Overview

Considering the needs of human practice, we used the method of mathematical model to make the processing devices in the project placed reasonably in the community.

Whether the device is reasonably placed in the community depends on the structural characteristics of the community, the population of the community, the maximum distance residents can accept when throwing drugs and other factors. Lacking consideration of these factors will result in unreasonable placement of the devices, low utilization of the devices and waste of resources. Therefore, we established a mathematical model to scientifically place the processing devices.

Inspired by the problem of layout of garbage cans in our daily life, we are ready to place the processing devices near the garbage cans to increase the possibility of residents throwing expired drugs into the processing devices. In order to place the processing device reasonably, we need to keep garbage cans and processing devices as close as possible. Considering the unscientific placement of the trash cans[1], it is necessary to place trash cans firstly.
Considering the lack of scientificity in the layout scheme of garbage cans nowadays, we took 50 meters as the maximum coverage radius, and then set up a multi-objective optimization model based on the interests of maintenance personnel, residents and us. We took the vertical distance between the garbage can and the edge of the main road, the closest distance between the residents and the garbage can, and the cost as the optimization objective, and took the capacity of the garbage can as the constraint condition. As a result, the problem of dustbins' distribution in the community is solved scientifically.

Then we improved the dustbin layout model and established the device layout model. Considering the maximum acceptable distance of residents of different age groups and different educational levels, a formula for calculating the coverage radius of the device was established. We changed the constraint condition to
make the needs of the residents satisfied, and required that the distance between the device and the dustbin should be less than or equal to 20 meters. Then the model of processing device layout was established. Then the distribution result of the garbage can and the device is fine-tuned to make the processing devices as close as possible to the garbage bin. Finally, we got our optimal layout scheme, which promotes the human practice. The final step in the application of our project, device layout, can be completed.

Model Assumptions

1. There is no unoccupied housing in the community.

2. All processing devices work properly.

3. The number of garbage removals in the community is three times a day.

2. All processing devices work properly.

3. The number of garbage removals in the community is three times a day.

Symbol and variable description

Symbol | Description |
---|---|

R | Number of groups of dustbins |

R_{i} |
Number of dustbins in group i |

W_{i} |
Weight of garbage produced within the coverage of group i per day |

ρ_{i} |
Number of households covered by group i of dustbins |

P_{i} |
The closest distance between the main road and the dustbin i |

p_{i} |
The i_{th} group of dustbins coverage range |

M | Total number of units in the community |

m_{i} |
Number of units covered by the i_{th} group of dustbins |

Q_{i} |
The vertical distance from the i_{th} device to the main road |

q_{i} |
Distance from the i_{th} unit to the nearest device |

x_{i} |
Total number of households covered by the i_{th} device |

N | Total number of devices |

L_{ij} |
Straight line distance between the i_{th} device and the j_{th} device |

L_{min} |
Minimum distance between two devices |

s_{i} |
The distance from the i_{th} device to the nearest dustbin |

\(\bar{s}\) | Average value of the distance from each device to the nearest dustbin |

\(\bar{d}\) | Average value of the distance from total units to the nearest device |

η_{i} |
Utilization of the i_{th} device |

n_{i} |
Number of units covered by the i_{th} device |

the Values of Parameter

Symbol | Description | Value | Units |
---|---|---|---|

W | Average daily garbage production per person in Chengdu | 1.04 | kg/d |

α | Average number of people per household | 2.57 | person/household |

ω | Frequency of the dustbin clearance | 3 | times/day |

ε | Garbage density | 0.2 | t/m^{3} |

φ | Filling coefficient of the dustbin | 0.8 | |

B | Volume of the dustbin | 240 | L |

A_{1} |
Daily garbage discharge weight unevenness coefficient | 1.1 | |

A_{2} |
Residential population coefficient of change | 1.05 | |

B_{1} |
Monthly discharge weight unevenness coefficient of abandoned drugs | 1.1 | |

m | Ratio of maximum person flow to average person flow | 1.2 | |

β | Contribution of the 0-14 years old population to the determination of the maximum coverage radius | 0.1 | |

v | Contribution of the 15-30 years old population to the determination of the maximum coverage radius | 0.2 | |

