Introduction

In recent years, with the rapid economic development and the accelerated urbanization, China is facing an increasing pressure on resource, environment, and ecology caused by a rapid industrial development and revolution (Su et al. 2016). The rise of the frequency and intensity of river chemical spills caused by potential industrial pollution risks requires a strong emphasis on improving our capabilities to make a quick response and alleviate the negative impacts on the environment (Jiang et al. 2012; Liu et al. 2016b; Shi et al. 2014). Emergency material allocation (EMA), which mainly deals with how to efficiently allocate emergency materials to pollution accident sites and satisfy the requirements of emergency rescues before the pollutants disperse in large-scale, plays a fundamental and essential role in decision-making process dealing with river chemical spills in environmental emergency management (Liu et al. 2016a). Therefore, a scientific and reasonable EMA scheme becomes critical for an effective actual emergency decision-making process.

Research on EMA has gained much attention recently in order to obtain optimizing emergency response schemes for emergency rescue to minimize the emergency response time or system cost (Huang and Fan 2010; Liu et al. 2016c). Barbarosoğlu and Arda (2004) developed a two-stage stochastic programming model to plan the first-aid commodity allocation and transportation scheme for disaster-affected areas; Chang et al. (2007) developed two stochastic programming models to allocate rescue resource for flood emergency logistics under scenario planning; Tzeng et al. (2007) constructed a multi-objective relief-distribution model to allocate relief material effectively; Zhang et al. (2012) presented a heuristic algorithm to efficiently solve emergency resource allocation problem dealing with possible secondary disasters; Abounacer et al. (2014) proposed an epsilon-constraint method to deal with the allocation of aid from aid distribution centers to demand points for disaster response; Wex et al. (2014) developed a corresponding decision support model to allocate rescue units and identify routing scheme based on different incident severities in disaster management; Su et al. (2016) proposed an emergency resource allocation model for multiple concurrent incidents caused by natural disaster; Fontem et al. (2016) developed a decomposition-based heuristic method to allocate emergency supplies and identify routing scheme according to the increased threat of natural disasters.

In general, EMA for natural disasters, such as earthquakes and floods, mainly deals with how to efficiently and quickly allocate emergency resources and equipment from supportive centers to disaster incident sites in order to reduce casualties, protect economic property, and maintain social stability (Khayal et al. 2015; Zhang et al. 2016). Therefore, most of the studies on EMA for natural disasters focus on routing selection and vehicle deployment considering various incident risk/severity scenarios to find a shortest path and available vehicle according to road situations from supportive centers to an incident site. However, EMA for river chemical spills primarily aims at identifying optimizing emergency material allocation schemes to minimize the response time and providing differentiated emergency materials to satisfy the requirements of emergency rescues before the pollutants disperse in large-scale. In general, it is not influenced by road situations and vehicle constraint. Moreover, with the unpredictability of river chemical spills, decision-making process for EMA is generally characterized by inherent uncertainty and complexity, which poses further challenge as emergency material demand and supply may vary over time in terms of material type and quantity (Zhou and Reniers 2016). Therefore, a reasonably dynamic EMA is an indispensable decision-making process to improve emergency rescue capability and satisfy the material requirements, especially under the time-varying features of the environmental conditions, in order to reduce environmental negative impacts and economic losses caused by river chemical spills.

Appropriate regular resource allocation is a dynamic optimization decision-making process considering the balance between resource demand and supply relationship. In recent years, various dynamic optimization methods have been successfully applied in many applications. Sheu (2007) proposed a logistics distribution method for the urgent requirements for disaster-affected areas considering time-varying relief demand and supply; Wang et al. (2011) developed a genetic algorithm to address task allocation problem of a two-echelon supply chain against stochastic demand; Omar et al. (2013) proposed a just-in-time (JIT) manufacturing system to obtain the supply and the delivery scheme of raw materials considering customer demand rate is linearly decreasing and time-varying; Govindan (2015) developed a mathematical model to schedule purchase orders for inventory replenishment with a time-varying stochastic demand in a two-echelon supply chain system for a minimum system cost objective; Krishna Priya and Bandyopadhyay (2015) identified an optimum mix of various supply equipment for meeting the time-varying demand to reduce the overall cost of the system; Amini Salehi et al. (2016) defined a stochastic robustness method to facilitate resource allocation in a dynamic environment, maximizing the number of tasks to meet their individual deadlines; Luscombe and Kozan (2016) proposed a dynamic scheduling framework to provide real-time support in order to manage and allocate the scarce resources for health care service.

