# A New Global Optimization Algorithm for a Class of Linear Fractional Programming

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## Abstract

**:**

## 1. Introduction

## 2. The Equivalence Problem of LFP

**Theorem**

**1.**

**Proof of Theorem**

**1.**

## 3. A New Linear Relaxation Technique

## 4. Branching Process

**Remark**

**1.**

**Remark**

**2.**

## 5. Output-Space Branch-and-Bound Algorithm and Its Convergence

**Step 1**. Set the tolerance $\u03f5>0$. Construct the initial hyper-rectangle ${H}^{0}=H=[\underline{t},\overline{t}]$. Solve the linear programming problem $LR{P}^{0}$ on super-rectangular ${H}^{0}$. The corresponding optimal solution and optimal value are recorded as $({x}^{0},{t}^{0})$ and $L\left({H}^{0}\right)$, respectively. Then, ${L}^{0}=L\left({H}^{0}\right)$ is the initial lower bound of the global optimal value of $LFP$. The initial upper bound is ${U}^{0}=f\left({x}^{0}\right)$. If ${U}^{0}-{L}^{0}\le \u03f5$, then stop, a $\u03f5$-global optimal solution ${x}^{*}={x}^{0}$ of problem $LFP$ is found. Otherwise, set $\Omega =\left\{{H}^{0}\right\}$, $F=\varnothing $, the initial iteration number $k=1$, and transfer to Step 2.

**Step 2**. If ${U}^{k}-{L}^{k}\le \u03f5$, then stop the iteration of the algorithm, output the current global optimal solution ${x}^{*}$ of the $LFP$ problem and the globally optimal value $f\left({x}^{*}\right)$; Otherwise, go to Step 3.

**Step 3**. The super-rectangle ${H}^{k}$, which corresponds to the current lower bound ${L}^{k}$, is selected, in $\Omega $, i.e., ${L}^{k}=L\left({H}^{k}\right)$.

**Step 4**. Using the rectangular branching process in Section 3, ${H}^{k}$ is divided into two sub-rectangles: ${H}^{k1}$ and ${H}^{k2}$ that satisfy ${H}^{k1}\cap {H}^{k2}=\varnothing $. For all $L\left({H}^{ki}\right)<{U}^{k}$, set $F=F\cup \left\{{H}^{ki}\right\}$, $Q=Q\cup \left\{{x}^{i}\right\}$$(i\in \{1,2\left\}\right)$. If $F=\varnothing $, go to Step 3. Otherwise, set $\Omega =\Omega \setminus {H}^{k}\cup F$, and continue.

**Step 5**. Let ${U}^{k}=min\{{U}^{k},min\{f\left(x\right):x\in Q\}\}$. If ${U}^{k}=min\{f\left(x\right):x\in Q\}$, the current optimal solution is ${x}^{*}\in argmin\{f\left(x\right),x\in Q\}$; Let ${L}^{k}=min\{L\left(H\right):H\in \Omega \}$; Set $k:=k+1$, $F=\varnothing $, $Q=\varnothing $, and go to Step 2.

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Theorem**

**2.**

**Proof of Theorem**

**2.**

**Theorem**

**3.**

**Proof of Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof of Theorem**

**5.**

**Remark**

**7.**

## 6. Numerical Examples

**Example**

**1**

**Example**

**2**

**Example**

**3**

**Example**

**4**

**Example**

**5**

**Example**

**6**

**Example**

**7**

**Example**

**8**

**Example**

**9**

**Example**

**10**

**Example**

**11**

**Example**

**12.**

**Example**

**13.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LFP | linear fractional programming |

EOP | equivalent nonlinear programming |

LRP | linear relaxation programming |

T | transpose of a vector or matrix |

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Example | Methods | ${\mathit{x}}^{*}$ | $\mathit{f}\left({\mathit{x}}^{*}\right)$ | $\mathit{Iter}$ | $\mathit{Time}$ | $\mathit{\u03f5}$ | |
---|---|---|---|---|---|---|---|