γ | Contribution of the 31-64 years old population to the determination of the maximum coverage radius | 0.3 | |

κ | Contribution of the population over 65 years of age to the determination of the maximum coverage radius | 0.4 | |

A | Processing capacity of each device | 1.1 | g/h |

F | Average amount of discarded antibiotics per household | 1.8 | g/month |

L_{min} |
Minimum distance between two devices | 20 | m |

Model establishment

## 5.1 Preparation before modeling：

### 5.1.1 Calculation of the number of the i_{th} group of dustbins

\[
R_{i}=\frac{W_{i}}{\omega\epsilon\varphi B}
\]
\[
W_{i}=\frac{mA_{1}\alpha\rho_{i}W}{A_{2}}
\]

R

_{i}is the number of dustbins in group i. W_{i}is weight of garbage produced within the coverage of group i per day. ω is the frequency of the dustbin clearance. ε is garbage density. Considering the type of different garbage, ε usually takes 0.2 t/m^{3}.φ is the filling coefficient of the dustbin which indicates that the clearing is the filling level of the trash can. φ usually takes 0.75~0.9 [2]. B is the volume of the dustbin. α is the average number of people per household, according to the survey, taking 2.57 people per household. ρ_{i}is the number of households covered by the i_{th}group of dustbins. W is the average daily garbage production per person in Chengdu. A_{1}is daily garbage discharge weight unevenness coefficient. A_{2}is residential population coefficient of change. m is the ratio of maximum person flow to average person flow.
The number of garbage containers at the garbage collection point is related to the number of people living in the collection area, the daily discharge of domestic garbage and frequency of the dustbin clearance. The amount of drug produced within the coverage is related to daily drug produced, the resident population and the volatility of the flow of people [1].

### 5.1.2 Determination of Average amount of discarded antibiotics per household F

There are 348 million 370 thousand households in 34 provinces, autonomous regions and municipalities directly under the central government producing 15 thousand tons of waste drugs annually. China requires that the use rate of antibiotics in drug prescriptions should not exceed 50%. On average, 3.6g of abandoned drugs are produced per household per month, and 1.8g of abandoned antibiotics are produced per household per month.

### 5.1.3 Determination of the processing capability of each device A

Since we has only studied engineered bacteria for CIP now, we are unable to obtain processing capability of other antibiotic drugs, so we estimated the processing capacity of the device.
We used the degradation rate of CIP by CrpP as a criterion for determining the processing capacity of the device. Then we can improve its processing ability subsequently.According to the literature [3], the maximum rate of degradation of ciprofloxacin by 5 ug/ml CrpP enzyme is 2.071 umol/min. According to the specific conditions of our device, the maximum enzyme concentration of all the engineered bacteria in the device is 135mg/L. So we calculated that the corresponding processing rate is 1.1g/h.

## 5.2 Model establishment：

### 5.2.1 Establishment of the dustbin layout model

### 5.2.1.1 Determination of the i_{th} group of dustbins coverage p_{i}

The city is divided into four areas based on the function of the region, which are residential areas, cultural areas, commercial areas and urban main roads. Due to the different composition and educational levels of population in different regions, the results of the survey are different, which is the maximum distance for residents' satisfaction when throwing rubbish[4].The characteristics of the residential area are the complex population age, the high proportion of the elderly and many family activities. The residents do not want to take up too much time to throw garbage deliberately, so the residents have little tolerance for carrying garbage (the same is true for drug delivery). According to the survey literature [4], the coverage of the i

_{th}group of dustbins is p_{i}≤50m.### 5.2.1.2 Dustbin layout optimization model

(1) optimization objective: Cover all units at minimal cost, and the sum of the vertical distance of the dustbin to the main road is minimized, so that the cleaning personnel can clean the garbage can conveniently.

