# The Optimization of Cyclic Links of Live Pig-Industry Chain Based on Circular Economics

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## Abstract

**:**

## 1. Introduction

## 2. Circular Economic System for Live Pig Industry

## 3. Cyclic Model for Live Pig-Industry Chain

#### 3.1. Problem Description

#### 3.2. Symbol Description

$X$ | Planting capacity (The unit is kg.) |

$Y$ | Production capacity of pig-breeding industry. (The unit is kg.) |

$Z$ | Production capacity of pig-slaughtering industry. (The unit is kg.) |

$W$ | Capacities of other related breeding industries. (The unit is kg.) |

${a}_{1}$ | Water demand of agricultural products in planting industry. (The unit is liter per kg.) |

${a}_{2}$ | Water demand of pig-breeding units. (The unit is liter per kg.) |

${a}_{3}$ | Water demand of pig-slaughtering units. (The unit is liter per kg.) |

${a}_{4}$ | Water demand of other related breeding units. (The unit is liter per kg.) |

$b$ | Feed demand of pig-breeding units, that is, the pig-breeding coefficient after planted products become feed. (The unit is kg.) |

$c$ | Feed demand of other breeding units, that is, the other breeding coefficient after planted products become feed. (The unit is kg.) |

$d$ | Pig demand of slaughtering units to produce pork products. (The unit is kg.) |

$e$ | Manure demand of planting units to produce agricultural products, that is, the input/output coefficient of waste manure to planting industry. (The unit is kg.) |

$f$ | Feed demand of other breeding units, that is, the breeding coefficient of other breeding industries after the waste becomes feed. (The unit is kg.) |

$g$ | Pig-slaughter capacity needed to meet the market demands. (The unit is kg.) |

$\mathsf{\alpha}$ | Average amount of excrement produced by pig-breeding units. (The unit is kg.) |

${\mathsf{\beta}}_{1}$ | Average amount of wastewater produced by pig-slaughtering units. (The unit is liter.) |

${\mathsf{\beta}}_{2}$ | Average amount of waste produced by pig-slaughtering units. (The unit is kg.) |

${\mathsf{\gamma}}_{1}$ | Conversion rate of wastewater treated to become reclaimed water. |

${\mathsf{\gamma}}_{2}$ | Conversion rate of waste treated to become manure. |

${\mathsf{\gamma}}_{3}$ | Conversion rate of waste treated to become feed. |

${Q}_{1}$ | Wastewater-treatment capacity. (The unit is liter.) |

${Q}_{2}$ | Waste-disposal capacity. (The unit is kg.) |

${v}_{1}$ | Construction cost of wastewater-treatment unit capacity. (The unit is yuan per liter.) |

${v}_{2}$ | Construction cost of waste-disposal unit capacity. (The unit is yuan per kg.) |

${\mathsf{\eta}}_{1}$ | Rated target utilization rate of wastewater-treatment capacity. |

${\mathsf{\eta}}_{2}$ | Rated target utilization rate of waste-disposal capacity. |

${q}_{1}$ | Use amount of circulating water in planting industry. (The unit is liter.) |

${q}_{2}$ | Use amount of circulating water in pig-breeding industry. (The unit is liter.) |

${q}_{3}$ | Use amount of circulating water in pig-slaughtering industry. (The unit is liter.) |

${q}_{4}$ | Use amount of circulating water in other livestock-breeding industries. (The unit is liter.) |

${q}_{5}$ | Use amount of recycling manure in planting industry. (The unit is kg.) |

${q}_{6}$ | Use amount of recycling feed in other breeding industries. (The unit is kg.) |

${\mathsf{\xi}}_{1}$ | Market demand for agricultural products. The random variables and distribution function can be determined by the empirical distribution of historical data. |

${\mathsf{\xi}}_{2}$ | Market demand for pig-slaughtering products. The random variables and distribution function can be determined by the empirical distribution of historical data. |

${\mathsf{\xi}}_{3}$ | Market demand for other breeding industries. The random variables and distribution function can be determined by the empirical distribution of historical data. |

#### 3.3. Model Construction

#### 3.4. Model Solution

**The first step:**Determine the input and output variables of the uncertainty function. According to constraints in Formulas (5)–(8) associated with the processing capacity, construct the uncertainty function as follows [41]:

**The second step:**Generate the input and output samples of the uncertainty function. By random simulation, generate input and output data for the uncertain function and form the training sample. The specific process is as follows: According to the largest values of market demand in previous years, determine the value ranges of ${Q}_{1}$ and ${Q}_{2}$; by random simulation, generate input and output data for the uncertain function within their value ranges. Take ${Q}_{1}$ as an example: After randomly generating ${Q}_{1}$, generate random simulation values according to market-demand distribution ${\mathsf{\xi}}_{1}$, ${\mathsf{\xi}}_{2}$, and ${\mathsf{\xi}}_{3}$; place them into the uncertainty function to calculate ${\mathsf{\xi}}_{2}g{\mathsf{\beta}}_{1}$ and to compare it with ${Q}_{1}$; count the ratio of the ${Q}_{1}\ge {\mathsf{\xi}}_{2}g{\mathsf{\beta}}_{1}$ times in total simulations; and take this ratio as the corresponding output result of input ${Q}_{1}$.

