Abstract
Due to the lack of medical materials in some emergency public events, for example, the outbreak of COVID-19, it is urgent to establish a medical emergency material warehouse. Taking Xi’an, China, as an example, this study aims to select suitable sites of Xi’an medical emergency material warehouse. In this study, the problem of site selection models as a multiobjective optimization problem. The coverage function and comprehensive efficiency function are designed as two conflicting objectives. Then, a multiobjective evolutionary algorithm based on multiple memetic direction is proposed to optimize the two objectives concurrently. The crossover and mutation operators are designed for evolutionary multiobjective site selection. The proposed crossover operator is able to balance the global and local search abilities, and the proposed mutation operator fuses the distribution information of hospital location, service population, and the overall coverage. Experiments on real dataset verify the superiority of the proposed evolutionary multiobjective site selection method.
1. Introduction
Xi’an is the capital city of Shaanxi Province, China. The construction of emergency material warehouse can effectively improve the ability to prevent and deal with various medical and health emergencies of Xi’an and even the whole Shaanxi Province. At present, most of the medical reserves are traditional Chinese medicine, chemicals, antibiotics, and biochemical drugs, and there is no special medical material warehouse for public health emergencies in Xi’an. In case of public health emergencies, the reserve of emergency materials is prone to be in short supply. For example, the masks were in short supply when COVID-19 broke out [1]. Therefore, a medical emergency material warehouse urgently needs to be established for emergencies in Xi’an. Moreover, due to the different attributes of emergencies and the different types or quantities of emergency relief materials, the warehouse should have the clear and stable characteristics of service functions, so as to facilitate site selection according to its specific service functions. Therefore, the key to improve the emergency supply capacity of medical materials in Xi’an is to comprehensively consider the types of emergencies, population distribution, and the matching relationship of required emergency materials, so as to realize the optimal site selection of emergency material warehouse.
A warehouse for emergency supplies is a crucial component of the emergency aid system. A specific amount and variety of disaster relief materials are kept in China’s network of emergency material warehouses. The design of warehouses for relief supplies still has several issues despite their crucial role in disaster rescue. Therefore, it is crucial to investigate the site selection of the emergency material warehouse [2]. Since Hakimi et al. studied the site selection of numerous facilities in a network [3], academics have conducted a great deal of research on facility site selection, propelling the theoretical study of facility site selection into prosperity. How to choose the warehouse location for time-sensitive aeronautical emergency material was examined by Liang and Tu [4]. Based on the set covering principle, a warehouse placement selection model for aviation emergency supplies has been developed. Demirel et al. [5] used a multicriteria approach based on Choquet integral for site selection. Choquet is essential to a large Turkish logistic company’s actual problem of choosing a warehouse location. In [6], the authors compared the outcomes of the analytic hierarchy process (AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Elimination and Choice Expressing Reality (ELECTRE), and generalized regression analysis (GRA) for the warehouse selection problem. A new balancing and ranking method along with an interval data approach was proposed by Malmir et al. to solve the warehouse location selection problem [7]. This method uses a three-step process to determine the overall complete final order of the warehouses that have already been chosen for the decision-making process.
In this study, the problem of site selection is first modeled as a multiobjective optimization problem (MOP) [8–12]. The coverage is considered as the first objective, which means that only a limited number of emergency warehouses can be established due to cost constraints and material aggregation effect. Furthermore, the comprehensive efficiency of the use of emergency medical materials is considered as the second objective. In the previous research studies, an MOP can be converted into a single objective optimization problem by the aggregation method with some regularization parameters. Multiobjective evolutionary algorithms (MOEAs) are able to optimize these objectives concurrently and achieve a group of nondominated solutions [13–21]. It has been demonstrated that MOEAs can achieve the best-so-far theoretically guaranteed approximation performance in some NP-hard problems, such as minimum set cover problem [22], change detection [23, 24], minimum cost coverage problem [25], ensemble pruning [26], sparse optimization [27], water reservoir management [28], and subset section [29].