(2) Constraint condition: Considering the amount of generated garbage, the number of dustbins in a group must be able to handle the amount of garbage generated within the coverage of a group of dustbins. Considering the satisfaction of residents, the covering radius of the i

_{th}group of dustbins is less than or equal to 50 meters. The total number of units covered by all dustbins must be greater than or equal to the total number of units in the community.
Objective function:
\[
min\sum_{i=1}^{R}R_{i}
\]
\[
min\sum_{i=1}^{R}P_{i}
\]
s.t.
\[
W_{i}\leq R_{i}\omega\epsilon\varphi B
\]
\[
p_{i}\leq 50
\]
\[
\sum_{i=1}^{R}m_{i}-M\geq 0
\]

R

_{i}is the number of dustbins in i_{th}group . R is the number of groups of dustbins. P_{i}is the closest distance between the main road and the dustbin i. m_{i}is the number of units covered by the i_{th}group of dustbins. M is the total number of units in the community.### 5.2.2 Establishment of processing device layout model

### 5.2.2.1 Determination of maximum coverage radius of the device d_{max}

The degree of satisfaction with the placement of the device depends on the distance between the device and the residences. For the requirements of distance, the requirements of people of different ages and different educational levels are different. In order to weigh the needs of different residents for the placement of devices, the proportion of people in each age group and the proportion of education in the corresponding age groups should be considered. Then the maximum coverage radius calculation formula is constructed.

\begin{equation}
\begin{split}
d_{max}=[\beta(p_{1}d_{1}+p_{2}d_{2}+p_{3}d_{3})
+v(q_{1}d_{4}+q{2}d_{5}+q_{3}d_{6})\\
+\gamma(t_{1}d_{7}+t_{2}d_{8}+t_{3}d_{9})+k(k_{1}d_{10}+k_{2}d_{11}+k_{3}d_{12})]
\end{split}
\end{equation}

β、v、γ、κ are the contributions of different age groups to the determination of the maximum coverage radius. p

_{i}、q_{i}、t_{i}、k_{i}are the proportions of residents with different educational levels in the four age groups. d_{i}is the average value of maximum distance that residents of different ages and education levels can accept. We obtained the proportion of residents with different educational levels in the four age groups and the average acceptable maximum distance of residents with different educational levels in different age groups from the questionnaire. For the elderly over 65 years old, the total amount of medicines produced is large and it is not convenient for them to walk. Therefore, we stipulated that the contribution rate of the elderly is the largest. We defined the contributions of 0-14, 15-30, 31-64, and over 65 years old residents to the maximum coverage radius are 0.1, 0.2, 0.3, and 0.4, respectively.### 5.2.2.2 Device layout optimization model

(1) optimization objective: Cover all units at minimal cost, and the deployment of the device should be as close as possible to the main road (facilitate regular replacement of bacteria in the equipment).

(2) Constraint condition: Considering the aesthetics of the layout, the distance between the two devices should be greater than the specified minimum spacing. The total number of units that are covered by all devices must be greater than the total number of units in the community. Considering the practicability of the device, it is necessary to satisfy the processing capacity of the device to meet the amount of abandoned drugs of the users within its coverage. In order to increase the possibility of people discarding drugs reasonably, the distance between the device and the nearest bin can not exceed 20 meters. For aesthetic reasons, the distance between the two processing devices is preferably not less than 20m.

Objective function:
\[
minN
\]
\[
min\sum_{i=1}^{N}Q_{i}
\]
s.t.
\[
L_{ij}\geq L_{min}
\]
\[
\sum_{i=1}^{N}n_{i}-M\geq 0
\]
\[
A-\frac{mB_{1}x_{i}F}{A_{2}}\geq 0
\]
\[
s_{i}\leq 20
\]

N is the total number of devices. Q

_{i}is the vertical distance from the i_{th}device to the nearest edge of the main road. L_{ij}is the straight line distance between the i_{th}device and the j_{th}device. L_{min}is the minimum distance between two devices. n_{i}is the number of units covered by the i_{th}device. A is the processing capacity of each device. F is the average amount of discarded antibiotics per household. s_{i}is the distance from the i_{th}device to the nearest dustbin.### 5.2.3 Establishment of evaluation model for device layout

Corresponding to our distribution principle, several relatively optimal solutions can be obtained. We analyzed the quality of the results of different distribution points and carried out the comprehensive evaluation of the results from four aspects. They are the total cost of the device, residents' satisfaction with the location of the device, the utilization rate of the device in the coverage area and the average distance from each device to the nearest dustbin. For the four different evaluation indicators, we mainly considered the aspects of residents, planners and us. Finally, we determined the weight ratio of the four factors, obtained the evaluation function of the device layout scheme and established the evaluation system to find the optimal layout scheme.

