Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator
Abstract
:1. Introduction
2. Preliminaries
2.1. Fuzzy Sets and Fuzzy Numbers
2.2. The Extension Principle
2.3. The Generalized “min” Operator
3. Problem Formulation and Solution Approach
3.1. First Variant
3.2. Second Variant
Algorithm 1 The Monte Carlo simulation with left–right endpoints 

4. Numerical Illustration
4.1. First Numerical Example
 The empiric solution provided by the general MC algorithm (in gray) better covers the exact fuzzy set representing the optimal solution when compared to the left–rightendpointbased MC algorithm (in red);
 The empiric fuzzy set solution has a quite large support, but the level set corresponding to $\alpha =0.1$ is already $47\%$ of the level set of $\alpha =0$, and is thus considerably narrower;
 Even though the FFLP problem has only nonnegative coefficients, no relevant solutions can be obtained either by setting all coefficients to the left endpoints of their $\alpha $cuts (in blue) or on their right endpoints (in green), since such derived solutions are too far from the exact ones derived by direct optimization (in gray).
4.2. Second Numerical Example
 The left–rightendpointbased MC algorithm and the general MC algorithmderived solutions (in red and gray, respectively) have a very similar accuracy; thus, the former algorithm performs better due to its higher running speed;
 The empirical solution very well covers the exact fuzzy set solution (either in red or gray); thus, the proposed methodology can provide good approximate solutions when the exact ones are hard to obtain;
 The $\alpha $cut interval is reduced to $55\%$ after a small increase in $\alpha $ from 0 to $0.1$;
 As in the previous example, the solutions derived by setting all coefficients to the same side of their $\alpha $cuts (left in blue or right in green) are not relevant to the decision maker.
5. Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
LP  Linear programming 
FF  Full fuzzy 
FN  Fuzzy number 
TFN  Triangular fuzzy number 
TrFN  Trapezoidal fuzzy number 
EP  Extension principle 
MC  Monte Carlo 
DEA  Data Enveloped Analysis 
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Authors  Year  Ref.  Problem  Full Compliance to the EP 

Stanojević et al.  2021  [3]  review fuzzy LP and LFP  N/A 
Wang et al.  2019  [4]  FFLP  yes 
Ezzati et al.  2015  [5]  FFLP  no 
Anukokila et al.  2019  [8]  transportation FFLFP  no 
Ebrahimnejad et al.  2018  [9]  transportation FFLFP  no 
Liu et al.  2004  [10]  transportation FFLFP  yes 
Kao et al.  2000  [17]  fuzzy DEA  yes 
Kao et al.  2011  [18]  fuzzy DEA  yes 
Soltanzadeh et al.  2018  [19]  fuzzy DEA  no 
Ghanbari et al.  2019  [22]  review fuzzy LP  N/A 
Problem  $\mathit{\alpha}$  min $\mathit{\alpha}$Cuts and Lengths  Product $\mathit{\alpha}$Cuts and Lengths  Lengths Ratio ^{1}  

(19)  $0.9$  $\left[535.17,618.46\right]$  $83.29$  $\left[558.77,601.61\right]$  $42.84$  $0.51$ 
$0.7$  $\left[456.13,704.91\right]$  $248.78$  $\left[519.78,635.58\right]$  $115.80$  $0.47$  
$0.5$  $\left[371.47,799.38\right]$  $427.92$  $\left[476.90,666.42\right]$  $189.53$  $0.44$  
$0.1$  $\left[236.51,1015.57\right]$  $779.06$  $\left[366.90,777.86\right]$  $410.95$  $0.53$  
$0.0$  $\left[209.04,1085.0\right]$  $875.96$  $\left[209.04,1085.0\right]$  $875.96$  1  
(27)  $0.9$  $\left[296.6,778\right]$  $481.4$  $\left[312.78,732.73\right]$  $419.94$  $0.87$ 
$0.7$  $\left[241.4,914\right]$  $672.6$  $\left[284.74,767.88\right]$  $483.14$  $0.72$  
$0.5$  $\left[192.7,1066\right]$  $873.3$  $\left[256.70,807.45\right]$  $550.75$  $0.63$  
$0.1$  $\left[114.2,1431\right]$  $1316.8$  $\left[185.86,973.82\right]$  $787.96$  $0.60$  
$0.0$  $\left[98.36,1537.5\right]$  $1439.14$  $\left[98.36,1537.5\right]$  $1439.14$  1 
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Stanojević, B.; Nǎdǎban, S. Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator. Mathematics 2023, 11, 4864. https://doi.org/10.3390/math11234864
Stanojević B, Nǎdǎban S. Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator. Mathematics. 2023; 11(23):4864. https://doi.org/10.3390/math11234864
Chicago/Turabian StyleStanojević, Bogdana, and Sorin Nǎdǎban. 2023. "Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator" Mathematics 11, no. 23: 4864. https://doi.org/10.3390/math11234864