In general, MOEAs can be divided into three main categories. The first one is based on Pareto dominance [30–35]. Pareto dominance-based MOEAs utilize the Pareto dominance concept together with the crowding distance or clustering methods to select the next population [31, 36]. The well-known Pareto dominance-based MOEAs consists of NSGA-II [31], SPEA2 [35], PAES [32], and AMOSA [30]. Deb and Jain designed a fast and elitist multiobjective genetic algorithm (NSGA-II). In NSGA-II, a fast nondominated sorting method is proposed to reduce the computational complexity. Then, an improve version of NSGA-II called NSGA-III has been proposed to deal with multiple/many objectives with reference points [37]. The second one is based on the performance indicator. Among these performance indicator-based methods, the hypervolume is the most popular indicator to evolve the population. In [38], SMS-EMOA is designed to measure exact hypervolume contributions of different solutions. However, it consumes a lot of time to calculate the hypervolume values of different solutions. Jing et al. deleted irrelevant solutions to quickly update the exact hypervolume contributions of different solutions [39]. The third one is based on decomposition [40–43]. The decomposition MOEAs decompose an MOP into a number of single objective subproblems, and these subproblems are solved in a collaborative way. Zhang and Li proposed a multiobjective evolutionary algorithm based on decomposition (MOEA/D) [42]. The subproblems are optimized by only using information from its corresponding neighboring subproblems. The comparison between these algorithms can be found in [44, 45].
In this study, the coverage function and comprehensive efficiency function are designed as two conflicting objectives. Then, a multiobjective evolutionary algorithm based on decomposition is proposed to simultaneously optimize these two objectives. In the proposed algorithm, novel crossover and mutation operators based on multiple memetic directions are proposed to speed up the convergence of the proposed method [46–48]. In the experiments, the real location data of various hospitals and clinics in Xi’an are collected. The experimental results have demonstrated the effectiveness of the proposed multiobjective site selection method.
The rest of this study is organized as follows. Section 2 gives the background of current situation of study area and evolutionary multiobjective optimization. In Section 3, the proposed method for multiobjective site selection is presented. In Section 4, the performance of the proposed algorithm is validated, and we also compare our algorithm with other approaches. The conclusion is finally summarized in Section 5.
2. Background and Motivation
2.1. Current Situation of Study Area
Xi’an is an important capital city in Western China. There are 11 districts and two counties, with a total area of 10,752 square kilometers and a permanent resident population of 10.2035 million. There are more than 7,000 health institutions in Xi’an, including 359 hospitals, nearly 300 community health service centers and 118 clinics. There are 112,200 health technicians and 72,500 beds in various health institutions.
At the beginning of the twenty-first century, Shaanxi province established a medical material warehouse of Guanzhong Area in Xianyang. It has improved the provincial health emergency logistics support system. It improved the service quality of warehousing and distribution, and more effectively ensured the efficient utilization of emergency medical materials stored in the whole Guanzhong area. The warehouse is located in the traditional Chinese medicine health industrial park of Xunyi, covering an area of 1600 square meters. At present, the warehouse has stored 2.48 million Chinese yuan worth of emergency drugs, 7.19 million Chinese yuan worth of masks and other epidemic prevention materials, and 3 million Chinese yuan worth of respirators and other medical equipment. There is no emergency material warehouse in Xi’an, and the emergency materials are scattered among companies with medical material reserve qualification.
2.2. Evolutionary Multiobjective Optimization
In general, a multiobjective optimization problem (MOP) with decision variables and objectives can be stated aswhere is the decision vector and consists of real-valued objective functions. In multiobjective optimization, as shown in Figure 1, the objective function is a vector and not scalar value. In the feasible region, usually there is no point that can minimize all objectives simultaneously. Multiobjective optimization is used to obtain a set of Pareto optimal solutions.