However, most present studies on dynamic optimization decision-making models concentrate more on resource allocation for regular management than emergency material allocation for environmental emergency management. Generally, it is more sensible for EMA with an objective of minimizing emergency response time, rather than minimizing total system cost for river chemical spills (Liu et al. 2017; Mohamadi and Yaghoubi 2017; Quinn and Jacobs 2007; Zhao and Chen 2015). Meanwhile, the emergency material allocation scheme for emergency decision-making should be characterized by flexibility and diversity in order to deal with different pollution conditions. Therefore, developing a dynamic optimization EMA model to accommodate the special characteristic of river chemical spills has more practical significance.

Therefore, the objective of this study is to develop an emergency material allocation framework based on time-varying supply-demand constraint to appropriately allocate emergency material and effectively minimize the emergency response time. The proposed emergency material allocation algorithm (EMAA) can help decision-makers identify optimizing emergency material allocation scheme under time-varying and scarce emergency material supply-demand conditions for the actual emergency rescue decision-making. The paper is organized as follows: the Methodology section presents the development of emergency material allocation system and the corresponding algorithm. The Application of emergency material allocation framework section describes a computational experiment and illustrates results and discussions, where emergency material allocation schemes based on a time-varying allocation process are analyzed. The Conclusions section gives some conclusions.

Methodology

The emergency rescue for emergency pollution accidents should spare no effort to coordinate emergency material allocation in a time-effective manner rather than in an economical effective manner (Fontem et al. 2016; Govindan 2015; Wang et al. 2011). Meanwhile, emergency material allocation for emergency pollution accidents is also a complicated task due to the diversity of pollution conditions and time-varying material supply and demand constraints. Therefore, the developed model can have a certain dynamic characteristic, that is, the supply amount of emergency material in the supportive center and the demand amount of emergency material in the demand point increase with the pollution condition changes. This dynamic structure can easily analyze the impacts of the different conditions on emergency pollution accidents.

Emergency material allocation framework under time-varying supply-demand constraint

In this study, an extremely harsh pollution condition is considered that the pollution source of a river chemical spill is not intercepted timely. The demand amount of emergency material in a pollution-affected site is characterized by a time-varying linear increase based on the chemical spill rate. Hence, emergency material, reserved in the pre-planned warehouses for emergency incident, is inevitably in such a scarce supply state. And the supply amount of emergency material requires a dynamic additional supplement. The replenishment policy for emergency material in the pre-planned warehouses with time-varying supply also approximately fits a linear characteristic based on the chemical spill rate. In this study, the system objective for emergency material allocation is minimizing the emergency response time to make sure that the whole emergency rescue system can support demand point B enough emergency material with a dynamic material replenishment process. The emergency material allocation pattern with single demand point and multi-supportive center is appropriate for only when a single spill incident occurs in water systems, such as slow-flowing rivers, lakes, and reservoirs.

As an emergency material scheduling problem with single demand point and multi-supportive center based on time-varying supply and demand constraint, let A = {A1,  A2, …,  An} be a finite set of potential supportive centers. And both the supply amount xi of emergency material in the supportive center i (i = 1,  2,  …,  n) and the demand amount y of emergency material in the demand point B increase over time. Hence, emergency rescue capability of the supportive center i can be expressed as xi = ai + biti, where ai is the initial storage amount of emergency material in the supportive center i and bi is the change rate of emergency material. Meanwhile, pollution condition of the demand point B can be expressed as y = c + et, where c is the initial demand amount of emergency material in the demand point B and e is the change rate of emergency material. In fact, the initial storage amount of emergency material for all the supportive centers can be expressed as \( \sum \limits_{i=1}^n{a}_i \), if \( \sum \limits_{i=1}^n{a}_i<c \); all the supportive centers cannot guarantee the requirement of emergency material for the demand point B and need some time to replenish the related emergency material.