**The third step:**Train the neural network to approximate each uncertainty function. According to the training sample generated, train a neural network to approximate the above uncertain function.

**The fourth step:**Produce the initial population of the outer genetic algorithm. Initially generate $N$ chromosomes $\left({Q}_{1},{Q}_{2}\right)$ as the initial population, and use the trained neural network to test the feasibility of the chromosome. Calculate each viable ${F}_{2}$ and ${F}_{4}$ corresponding to the chromosome, and take the smallest ${F}_{2}$ and ${F}_{4}$ of all chromosomes as the initial ${F}_{2}^{*}$ and ${F}_{4}^{*}$.

**The fifth step:**Calculate the corresponding objective function values of each chromosome of the initial population by constructing an inner random expectation model. For each feasible chromosome $\left({Q}_{1},{Q}_{2}\right)$, solve the random expectation planning. Taking ${Q}_{1}$ as an example, construct the random expectation planning as follows:

- (1)
- Construct the uncertainty function as follows:$${U}_{5}:\left({q}_{1},{q}_{2},{q}_{3},{q}_{4}\right)\to E\left(\mathrm{min}\left(\frac{{q}_{1}+{q}_{2}+{q}_{3}+{q}_{4}}{\left(c{\mathsf{\xi}}_{3}+b{\mathsf{\xi}}_{2}gd+{\mathsf{\xi}}_{1}\right){a}_{1}+\left({\mathsf{\xi}}_{2}gd\right){a}_{2}+\left({\mathsf{\xi}}_{2}g\right){a}_{3}+{\mathsf{\xi}}_{3}{a}_{4}},1\right)\right)$$
- (2)
- Generate input and output data for the uncertainty function. Generate the input and output data, of which the input data are produced as below: Randomly generate ${q}_{1}$ in accordance with the uniform distribution in $[0,{Q}_{1}{\mathsf{\gamma}}_{1}]$. Then, randomly generate ${q}_{2}$ in accordance with the uniform distribution in $[0,{Q}_{1}{\mathsf{\gamma}}_{1}-{q}_{1}]$. Similarly, randomly generate ${q}_{3}$ in accordance with the uniform distribution in $[0,{Q}_{1}{\mathsf{\gamma}}_{1}-{q}_{1}-{q}_{2}]$, and finally, calculate ${q}_{4}={Q}_{1}{\mathsf{\gamma}}_{1}-{q}_{1}-{q}_{2}-{q}_{3}$. The corresponding output data are the expectations for the resource-utilization rate. Generate random samples according to market-demand distribution; then, calculate the corresponding uncertain function values of these samples. Finally, average the values according to the number of samples to obtain the output data of the input data under the group.
- (3)
- Use the input and output data to train the neural network to approximate ${U}_{5}$. Utilize the input and output data to train the neural network.
- (4)
- Randomly generate the initial population of the genetic algorithm in the inner layer. Randomly generate sub-chromosomes to form the initial population according to step (2), and use the trained neural network to test its feasibility.
- (5)
- Crossover and mutation operation of inner-layer genetic algorithm. Conduct cross-operation and mutation operation on dub-chromosomes, generate new chromosomes, and use the trained neural network to test its feasibility.
- (6)
- Calculate the target value of the genetic algorithm in the inner layer. Calculate the corresponding objective value of each sub-chromosome, and calculate the fitness of each sub-chromosome according to the objective value.
- (7)
- Roulette selection of sub-chromosome for inner-layer optimization and iteration. Select the sub-chromosomes with roulette. Repeat (5)–(6) until the pre-set optimal number of cycles is reached.
- (8)
- Output the optimal solution of the inner genetic algorithm. Obtain the corresponding sub-chromosome of the optimal objective values.

**The sixth step:**Calculate the final objective function value. After obtaining the optimal recycling resource-distribution schemes of given ${Q}_{1}$ and ${Q}_{2}$, combine them with the results of the fourth step to calculate the fitness of the comprehensive evaluation, that is, Formula (13).

**The seventh step:**Calculate the target value of the genetic algorithm in the outer layer: Conduct cross-operation and mutation operation to produce new chromosomes, and use a trained neural network to test the feasibility of chromosomes. Cross-operation operators are composed of (1) the summation of two corresponding chromosomes to produce new chromosomes; (2) the averaging of two corresponding chromosomes to produce new chromosomes; (3) the differential of two corresponding chromosomes to produce new chromosomes; and (4) the alternation of two corresponding chromosomes to produce new chromosomes. The four types of operations are random. The mutation operation is (1) add or subtract a small random number on each part of a chromosome; and (2) add or subtract a larger random number on each part of a chromosome. The mutation operations are also random.