### 5.2.3.1 Total cost of devices

When establishing the optimization model, we used the total number of devices to represent the total cost of devices. The total cost of devices is a very important factor for us to consider. The smaller total number of devices are, the smaller total cost of the device are. It is more advantageous for us.

### 5.2.3.2 Residents' satisfaction with the placement of the devices

Residents' satisfaction with equipment placement can be represented by the average distance from the unit building to the nearest equipment.

Average of the closest distance from all units to the device:
\[
\bar{d}=\frac{\sum_{i=1}^{M}q_{i}}{M}
\]

For the residents of the community, the smaller the average distance of all units to the nearest the device are, the smaller the average distance walked by the residents are when throwing expired drugs. The residents will have the higher satisfaction with the devices. The average distance of all units to the nearest the device is an important indicator of the satisfaction of the community residents with the device.

### 5.2.3.3 Device utilization rate

In order to describe the usage of each device, we established the calculation formula for the utilization rate of the devices.

\[
\eta_{i}=\frac{x_{i}F}{A}
\]

For any device, 0≤η

_{i}≤1. When η_{i}=1, the utilization rate of any device is the highest, and the use of a single device is the best.
In order to evaluate the utilization of all devices, we established a calculation formula for average utilization of all devices.

\[
\eta=\frac{\sum_{1}^{N}\eta_{i}}{N}
\]

For any device, 0≤η≤1. The larger η is, the higher the utilization rate of the device is.

### 5.2.3.4 Average distance from each device to its nearest dustbin

\[
\bar{s}=\sum_{i=1}^{N}s_{i}
\]

If the average distance from each device to the nearest dustbin is smaller, it's much more possible that the residents will throw drugs into our devices when throwing garbage. At the same time, it is more convenient for residents.

### 5.2.3.5 Evaluation model established by Analytic hierarchy process

The hierarchy diagram is shown below [5]:

(1) Pairwise comparison matrix

Regarding the determination of a

_{ij}, the numbers 1 to 9 and their reciprocals are used as scales:Scale | Meaning |
---|---|

1 | Two factors have the same importance |

3 | One factor is slightly more important than the other |

5 | One factor is significantly more important than the other |

7 | One factor is more important than the other |

9 | One factor is extremely more important than the other |

2,4,6,8 | The median of the above two adjacent judgements |

1,1/2…,1/9 | The importance of changing the order of the two factors |

C

_{1}、C_{2}、C_{3}、C_{4}are residents' satisfaction with the placement of the device, the total cost of the device, the distance between the device and the nearest dustbin, and the utilization of the device for coverage.
Establish a pairwise comparison matrix of function evaluation systems.

\[
C=\begin{bmatrix}
1&2&3&4\\
1/2&1&2&3\\
1/3&1/2&1&2\\
1/4&1/3&1/2&1
\end{bmatrix}
\]

When the actual situation is taken into account, assigning weights for the four factors is reasonable. The queue of C

_{1}, C_{2}, C_{3}and C_{4}is the descending ordering sequence of importance of four factors. In the above matrix, a_{12}indicates that the ratio of the importance of C_{1}to C_{2}is 2:1. a_{13}indicates that the ratio of the importance of C_{1}to C_{3}is 3:1. a_{14}indicates that the ratio of the importance of C_{1}to C_{4}is 4:1. a_{23}indicates that the ratio of the importance of C_{2}to C_{3}is 2:1. a_{24}indicates that the ratio of the importance of C_{2}to C_{4}is 3:1. a_{34}indicates that the ratio of the importance of C_{3}to C_{4}is 2:1.
(2) Judge the consistency of the matrix

