Team:UESTC-China/Model4
Overview
Considering the needs of human practice, we used the method of mathematical modeling to make the treatment device 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 longest distance residents can accept to walk 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 have established a mathematical model to scientifically place our processing devices.
Inspired by the problem of layout of garbage cans in our daily life, we are ready to place the treatment devices near the garbage cans to increase the probability of residents throwing expired drugs into the treatment devices. In order to distribute the points for our disposal equipment, it is necessary to distribute the points for the garbage cans first.
Considering the lack of scientificity in the layout scheme of garbage cans nowadays, we take 50m as the maximum coverage radius, and set up a multi-objective optimization layout model based on the interests of maintenance personnel, residents and us. Taking 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 taking the capacity of the garbage can as the constraint condition, the problem of garbage can 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. Change the constraint condition to that the processing capacity of devices needs to meet the needs of residents, and require that the distance between the device and the dustbin be less than or equal to 20m. Then the model of treatment device layout was established. Then the distribution result of the garbage can and the distribution device is fine-tuned to make the treatment device as close as possible to the garbage bin. Finally, we get our optimal layout scheme, which promotes the HP process, so that the final step in the application process of our project, device layout, can be completed.
Model Assumptions
1. Assume that 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 |
| Ri | Number of dustbins in group i |
| Wi | Volume of garbage produced within the coverage of group i per day |
| ρi | Number of households covered by group i of dustbins |
| Pi | The vertical distance from the closest edge of the main road to the dustbin i |
| pi | The closest distance between the ith unit and the dustbin |
| M | Total number of units in the community |
| mi | Number of units covered by the ith group of dustbins |
| Qi | The vertical distance from the ith device to the nearest edge of the main road |
| qi | Distance from the ith unit to its nearest device |
| xi | Total number of households covered by the ith device |
| N | Total number of devices |
| Lij | Straight line distance between the ith device and the jth device |
| Lmin | Minimum distance between two devices |
| si | The distance from the ith 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 |
| ηi | Utilization of the ith device |
| ni | Number of units covered by the ith device |
the Values of Parameter
| Symbol | Description | Value | Units |
|---|---|---|---|
| W | Total amount of garbage generated per capita per day 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/m3 |
| φ | Filling coefficient of the dustbin | 0.8 | |
| B | Volume of the dustbin | 240 | L |
| A1 | Daily garbage discharge weight unevenness coefficient | 1.1 | |
| A2 | Residential population coefficient of | 1.05 | |
| B1 | 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 |
| Lmin | Minimum distance between two devices | 20 | m |
Model Establishment
5.1 Preparation before modeling：
5.1.1 Calculation of the number of the ith group of dustbins
\[
R_{i}=\frac{W_{i}}{\omega\epsilon\varphi B}
\]
\[
W_{i}=\frac{mA_{1}\alpha\rho_{i}W}{A_{2}}
\]
Ri is the number of dustbins in group i. Wi is Volume 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/m3 .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[1]. 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. ρiis the number of households covered by the ith group of dustbins. W is the total amount of garbage generated per capita per day in Chengdu. A1 is daily garbage discharge weight unevenness coefficient. A2 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, and the amount of drug produced within the coverage is related to daily drug produced, the resident population, the volatility of the flow of people[2].
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, and 15 thousand tons of waste drugs are produced annually. Our country requires that the use rate of antibiotics 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 our project has only studied engineering bacteria for CIP for the time being, we are unable to obtain processing power for other antibiotic drugs, so we estimate the processing capacity of the device. Only the rate at which CrpP degrades CIP is used as a criterion for determining the processing capacity of the device, and its processing ability can be subsequently improved. 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 engineering bacteria in the device is 135mg/L, and the corresponding treatment 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 ith group of dustbins coverage pi
The city is divided into four areas based on the leading function of the region, and the city can be divided into residential areas, cultural areas, commercial areas and urban main roads. Due to the different population composition of different regions and the educational level of the population, the results of the survey that can be tolerated by the farthest distance of garbage are different [4]. The age of the population in the residential area is complex, the proportion of the elderly and children is high, and the household activities are the main ones. The residents do not want to take up too much life to throw garbage deliberately and leisure time, so the residents have a small tolerance for carrying garbage. (The same is true for drug delivery) According to the survey literature [4], the coverage of the ith group of dustbins is pi≤50m.
5.2.1.2 Dustbin layout optimization model
(1) Optimize the target: 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.
(2) Constraint condition: Considering the amount of garbage generated, 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 coverage of the i_th group of dustbins is less than or equal to 50m. 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.
\[
W_{i}\leq R_{i}\omega\epsilon\varphi B
\]
\[
p_{i}\leq 50
\]
\[
\sum_{i=1}^{R}m_{i}-M\geq 0
\]
Ri is the number of dustbins in group ith. R is the number of groups of dustbins. Pi is the vertical distance from the closest edge of the main road to the dustbin i. mi is the number of units covered by the ith 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 Determining model of maximum coverage radius of the device dmax
The degree of satisfaction with the placement of the device depends on the distance between the device and the residents' 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 are 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. pi、qi、ti、ki are the proportions of residents of different educational levels in the four age groups. di is the average value of maximum distance that residents of different ages and education levels can accept. 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 are obtained from the questionnaire. For the elderly over 65 years old, the total amount of medicines produced is large and it is not convenient to walk. Therefore, we stipulate that the contribution rate of the elderly is the largest. We define 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) Optimize the target: Cover all units at minimal cost, and the deployment of the device should be as close as possible to the main road. (Facilitate the staff to regularly change the bacteria in the device)
(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 can be 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 user within its coverage. In order to increase the probability 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 treatment devices must not be less than 20m.
