1.3909 | 1.3909 | 1.0663 | 1.0663 | 1.2424 | 1.303 | 1.303 | 1.303 | 2.009 |

Model 2 | 1.3591 | 1.3345 | 1.3345 | 1.1789 | 0.9084 | 0.9084 | 1.1865 | 1.1865 | 1.1865 | 1.1865 | 2.009 | |

PRT4 | Model 1 | 1.6285 | 1.6285 | 1.6285 | 1.6285 | 1.3091 | 1.3111 | 1.3111 | 1.3111 | 1.3111 | 1.3111 | 2.009 |

Model 2 | 1.5657 | 1.5657 | 1.4032 | 1.2258 | 1.069 | 1.1949 | 1.1949 | 1.1949 | 1.1949 | 1.1949 | 2.009 | |

PRT5 | Model 1 | 1.9378 | 1.4626 | 0.838 | 0.838 | 0.838 | 0.8475 | 0.8475 | 0.8475 | 0.8475 | 0.8475 | 2.0056 |

Model 2 | 1.8815 | 1.171 | 0.6339 | 0.6339 | 0.6339 | 0.6423 | 0.6423 | 0.6423 | 0.6423 | 0.6423 | 2.0056 |

PR Type | Distance Measure for Model 3 and 4 | ||||||
---|---|---|---|---|---|---|---|

(${\mathit{\gamma}}_{\mathbf{1}},{\mathit{\gamma}}_{\mathbf{2}},{\mathit{\gamma}}_{\mathbf{3}}$) | c1 | c2 | c3 | c4 | c5 | c6 | |

PRT1 | Model 3 | 1.5905 | 1.2172 | 1.3147 | 1.3147 | 1.3147 | 1.3395 |

Model 4 | 1.5151 | 0.982 | 1.1943 | 1.1943 | 1.2143 | 1.2143 | |

PRT2 | Model 3 | 1.5769 | 0.9864 | 1.0079 | 1.0079 | 1.0151 | 1.0322 |

Model 4 | 1.2682 | 0.7518 | 0.7687 | 0.7687 | 0.7754 | 0.7912 | |

PRT3 | Model 3 | 1.3909 | 1.0663 | 1.303 | 1.2545 | 1.2953 | 1.2953 |

Model 4 | 1.1928 | 1.1918 | 1.1865 | 1.186 | 1.186 | 1.186 | |

PRT4 | Model 3 | 1.6285 | 1.2284 | 1.3111 | 1.3111 | 1.3111 | 1.3422 |

Model 4 | 1.4763 | 1.0258 | 1.1949 | 1.1949 | 1.1949 | 1.2217 | |

PRT5 | Model 3 | 1.258 | 0.838 | 0.8475 | 0.8519 | 0.8519 | 0.8519 |

Model 4 | 0.6669 | 0.6339 | 0.6423 | 0.6461 | 0.6461 | 0.6472 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Khan, M.F.; Hasan, M.G.; Quddoos, A.; Fügenschuh, A.; Hasan, S.S.
Goal Programming Models with Linear and Exponential Fuzzy Preference Relations. *Symmetry* **2020**, *12*, 934.
https://doi.org/10.3390/sym12060934

**AMA Style**

Khan MF, Hasan MG, Quddoos A, Fügenschuh A, Hasan SS.
Goal Programming Models with Linear and Exponential Fuzzy Preference Relations. *Symmetry*. 2020; 12(6):934.
https://doi.org/10.3390/sym12060934

**Chicago/Turabian Style**

Khan, Mohammad Faisal, Md. Gulzarul Hasan, Abdul Quddoos, Armin Fügenschuh, and Syed Suhaib Hasan.
2020. "Goal Programming Models with Linear and Exponential Fuzzy Preference Relations" *Symmetry* 12, no. 6: 934.
https://doi.org/10.3390/sym12060934