The feature vector \(\vec{w}=[(w_{1},w_{2},…,w_{n})]^{T}\) corresponding to the largest eigenvalue \(\lambda_{max}\) of the matrix C is used as the weight vector.

\[
C\cdot \lambda_{max}=\lambda_{max}\cdot \vec{w}
\]
\[
\sum_{i=1}^{n}w_{i}=1
\]

When a

_{ij}is not far from the consistency requirement, the eigenvalue and eigenvector of C are also not much different from the consistency matrix. When \(\lambda_{max}\) is larger than n, the degree of inconsistency of C is greater, and the error caused by using the feature vector as the weight vector is larger. Therefore, the size of \(\lambda_{max}-n\) can be used to measure the degree of inconsistency of A.
Calculate the consistency indicator CI:
\[
CI=\frac{\lambda_{max}-n}{n-1}
\]

For a consistent positive reciprocal matrix, the consistency indicator CI is equal to zero.

Define the average random consistency indicator RI:
\[
RI=\frac{\lambda'_{max}-n}{n-1}
\]

For the judgment matrix of 1~9 order, the RI value is shown in the following table.

N | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

RI | 0.00 | 0.00 | 0.58 | 0.90 | 1.12 | 1.24 | 1.32 | 1.41 | 1.45 | 1.49 | 1.51 |

Calculate the consistency ratio CR:
\[
CR=\frac{CI}{RI}
\]

When CR<0.1, it is considered that the degree of inconsistency of the paired discriminant matrix C is within the allowable range, and its feature vector can be used as the weight vector. Otherwise, it is necessary to adjust the pairwise discriminant matrix to make it satisfying consistency.

If the consistency of the matrix passes, the eigenvector \(\vec{w}\) corresponding to \(\lambda_{max}\) is the weight of the four factors.

For matrix C, CR = 0.0115, the matrix C passes consistency test. It is easy to find the maximum eigenvalue \(\lambda_{max}=4.031\), and the corresponding normalized feature vector is
\[
\vec{w}=(0.4673,0.2772,0.1601,0.0954)
\]

(3) Establishment of evaluation function Z

Since the units of each evaluation index are different, it is necessary to perform dimensionless processing on three indicators other than η.

Normalize N:

Normalize N:

Regardless of the spatial distribution of the community, only the total number of households in the community can be considered, and we can obtain a theoretical minimum of device demand.( The actual situation has the influence of spatial distribution, so the total number of actual devices obtained by the solution must be greater than this minimum.)
\[
N_{min}=\frac{XF}{A}
\]

X represents the total number of households. F is the average amount of abandoned antibiotics produced per household per month. A is the processing capacity of each device.
\[
N'=\frac{N_{min}}{N}
\]

The higher the cost is, the smaller N' is.

Normalize \(\bar{d}\): \[ \bar{d'}=\frac{\bar{d}-d_{min}}{d_{max}-d_{min}} \] Normalize \(\bar{s}\): \[ \bar{s'}=\frac{\bar{s}-s_{min}}{s_{max}-s_{min}} \] \[ Z=w_{1}\cdot N' -w_{2}\cdot \bar{d}+w_{3}\cdot \eta-w_{4}\bar{s} \]

Normalize \(\bar{d}\): \[ \bar{d'}=\frac{\bar{d}-d_{min}}{d_{max}-d_{min}} \] Normalize \(\bar{s}\): \[ \bar{s'}=\frac{\bar{s}-s_{min}}{s_{max}-s_{min}} \] \[ Z=w_{1}\cdot N' -w_{2}\cdot \bar{d}+w_{3}\cdot \eta-w_{4}\bar{s} \]

Normalize the indicators of the better schemes and bring them into the formula Z. The larger the evaluation function Z is, the better the layout scheme of the result is, so that we can find the optimal layout scheme. Others can also apply our evaluation function to their placement results to find the best solution they want.

Of course, others can change the importance of different factors according to their own or actual needs. They can also change the weight of different factors, and construct a suitable evaluation function according to their own needs.

