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A covering model application on Chinese industrial hazardous waste management based on integer program method
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A covering model application on Chinese industrial hazardous waste management based on integer program method Lei Li ∗ , Shi Wang, Yanting Lin, Wenting Liu, Ting Chi School of Management and Economics, Tianjin University, Tianjin, China
a r t i c l e
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Article history: Received 29 December 2013 Received in revised form 28 March 2014 Accepted 8 May 2014 Keywords: Prospective optimization Industrial hazardous waste Integrated treatment and disposal center Covering location model Integer program method
a b s t r a c t Industrial hazardous waste is a special kind of toxic substance, which poses risks to the environment as well as human health. With the speeding up of industrialization in China, the collecting, recycling, treatment and disposal of industrial hazardous waste gradually become a severe problem to both the environmental protection and the resource management. In this paper, we laid an emphasis on prospective optimization, used integer program method, and selected the optimal locating approach to the collecting and handling of industrial hazardous waste on the basis of covering location model. We also selected an industrial intensive district in Hebei Province in China as an empirical object and examined our model result. Compared to previous results, this one bears the characteristics of immediacy, dynamism and predictability. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Industrial waste can be characterized as hazardous if they are on the national hazardous waste list or are identified as solid waste with hazardous properties according to national standards and approaches to identifying hazardous waste. It is a special kind of waste, which could be toxic, flammable, explosive, corrosive, chemical reactive, infectious and radioactive. It not only poses risks to the surrounding air, water and soil, but also do harm to the ecological environment and human health through diversified channels. Since the industrial and technical revolution, the increasing amount of industrial hazardous waste had brought extreme pressure to the environment as well as waste and damage to resources. These industrial wastes all bear the characteristics of chronicity, latency, invisibility, non-dilution and sensibility. This paper firstly analyzed some research approaches to the integrated treatment and disposal of industrial hazardous waste, then introduced in the covering model based on this analysis, and then conducted an empirical research in Hebei Province of China (Hebei Province is a vital industrial area of China and there are some heavy pollution plants in Hebei Province such as chemistry, Auto and paper. Here we selected an industrial intensive district in Hebei Province which could be a typical object for our study and be well applied our model on) and made an analysis of the result on the
∗ Corresponding author. Tel.: +86 138 21260061; fax: +86 022 27401815. E-mail address: [email protected] (L. Li).
basis of integer program (KPP-POS), then gave some advices to the local government about the location of industrial hazardous waste treatment and disposal centers, and at last made conclusions and improvements based on the above analysis. 2. Previously relevant researches review 2.1. Review of researches in industrial hazardous waste management Industrial hazardous waste processing (generation, collection, transportation, storage, integrated utilization and ultimate disposal) is indefinite in both temporal and spatial respects, which makes the pollution control a big problem for environmental management. Developed countries (such as the USA, Japan and some European countries) are the main producers of industrial hazardous waste of the world. Their researches to industrial hazardous management are the most advanced, effective and have the longest history, which are the best references for China. For instance, Samanlioglu (2013) divided the integrated disposal center into collection center, initially processing center, recycling center and ultimate disposal center, and then selected the best route among centers and the optimal disposal approach using multiobjective linear programming model, which minimized the total cost and the transportation risk; he distributed the weights of risk by the importance of each district and selected the most reasonable transportation solution to minimize the total risk (Taha, 1971); in order to solve the connection problem
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between disposal node and generation node (make sure every generation node could connect to at least one disposal node), Gollowitzer and Ljubic´ (2011) used mixed integer programming model and denoted each node (disposal node and generation node) as a node in a mathematical tree to attain the optimal configuration solution; Church and Arfinkel (Zografos and Samara, 1989) introduced in integrated model, which gave an emphasis on simultaneously enlarging the distance between integrated disposal center and residential area while minimizing transportation cost. The above studies are all devoted in looking for the balance among high efficiency, low cost, and low risk, and the balance between connectivity and synthesis in transportation, or in other words, to select the best model under the restriction of a fixed environment or a rigid condition. But these studies all neglect the predetermined variable which controls the whole disposal process: the location of integrated disposal center, namely prospective optimization (Taha, 1971; Alidi, 1992). For instance, Ibel Alumer (Alumur and Kara, 2007) selects 13 relatively centric nodes from 92 waste generation nodes as disposal nodes on the basis of geological centrality, but the result is not optimal. If either the rigid condition (location problem of disposal node in this paper) or the mathematical approach of locating is improved, twice the result will be produced with half the effort.
2.3. The applicability of covering model in industrial hazardous waste disposal When it comes to public source and environmental management, what people care about is not the most applicable solution but the most desirable one, and the solution is measured by people’s desirability. Therefore, covering location model is often applied on public facility location problem, such as the location of firefighting point, emergency evacuation center, service center, public school, police office, library, hospital, and public buildings (Francis et al., 1974). In fact, when we construct an integrated industrial hazardous waste disposal center, what people care most is that all the hazardous waste in a given distance could be disposed immediately and meet their basic demand. Next, is looking for the optimal solution that minimize the distance and cost (Schilling et al., 1993). In addition, covering location model is more applicable to large-scaled, complex and dynamical location problems (Owen and Daskin, 1998; Schilling et al., 1993). All in all, this paper used the covering location model for industrial hazardous waste management because it is the best when applied on public facility location problem and can better measure people’s desirability when compared to other models mentioned above. 3. A covering model based on integer program (KPP-POS)
2.2. Location model Location model problem began in 1909. Alfred Weberk considered about how to select the location of a warehouse to minimize the total distance from customers; on the ground of that, Hakimi (1964) explored where to set up contact and help centers in the highway system to minimize the total distance between people and the nearest center. After him, the selection approach about static center location is classified into three models: median model, covering location model (Church and Velle, 1974) and center model (Owen and Daskin, 1998).
2.2.1. Median model Using multiobjective program or linear objective integer program to set up P disposal nodes in order to minimize the total distance between demands and disposal nodes, e.g., Church and Velle (1974) proposed that the efficiency of a disposal node is determined by the average distance from the demand to this very node; ReVelle refined the P-center model to either maximize the number of customers a disposal node could handle or maximize profits. This model is applicable to integer program or moderate-scaled locating cases.
2.2.2. Covering location model This model could be mainly divided into two directions: (1) when it is guaranteed that the given demand range is covered, how to configure the location and number of disposal nodes to minimize the cost (Schilling et al., 1993; White and Case, 1974); (2) when the resource is limited, how to configure the given number of disposal nodes to maximize the demand range (White and Case, 1974; Li et al., 2013).
2.2.3. Center location model On the basis of covering location model, in the condition of limited resource and uncertainty of distance that can be handled, how to locate to cover all the demand range and simultaneously minimize the total distance.