2.3. Motivation of Using MOEAs for Site Selection
Numerous variables need to be taken into account when deciding where to locate the warehouse for emergency supplies. In order to safeguard the life safety of the majority of residents, the emergency material warehouse’s objective is to swiftly and effectively offer adequate material support in various emergencies. People who are not directly served by the hospital should also be taken into account, so try to cover as much ground as you can. As a result, we focus primarily on the distribution of the direct service population, the accessibility of medical facilities, and coverage. The coverage function and comprehensive efficiency function are two competing aims in this work. To simultaneously optimize these two goals, a multiobjective evolutionary algorithm based on decomposition is then suggested.
3. Methodology
In this section, we introduce the proposed multiobjective evolutionary algorithm based on multiple memetic directions for site selection. First, the multiobjective model for site selection is described in detail. Next, the framework of the proposed multiobjective evolutionary algorithm is depicted. Finally, novel crossover and mutation operators are proposed to speed up the convergence.
3.1. Data Modeling
In this study, the coordinates of hospitals in Xi’an with complete medical service qualifications are collected, and the number of service population in the corresponding area is calculated by their beds and employees. As medical emergency materials are relatively professional materials, which are usually used by professional medical personnel, the population in terms of efficiency is within the scope of hospital services. At the same time, the specific distribution is the sum of the average number of people served by multiple hospitals within their radiation range, and the sum of hospital beds and the number of employees is taken as the emergency handling capacity of the hospital. Both of them are used as indicators of the application capacity of the emergency warehouse in this study, and the specific data are shown in Table 1 of Appendix.
3.2. Multiobjective Site Selection Model
As shown in Figure 2, the location of emergency material warehouse is a problem that needs to consider many factors [30, 38, 49]. The purpose of the emergency material warehouse is to timely and effectively provide sufficient material support in various emergencies, so as to ensure the life safety of most residents in Xi’an as far as possible. At the same time, people who are not directly covered by the hospital should also be considered, that is, to cover a large area as much as possible. Therefore, in this paper, we mainly consider the following core factors: the distribution of direct service population, the distance of medical institutions, and the coverage. The distribution of direct service population is reflected in its proximity to the hospital and its ability to fully consume medical resources. The distance of medical institutions affects the efficiency and the effective use of emergency materials. The emergency material warehouse closer to various hospitals at all levels can cooperate more quickly with hospitals to achieve more efficient emergency treatment. The coverage is that only a limited number of emergency warehouses can be established due to cost constraints and the material aggregation effect. The emergency warehouse must also be able to quickly deploy with hospital resources to nondirect service areas, which leads to the inevitable existence of vacuum areas that cannot be covered by emergency services.
According to the above analysis, we establish a multiobjective optimization model, which is shown aswhere is the coverage function and is the comprehensive efficiency function of emergency warehouse.
For , is the indication function, indicating whether the current local block can be effectively be covered by emergency resources, which is shown aswhere is the number of people in block and the whole target area is divided into areas of size .
For , is a constant and is the number of hospitals. is the distance from the th hospital to the emergency warehouse. and are the number of beds and employees of the th hospital, respectively. is the number of special service population in the block .
According to formula (2), makes multiple emergency warehouses tend to be distributed evenly and pay attention to the balance of services. However, makes the emergency warehouse close to the hospital and pays attention to the efficiency of service. Therefore, these two important objectives are contradictory and constitute a two-objective optimization problem.
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In order to solve formula (2), a multiobjective evolutionary algorithm based on decomposition is designed to optimize and concurrently. The proposed algorithm is shown in Algorithm 1. In this study, it is assumed that the number of emergency warehouses is , and then the decision vector is described aswhere is the position of the emergency warehouse. The ranges of search space are for the variable and for the variable .
In this paper, the Tchebycheff technique is used to convert an MOP into a set of scalar objective subproblems because it is less sensitive to the shape of PF and can be used to find Pareto optimal solutions in both convex and nonconvex PFs. is the offspring generated by Algorithm 2, which will be introduced in the next section. For each subproblem , we calculate its neighbors . For all individuals in , we calculate the decomposition objective function value of under this subproblem and select subproblem which can be replaced by and has the best decomposition objective function value.