1 | [15] | (0.00,0.2817) | 1.6232 | 43 | 4.3217 | ${10}^{-8}$ | |

1 | [18] | (0.00,0.2839) | 1.6232 | 65 | 0.9524 | ${10}^{-8}$ | |

1 | $OSBBA$ | (0.00,0.2839) | 1.6232 | 1983 | 40.3528 | ${10}^{-8}$ | |

2 | [18] | (0.0,1.0) | 3.5750 | 1 | 0.0561 | ${10}^{-9}$ | |

2 | $OSBBA$ | (0.0,1.0) | 3.5750 | 12 | 0.1626 | ${10}^{-9}$ | |

3 | [17] | (5.0000,0.0000,0.0000) | 2.8619 | 16 | 0.1250 | ${10}^{-3}$ | |

3 | [21] | (5.0000,0.0000,0.0000) | 2.8619 | 12 | 28.2943 | ${10}^{-4}$ | |

3 | $OSBBA$ | (5.0000,0.0000,0.0000) | 2.8619 | 379 | 8.4163 | ${10}^{-4}$ | |

4 | [17] | (1.1111,0.0000,0.0000) | −4.0907 | 185 | 3.2510 | ${10}^{-2}$ | |

4 | [22] | (1.0715,0,0) | −4.0874 | 17 | - | ${10}^{-6}$ | |

4 | $OSBBA$ | (1.11111111,0,0) | −4.0907 | 70 | 1.8196 | ${10}^{-6}$ | |

5 | [17] | (0.0000,1.66666667,0.0000) | 3.7109 | 8 | 0.1830 | ${10}^{-3}$ | |

5 | [21] | (0.0000,1.6667,0.0000) | 3.7087 | 5 | 4.1903 | ${10}^{-4}$ | |

5 | [24] | (0.0000,0.625,1.875) | 4.0000 | 58 | 2.9686 | ${10}^{-4}$ | |

5 | $OSBBA$ | (0.0000,1.66666667,0.0000) | 3.7109 | 169 | 4.2429 | ${10}^{-6}$ | |

6 | [17] | (0,0.333333,0) | −3.0029 | 17 | 0.1290 | ${10}^{-3}$ | |

6 | [22] | (0,0.33329,0) | −3.0000 | 30 | ${10}^{-6}$ | ||

6 | $OSBBA$ | (0,0.333333,0) | −3.0029 | 2090 | 50.8226 | ${10}^{-6}$ | |

7 | [17] | (1.5000,1.5000) | 4.9125 | 56 | 1.0870 | ${10}^{-3}$ | |

7 | [21] | (1.5000,1.5000) | 4.9125 | 113 | 201.6260 | ${10}^{-4}$ | |

7 | $OSBBA$ | (1.5000,1.5000) | 4.9125 | 460 | 8.7944 | ${10}^{-4}$ | |

8 | [12] | (1.1111, 0, $-3.33067\times {10}^{-5}$) | 4.0907 | 3 | 0.0000 | ${10}^{-6}$ | |

8 | [14] | (1.11111, 0.00000, 0.00000) | 4.0907 | 2 | 0.0020 | ${10}^{-6}$ | |

8 | $OSBBA$ | (1.11111, 0.00000,0.00000) | 4.0907 | 42 | 1.1433 | ${10}^{-6}$ | |

9 | [12] | (3.0, 4.0) | 3.2916 | 9 | 0.0000 | ${10}^{-6}$ | |

9 | [14] | (3.0, 4.0) | 3.2916 | 2 | 0.0017 | ${10}^{-6}$ | |

9 | [23] | (3.0, 4.0) | 3.2916 | 78 | 0.0000 | ${10}^{-6}$ | |

9 | $OSBBA$ | (3.0,4.0) | 3.2916 | 693 | 16.5359 | ${10}^{-6}$ | |

10 | [14] | (5.0,0.0,0.0) | 4.4285 | 2 | 0.0018 | ${10}^{-6}$ | |

10 | [23] | (5.0,0.0,0.0) | 4.4285 | 35 | 0.0000 | ${10}^{-4}$ | |

10 | [24] | (0.0, 0.625, 1875) | 4.0000 | 58 | 2.9686 | ${10}^{-4}$ | |

10 | $OSBBA$ | (5.0,0.0,0.0) | 4.4285 | 61 | 1.6153 | ${10}^{-6}$ | |

11 | [18] | (0.0,3.3333,0.0) | 1.9000 | 8 | 0.1389 | ${10}^{-6}$ | |

11 | $OSBBA$ | (0.0,3.3333,0.0) | 1.9000 | 402 | 6.8145 | ${10}^{-6}$ |

**Table 2.**The results of random calculations for Example 12. Ave.Iter, the average number of iterations on problems 12–13; Ave.Time, the average CPU running time of problems 12–13; $SR$, the success rate of the algorithm in calculating problem 12.