Before we introduce the integer programming method used to construct the covering model, we must firstly introduce the spirit of KPP-POS, which is the base of our integer programming mathematical model. 3.1. Principles of KPP-POS method Regard collection nodes as a network and identify the positional relationship among generation nodes by their connective relationship and geological positions. To measure the importance of the nodes is to identify their centrality. Centrality is a way to measure the relevant importance of a vertex in a network. Common centrality includes: Degree Centrality (Hakimi, 1964), Betweenness Centrality (Freeman, 1977), Closeness Centrality (Stephenson and Zelen, 1989), Eigenvector Centrality (Bonacich, 1972), Katz Centrality (Katz, 1953; Bonacich and Lloyd, 2001) and Pagerank Centrality (Page et al., 1998). However, these centralities are all measures of importance for nodes while the importance for node groups (Group Centrality) are somehow neglected. Realized the limitations of nodes centrality, Everett and Borgatti (1999) developed a new measure to group centrality for individuals, groups and classes by extending the standard network centrality measures. On the basis of the measures to the group centrality for the above objects, we can conclude two measures to the importance for node groups. In addition, Borgatti (2003) defined two new group measures for the identification of key players: Key Player Problem/Negative (KPP-NEG) and Key Player Problem/Positive (KPP-POS), which are used to measure the contribution of a node set in two different respects. KPP-POS could be defined as: given a node network, find out a set of n key players which are maximally connected to other nodes outside the set. This measure could be used to optimize the collection of waste by selecting the key players as seeds. While KPP-NEG on the contrary, removes the selected key nodes in a network to disrupt or fragment the network, and finally identifies and verifies the key players (Wen, 2012). In this paper, we will focus on the positive key problem. The figure below is an example of KPP-POS problem, which is totally inconsistent to intuition. If we select two nodes as a set to maximize
Please cite this article in press as: Li, L., et al., A covering model application on Chinese industrial hazardous waste management based on integer program method. Ecol. Indicat. (2014), http://dx.doi.org/10.1016/j.ecolind.2014.05.001
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D
A B
Because Aij is a binary matrix, i Aij ∧ 1 has the same value with maxi Aij . In mathematical programming, the first form is more popular because it could be denoted by linear programming constraints after we add some auxiliary variables, which we will discuss later. In order to remove the constraints the selected node set S, we now define a new auxiliary binary variable xi as follows:
J
G H
E
xi =
C
F
I
K
1 when i ∈ S 1 when i ∈ S¯
(3)
¯ then xj = 0 and 1 − xj = 1. If j ∈ S, then xj = 1 and 1 − xj = 0; if j ∈ S, With the help of auxiliary variable, we are able to remove the constraints of set S and rewrite the formula in the following forms:
L
Maximize Fig. 1. Example of KPP-POS problem.
S.T. the neighboring nodes in this set, then which two should we select? From the first sight, we should make the selection from Nodes D, E and F because they all have the maximum number of individual node degree (6 node degrees). However, these three nodes share the same neighboring nodes and select any two nodes from these three nodes will not increase their neighboring nodes. Nevertheless, though node I only has 5 nodes, the set of node I and node E has 8 neighboring nodes, which is more than the number of neighboring nodes formed by the set of Node D and Node E (Fig. 1).
i∈V
x i i
(1 − xi )(
The objective function could be written in the above form because: if i ∈ S, we could know that 1 − xi = 0. The first line of the ¯ objective function only considers nodes that are not in set S. If i ∈ S, then xi = 0. The second line of the objective function only covers nodes that are in set S. i xi is the number of nodes in set S. So it agrees with the value constraints. Plus, xi is binary. To simplify it, we use a matrix to describe the optimal problem as follows: (1 − x)t (Ax ∧ 1)
(1) G = (V, E) notates the network or figure; (2) V = (V1 , V2 , . . ., Vn ) notates the set of nodes; (3) E = (E1 , E2 , . . ., En ) notates the set of sides, among them, Ek = Vi , Vj is the connection between Node Vi and Node Vj . Here we assume the network figure is scalar, i.e., the side is shared:
S.T.
xt 1 ≤ s,
Or denote as: Maximize
|S| ≤ s
i
Aij ∧ 1 =
Maximize S.T.
0
when j and S are not connected
i∈V
(6)
mi
mi ≤ (1 − xi ), mi ≤ (1 − xi )Ai x,
(7)
(1)
This is an integer programming problem. To be clearer, S variable xi , mi are all binary. This problem could be well solved by CPLEX, and find the optimal solution in theory. What is more, if we loosen the last constraint, we get: Maximize S.T.
1 when j and S are connected
≤ s,
xi is 1 or 0
In this formula, Aij is an adjacent matrix, S is the selected node set. |S| is the base or value of set S. We will introduce mathematical programming methods based on this KPP-POS problem. This time the above problem could be denoted by formula as an integer programming problem (0–1 integer programming problem), defined as follows: ¯ which means j = maxi∈S Aij . We could describe For all nodes j ∈ S, this terminology as i Aij ∧ 1 because we are given that:
(1 − xi )(Ai x ∧ 1)
xt 1 ≤ s,
Formally, KPP-POS problem could be defined as: a set of size s is maximally connected to nodes outside the set. Therefore, we could describe KPP-POS problem again using the following Maximum target function.
S.T.
i∈V
xt 1
Change the objective function and add more constraints. The above problem could be concluded as:
3.3. Mathematical model
(maxi ∈ S Aij )
xi is 1 or 0
A = {aij , i = 1, 2, . . ., n, j = 1, 2, . . ., n} denotes the adjacent matrix of network figure G. Element {aij } in A is a binary variable. aij = 1 denotes that there is a side between Vi and Vj , aij = 0 denotes there is no side between Vi and Vj . Diagonal element in matrix A is 0.
j ∈ S¯
(5)
xi is 1 or 0
S.T.
(4) S notates the selected node set (5) v = |V |, notates the base or value of Set V (6) s = |S|, notates the base or value of Set S
(4)
xi ∈ {0, 1}
Maximize
Maximize
Aij xj ∧ 1)
≤ s,
3.2. Notation
Ek = Vi , Vj = Vj , Vi
j∈V
(2)
i∈V
mi
mi ≤ (1 − xi ), mi ≤ (1 − xi )Ai x, xt 1 xi2
(8)
≤ s,
− xi = 0
We could use Semi-Definite Programming (SDP) to estimate the upper bound of this problem. This is a slack problem we use when the calculation costs is very high. First, we will introduce the basic principles of SDP problem. Define Y = xxt , x = {x0 , x1 , . . ., xn } t (because x is the feasible region of formula (8) so any Y in conv(xxt ) is a feasible solution to formula
Please cite this article in press as: Li, L., et al., A covering model application on Chinese industrial hazardous waste management based on integer program method. Ecol. Indicat. (2014), http://dx.doi.org/10.1016/j.ecolind.2014.05.001
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(9)), and then we could treat this problem as a SDP problem to loosen: Maximize S.T.