Model Solving

It is impossible to directly solve the optimization model, because the multi-objective optimization model of the problem is complex and affected by the specific spatial conditions of the community. Therefore, we need to establish the layout principle and combine the specific spatial structure of the community to find the optimal layout scheme (non-inferior solution).

The principle of the optimization model is as follows:

Step1: Carry out the community layout of the garbage can according to the optimization model. When the constraint condition is met, the garbage can should be close to the main road of the community and cover all buildings with as few garbage cans as possible.

Step 2: Considering the intensiveness of people at the intersection, we should place the device at the intersections with multiple channels and more surrounding residents firstly, and the service range of the device will be drawn according to the service radius of 75 meters.

Step 3: Then the device is deployed in other uncovered areas of the community, so that the service range of all devices covers all the buildings in the community, and the overlap ratio between the device coverages is minimized. Taking into account the cleaning jobs by the staff on the main road and the convenience of residents throwing expired drugs, we need to give a priority to laying the device on the main road.

Step4: Check whether the device processing capability within the service range of each device can meet the needs of users. Then, we should redistribute the parts that cannot meet the constraint conditions, appropriately increase the number of devices or adjust the device position to meet the requirements of the constraint conditions.

Step5: Adjust the relative position of the garbage can and the device, and make the garbage can and the device as close as possible without changing the constraints of the constraint. Therefore, we can increase the possibility that the resident throws the medicine into the device while throwing the daily garbage.

An example is as follows:

We selected a community in Chengdu, China and analyzed its spatial structure and population distribution. Combined with the multi-objective optimization model and the above-mentioned layout principle, we distributed our devices.

Step1: Use the principle one to arrange garbage cans. The results are as follows:

Step2: To deploy the device, we found the most important three intersections in the community which have many residents around. Then we placed the devices at the intersections. The results of the first deployment are as follows.

Step3: According to principle three, we distributed the uncovered parts, and the results are as follows:

Step 4: Redistribute the part that cannot meet the constraint condition according to principle four. The results are as follows:

Step 5: Adjust the relative position of the garbage can and the device. The results of the final garbage bin and the processing device are as follows:

References

[1]Zejing Jia. (2017). Study on reasonable set-up of garbage containers in residential area.

[2]Yumei Sui, Zhenshan Li, Xiaoyan Qu, & Lei Yang. (2010). Simulation calculation of garbage bin configuration for domestic waste classification community in Beijing.

[3]Chávez-Jacobo, V. M., Hernández-Ramírez, K. C., et.al. (2018). CrpP is a novel ciprofloxacin-modifying enzyme encoded by the Pseudomonas aeruginosa pUM505 plasmid.

[4]Huiming Zong. (2014). Research on Road Garbage Layout Based on Urban Functional Area-Taking Beibei District of Chongqing as an example.

[5]Xue Deng, Jiaming Li, Haojian Zeng, Junyang Chen, & Junfeng Zhao. (2012). Analysis and Application of Weight Calculation Method of Analytic.

Green Building

(2), 63-67.[2]Yumei Sui, Zhenshan Li, Xiaoyan Qu, & Lei Yang. (2010). Simulation calculation of garbage bin configuration for domestic waste classification community in Beijing.

Journal of Peking University (Natural science edition), 46

(2), 265-270.[3]Chávez-Jacobo, V. M., Hernández-Ramírez, K. C., et.al. (2018). CrpP is a novel ciprofloxacin-modifying enzyme encoded by the Pseudomonas aeruginosa pUM505 plasmid.

Antimicrobial agents and chemotherapy, 62

(6), e02629-17.[4]Huiming Zong. (2014). Research on Road Garbage Layout Based on Urban Functional Area-Taking Beibei District of Chongqing as an example.

Journal of Southwest University(Natural science edition), 36

(10), 124-129.[5]Xue Deng, Jiaming Li, Haojian Zeng, Junyang Chen, & Junfeng Zhao. (2012). Analysis and Application of Weight Calculation Method of Analytic.

Practice and understanding of mathematics,42

(7), 93-100.