p | m | n | $\mathit{OSBBA}$ | $\mathit{Reference}\phantom{\rule{3.33333pt}{0ex}}$ [18] | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | $\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | |||

5 | 30 | 30 | 3.3 | 0.1636 | 100% | 2.5 | 0.0866 | 100% |

5 | 50 | 50 | 4.2 | 0.2090 | 100% | 2.7 | 0.1818 | 100% |

5 | 100 | 100 | 4.9 | 0.3075 | 100% | 3.3 | 1.7679 | 100% |

10 | 30 | 30 | 7.7 | 0.4194 | 100% | 2.8 | 0.1440 | 100% |

10 | 50 | 50 | 8.2 | 0.6288 | 100% | 4.7 | 0.3852 | 100% |

10 | 100 | 100 | 12.8 | 0.8931 | 100% | 6.3 | 2.7120 | 100% |

5 | 50 | 100 | 5.1 | 0.2191 | 100% | 2.8 | 0.4551 | 100% |

5 | 100 | 300 | 18.4 | 2.3497 | 100% | 72.5 | 49.1801 | 100% |

5 | 200 | 500 | 16.7 | 11.1795 | 100% | 19.9 | 328.1308 | 100% |

10 | 50 | 100 | 480.7 | 11.4817 | 100% | 50.6 | 17.0826 | 100% |

10 | 100 | 300 | 818.5 | 95.7513 | 100% | 1064.4 | 836.2154 | 100% |

10 | 200 | 500 | 435.8 | 281.8541 | 100% | 1784.5 | 1126.2541 | 100% |

p | m | n | $\mathit{OSBBA}$ | $\mathit{Reference}$ [18] | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | $\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | |||

5 | 100 | 1000 | 7.5 | 5.7765 | 80% | – | – | 0% |

5 | 200 | 2000 | 14.0 | 96.2968 | 80% | – | – | 0% |

5 | 300 | 3000 | 18.5 | 555.5875 | 80% | – | – | 0% |

10 | 100 | 1000 | 19.2 | 127.5658 | 60% | – | – | 0% |

10 | 200 | 2000 | 32.1 | 649.6260 | 60% | – | – | 0% |

10 | 300 | 3000 | 39.4 | 964.1816 | 60% | – | – | 0% |

5 | 300 | 5000 | 6.1 | 594.2337 | 60% | – | – | 0% |

5 | 400 | 8000 | 2.3 | 785.0338 | 60% | – | – | 0% |

5 | 500 | 10,000 | 1.4 | 913.8774 | 40% | – | – | 0% |

10 | 300 | 5000 | 2.1 | 216.3840 | 40% | – | – | 0% |

10 | 400 | 8000 | 2.4 | 1095.3918 | 40% | – | – | 0% |

10 | 500 | 10,000 | 2.3 | 1180.6531 | 20% | – | – | 0% |

p | m | n | $\mathit{OSBBA}$ | $\mathit{BMIBNB}$ | ||
---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | |||

5 | 30 | 30 | 0.1911 | 100% | 0.0840 | 100% |

5 | 50 | 50 | 0.2106 | 100% | 0.1117 | 100% |

5 | 100 | 100 | 0.9178 | 100% | 0.7412 | 100% |

10 | 30 | 30 | 4.7010 | 100% | 2.2465 | 100% |

10 | 50 | 50 | 5.8519 | 100% | 3.9418 | 100% |

10 | 100 | 100 | 33.9510 | 100% | 11.5741 | 100% |

5 | 50 | 100 | 0.5447 | 100% | 6.9531 | 100% |

5 | 100 | 300 | 4.1197 | 100% | 78.3507 | 100% |

5 | 200 | 500 | 69.9101 | 100% | 403.4631 | 100% |

10 | 50 | 100 | 108.1924 | 100% | 10.4275 | 100% |

10 | 100 | 300 | 115.5843 | 100% | 149.4162 | 100% |

10 | 200 | 500 | 238.9797 | 100% | 806.4061 | 100% |

p | m | n | $\mathit{OSBBA}$ | $\mathit{BMIBNB}$ | ||
---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{SR}$ | |||