Emergency material allocation algorithm

In the emergency rescue system, it is supposed that the requirements of the demand point B can be satisfied at time t = T, which is the theoretically critical emergency response time. Hence, supportive center i has (T − di) hour as the replenishment time to prepare additional emergency materials. Hence, the storage amount of emergency material xi in the supportive center i can be expressed as

$$ {x}_i={a}_i+{b}_i\cdot \left(T-{d}_i\right) $$
(1)

where ai is the initial storage amount of emergency material in the supportive center i, bi is the change rate of emergency material, di is the transportation time from supportive center i to the demand point B, and T is the emergency response time for whole emergency rescue system. For the demand point B, multiple supportive centers are supposed to allocate emergency material for demand point B. Hence, the time when the last supportive center arrives at demand point B and the requirement of emergency material for the demand point B can be guaranteed is the emergency response time for whole emergency rescue system.

The demand amount of emergency material y of the demand point B can be expressed as

$$ y=c+e\cdot T $$
(2)

where c is the initial demand amount of emergency material in the demand point B and e is the change rate of emergency material.

It is assumed that all the potential supportive centers are supposed to allocate emergency materials to the demand point B. In order to satisfy the requirements of the demand point, \( \sum \limits_{i=1}^n{x}_i=y \). Hence,

$$ \sum \limits_{i=1}^n\left[{a}_i+{b}_i\cdot \left(T-{d}_i\right)\right]=c+e\cdot T $$
(3)

Thus, the theoretically critical emergency response time T can be obtained and expressed as follows:

$$ T=\frac{\sum \limits_{i=1}^n{a}_i-\sum \limits_{i=1}^n{b}_i{d}_i-c}{e-\sum \limits_{i=1}^n{b}_i} $$
(4)

However, some potential supportive centers are distributed too far away from the pollution-affected site (demand point B) and yield to guaranteeing an effective emergency rescue for the demand point B in time T (di > T). Hence, the theoretically critical emergency response time T is not the practically critical emergency response time in the whole rescue system.

Then, an enumeration method is applied to obtain the emergency material allocation scheme. Let T′ represent the practically critical emergency response time (T′ < T). Step 1, sort the potential supportive centers i in ascending order according to their di values. And select a series of supportive centers which meet the relationship (di < T). Step 2, let \( {d}_i^{\prime } \) be the transportation time of the ith supportive center after the sort. In the new set of supportive centers, it is assumed that T′ and \( {d}_i^{\prime } \) meet the relationship (\( {d}_i^{\prime }<{T}^{\prime }<{d}_{i+1}^{\prime } \)). Hence, the number of supportive centers (j) which is qualified to support the emergency materials to the demand point can be obtained. Step 3, put the number of supportive centers (j) into Eq. (5) and calculate the practically critical emergency response time T′ according to Eq. (5).

$$ \left\{\begin{array}{c}{T}^{\prime }=\frac{\sum \limits_{i=1}^j{a}_i-\sum \limits_{i=1}^j{b}_i{d}_i-c}{e-\sum \limits_{i=1}^j{b}_i}\\ {}{d}_i^{\prime }<{T}^{\prime }<{d}_{j+1}^{\prime}\kern3.5em \end{array}\right. $$
(5)

If T′ does not satisfy the constraints (\( {d}_i^{\prime }<{T}^{\prime }<{d}_{i+1}^{\prime } \)), a new relationship (\( {d}_{i\hbox{-} 1}^{\prime }<{T}^{\prime }<{d}_i^{\prime } \)) would be built. And the iteration of Step 2 and Step 3 would be continued until the calculated T′ meets Eq. (5). And, if T′ satisfies the constraints (\( {d}_i^{\prime }<T<{d}_{i+1}^{\prime } \)), the assumption is valid. Then, according to Eq. (2), the practical supply amount of emergency material \( {x}_i^{\prime } \) in the supportive center i can be expressed as

$$ {x}_i^{\prime }={a}_i+{b}_i\cdot \left({T}^{\prime }-{d}_i\right) $$
(6)

Hence, the practical demand amount of emergency material in the demand point B can be expressed as

$$ \sum \limits_{i=1}^j{x}_i^{\prime }=\sum \limits_{i=1}^j\left[{a}_i+{b}_i\cdot \left({T}^{\prime }-{d}_i\right)\right] $$
(7)

The algorithm is presented in Fig. 1 and explained as follows:

Fig. 1
figure 1

The framework schema of emergency material allocation algorithm

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The pseudo-code for EMAA can be shown in Algorithm 1.

figure a

Application of emergency material allocation framework

Overview of the study region

The south-to-north water transfer is a large-scale national project to optimize regional water resource utilization and allocate water resource from Jiangsu province to Beijing due to the shortage and uneven distribution of water resources in China. However, various hazardous chemical industries are located in Jiangsu province and have threatened to the project channel and related rivers. Therefore, water supply security in Jiangsu province is critical to guarantee the effective implementation of the project. Meanwhile, appropriate EMA decision-making for Jiangsu province in order to carry out emergency rescue in a quick response dealing with river chemical spills can play an important role in ensuring water quality protection and supply security, reducing economic damage and maintaining social stability for the implementation of the project. In this study, the distribution of representative hazardous chemical industries (risk sources) and feasible emergency material warehouses in Jiangsu province is shown in Fig. 2.

Fig. 2
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The distribution of risk source and emergency material warehouse in study area

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The information of representative potential risk sources and pre-planning emergency material warehouses are shown in Tables 1 and 2 (Liu et al. 2016a). WZ12, WZ19, XJ1, XJ2, and XJ3 are emergency material warehouses in level I and would give priority to emergency rescue to risk sources within 2 h, and WZ2, WZ9, WZ15, WZ18, and WZ22 are emergency material warehouses in level II, guaranteeing to allocate emergency materials to risk sources within 3 h. And related emergency materials and equipment, such as activated carbon, activated alumina, ferrous sulfate, sacks, pontoons, oil containment boom, and so on, are pre-stored in these ten emergency material warehouses and kept in a ready-to-be-used state.

Table 1 The information of potential risk sources
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Table 2 The distribution of pre-planned emergency material warehouses
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Results and discussion

In this study, the proposed emergency material allocation framework is applied to a computational experiment for EMA decision-making in case of river chemical spills in Jiangsu province according to the above information. As a computational experiment in this study, it is supposed that risk source FXY6 occurs river chemical spill and the pollution source of a river chemical spill is not intercepted timely. Hence, effective decision-making for emergency rescue before the pollutants disperse in large-scale is particularly important. The system objective for emergency material allocation is minimizing the emergency response time to make sure that the whole emergency rescue system can allocate risk source FXY6 enough emergency material with a dynamic material replenishment process. The demand amount of emergency material in pollution-affected site is characterized by a time-varying linear increase based on the chemical spill rate, which is expressed as y = 1800 + 3000T. And the supply amount of emergency material requires a dynamic additional replenishment. The supply amount of emergency material in supportive center Ai also approximately fits a linear relationship based on the chemical spill rate, which is expressed as xi = ai + 1500(T − di). For emergency material sacks, as an example, the initial demand (c) in risk source FXY6, the initial storage amount (ai) in emergency material warehouses, and the transportation time (di) from risk source FXY6 to emergency material warehouses are shown in Table 3.

Table 3 The input information of computational experiment
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The results of emergency material allocation scheme

In this study, the theoretically critical emergency response time T is obtained for the emergency material allocation system based on Eq. (4) and T = 1.932 h. Some potential supportive centers are eliminated which are distributed too far away from demand point J = 6 and yield to guaranteeing an effective emergency rescue for the demand point J = 6 in time T. The emergency material warehouses which can be selected as supportive centers and allocate emergency materials to the incident should meet the relationship di < T. Then, a new series dnewj is built, sorting the potential supportive centers i in ascending order according to their transportation times di. And the new series \( {d}_i^{\prime } \) contains emergency material warehouses WZ12, WZ18, WZ15, and WZ9. Finally, the practically critical emergency response time T′ is calculated by the enumeration method according to Eq. (5) and T′ = 1.7425 h. Finally, the emergency material warehouses selected as supportive centers and the supply amount of the sacks can be calculated by Eqs. (6) and (7). Hence, emergency material allocation Scheme 1, as the optimizing scheme in the allocation system, is obtained and shown in Table 4, with an emergency response time at 1.7425 h. Due to relatively distant location and low storage of WZ9, WZ9 are supposed to play only a minor role in the whole allocation system. Therefore, emergency material allocation Scheme 2 is calculated by EMAA without the participation of WZ9, obtaining an emergency response time at 2.1 h. Meanwhile, if a river chemical spill is rather serious, the nearest supportive center WZ12 can be selected and allocate emergency materials for an initial system rescue based on its initial storage amount of emergency material without a dynamic material replenishment which results in a relatively high emergency response time at 2.54 h shown in emergency material allocation Scheme 3.