i∈V
mi
mi ≤ 1 − Y0i , mi ≤ 1 − Yij − Y0i + Y0j ,
Y i 0i
≤ s,
Yii = Y0i ,
(9)
i = 0, 1, 2, . . ., n
0 ≤ mi ≤ 1,
i = 0, 1, 2, . . ., n
4. Demonstration 4.1. Background and assumption Realities of China at the moment could help better understand Chinese industrial hazardous waste management condition and we can accordingly adjust our model to make it more practically significant. Firstly, for long terms, Chinese economy development is an extensive development mode of high consumption and heavy pollution. Along with the further of industrialization, hazardous waste has become one of the main environmental pollutions and a significant contradiction for economy and society development. In 2005, China generated 11.62 million tons of industrial wastes in total, of which 43.4% is recycled, 33% is stored and 23% is safely discarded (Duan et al., 2008). Next, the amount of generation and emission of industrial waste are both increasing rapidly. Its large varieties, complex properties, wide generation source range and difficulties for management all make the industrial hazardous waste management and disposal a significant problem to China’s environmental management. What is more, hazardous waste management in China starts late. By 2005, there are only 177 synthesis disposal center in China and just 416,000 tons of industrial wastes had been disposed. The current problems of hazardous waste management in China mainly concentrate on knowing little about its hazard, nonstandard disposal, low efficiency centralized disposal, etc. A more efficient, rapid, systematical system of industrial hazardous waste management is needed. In the actual conditions, consider from the following five aspects: (1) The range and number of corporations that generate hazardous waste is huge and the distribution is scattered. The waste is hard for centralized disposal, and the environmental risks will inevitably increase if sent to qualified disposal site; (2) The equipment in producing units which are qualified for disposal is out of date. Their ability of disposing, processing and integrated utilization of industrial hazardous waste is limited, which could not realize the environmental protection and the fully utilization of resource; (3) Not all generation units are qualified for disposal and the level of disposal is low, which may cause problems of non-standard; (4) The high costs of handle and reuse, the complexity of management are all hard to realize scale economy; (5) The generation of hazardous waste is dynamic, which is hard to predict. We consider the centralized disposal as the optimal disposal method as well as immediately, swiftly and accurately identify the location of temporary disposal node according to the circumstances at that time, in order to realize immediately and efficiently disposing the hazardous waste and saving resource.
Problems need to be solved: (1) assume that all of the wastes are supposed to be disposed or, in other words, under the circumstance of all the generation nodes are desired to be covered, then how to determine the optimal site selection; (2) assume that the resource is limited, then how to determine the number and location of disposal nodes to cover a larger range of generation nodes. Based on the situation of industrial hazardous waste in an industrial intensive district in Hebei Province of China, we construct our model follow main principles as below: (1) Firstly count the number of nodes that generate wastes within a certain period of time; (2) Select hazardous waste generation nodes as nodes in the network figure; (3) Compromise the transportation costs and risks. In case of the leak when exceeding certain distances, we recommend to use 20-km as the safety distance and construct the sides of fundamental structural nodes on the basis of safety distance; (4) If the distance between two hazardous waste generation nodes is less than 20-km, there are attachable sides between nodes, and the transportation safety of hazardous waste could be guaranteed, or we say its transportation distance is within the allowable range of safety distance; (5) In another time period, if one node has no hazardous waste generation, then this node will not be considered as a hazardous waste generation node; (6) Consider the costs and control problem, temporary disposal nodes could help with centralized dispose, and at this time the location of disposal node can only be selected from hazardous generation nodes; 4.2. Result analysis and demonstration The calculating result of KPP-POS could be different because every time the nodes that generate hazardous waste are dynamically changing. 4.2.1. Result analysis of the first time period Assume in the first time period there are 108 corporations generating hazardous waste in total, i.e., there are 108 hazardous waste generation nodes, the distribution and reach ability are denoted in Fig. 2: If sorted by the node degree of each node, the top 10 nodes with the largest node degree are: Node 1, with 18 node degrees; Node 3, with 12 node degrees; Node 6, 10 node degrees; Node 2, 7 node degrees; Node 10, 6 node degrees; Node 13, 4 node degrees; and Nodes 45, 67, 82, 89 all have 4 node degrees. The calculation results of KPP-POS by mathematical programming are concluded in the following table: By observing and analyzing the data in Table 1, we could draw the following conclusions: (1) Under the circumstance of limited disposal resource, when k = 1, only one temporary disposal node could be set up. Obviously nodes with the largest node degree should be selected, which means Node 1 should be selected or the temporary disposal node should be set up in the location of Node 1. In this way, we could centrally collect and dispose 18 + 1 = 19 nodes which generate hazardous waste and accomplish the disposal of 19/108 = 17.6% of the total generation. When k = 2, all the disposal resource we have could only afford to set up 2 temporary disposal node. According to the result indicated in Table 1, if select Node 1 and Node 3, the number of reachable nodes is up to 28 + 2 = 30, which means if set up these two temporary disposal nodes in the location of Node 1 and Node 3, then we
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Fig. 2. 108 hazardous waste generation nodes and their sides.
Fig. 3. 87 hazardous waste generation nodes and their connected sides.
could accomplish the hazardous waste disposal of 30 generation nodes or to say we could accomplish 27.8% of the disposal. If we consider about adding one more disposal node on the basis of k = 2 in the future, we could make the prediction that this one more node will be in the location of Node 6 and then make relevant preparation based on this prediction. The rest can be done in the same manner. Consider the cost of resource by disposal nodes and the disposal radio for hazardous waste that could be accomplished, assume add one more disposal node, if the disposal ratio increase is not less than 8% compared to the previous ratio, we consider it is worthy to add this one more node; if the increase of disposal ratio is less than 8% but not less than 4.7% compared to the previous ratio, we consider it is worthy to think about adding this one more node; if the increase of disposal ratio is less than 5% compared to the previous ratio, we do not recommend to add this one more node. According to the data in Table 1, we consider it is worthy to add at most 3 more disposal nodes; it is worthy to think about adding at most 6 disposal nodes and we do not recommend adding more nodes because the utilization efficiency of the additional disposal resource is low. (2) Under the circumstance of completely disposal, If it is desired to completely dispose all the hazardous waste that corporations have generated and accomplish 100% disposal, which means to cover all the generation nodes, we need to set up 32 disposal nodes
and in the location of Nodes 1, 2, 3, 6, 10, 13, 24, 37, 41, 42, 43, 45, 46, 49, 53, 58, 65, 67, 69, 71, 77, 78, 81, 82, 86, 87, 89, 90, 100, 103, 104 and 106. In this case, we should use the strategy of setting up 32 disposal nodes. If the number of disposal nodes is more than 32, it will lead to the waste of disposal resource; if the number is less than 32, it will not meet the desirability of 100% disposal. 4.2.2. Comparison and analysis of the results of the second time period and the first time period Assume in the second time period, there are 87 corporations generating hazardous waste, which means there are 87 generation nodes. The distribution is indicated in Fig. 3. The calculating results of KPP-POS by mathematical programming are concluded in the following table. Assume the number of hazardous waste generation nodes decline from 108 (the first time period) to 87 (the second time period), which means the circumstance transfers from Figs. 2 and 3. In this case, by comparing the data in Tables 1 and 2, we could draw the following conclusions: (1) Under the circumstance of limited disposal resource, when k = 8, besides Nodes 1, 2, 3, 6, 10, 13, 89 which are among the top 8 nodes with the largest node degree, we select Node 67 instead of Node 45 which is also one of the top 8 nodes. Because in Fig. 2 we could see that Node 45 and Node 6 share some neighboring nodes and selecting Node 45 will not dramatically increase the
Table 1 Number of nodes that could be reached within a length of 1 from set 1 to set 10 (108 generation nodes).
Table 2 Number of nodes that could be reached within a length of 1 from set 1 to set 10 (87 generation nodes).
