1. Introduction
COVID-19 is a severe global public health emergency that has had a profound impact on medical systems and social economies [
1]. During the outbreak of large-scale infectious diseases, the scheduling of emergency supplies is necessary to ensure medical treatment and the continuation of normal life. Thus, it is important to establish an emergency resources supply system fully tailored to the epidemic process. Among the issues linked with emergencies, methods of efficiently dispatching resources require attention. There are many factors affecting dispatching, including external factors, such as region and time, and internal factors, such as material supply and demand. A state of uncertainty and emergency increases the difficulty of dispatching materials. Therefore, the first factor that must be considered is the prediction of the possible demand through scientific methods. A two-stage location-routing model has been proposed for guiding resource allocation when the requirements and infrastructure are unknown [
2]. The model has a lower computational cost because of its simple calculation process. Then, case-based reasoning (CBR) and the Dempster–Shafer theory have been employed to improve the accuracy in forecasting emergency material demand [
3]. A good method is necessary not only for estimating demand but also for the organization of the supply chain and the coordination of the relationship between the parties in order to enhance the effectiveness of the material distribution. A two-stage MADA-B mechanism was designed to research the supply and demand of multi-attribute emergency materials, which combines a multi-attribute double auction (MADA) with bargaining and can perfectly match buyers with sellers through the game playing of the transaction price and quantity [
4]. After demand matching is completed, the subsequent production plan becomes the new focus. Then, a fuzzy linear programming model was provided to solve the aggregate production planning problem. Its advantage is the incorporation of uncertainty of the customer demands, and unit holding and backordering costs of the production plan [
5]. In one work, a method based on the timed-colored Petri net (TCPN) model was proposed to model the cooperation of actions with time analysis [
6]. After the production of materials, timeliness needs to be considered in the selection of transportation methods. After an in-depth discussion of the cold chain model selection problem, taking into account economic and environmental objectives from both business and financial aspects, a value-based management method is provided as a new shipping approach [
7]. The method effectively solves material planning by cutting out unnecessary actions. Other methods focus on the quick construction of the supply chain according to the criterion of reaction speed. Based on this idea, a hybrid algorithm combining artificial immunity with ant colony optimization has been developed, the transportation scheme of which has a shorter response time and covers more demand points [
8]. With the hierarchical timed color Petri net (HTCPN) model and the skyline operator, a multi-objective optimization (MOO) model for a fire emergency response was established, which not only shortened the response time but also reduced resource consumption [
9]. It must be noted that the above methods assume that materials are directly transported from the supply side to the demand side. They do not take into account cross-regional transportation, which is more likely in epidemic situations. To overcome this disadvantage, an inter-regional emergency cooperation network that includes system construction, organization and coordination, and mechanism design is proposed to offer an optimal countermeasure for city cooperation [
10]. Transit points need to be considered when cross-regional issues are involved. The location of transit points will affect transportation efficiency. Considering this, a multi-objective optimization model for the selection of rescue stations has been established to improve efficiency [
11]. In the research into transit points for cross-regional issues, the requirement for warehouses becomes obvious because it is nearly impossible to match the rate of supply with the rate of consumption. A mixed-integer programming model for uncertain requirements controlled by time and cost provides a helpful solution for emergency warehouse location and distribution [
12]. Additionally, when stocks are available, a simulation–optimization approach based on the stochastic counterpart or sample path has been shown to optimize the pharmaceutical supply chain by managing the records of the stocks [
13]. Due to the uncertainty of epidemics and the timeliness of drugs, medical demand is difficult to predict and handle. For that, a deterministic MILP model and a robust optimization model are used to deal with the demand uncertainty while integrating warehouse selection, inventory strategy and delivery route optimization of the VMI [
14].
The above examples from the literature show different solutions for emergency events. However, all of them ignore the fact that the degrees of urgency of different requirements play a role in the response, especially when the emergency supplies are not enough to meet all of the requirements. In this situation, the distribution of materials has to consider the degree of urgency. An optimization model combines the location hazard index (LHI) with the response time; the LHI measures the potential hazard of a location, while the response time provides resource allocation in response to an emergency situation [
15]. From the observation of multiple independent emergency events, a deep ensemble multitask model integrating four subnetworks has been proposed. It can improve the medical dispatch process by classifying the degree of emergency based on clinical data, environmental data and other factors [
16]. In the case of an epidemic outbreak, a hybrid multi-verse optimizer algorithm based on the multi-verse optimization algorithm and the differential evolution algorithm can effectively reduce the distribution cost by considering the urgency of the demand for emergency supplies [
17]. Numerous studies have comprehensively discussed good solutions for dispatching materials by recreating the scene of the emergency. The pre-emergency warning process has become another research hotspot. A study has formulated a multi-objective mixed-integer non-linear programming model to determine the location and number of relief centers, with their prepositioned inventory level, in the pre-emergency stage. The decision provided by the model can minimize costs and transportation distances [
18].
The above literature examples discuss the various factors that support a reasonable resource-scheduling solution to advance the development of emergency management. However, most of the studies concentrate on static analysis to optimize resource scheduling, which means that the variations in the requirements and degrees of emergency are totally ignored in the process. In addition, the works mainly focus on the post-stage response, and the pre-stage early warning mechanism is rarely involved. In order to offer a solution incorporating all factors, a coordinated allocation model of multiple materials based on the gray prediction model is proposed in this work and is named the Multi-catalog Schedule Considering Costs and Requirements Under Uncertainty. If the number of infectious members of the population can be forecast, then the materials that will subsequently be required can be prepared. Thus, by collecting information on historical infectious diseases, the model uses a gray prediction algorithm to predict the number of infectious diseases in the future. According to the prediction results, the demand relative to infectious disease is determined, and this includes both medical materials and general goods. At the same time, the cost is also considered. With the goal of reducing the cost and meeting demand, a multi-object function is defined and takes into consideration the type of relief material, the time difference, and trans-regional coordination. This model contains numerous variables from different angles, meaning it is difficult to set the initial solutions. The ant colony model does not require much for the initial solutions and has few parameters, meaning that it is suitable for combinational optimization problems such as material dispatch. Therefore, an ant colony model is designed to solve our problem. The contributions of this work are as follows:
(1) The gray prediction algorithm is used to predict the number of confirmed cases at various times. Then, the degree of emergency can be estimated, and the predicted data can be used to guide material scheduling. The application of this prediction module means that our model can play a certain role in early warning systems.
(2) Both external and internal factors are considered in order to expand the scope of the model’s application and improve the satisfaction of the solution provided by the proposed model. External factors include distances and the time of transportation. The internal factor comprises the maximum level of production. Then, an objective function for cross-regional scheduling is defined, in which the uncertainty of requirements and different types of goods in a period of time are taken into account.
(3) In order to obtain the final schedule, the model uses the ant colony algorithm to solve the objective function. There are numerous integer variables in the function, and the initial solution is a three-dimensional matrix. Thus, the model records the directions of each ant’s action in each dimension in the matrix and defines a utility function, which is used to calculate the effect of the ant’s every choice. Unlike the pheromone, the calculated results will help shorten the time required to obtain the result of the model by adjusting the probability of picking the direction in the course of each ant’s actions.
This paper consists of five sections.
Section 1 mainly describes the latest achievements regarding the research issues in this paper and discusses their advantages and disadvantages. Then, the model and the research value proposed in this paper are briefly introduced.
Section 2 provides a detailed introduction to the theories used in the model.
Section 3 consists of the building and solving processes of the model. The results of the model are verified and presented using examples in
Section 4. Finally, the conclusions are discussed in
Section 5.