Table 4 The results of emergency material allocation schemes
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The comparison for each supportive center

On the whole, Fig. 3 compares the current condition in the studied area with the solution that the model has proposed, such as the comparison between the initial storage amount (ai) of emergency material and supply amount (xi) of emergency material in different emergency material warehouses and the comparison between the transportation time (di) and the practically critical emergency response time (T′). Meanwhile, the comparison of emergency material allocation for each supportive center among three schemes is also shown in Fig. 3. The emergency material warehouses WZ12 and WZ 15 play an important role in the actual emergency rescue process with relatively supply amount of emergency materials. However, the emergency material warehouses WZ12 and WZ18 have significant advantages in replenishing emergency materials based on their excellent locations. Meanwhile, the comparison of three emergency material allocation schemes is shown in Fig. 4. In terms of emergency response time and total supply amount of emergency materials, emergency material allocation Scheme 1, as the optimizing scheme in the allocation system, minimizes the emergency response time to make sure that the whole emergency rescue system can meet the time-varying requirements of the emergency materials with a dynamic emergency decision-making process. However, although emergency material allocation Scheme 2 requires a higher emergency response time and supply amount of emergency materials for the actual emergency decision-making, it only needs three emergency material warehouses (WZ12, WZ15, and WZ18) as supportive centers to meet the emergency material requirements of demand point J = 6 and can, in some extent, reduce the total rescue system cost and avoid potential uncertain factors in the rescue process. As for emergency material allocation Scheme 3, the emergency response time and supply amount of emergency materials compared to Scheme 1 inevitably are affected. Nonetheless, the emergency materials allocated by WA12 to risk source J = 6 can be supplied at the first time and emergency treatment technology, such as adsorption dam, can be built to prevent the pollutions from dispersing in large-scale, waiting for the follow-up emergency material support from other supportive centers. Therefore, the decision-makers can make a flexible selection among the three emergency material allocation schemes according to the different emergency pollution conditions and external environment influences.

Fig. 3
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The comparison of emergency material allocation for each supportive center

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Fig. 4
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The comparison of three emergency material allocation schemes

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Conclusions

In this study, an emergency material allocation framework is developed for the actual emergency rescue decision-making in response to river chemical spills. The proposed approach considers dynamic features of pollution environments so as to meet the requirements of emergency material allocation in river chemical spills and help the decision-makers to find the most suitable task allocation scheme in a quick response. The proposed framework is then applied to a computational experiment for emergency material allocation decision-making in Jiangsu province. And the results put forward the three optimizing emergency material schemes to assist decision-makers in implementing different optimizing emergency material allocation strategies coping with varying emergency pollution conditions in keeping a balance between the response time and the emergency rescue cost.

The results suggested that the developed framework was effective in reflecting dynamic and uncertainty characteristics in the actual emergency rescue decision-making process and demonstrated that (a) the developed framework can tackle the dynamic emergency material allocation problem and obtain emergency material allocation schemes under a time-varying supply-demand constraint for the actual emergency rescue decision-making; (b) the developed framework minimizes the emergency response time, satisfying time-varying emergency material demand for the whole emergency rescue system in an effective emergency response; and (c) the developed framework can help decision-makers to identify an appropriate emergency material allocation scheme with flexible decision-making according to different emergency pollution conditions with time-effective and cost-effective manners.

However, two following improvements are recommended for future studies. First, the linear characteristic to describe time-varying supply-demand relationship may require future improvements in considering more complexities expressed as fuzzy set, possibilities, and stochastic to support the developed framework for satisfying real-world applications. Thus, it can be used as an efficient tool for describing a nonlinear characteristic dealing with river chemical spill problems. Second, the developed framework has been proposed for a single pollution-affected site. It is also necessary to advance a dynamic optimization method to tackle these emergency material allocation problems for emergency rescue among multiple pollution-affected sites which may be influenced by pollution accidents in the same time.