K 1 2 3 4 5 6 7 8 9 10
Reachable nodes (m) 18 + 1 28 + 2 36 + 3 42 + 4 47 + 5 51 + 6 55 + 7 58 + 8 61 + 9 64 + 10
Selected nodes
Completed processing ratio (m/108)
K
{1} {1, 3} {1, 3, 6} {1, 2, 3, 6} {1, 2, 3, 6, 10} {1, 2, 3, 6, 10, 82} {1, 2, 3, 6, 10, 67, 82} {1, 2, 3, 6, 10, 13, 67, 82} {1, 2, 3, 6, 10, 13, 45, 67, 82} {1, 2, 3, 6, 10, 13, 45, 67, 82, 89}
17.59% 27.78% 36.11% 42.59% 48.15% 52.78% 57.41% 61.11% 64.81% 68.52%
1 2 3 4 5 6 7 8 9 10
Reachable nodes (m) 14 + 1 24 + 2 33 + 3 39 + 4 43 + 5 47 + 6 50 + 7 52 + 8 54 + 9 56 + 10
Selected nodes
Completed processing ratio (m/87)
{1} {1, 6} {1, 3, 6} {1, 2, 3, 6} {1, 2, 3, 6, 10} {1, 2, 3, 6, 10, 13} {1, 2, 3, 6, 10, 67, 89} {1, 2, 3, 6, 10, 13, 67, 89} {1, 2, 3, 6, 10, 13, 67, 78, 89} {1, 2, 3, 6, 10, 13, 25, 67, 78, 89}
17.24% 29.89% 41.38% 49.43% 55.17% 60.92% 65.52% 68.97% 72.41% 75.86%
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neighboring nodes. Consider the interruption of repeated reachable nodes, it will actually increase 3 more neighboring nodes, which is the same if we select Node 67. So we could see it is the same thing to select either of these two nodes. Assume there is only one disposal node now, then the number of disposal nodes that we consider is worthy to add increases from 2 to 3, the number of disposal nodes that we consider is worthy to think about adding increases from 5 to 6. Assume the government determines whether to set up disposal nodes by whether it is worthy to do it, and requires to dispose as much hazardous waste as possible, if the status changes in the process of setting up disposal nodes, then the optimal plan changes from the previous Nodes 1, 3 to Nodes 1, 3, 6, the strategy we use should be the construction of nodes in the location of Node 6, which could help make the optimal decision and accomplish our original target swiftly and efficiently. If it is desired to accomplish at least 50% disposal of the hazardous waste, then the number of disposal nodes that we consider is worthy to add will decline from 6 to 5, the selected nodes changes from Nodes 1, 2, 3, 6, 10, 82 to Nodes 1, 2, 3, 6, 10, if this status changes in the process of setting up disposal nodes, then we could immediately make the strategy and the give up of construction in Node 10, which will effectively avoid the waste of resource. Assume the disposal resource is enough to set up 4 disposal nodes, according to data in Tables 1 and 2, we could make the same selection this time, which means to select Nodes 1, 2, 3, 6. Assume the disposal resource is enough to set up 6 disposal nodes, according to data in Table 2, we should select Nodes 1, 2, 3, 6, 10, 13 as the location of disposal nodes instead of Nodes 1, 2, 3, 6, 10, 82, which is recommended in the data of Table 1 (because in the second status, Node 82 does not generate waste and we do not consider it as a generation node, therefore still refer to Table 1 is not sensible). In this case, we could accomplish the hazardous waste disposal of 43 + 5 = 48 generation nodes, i.e., accomplish 55.17% of the waste disposal. At this time, the nodes that are supposed to be selected have changed. The reason of this dynamic change is that the connections among nodes will change as the dynamic change of the distribution of generation nodes. Even the calculating result of this model, i.e., the location of disposal nodes, will change, either. Thus it can be seen that when the number and distribution of hazardous waste generation nodes change, according to the principle of this model, we could make the optimal decision in immediately, swiftly and accurately locating the disposal nodes. In the cases of generation nodes dynamic change, we realized the effective and immediate change of strategy, which assists in realizing making the optimal decision at all time. (2) Under the circumstance of completely disposal, If it is desired to accomplish 100% disposal, which means to cover all the generation nodes, then the minimum number of disposal nodes will decline from 32 to 23 (Nodes 1, 2, 3, 6, 7, 10, 13, 24, 40, 42, 45, 49, 64, 67, 71, 77, 78, 81, 87, 88, 89, 102, 106), which helps make the optimal decision immediately as well as avoid the waste of disposal resource. To sum up, this model realizes swiftly and accurately centralized disposal of industrial hazardous waste, we could see the advantages as follows: (1) It is good for saving the disposal resource and avoiding wastes of resource. (2) It guarantees the efficiency of hazardous waste disposal. The hazardous waste disposal level of temporary nodes is certified, which lower the possibility of inappropriate disposal of hazardous waste. (3) It is immediate and dynamic. The nodes which generate toxic wastes change from time to time, which is dynamic. This model
could swiftly and immediately identify the location of the optimal disposal node based on the distribution of hazardous waste generation nodes and the number of disposal nodes. (4) It is predictable. We could predict the location of the next disposal node that we need to add, which is good for making relevant preparation forward. 5. Conclusions In this paper, we used integrated programming (KPP-POS) and solved two aspects of problems based on the distribution of hazardous generation nodes in a certain time period: (1) assume only K temporary disposal nodes can be set up, we identified the optimal location of K temporary disposal nodes which maximum the covering range (i.e., the number of covered generation nodes); (2) assume it is desired to cover all the generation nodes, we identified the number of minimum disposal nodes and the optimal location. We solved the problem about how to swiftly, efficiently and accurately make the optimal temporary nodes construction decision under the circumstance of limited disposal resource or desirable complete disposal according to the practical situation of dynamic changes. Our study separately discussed the circumstance of limited disposal resource and desirable complete disposal, and both of them consistent to practical situation. The predictability of this model makes previous preparation for the construction of the next disposal node possible. Under the circumstance of dynamically changing hazardous waste generation nodes, the model in this paper could immediately consider the current situation and accurately make the optimal decision in a quickly response, which has strong practical significance. The limitation of this paper is that this model focuses on the prospective optimization of industrial hazardous waste disposal system, which is supposed to be combined with subsequent optimization model in order to optimize the entire management system. Next, this model is applicable to every disposal node and in condition of non-difference. In the future we should have further discussion under different situations of each node (output, disposal amount, disposal tech, etc.). In addition, it will have stronger practical significance if we discuss on the basis of specific practical situations and specific data, e.g., bring the factors such as the output of each hazardous waste generation node, the scale of risks of different hazardous waste, the costs of transportation distance, the disposal ability of each temporary disposal node (including the disposal amount and the sorts of hazardous waste that could be disposed, etc.) into consideration, under this circumstance, the result of the simulation will has stronger practical significance. Acknowledgments This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 71203153 and 70871085); Independent Innovation Foundation of Tianjin University (Grant No. 2013XS-0039); Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100032120031); Research Fund for the Public Resource Center of Tianjin University. The authors are grateful for these supports. References Alidi, A.S., 1992. An integer goal programming model for hazardous waste treatment and disposal. Appl. Math. Model. 16, 645–651. Alumur, S., Kara, B.Y., 2007. A new model for the hazardous waste location-routing problem. Comput. Oper. Res. 34, 1406–1423. Bonacich, P., 1972. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol. 2, 113–120.