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3.3. Genetic Operator Based on Multiple Memetic Directions
According to the previous analysis, it can be observed that different hospitals, population distribution, and coverage contribute to the emergency resources. Therefore, we need to make full use of these information to guide the site selection. In this study, we propose crossover and mutation operators driven by multiple memetic directions, which is shown in Figure 3. In the proposed algorithm, the core is to integrate the problem information and population information.
3.3.1. Crossover
For two individuals and randomly selected from the population, the exchange of local population information are realized through different parent individuals. In Figure 3, it is realized through individual 1 and individual 2.
The process is to realize the interaction of population information by exchanging each position in the two individuals. The offspring individuals are generated as follows:where . is the candidate site pool, which consists of emergency warehouse sites and is randomly selected from the . The value of determines the sensitivity of the algorithm to different emergency warehouse sequences among different individuals. A small value of can make the algorithm pay more attention to the differences between different individuals and have strong local optimization ability [50]. On the contrary, a large value of makes the algorithm pay attention to the recombination efficiency between different individuals and have better global optimization ability.
3.3.2. Mutation
After crossover, the difference information between individuals in the population can be used. In order to search the distribution information of hospital location, service population, and the overall coverage, a multidirectional mutation operator based on memetic algorithm is proposed. In Figure 3, the offspring obtained after crossover is . The mutation operation is described as follows:where is the probability of mutation and is a random vector, which is represented aswhere and are the search direction related to the location of the hospital. In this study, we consider the location of the two nearest hospitals to provide corresponding heuristic information, which is defined as follows:and
In order to fuse different information effectively, the algorithm maintains an information fusion weight vector , which satisfies
4. Experimental Studies
In order to deeply study the performance of the proposed algorithm, the experimental part is divided into two parts. The first part is to investigate the influence of parameter on the performance of the proposed algorithm. The second part mainly studies the differences between the proposed algorithm and several mainstream algorithms. In order to get closer to reality, the dataset used in this study comes from the real location data of various hospitals and clinics in Xi’an. The corresponding number of service population is simulated in combination with the own positioning and level of hospitals. The number of emergency warehouses set in all experiments is four. The specific distribution is shown in Figure 4.
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4.1. Test of Parameter
According to formulas (3) and (4), the parameter controls the trade-off relationships between problem information and population information in the whole evolutionary process. It is an important problem to balance these two key information, which have a great impact on the performance of the proposed algorithm. In order to deeply study this problem, this study sets a set of weights as , , , , ,, , , , and . The above experiment is tested on the deployment optimization of emergency warehouse in Gaoxin and Yant districts, and the hypervolume is used as the evaluation index. The experimental results are shown in Figure 5.
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It can be found from the results in Figure 5 that the hypervolume values obtained by , , and are much better than those of others. It can be observed that these three weight values are evenly distributed in different types of information, which comprehensively considers the problem information and the evolutionary information of the population. However, other weights are focused on one aspect of information, so they cannot obtain satisfactory results. From the above results, we can conclude that it plays an important role to employ the heuristic information of the problem or the heuristic information obtained in the population evolutionary process in the deployment of emergency material warehouse. Therefore, it is encouraged to balance the problem information and the evolutionary information of the population. The preference for any type of information will lead to the degradation of algorithm performance. Based on the above analysis, this study uses as the weight of heuristic information in the following experiments.