5 | 100 | 1000 | 38.6369 | 100% | – | 0% |

5 | 200 | 2000 | 161.0308 | 80% | – | 0% |

5 | 300 | 3000 | 250.7251 | 80% | – | 0% |

10 | 100 | 1000 | 167.5179 | 80% | – | 0% |

10 | 200 | 2000 | 369.3976 | 60% | – | 0% |

10 | 300 | 3000 | 603.8334 | 20% | – | 0% |

5 | 300 | 5000 | 484.2731 | 60% | – | 0% |

5 | 400 | 8000 | 697.9532 | 60% | – | 0% |

5 | 500 | 10,000 | 907.1948 | 40% | – | 0% |

10 | 300 | 5000 | 716.1021 | 40% | – | 0% |

10 | 400 | 8000 | 1105.1437 | 40% | – | 0% |

10 | 500 | 10,000 | 1177.9841 | 20% | – | 0% |

m | n | p | $\mathit{OSBBA}$ | $\mathit{BARON}$ | ||
---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | |||

5 | 10 | 2 | 36 | 1.1612 | 2 | 0.7473 |

5 | 20 | 2 | 79 | 1.9147 | 4 | 0.6075 |

5 | 30 | 2 | 114.5 | 2.7861 | 9 | 0.6482 |

5 | 50 | 2 | 75 | 1.9192 | 5 | 0.6511 |

5 | 70 | 2 | 286 | 5.9812 | 14 | 0.7949 |

5 | 90 | 2 | 67.5 | 1.6454 | 13 | 0.8312 |

5 | 100 | 2 | 156.5 | 3.0462 | 6 | 0.7616 |

5 | 10 | 3 | 203.5 | 5.1221 | 7 | 0.5021 |

5 | 20 | 3 | 624.5 | 14.6705 | 9 | 0.6231 |

5 | 30 | 3 | 246 | 6.9570 | 58 | 0.9305 |

5 | 50 | 3 | 466 | 11.2603 | 16 | 0.9116 |

5 | 70 | 3 | 97 | 4.0403 | 4 | 0.7513 |

5 | 90 | 3 | 815.4 | 15.5686 | 638.8 | 3.5839 |

5 | 100 | 3 | 753 | 13.8953 | 26.2 | 0.9445 |

5 | 10 | 4 | 896 | 18.8903 | 197 | 0.7966 |

5 | 10 | 5 | 138.5 | 3.4184 | 1 | 0.5202 |

5 | 10 | 6 | 245.5 | 5.6823 | 3 | 0.4332 |

5 | 20 | 4 | 472.5 | 9.0087 | 4 | 0.4629 |

5 | 20 | 5 | 269.5 | 5.8828 | 4 | 0.6038 |

5 | 20 | 6 | 4482 | 96.0794 | 90 | 0.8938 |

5 | 30 | 4 | 140.5 | 3.2252 | 3 | 0.4684 |

5 | 30 | 5 | 205 | 4.7040 | 3 | 0.4701 |

5 | 30 | 6 | 1404.5 | 31.7456 | 18 | 0.6425 |

10 | 30 | 5 | 2401 | 50.5017 | 32 | 0.6387 |

10 | 50 | 5 | 444 | 9.6959 | 3 | 0.5758 |

10 | 70 | 5 | 777 | 16.9510 | 36 | 0.9000 |

10 | 90 | 5 | 2187.5 | 49.8752 | 595 | 5.9809 |

10 | 100 | 2 | 64.4 | 1.2717 | 7.2 | 0.7844 |

10 | 100 | 3 | 223.1 | 4.4559 | 14.2 | 0.9466 |

10 | 100 | 4 | 594.2 | 12.1786 | 9651.2 | 53.4707 |

10 | 200 | 4 | 2009.2 | 44.0950 | 4363 | 49.0320 |