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A covering model application on Chinese industrial hazardous waste management based on integer program method Lei Li ∗ , Shi Wang, Yanting Lin, Wenting Liu, Ting Chi School of Management and Economics, Tianjin University, Tianjin, China
a r t i c l e
i n f o
Article history: Received 29 December 2013 Received in revised form 28 March 2014 Accepted 8 May 2014 Keywords: Prospective optimization Industrial hazardous waste Integrated treatment and disposal center Covering location model Integer program method
a b s t r a c t Industrial hazardous waste is a special kind of toxic substance, which poses risks to the environment as well as human health. With the speeding up of industrialization in China, the collecting, recycling, treatment and disposal of industrial hazardous waste gradually become a severe problem to both the environmental protection and the resource management. In this paper, we laid an emphasis on prospective optimization, used integer program method, and selected the optimal locating approach to the collecting and handling of industrial hazardous waste on the basis of covering location model. We also selected an industrial intensive district in Hebei Province in China as an empirical object and examined our model result. Compared to previous results, this one bears the characteristics of immediacy, dynamism and predictability. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Industrial waste can be characterized as hazardous if they are on the national hazardous waste list or are identified as solid waste with hazardous properties according to national standards and approaches to identifying hazardous waste. It is a special kind of waste, which could be toxic, flammable, explosive, corrosive, chemical reactive, infectious and radioactive. It not only poses risks to the surrounding air, water and soil, but also do harm to the ecological environment and human health through diversified channels. Since the industrial and technical revolution, the increasing amount of industrial hazardous waste had brought extreme pressure to the environment as well as waste and damage to resources. These industrial wastes all bear the characteristics of chronicity, latency, invisibility, non-dilution and sensibility. This paper firstly analyzed some research approaches to the integrated treatment and disposal of industrial hazardous waste, then introduced in the covering model based on this analysis, and then conducted an empirical research in Hebei Province of China (Hebei Province is a vital industrial area of China and there are some heavy pollution plants in Hebei Province such as chemistry, Auto and paper. Here we selected an industrial intensive district in Hebei Province which could be a typical object for our study and be well applied our model on) and made an analysis of the result on the
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basis of integer program (KPP-POS), then gave some advices to the local government about the location of industrial hazardous waste treatment and disposal centers, and at last made conclusions and improvements based on the above analysis. 2. Previously relevant researches review 2.1. Review of researches in industrial hazardous waste management Industrial hazardous waste processing (generation, collection, transportation, storage, integrated utilization and ultimate disposal) is indefinite in both temporal and spatial respects, which makes the pollution control a big problem for environmental management. Developed countries (such as the USA, Japan and some European countries) are the main producers of industrial hazardous waste of the world. Their researches to industrial hazardous management are the most advanced, effective and have the longest history, which are the best references for China. For instance, Samanlioglu (2013) divided the integrated disposal center into collection center, initially processing center, recycling center and ultimate disposal center, and then selected the best route among centers and the optimal disposal approach using multiobjective linear programming model, which minimized the total cost and the transportation risk; he distributed the weights of risk by the importance of each district and selected the most reasonable transportation solution to minimize the total risk (Taha, 1971); in order to solve the connection problem
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between disposal node and generation node (make sure every generation node could connect to at least one disposal node), Gollowitzer and Ljubic´ (2011) used mixed integer programming model and denoted each node (disposal node and generation node) as a node in a mathematical tree to attain the optimal configuration solution; Church and Arfinkel (Zografos and Samara, 1989) introduced in integrated model, which gave an emphasis on simultaneously enlarging the distance between integrated disposal center and residential area while minimizing transportation cost. The above studies are all devoted in looking for the balance among high efficiency, low cost, and low risk, and the balance between connectivity and synthesis in transportation, or in other words, to select the best model under the restriction of a fixed environment or a rigid condition. But these studies all neglect the predetermined variable which controls the whole disposal process: the location of integrated disposal center, namely prospective optimization (Taha, 1971; Alidi, 1992). For instance, Ibel Alumer (Alumur and Kara, 2007) selects 13 relatively centric nodes from 92 waste generation nodes as disposal nodes on the basis of geological centrality, but the result is not optimal. If either the rigid condition (location problem of disposal node in this paper) or the mathematical approach of locating is improved, twice the result will be produced with half the effort.
2.3. The applicability of covering model in industrial hazardous waste disposal When it comes to public source and environmental management, what people care about is not the most applicable solution but the most desirable one, and the solution is measured by people’s desirability. Therefore, covering location model is often applied on public facility location problem, such as the location of firefighting point, emergency evacuation center, service center, public school, police office, library, hospital, and public buildings (Francis et al., 1974). In fact, when we construct an integrated industrial hazardous waste disposal center, what people care most is that all the hazardous waste in a given distance could be disposed immediately and meet their basic demand. Next, is looking for the optimal solution that minimize the distance and cost (Schilling et al., 1993). In addition, covering location model is more applicable to large-scaled, complex and dynamical location problems (Owen and Daskin, 1998; Schilling et al., 1993). All in all, this paper used the covering location model for industrial hazardous waste management because it is the best when applied on public facility location problem and can better measure people’s desirability when compared to other models mentioned above. 3. A covering model based on integer program (KPP-POS)
2.2. Location model Location model problem began in 1909. Alfred Weberk considered about how to select the location of a warehouse to minimize the total distance from customers; on the ground of that, Hakimi (1964) explored where to set up contact and help centers in the highway system to minimize the total distance between people and the nearest center. After him, the selection approach about static center location is classified into three models: median model, covering location model (Church and Velle, 1974) and center model (Owen and Daskin, 1998).
2.2.1. Median model Using multiobjective program or linear objective integer program to set up P disposal nodes in order to minimize the total distance between demands and disposal nodes, e.g., Church and Velle (1974) proposed that the efficiency of a disposal node is determined by the average distance from the demand to this very node; ReVelle refined the P-center model to either maximize the number of customers a disposal node could handle or maximize profits. This model is applicable to integer program or moderate-scaled locating cases.
2.2.2. Covering location model This model could be mainly divided into two directions: (1) when it is guaranteed that the given demand range is covered, how to configure the location and number of disposal nodes to minimize the cost (Schilling et al., 1993; White and Case, 1974); (2) when the resource is limited, how to configure the given number of disposal nodes to maximize the demand range (White and Case, 1974; Li et al., 2013).
2.2.3. Center location model On the basis of covering location model, in the condition of limited resource and uncertainty of distance that can be handled, how to locate to cover all the demand range and simultaneously minimize the total distance.