Objective function:
\[
minN
\]
\[
min\sum_{i=1}^{N}Q_{i}
\]
s.t.
\[
L_{ij}\geq L_{min}
\]
\[
\sum_{i=1}^{N}-M\geq 0
\]
\[
A-\frac{mB_{1}x_{i}F}{A_{2}}\geq 0
\]
\[
s_{i}\leq 20
\]
N is the total number of devices. Qi is the vertical distance from the ith device to the nearest edge of the main road. Lij is the straight line distance between the i_th device and the jth device. Lmin is the minimum distance between two devices. ni is the number of units covered by the ith device. A is the processing capacity of each device. F is the average amount of discarded antibiotics per household. si is the distance from the ith 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. The quality of the results of different distribution points is analyzed, and the comprehensive evaluation of the results is carried out 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, mainly consider the aspects of residents and us, planners. Finally, the weight ratio of the four factors is determined, the evaluation function of the device layout scheme is obtained, and the evaluation system is established to find the optimal layout scheme.
5.2.3.1 Total cost of devices
When establishing the optimization goal, the minimum cost of the devices is replaced by the minimum total number of devices. The total cost of devices is a very important consideration for us. The smaller the total number of devices, the smaller the total cost of the device, and the more advantageous it is for us.
5.2.3.2 Residents' satisfaction with the placement of the devices
Residents' satisfaction with the placement of the device can be expressed as the average of the distance from all units to the nearest device.
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 distance of the device, the smaller the average distance traveled by the residents when throwing expired drugs, and the higher the satisfaction of the residents with the devices. The average number of units to the nearest 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, the calculation formula for the utilization rate of the devices is established.
\[
\eta=\frac{x_{i}F}{A}
\]
For any device, 0≤ηi≤1, and when ηi=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, establish an average utilization calculation formula for all devices.
\[
\eta=\frac{\sum_{1}^{N}\eta_{i}}{N}
\]
For any device, 0≤ηi≤1, and the larger the η, the higher the utilization rate of the device, the better the device is used.
5.2.3.4 Average distance from each device to its nearest dustbin
\[
\bar{s}=\sum_{i=1}^{N}s_{i}
\]
The smaller the average distance from each device to its nearest dustbin, the closer the distance between all devices and the dustbins, the greater the probability that the residents will throw drugs when throwing garbage, and the more convenient the device layout scheme is for residents.
5.2.3.5 Evaluation model established by Analytic hierarchy process[5]
The hierarchy diagram is shown below:
(1) Pairwise comparison matrix
Regarding the determination of aij, 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 |
C1 、C2 、C3 、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.
\[
C=\begin{bmatrix}
1&2&3&4\\
1/2&1&2&3\\
1/3&1/2&1&2\\
1/4&1/3&1/2&1
\end{bmatrix}
\]
According to the actual situation, it is reasonable to assign importance for the four factors. Among the four factors C1, C2, C3, and C4, C1 is the most important, C2 is the second, C3 is again, and C4 is the least important. In the above matrix, a12 indicates that the ratio of the importance of C1 to C2 is 2:1. a13 indicates that the ratio of the importance of C1 to C3 is 3:1. a14 indicates that the ratio of the importance of C1 to C4 is 4:1. a23 indicates that the ratio of the importance of C2 to C3 is 2:1. a24 indicates that the ratio of the importance of C2 to C4 is 3:1. a34 indicates that the ratio of the importance of C3 to C4 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 aij 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 three 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 a theoretical minimum of device demand can be obtained.( 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 larger N 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 the optimal layout scheme is found. 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 not possible 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: The community layout of the garbage can is carried out 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 probability 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.
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:
(Purple: dustbins, Red: treatment devices)
References
[1] Yumei Sui, Zhenshan Li, Xiaoyan Qu, Lei Yang. Simulation calculation of garbage bin configuration for domestic waste classification community in Beijing[J]. Journal of Peking University (Natural science edition), 2010, 46(02):265-270.
[2] Zejing Jia. Study on reasonable setting of garbage containers in residential quarters[J]. Green Building, 2017(2).
[3] Chávez-Jacobo V M, Hernández-Ramírez K C, Romo-Rodríguez P, et al. CrpP is a novel ciprofloxacin-modifying enzyme encoded by the Pseudomonas aeruginosa pUM505 plasmid[J]. Antimicrobial agents and chemotherapy, 2018, 62(6): e02629-17.
[4] Huiming Zong. Research on Road Garbage Layout Based on Urban Functional Area- Taking Beibei District of Chongqing as an example [J]. Journal of Southwest University(Natural science edition), 2014, 36(10):124-129.
[5]Xue Deng, Jiaming Li, Haojian Zeng, Junyang Chen, Junfeng Zhao. Analysis and Application of Weight Calculation Method of Analytic [J]. Practice and understanding of mathematics, 2012,42(07):93-100.
[2] Zejing Jia. Study on reasonable setting of garbage containers in residential quarters[J]. Green Building, 2017(2).
[3] Chávez-Jacobo V M, Hernández-Ramírez K C, Romo-Rodríguez P, et al. CrpP is a novel ciprofloxacin-modifying enzyme encoded by the Pseudomonas aeruginosa pUM505 plasmid[J]. Antimicrobial agents and chemotherapy, 2018, 62(6): e02629-17.
[4] Huiming Zong. Research on Road Garbage Layout Based on Urban Functional Area- Taking Beibei District of Chongqing as an example [J]. Journal of Southwest University(Natural science edition), 2014, 36(10):124-129.
[5]Xue Deng, Jiaming Li, Haojian Zeng, Junyang Chen, Junfeng Zhao. Analysis and Application of Weight Calculation Method of Analytic [J]. Practice and understanding of mathematics, 2012,42(07):93-100.