4.2. Comparison against Other Methods
After studying the parameter , this section verifies the performance of the proposed multiobjective evolutionary algorithm driven by multiple memetic directions based on decomposition, which is termed as MMD-MOEA/D. In order to compare the proposed algorithm, several typical multiobjective evolutionary algorithms are selected as the comparison algorithms, such as MOEA/D with differential operator (MOEA/D-DE) [45, 51, 52], MOEA/D with simulated binary crossover (MOEA/D-SBX) [42], MOEA/D with blend crossover (MOEA/D-BLX) [49], and MOEA/D with geometrical crossover (MOEA/D-GC) [53]. MOEA/D-DE has two additional measures for preserving population diversity in addition to using the polynomial mutation operator and the differential evolution operators to generate new solutions [54]. MOEA/D-SBX and MOEA/D-BLX refer to a straightforward implementation of MOEA/D with SBX and BLX, in which the SBX and BLX operators were applied to produce new solutions. To reproduce new individuals, GC is also considered in the experiments, which is called MOEA/D-GC. The comparison algorithms adopt the polynomial mutation method [42]. The relevant parameters of the comparison algorithms are shown in Table 2.
The Pareto fronts obtained on the four tests are shown in Figure 6. In the results shown in Figure 6, it can be found that the MMD-MOEA/D proposed in this paper can obtain better results than those of other methods. Because the total number of iterations of all algorithms is limited, it is difficult for other algorithms to converge to a good Pareto front. Note that the results obtained by these algorithms have large differences with respect to the first objective function. The performance of these comparison algorithms is poor in terms of the comprehensive efficiency of the use of emergency medical materials. This is mainly because the first objective function is more difficult to optimize compared with the overall coverage. In order to optimize the first objective function effectively, population information and the problem information should be considered to enhance the search ability.
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The distribution of emergency warehouse obtained by the proposed algorithm for over 50 runs is shown in Figure 7. According to the results shown in Figure 7, it can be seen that the proposed algorithm can track different hospitals and key service groups well. In these areas and nearby locations, a large number of reasonable locations are searched by the algorithm to deploy the emergency warehouse. Due to the different processing capacity of different hospitals, the location near them is also different. In the area between hospitals, many locations have also been selected to deploy emergency warehouses. Because the number of emergency warehouses set in the experiment is relatively insufficient, the algorithm takes more consideration of the space between different hospitals when considering the factor of coverage, which is also determined by the iterative characteristics of the algorithm.
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In order to comprehensively analyze the experimental results, the hypervolume values obtained by all algorithms are compared in this study. The significance level of comparison is 0.05, and the results are shown in Table 3. According to the results in Table 3, the MMD-MOEA/D algorithm proposed in this study has achieved the best average hypervolume results on the four test problems, and its average hypervolume is significantly better than other algorithms at the significance level of 0.05. It can be seen from the results that the algorithm proposed in this study has significantly better performance. However, combined with the results in Table 3 and the Pareto front results in Figure 6, we can observe that there are still many Pareto optimal solutions that have not been found, so we need to improve the algorithm later.
5. Conclusion
This study makes an in-depth study on the site selection of emergency material warehouse in Xi’an. Considering the number of emergency material warehouses, the distribution of hospitals, the resources of hospitals, the distribution of populations, and the distribution of the whole region, the deployment of emergency material warehouse is accurately constructed as a multiobjective optimization problem. The coverage function and the comprehensive efficiency function have been constructed and optimized by a multiobjective evolutionary algorithm simultaneously. In order to solve this problem effectively, this study proposed a multiobjective evolutionary algorithm driven by multiple memetic directions to solve this problem. Through experiments, the cooperative relationship between different heuristic information is deeply explored, and the advantages of the proposed algorithm compared with other major mainstream algorithms are verified. In the future, we aim to improve the local search ability of the proposed algorithm.
Data Availability
The (DIGITAL) data used to support the findings of this study are included in the Appendix.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (62003279 and 61973249), the Key R&D Programs of Shaanxi Province, China (2021ZDLGY02-06), the Qinchuangyuan Project (2021QCYRC4-49), the National Defense Science and Technology Key Laboratory Fund Project (6142101210202), the Qinchuangyuan Scientist + Engineer (2022KXJ-169), and the Shaanxi Association for Science and Technology Young Talent Lifting Program (XXJS202242).