10 | 100 | 5 | 4414 | 88.7048 | 30,905.4 | 205.4919 |

10 | 200 | 5 | 3899 | 91.1614 | 35,481 | 502.3583 |

10 | 300 | 5 | 6591 | 176.2488 | 22,469 | 505.1200 |

20 | 300 | 5 | 1668.2 | 45.0485 | 14,140.1 | 307.2198 |

20 | 500 | 5 | 5091.8 | 165.9364 | 4878.2 | 264.4288 |

20 | 500 | 6 | 14,975.1 | 577.1660 | 6854.1 | 416.4063 |

20 | 500 | 7 | 62,930.1 | 3012.5028 | 7133.7 | 487.2844 |

20 | 500 | 8 | 3570 | 179.9689 | 36 | 62.7567 |

20 | 500 | 9 | 75,222 | 3909.7296 | 6740.5 | 637.1516 |

20 | 500 | 10 | 30,274 | 1504.8824 | 32 | 74.1234 |

m | n | p | $\mathbf{OSBBA}$ | $\mathit{BARON}$ | ||
---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | |||

50 | 300 | 2 | 20 | 1.1057 | 5 | 4.8298 |

50 | 300 | 3 | 27 | 2.9405 | 5 | 5.6102 |

50 | 300 | 4 | 105 | 5.2798 | 25 | 28.8512 |

50 | 500 | 2 | 16 | 1.8027 | 3 | 12.2832 |

50 | 500 | 3 | 51 | 2.6929 | 17 | 23.8195 |

50 | 500 | 4 | 68 | 8.9815 | 4381 | 356.2157 |

50 | 700 | 2 | 37 | 3.8667 | 11 | 67.7368 |

50 | 700 | 3 | 58 | 4.9608 | 7738 | 1002.5904 |

50 | 700 | 4 | 41 | 12.7235 | 3 | 51.0155 |

50 | 900 | 2 | 36 | 5.3660 | 7 | 108.5627 |

50 | 900 | 3 | 34 | 7.27312 | 3 | 142.4374 |

50 | 900 | 4 | 40 | 14.9773 | 3 | 178.2181 |

50 | 1000 | 2 | 46 | 10.6561 | 1 | 131.5696 |

50 | 1000 | 3 | 82 | 22.04153 | 3 | 162.2937 |

50 | 1000 | 4 | 636 | 105.3631 | 19 | 315.2514 |

50 | 2000 | 2 | 50 | 36.0024 | 1 | 779.5139 |

50 | 2000 | 3 | 169 | 67.2509 | 6 | 1007.5297 |

50 | 2000 | 4 | 436 | 113.1053 | 29 | 1008.2303 |

50 | 3000 | 2 | 77 | 82.6399 | 3 | 304.8984 |

50 | 3000 | 3 | 123 | 96.3323 | – | – |

50 | 4000 | 2 | 42 | 85.5768 | – | – |

50 | 4000 | 3 | 156 | 129.1936 | – | – |

50 | 5000 | 2 | 81 | 93.9734 | – | – |

50 | 2000 | 5 | 370 | 215.7799 | – | – |

50 | 2000 | 6 | 1274 | 283.4462 | – | – |

50 | 2000 | 7 | 2867 | 366.5079 | – | – |

50 | 2000 | 8 | 3032 | 414.5711 | – | – |

m | n | p | $\mathit{OSBBA}$ | $\mathit{BARON}$ | ||
---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | |||