Before we introduce the integer programming method used to construct the covering model, we must firstly introduce the spirit of KPP-POS, which is the base of our integer programming mathematical model. 3.1. Principles of KPP-POS method Regard collection nodes as a network and identify the positional relationship among generation nodes by their connective relationship and geological positions. To measure the importance of the nodes is to identify their centrality. Centrality is a way to measure the relevant importance of a vertex in a network. Common centrality includes: Degree Centrality (Hakimi, 1964), Betweenness Centrality (Freeman, 1977), Closeness Centrality (Stephenson and Zelen, 1989), Eigenvector Centrality (Bonacich, 1972), Katz Centrality (Katz, 1953; Bonacich and Lloyd, 2001) and Pagerank Centrality (Page et al., 1998). However, these centralities are all measures of importance for nodes while the importance for node groups (Group Centrality) are somehow neglected. Realized the limitations of nodes centrality, Everett and Borgatti (1999) developed a new measure to group centrality for individuals, groups and classes by extending the standard network centrality measures. On the basis of the measures to the group centrality for the above objects, we can conclude two measures to the importance for node groups. In addition, Borgatti (2003) defined two new group measures for the identification of key players: Key Player Problem/Negative (KPP-NEG) and Key Player Problem/Positive (KPP-POS), which are used to measure the contribution of a node set in two different respects. KPP-POS could be defined as: given a node network, find out a set of n key players which are maximally connected to other nodes outside the set. This measure could be used to optimize the collection of waste by selecting the key players as seeds. While KPP-NEG on the contrary, removes the selected key nodes in a network to disrupt or fragment the network, and finally identifies and verifies the key players (Wen, 2012). In this paper, we will focus on the positive key problem. The figure below is an example of KPP-POS problem, which is totally inconsistent to intuition. If we select two nodes as a set to maximize
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D
A B
Because Aij is a binary matrix, i Aij ∧ 1 has the same value with maxi Aij . In mathematical programming, the first form is more popular because it could be denoted by linear programming constraints after we add some auxiliary variables, which we will discuss later. In order to remove the constraints the selected node set S, we now define a new auxiliary binary variable xi as follows:
J
G H
E
xi =
C
F
I
K
1 when i ∈ S 1 when i ∈ S¯
(3)
¯ then xj = 0 and 1 − xj = 1. If j ∈ S, then xj = 1 and 1 − xj = 0; if j ∈ S, With the help of auxiliary variable, we are able to remove the constraints of set S and rewrite the formula in the following forms:
L
Maximize Fig. 1. Example of KPP-POS problem.
S.T. the neighboring nodes in this set, then which two should we select? From the first sight, we should make the selection from Nodes D, E and F because they all have the maximum number of individual node degree (6 node degrees). However, these three nodes share the same neighboring nodes and select any two nodes from these three nodes will not increase their neighboring nodes. Nevertheless, though node I only has 5 nodes, the set of node I and node E has 8 neighboring nodes, which is more than the number of neighboring nodes formed by the set of Node D and Node E (Fig. 1).
i∈V
x i i
(1 − xi )(
The objective function could be written in the above form because: if i ∈ S, we could know that 1 − xi = 0. The first line of the ¯ objective function only considers nodes that are not in set S. If i ∈ S, then xi = 0. The second line of the objective function only covers nodes that are in set S. i xi is the number of nodes in set S. So it agrees with the value constraints. Plus, xi is binary. To simplify it, we use a matrix to describe the optimal problem as follows: (1 − x)t (Ax ∧ 1)
(1) G = (V, E) notates the network or figure; (2) V = (V1 , V2 , . . ., Vn ) notates the set of nodes; (3) E = (E1 , E2 , . . ., En ) notates the set of sides, among them, Ek = Vi , Vj is the connection between Node Vi and Node Vj . Here we assume the network figure is scalar, i.e., the side is shared:
S.T.
xt 1 ≤ s,
Or denote as: Maximize
|S| ≤ s
i
Aij ∧ 1 =
Maximize S.T.
0
when j and S are not connected
i∈V
(6)
mi
mi ≤ (1 − xi ), mi ≤ (1 − xi )Ai x,
(7)
(1)
This is an integer programming problem. To be clearer, S variable xi , mi are all binary. This problem could be well solved by CPLEX, and find the optimal solution in theory. What is more, if we loosen the last constraint, we get: Maximize S.T.
1 when j and S are connected
≤ s,
xi is 1 or 0
In this formula, Aij is an adjacent matrix, S is the selected node set. |S| is the base or value of set S. We will introduce mathematical programming methods based on this KPP-POS problem. This time the above problem could be denoted by formula as an integer programming problem (0–1 integer programming problem), defined as follows: ¯ which means j = maxi∈S Aij . We could describe For all nodes j ∈ S, this terminology as i Aij ∧ 1 because we are given that:
(1 − xi )(Ai x ∧ 1)
xt 1 ≤ s,
Formally, KPP-POS problem could be defined as: a set of size s is maximally connected to nodes outside the set. Therefore, we could describe KPP-POS problem again using the following Maximum target function.
S.T.
i∈V
xt 1
Change the objective function and add more constraints. The above problem could be concluded as:
3.3. Mathematical model
(maxi ∈ S Aij )
xi is 1 or 0
A = {aij , i = 1, 2, . . ., n, j = 1, 2, . . ., n} denotes the adjacent matrix of network figure G. Element {aij } in A is a binary variable. aij = 1 denotes that there is a side between Vi and Vj , aij = 0 denotes there is no side between Vi and Vj . Diagonal element in matrix A is 0.
j ∈ S¯
(5)
xi is 1 or 0
S.T.
(4) S notates the selected node set (5) v = |V |, notates the base or value of Set V (6) s = |S|, notates the base or value of Set S
(4)
xi ∈ {0, 1}
Maximize
Maximize
Aij xj ∧ 1)
≤ s,
3.2. Notation
Ek = Vi , Vj = Vj , Vi
j∈V
(2)
i∈V
mi
mi ≤ (1 − xi ), mi ≤ (1 − xi )Ai x, xt 1 xi2
(8)
≤ s,
− xi = 0
We could use Semi-Definite Programming (SDP) to estimate the upper bound of this problem. This is a slack problem we use when the calculation costs is very high. First, we will introduce the basic principles of SDP problem. Define Y = xxt , x = {x0 , x1 , . . ., xn } t (because x is the feasible region of formula (8) so any Y in conv(xxt ) is a feasible solution to formula
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(9)), and then we could treat this problem as a SDP problem to loosen: Maximize S.T.
i∈V
mi
mi ≤ 1 − Y0i , mi ≤ 1 − Yij − Y0i + Y0j ,
Y i 0i
≤ s,
Yii = Y0i ,
(9)
i = 0, 1, 2, . . ., n
0 ≤ mi ≤ 1,
i = 0, 1, 2, . . ., n
4. Demonstration 4.1. Background and assumption Realities of China at the moment could help better understand Chinese industrial hazardous waste management condition and we can accordingly adjust our model to make it more practically significant. Firstly, for long terms, Chinese economy development is an extensive development mode of high consumption and heavy pollution. Along with the further of industrialization, hazardous waste has become one of the main environmental pollutions and a significant contradiction for economy and society development. In 2005, China generated 11.62 million tons of industrial wastes in total, of which 43.4% is recycled, 33% is stored and 23% is safely discarded (Duan et al., 2008). Next, the amount of generation and emission of industrial waste are both increasing rapidly. Its large varieties, complex properties, wide generation source range and difficulties for management all make the industrial hazardous waste management and disposal a significant problem to China’s environmental management. What is more, hazardous waste management in China starts late. By 2005, there are only 177 synthesis disposal center in China and just 416,000 tons of industrial wastes had been disposed. The current problems of hazardous waste management in China mainly concentrate on knowing little about its hazard, nonstandard disposal, low efficiency centralized disposal, etc. A more efficient, rapid, systematical system of industrial hazardous waste management is needed. In the actual conditions, consider from the following five aspects: (1) The range and number of corporations that generate hazardous waste is huge and the distribution is scattered. The waste is hard for centralized disposal, and the environmental risks will inevitably increase if sent to qualified disposal site; (2) The equipment in producing units which are qualified for disposal is out of date. Their ability of disposing, processing and integrated utilization of industrial hazardous waste is limited, which could not realize the environmental protection and the fully utilization of resource; (3) Not all generation units are qualified for disposal and the level of disposal is low, which may cause problems of non-standard; (4) The high costs of handle and reuse, the complexity of management are all hard to realize scale economy; (5) The generation of hazardous waste is dynamic, which is hard to predict. We consider the centralized disposal as the optimal disposal method as well as immediately, swiftly and accurately identify the location of temporary disposal node according to the circumstances at that time, in order to realize immediately and efficiently disposing the hazardous waste and saving resource.