**The eighth step:**Calculate the fitness function of the outer genetic algorithm. Calculate the objective values of all chromosomes, and according to the objective values, calculate the fitness.

**The ninth step:**Roulette selection of sub chromosome for outer-layer optimization and iteration. Choose the chromosomes with roulette to repeat steps five to eight, until the pre-set optimal number of cycles is reached.

**The tenth step:**Output the optimal solution and the optimal objective function value. Output the corresponding chromosomes of the optimal objective value.

## 4. Case Analysis

#### 4.1. Case Design

^{3}, and the yield per mu is approximately 600 kg. Therefore, the average water use for the production of one kg of corn is ${a}_{1}=\frac{1}{150}$. Because the cycle of corn planting is approximately 100 days, to ensure that there is enough feed for pig breeding, the planting scale should be 100 × daily supply. In terms of manure, a complete cycle of planting requires approximately 40 kg of manure for each mu, with an average daily manure amount per kilogram of $e=\frac{40}{600\times 100}=\frac{1}{1500}$.

^{3}. Calculated as 120 kg per live pig, the average water demand per kg per pig is ${a}_{2}=\frac{127}{1,200,000}$. The pig-breeding cycle is approximately 150 days to ensure that a sufficient number of pigs be supplied daily to the slaughterhouses. The feeding size should be a 150 × daily supply. Each pig can produce 2.17 kg of excrement per day. Calculated as 120 kg per pig, the pig breeding can produce an average of $\mathsf{\alpha}=\frac{217}{12,000}$ per kilogram. There exists a certain rate of incidence and mortality in the pig-breeding process, so the real ratio of supply to slaughterhouses is approximately 96%, that is, $d=\frac{1}{96\%}=\frac{25}{24}$.

^{3}. Calculated by the weight of 120 kg per live pig, the average wastewater coefficient produced per kg is ${\mathsf{\beta}}_{1}=\frac{1}{200}$. After slaughtering, approximately 80% can eventually become consumer products; that is, to provide 1 unit of product to the market, the slaughtered pig amount is $g=1.25$. The proportion of waste generated in the process of slaughtering is approximately 20%, that is ${\mathsf{\beta}}_{2}=\frac{1}{5}$. In the process of pig slaughterhouse operation, the ratio of using recycling water is approximately 40%, the average recycling water demand for each slaughtered pig is 0.48 m

^{3}, and the average recycling water available per kg for pig slaughtering is ${a}_{3}=\frac{1}{625}$.

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

a_{1} | 1/150 | g | 1.25 | p_{1} | 0.95 |

a_{2} | 127/1,200,000 | α | 217/12,000 | p_{2} | 0.95 |

a_{3} | 1/625 | β_{1} | 1/200 | p_{3} | 0.90 |

a_{4} | 1/180 | β_{2} | 1/5 | p_{4} | 0.90 |

b | 2.5 | γ_{1} | 0.80 | ξ_{1} | N(8000,128) |

c | 3 | γ_{2} | 0.90 | ξ_{2} | N(96000,500) |

d | 25/24 | γ_{3} | 0.56 | ξ_{3} | N(6000,138) |

e | 1/1500 | η_{1} | 0.95 | ||

f | 2.2 | η_{2} | 0.90 |

#### 4.2. Results

Variable | Value | Variable | Value |
---|---|---|---|

${Q}_{1}$ | 591.73 m^{3} | ${Q}_{2}$ | 23,155.90 kg |

${q}_{1}$ | 186.34 m^{3} | ${q}_{2}$ | 71.12 m^{3} |

${q}_{3}$ | 207.25 m^{3} | ${q}_{4}$ | 45.33 m^{3} |

${q}_{5}$ | 83.62 kg | ${q}_{6}$ | 9639.10 kg |

^{3}and waste disposal capacity of 23,155.90 kg can meet the needs of water recycling, recycling feeds and manures. After treatment, 186.34 m

^{3}of the recycled water will be assigned to the planting industry, 71.12 m

^{3}will be assigned to the pig-breeding industry, 207.25 m

^{3}will be assigned to the pig-slaughtering industry, and 45.33 m

^{3}will be assigned to other livestock-breeding industries. After treatment the waste will be converted to 83.62 kg manures to be used in the planting industry, the others will be converted into 9639.10 kg feeds, for use in other breeding industries.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Liu, X.; Xiao, X.
The Optimization of Cyclic Links of Live Pig-Industry Chain Based on Circular Economics. *Sustainability* **2016**, *8*, 26.
https://doi.org/10.3390/su8010026

**AMA Style**

Liu X, Xiao X.
The Optimization of Cyclic Links of Live Pig-Industry Chain Based on Circular Economics. *Sustainability*. 2016; 8(1):26.
https://doi.org/10.3390/su8010026

**Chicago/Turabian Style**

Liu, Xing, and Xu Xiao.
2016. "The Optimization of Cyclic Links of Live Pig-Industry Chain Based on Circular Economics" *Sustainability* 8, no. 1: 26.
https://doi.org/10.3390/su8010026