100 | 300 | 3 | 19.3 | 4.1248 | 2 | 6.7865 |

100 | 300 | 5 | 105 | 6.6735 | 3.8 | 11.3750 |

100 | 300 | 8 | 676.4 | 41.5368 | 6.8 | 14.4418 |

100 | 500 | 3 | 21.9 | 5.9263 | 3.4 | 20.4406 |

100 | 500 | 5 | 162.3 | 16.4179 | 6.8 | 35.2466 |

100 | 500 | 8 | 2812.1 | 235.0307 | 13.4 | 69.6685 |

150 | 500 | 3 | 15.9 | 7.3081 | 2.4 | 21.4519 |

150 | 500 | 5 | 58.1 | 8.3569 | 3 | 35.9739 |

150 | 500 | 8 | 694.4 | 80.6513 | 7.2 | 51.4611 |

200 | 700 | 3 | 9.3 | 2.7830 | 2 | 47.8042 |

200 | 700 | 5 | 45.2 | 10.8633 | 3.2 | 84.6023 |

200 | 700 | 8 | 118.7 | 40.3079 | 4.2 | 122.5232 |

300 | 700 | 3 | 4.6 | 1.8944 | 1.6 | 60.0400 |

300 | 700 | 5 | 10.5 | 5.38701 | 1.8 | 78.3503 |

300 | 700 | 8 | 91.3 | 28.5988 | 4 | 119.6601 |

300 | 900 | 3 | 2.1 | 1.8728 | 1.2 | 93.1878 |

300 | 900 | 5 | 24.5 | 10.8376 | 3.2 | 158.5425 |

300 | 900 | 8 | 549 | 175.6600 | 3.8 | 231.1125 |

300 | 1000 | 3 | 9.1 | 6.8319 | 2.2 | 131.6551 |

300 | 1000 | 5 | 19.1 | 15.6941 | 2.2 | 144.4356 |

300 | 1000 | 8 | 98.3 | 85.0639 | 3.2 | 220.8366 |

m | n | p | $\mathit{OSBBA}$ | $\mathit{BARON}$ | ||
---|---|---|---|---|---|---|

$\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | $\mathit{Ave}.\mathit{Iter}$ | $\mathit{Ave}.\mathit{Time}$ | |||

500 | 1000 | 3 | 1.5 | 3.4675 | 1 | 99.9415 |

500 | 1000 | 5 | 7.2 | 9.7535 | 2.2 | 147.3751 |

500 | 1000 | 8 | 24.7 | 30.0108 | 2.8 | 187.4586 |

500 | 2000 | 3 | 3 | 16.0995 | 1.7 | 496.0448 |

500 | 2000 | 5 | 6.9 | 23.9408 | 2.6 | 741.5934 |

500 | 2000 | 8 | 22.3 | 93.7039 | 2.7 | 861.7061 |

500 | 3000 | 3 | 3.5 | 26.4755 | 3.0 | 920.6775 |

500 | 3000 | 5 | 4.5 | 56.1823 | – | – |

500 | 3000 | 8 | 30 | 125.0527 | – | – |

500 | 4000 | 3 | 2.4 | 54.7006 | – | – |

500 | 4000 | 5 | 9 | 136.5505 | – | – |

500 | 4000 | 8 | 10.8 | 196.5768 | – | – |

500 | 5000 | 3 | 2 | 71.7622 | – | – |

500 | 5000 | 5 | 5 | 126.6783 | – | – |

500 | 5000 | 8 | 17.3 | 330.9163 | – | – |

500 | 6000 | 3 | 1.8 | 72.6898 | – | – |

500 | 7000 | 3 | 2.6 | 238.4842 | – | – |

500 | 9000 | 3 | 2.3 | 209.8106 | – | – |

500 | 10,000 | 3 | 3.7 | 319.7487 | – | – |

© 2019 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**

Liu, X.; Gao, Y.L.; Zhang, B.; Tian, F.P.
A New Global Optimization Algorithm for a Class of Linear Fractional Programming. *Mathematics* **2019**, *7*, 867.
https://doi.org/10.3390/math7090867

**AMA Style**

Liu X, Gao YL, Zhang B, Tian FP.
A New Global Optimization Algorithm for a Class of Linear Fractional Programming. *Mathematics*. 2019; 7(9):867.
https://doi.org/10.3390/math7090867

**Chicago/Turabian Style**

Liu, X., Y.L. Gao, B. Zhang, and F.P. Tian.
2019. "A New Global Optimization Algorithm for a Class of Linear Fractional Programming" *Mathematics* 7, no. 9: 867.
https://doi.org/10.3390/math7090867