Problems need to be solved: (1) assume that all of the wastes are supposed to be disposed or, in other words, under the circumstance of all the generation nodes are desired to be covered, then how to determine the optimal site selection; (2) assume that the resource is limited, then how to determine the number and location of disposal nodes to cover a larger range of generation nodes. Based on the situation of industrial hazardous waste in an industrial intensive district in Hebei Province of China, we construct our model follow main principles as below: (1) Firstly count the number of nodes that generate wastes within a certain period of time; (2) Select hazardous waste generation nodes as nodes in the network figure; (3) Compromise the transportation costs and risks. In case of the leak when exceeding certain distances, we recommend to use 20-km as the safety distance and construct the sides of fundamental structural nodes on the basis of safety distance; (4) If the distance between two hazardous waste generation nodes is less than 20-km, there are attachable sides between nodes, and the transportation safety of hazardous waste could be guaranteed, or we say its transportation distance is within the allowable range of safety distance; (5) In another time period, if one node has no hazardous waste generation, then this node will not be considered as a hazardous waste generation node; (6) Consider the costs and control problem, temporary disposal nodes could help with centralized dispose, and at this time the location of disposal node can only be selected from hazardous generation nodes; 4.2. Result analysis and demonstration The calculating result of KPP-POS could be different because every time the nodes that generate hazardous waste are dynamically changing. 4.2.1. Result analysis of the first time period Assume in the first time period there are 108 corporations generating hazardous waste in total, i.e., there are 108 hazardous waste generation nodes, the distribution and reach ability are denoted in Fig. 2: If sorted by the node degree of each node, the top 10 nodes with the largest node degree are: Node 1, with 18 node degrees; Node 3, with 12 node degrees; Node 6, 10 node degrees; Node 2, 7 node degrees; Node 10, 6 node degrees; Node 13, 4 node degrees; and Nodes 45, 67, 82, 89 all have 4 node degrees. The calculation results of KPP-POS by mathematical programming are concluded in the following table: By observing and analyzing the data in Table 1, we could draw the following conclusions: (1) Under the circumstance of limited disposal resource, when k = 1, only one temporary disposal node could be set up. Obviously nodes with the largest node degree should be selected, which means Node 1 should be selected or the temporary disposal node should be set up in the location of Node 1. In this way, we could centrally collect and dispose 18 + 1 = 19 nodes which generate hazardous waste and accomplish the disposal of 19/108 = 17.6% of the total generation. When k = 2, all the disposal resource we have could only afford to set up 2 temporary disposal node. According to the result indicated in Table 1, if select Node 1 and Node 3, the number of reachable nodes is up to 28 + 2 = 30, which means if set up these two temporary disposal nodes in the location of Node 1 and Node 3, then we
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Fig. 2. 108 hazardous waste generation nodes and their sides.
Fig. 3. 87 hazardous waste generation nodes and their connected sides.
could accomplish the hazardous waste disposal of 30 generation nodes or to say we could accomplish 27.8% of the disposal. If we consider about adding one more disposal node on the basis of k = 2 in the future, we could make the prediction that this one more node will be in the location of Node 6 and then make relevant preparation based on this prediction. The rest can be done in the same manner. Consider the cost of resource by disposal nodes and the disposal radio for hazardous waste that could be accomplished, assume add one more disposal node, if the disposal ratio increase is not less than 8% compared to the previous ratio, we consider it is worthy to add this one more node; if the increase of disposal ratio is less than 8% but not less than 4.7% compared to the previous ratio, we consider it is worthy to think about adding this one more node; if the increase of disposal ratio is less than 5% compared to the previous ratio, we do not recommend to add this one more node. According to the data in Table 1, we consider it is worthy to add at most 3 more disposal nodes; it is worthy to think about adding at most 6 disposal nodes and we do not recommend adding more nodes because the utilization efficiency of the additional disposal resource is low. (2) Under the circumstance of completely disposal, If it is desired to completely dispose all the hazardous waste that corporations have generated and accomplish 100% disposal, which means to cover all the generation nodes, we need to set up 32 disposal nodes
and in the location of Nodes 1, 2, 3, 6, 10, 13, 24, 37, 41, 42, 43, 45, 46, 49, 53, 58, 65, 67, 69, 71, 77, 78, 81, 82, 86, 87, 89, 90, 100, 103, 104 and 106. In this case, we should use the strategy of setting up 32 disposal nodes. If the number of disposal nodes is more than 32, it will lead to the waste of disposal resource; if the number is less than 32, it will not meet the desirability of 100% disposal. 4.2.2. Comparison and analysis of the results of the second time period and the first time period Assume in the second time period, there are 87 corporations generating hazardous waste, which means there are 87 generation nodes. The distribution is indicated in Fig. 3. The calculating results of KPP-POS by mathematical programming are concluded in the following table. Assume the number of hazardous waste generation nodes decline from 108 (the first time period) to 87 (the second time period), which means the circumstance transfers from Figs. 2 and 3. In this case, by comparing the data in Tables 1 and 2, we could draw the following conclusions: (1) Under the circumstance of limited disposal resource, when k = 8, besides Nodes 1, 2, 3, 6, 10, 13, 89 which are among the top 8 nodes with the largest node degree, we select Node 67 instead of Node 45 which is also one of the top 8 nodes. Because in Fig. 2 we could see that Node 45 and Node 6 share some neighboring nodes and selecting Node 45 will not dramatically increase the
Table 1 Number of nodes that could be reached within a length of 1 from set 1 to set 10 (108 generation nodes).
Table 2 Number of nodes that could be reached within a length of 1 from set 1 to set 10 (87 generation nodes).
K 1 2 3 4 5 6 7 8 9 10
Reachable nodes (m) 18 + 1 28 + 2 36 + 3 42 + 4 47 + 5 51 + 6 55 + 7 58 + 8 61 + 9 64 + 10
Selected nodes
Completed processing ratio (m/108)
K
{1} {1, 3} {1, 3, 6} {1, 2, 3, 6} {1, 2, 3, 6, 10} {1, 2, 3, 6, 10, 82} {1, 2, 3, 6, 10, 67, 82} {1, 2, 3, 6, 10, 13, 67, 82} {1, 2, 3, 6, 10, 13, 45, 67, 82} {1, 2, 3, 6, 10, 13, 45, 67, 82, 89}
17.59% 27.78% 36.11% 42.59% 48.15% 52.78% 57.41% 61.11% 64.81% 68.52%
1 2 3 4 5 6 7 8 9 10
Reachable nodes (m) 14 + 1 24 + 2 33 + 3 39 + 4 43 + 5 47 + 6 50 + 7 52 + 8 54 + 9 56 + 10
Selected nodes
Completed processing ratio (m/87)
{1} {1, 6} {1, 3, 6} {1, 2, 3, 6} {1, 2, 3, 6, 10} {1, 2, 3, 6, 10, 13} {1, 2, 3, 6, 10, 67, 89} {1, 2, 3, 6, 10, 13, 67, 89} {1, 2, 3, 6, 10, 13, 67, 78, 89} {1, 2, 3, 6, 10, 13, 25, 67, 78, 89}
17.24% 29.89% 41.38% 49.43% 55.17% 60.92% 65.52% 68.97% 72.41% 75.86%
Please cite this article in press as: Li, L., et al., A covering model application on Chinese industrial hazardous waste management based on integer program method. Ecol. Indicat. (2014), http://dx.doi.org/10.1016/j.ecolind.2014.05.001
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neighboring nodes. Consider the interruption of repeated reachable nodes, it will actually increase 3 more neighboring nodes, which is the same if we select Node 67. So we could see it is the same thing to select either of these two nodes. Assume there is only one disposal node now, then the number of disposal nodes that we consider is worthy to add increases from 2 to 3, the number of disposal nodes that we consider is worthy to think about adding increases from 5 to 6. Assume the government determines whether to set up disposal nodes by whether it is worthy to do it, and requires to dispose as much hazardous waste as possible, if the status changes in the process of setting up disposal nodes, then the optimal plan changes from the previous Nodes 1, 3 to Nodes 1, 3, 6, the strategy we use should be the construction of nodes in the location of Node 6, which could help make the optimal decision and accomplish our original target swiftly and efficiently. If it is desired to accomplish at least 50% disposal of the hazardous waste, then the number of disposal nodes that we consider is worthy to add will decline from 6 to 5, the selected nodes changes from Nodes 1, 2, 3, 6, 10, 82 to Nodes 1, 2, 3, 6, 10, if this status changes in the process of setting up disposal nodes, then we could immediately make the strategy and the give up of construction in Node 10, which will effectively avoid the waste of resource. Assume the disposal resource is enough to set up 4 disposal nodes, according to data in Tables 1 and 2, we could make the same selection this time, which means to select Nodes 1, 2, 3, 6. Assume the disposal resource is enough to set up 6 disposal nodes, according to data in Table 2, we should select Nodes 1, 2, 3, 6, 10, 13 as the location of disposal nodes instead of Nodes 1, 2, 3, 6, 10, 82, which is recommended in the data of Table 1 (because in the second status, Node 82 does not generate waste and we do not consider it as a generation node, therefore still refer to Table 1 is not sensible). In this case, we could accomplish the hazardous waste disposal of 43 + 5 = 48 generation nodes, i.e., accomplish 55.17% of the waste disposal. At this time, the nodes that are supposed to be selected have changed. The reason of this dynamic change is that the connections among nodes will change as the dynamic change of the distribution of generation nodes. Even the calculating result of this model, i.e., the location of disposal nodes, will change, either. Thus it can be seen that when the number and distribution of hazardous waste generation nodes change, according to the principle of this model, we could make the optimal decision in immediately, swiftly and accurately locating the disposal nodes. In the cases of generation nodes dynamic change, we realized the effective and immediate change of strategy, which assists in realizing making the optimal decision at all time. (2) Under the circumstance of completely disposal, If it is desired to accomplish 100% disposal, which means to cover all the generation nodes, then the minimum number of disposal nodes will decline from 32 to 23 (Nodes 1, 2, 3, 6, 7, 10, 13, 24, 40, 42, 45, 49, 64, 67, 71, 77, 78, 81, 87, 88, 89, 102, 106), which helps make the optimal decision immediately as well as avoid the waste of disposal resource. To sum up, this model realizes swiftly and accurately centralized disposal of industrial hazardous waste, we could see the advantages as follows: (1) It is good for saving the disposal resource and avoiding wastes of resource. (2) It guarantees the efficiency of hazardous waste disposal. The hazardous waste disposal level of temporary nodes is certified, which lower the possibility of inappropriate disposal of hazardous waste. (3) It is immediate and dynamic. The nodes which generate toxic wastes change from time to time, which is dynamic. This model
could swiftly and immediately identify the location of the optimal disposal node based on the distribution of hazardous waste generation nodes and the number of disposal nodes. (4) It is predictable. We could predict the location of the next disposal node that we need to add, which is good for making relevant preparation forward. 5. Conclusions In this paper, we used integrated programming (KPP-POS) and solved two aspects of problems based on the distribution of hazardous generation nodes in a certain time period: (1) assume only K temporary disposal nodes can be set up, we identified the optimal location of K temporary disposal nodes which maximum the covering range (i.e., the number of covered generation nodes); (2) assume it is desired to cover all the generation nodes, we identified the number of minimum disposal nodes and the optimal location. We solved the problem about how to swiftly, efficiently and accurately make the optimal temporary nodes construction decision under the circumstance of limited disposal resource or desirable complete disposal according to the practical situation of dynamic changes. Our study separately discussed the circumstance of limited disposal resource and desirable complete disposal, and both of them consistent to practical situation. The predictability of this model makes previous preparation for the construction of the next disposal node possible. Under the circumstance of dynamically changing hazardous waste generation nodes, the model in this paper could immediately consider the current situation and accurately make the optimal decision in a quickly response, which has strong practical significance. The limitation of this paper is that this model focuses on the prospective optimization of industrial hazardous waste disposal system, which is supposed to be combined with subsequent optimization model in order to optimize the entire management system. Next, this model is applicable to every disposal node and in condition of non-difference. In the future we should have further discussion under different situations of each node (output, disposal amount, disposal tech, etc.). In addition, it will have stronger practical significance if we discuss on the basis of specific practical situations and specific data, e.g., bring the factors such as the output of each hazardous waste generation node, the scale of risks of different hazardous waste, the costs of transportation distance, the disposal ability of each temporary disposal node (including the disposal amount and the sorts of hazardous waste that could be disposed, etc.) into consideration, under this circumstance, the result of the simulation will has stronger practical significance. Acknowledgments This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 71203153 and 70871085); Independent Innovation Foundation of Tianjin University (Grant No. 2013XS-0039); Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100032120031); Research Fund for the Public Resource Center of Tianjin University. The authors are grateful for these supports. References Alidi, A.S., 1992. An integer goal programming model for hazardous waste treatment and disposal. Appl. Math. Model. 16, 645–651. Alumur, S., Kara, B.Y., 2007. A new model for the hazardous waste location-routing problem. Comput. Oper. Res. 34, 1406–1423. Bonacich, P., 1972. Factoring and weighting approaches to status scores and clique identification. J. Math. Sociol. 2, 113–120.
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