2. Estimation of Demand and Capacity

Estimating the demand is difficult. The questions to be answered are the following: Who needs the masks, and how many pieces do they need? Masks are not recommended for children under the age of 6 y. Thus, the theoretical answer is that every person above the age of 6 y needs 1 mask per d. However, persons who work in some sectors, like health care, public transport, and shops, may need more than 1 per d. Pandemics are expected in the future more frequently. A new pandemic will not occur in the EU at the same time. A surgical mask can be used once in principle; thus, every person who must wear 1 needs a new mask every day. However, the demand determined in this way is a rough overestimation of the real demand. Surgical masks can be sterilized in an autoclave, ⁴ and this document also discusses other options for sterilization that decrease the demand of medical and veterinary practices. Another reason is that many people keep to the curfews and do not need masks. Other people, unfortunately, do not change their masks.

The production capacity increased greatly in 2020 in China, which has complete technology for the automated production of masks. China even has a producer of this automated technology. ^{Reference Dumbrill5} Parker ^{Reference Parker3} stated that China’s production capacity is 200 million masks per d. Starting on May 10, the company BYD alone had a capacity of 50 million masks per d. ⁶ The consumption in the country was 8 million masks per d. ^{Reference Aguilar7} Thus, most of the production was exported. Increasing the capacity involves serious investment. The payback period of the capital should be very short as it is not known when the next pandemic will occur, making the prices high. The high cost of air transportation from China to Europe contributes to the high prices as well. ^{Reference Aguilar7}

Thus, the demand rises in different areas at different times. This is an advantage from the point of view of supply because a lower total capacity is sufficient. The population of China was estimated to be 1400 million in 2019. ⁸ The 8 million masks consumed daily account for less than 0.6% of the population. The gap between the theoretical demand and the real-life demand is too large to draw exact conclusions. The wearing of masks is very likely to be concentrated in cities, with masks tending not to be worn in large rural areas due to the lack of supply and people disobeying the rules.

It can be supposed that the relative demand in the EU is proportionally greater. The population is 440 million people. ⁹ Table 1 presents the daily demand if masks are provided to different percentages of the population.

Table 1.

Potential daily demand for masks in the EU

It is clear that the demand in the EU is lower than the production in China. Thus, it is possible to meet the need even in Europe. The production quantity depends on the available quantity of the raw material as well.

3. Structure of the Suggested System

There are 3 main sources of masks in the suggested system. The most important sector consists of the production units in the EU, which are not under the juridical supervision of the member states during the pandemic. The distribution of their production is planned and supervised by the Committee of the EU in accordance with the temporary situation. The production and distribution must be financed from a special fund by the Committee. This arrangement also means that the governments of the member states may not prohibit the exporting of the items produced by these factories from their own countries. Air and road transportation are used for distribution. Local governments are responsible for distributing the quantities allocated to the country by the Committee.

The selection of the production companies on the EU level must be made according to principles of supplier selection in supply chain management. ^{Reference Alkahtani and Kaid10} First, an invitation to tender must be announced to companies. The applicant companies must estimate the future capacity, the set-up time for starting production, the price, technological level of the production lines, service quality and safety, time of delivery, reliability, responsiveness, affordability, capability, flexibility and reliability. The applications must be accepted or rejected, and the selected companies must be located far from each other. The next section shows a simple mathematical model, like the capacitated plant location model, ^{Reference Sridharan11} for selecting companies from the accepted applications such that the total transportation cost is minimized.

The member states obtain the delivered products at a priori selected centers. A member state may have several centers because some states cover large areas. Each state is responsible for the distribution within the state.

The establishment of the factories needs investment at the EU level. It is more economical to create supplementary capacity in existing factories. Production lines must be purchased and maintained from time to time. Some raw materials must be stored to allow a rapid start of production if necessary. These raw materials must be replaced by new ones occasionally. The factory must have a plan to make space for labor-intensive operations.

Small local factories in the fashion industry are the second main source. There are many such companies, which can react flexibly. The necessary measures taken by governments may cause production to stop in these factories. The production of protective items can help to avoid their financial collapse. Even when there is a lack of polypropylene raw material, they can produce masks from the available textiles, although these masks are less effective. The total capacity in this sector is considerable. It can produce masks without investment but in a more labor-intensive way. This sector can also contribute to the production of protective items in another way. The production of gowns cannot be automated because of the complex structure and shape of the product. Companies in the fashion industry have complete technology for gown production. Gown production can be supported technically even on the EU level by providing cutting patterns for the raw material and patterns for gowns.

Imports are the third main source. Obviously, China, and even India, will be able to satisfy a large demand. ¹² Imports can be a great help at the beginning of a pandemic when the factories in the EU are still switching their production to protective items. Imports can also help to resolve critical situations when a factory experiences a breakdown.

This sector can satisfy the local demand for masks or the demand for gowns in a larger area. Accordingly, the cost and required transportation capacity for the distribution of the items obtained from the first and second sources can be decreased. It is difficult to estimate the capacity of the sector as the structure of the fashion industry is variable in EU countries. Based on the example given by Szombathely, ¹ there is 1 facility for every million people, which is able to produce 30,000 masks per day. Thus, the capacity of the sector can be huge. However, only part of this capacity can be converted for the production of masks, gowns, or both.

The fourth sector in the system is the chemical industry, which produces polypropylene, the most important raw material for the textiles used for protective items, and the textile itself, which is the direct raw material for the safety items. Thus, the system extends back at least 2 technological steps. There is the supplier of the textile-type raw material and the supplier’s supplier, which provides the necessary chemicals.

An important principle of the system is that no single capacity can satisfy the demand. If a production facility is located in an infected area, it can easily happen that its products cannot be used. Each facility must have a higher capacity than would suffice if all facilities can work.

Companies join the system of emergency production on both the union level and the national level by bidding for contracts. They must have the necessary technology and a certain amount of raw material. If investment is necessary to purchase the technology, they must obtain financial support. The source of the financial support depends on the level at which the company joins the system. However, each member state of the EU must have a certain amount of emergency capacity depending on its size and population. Obviously, the member states must agree on the principles and parameters of the system. Let us assume that a potential pandemic is detected outside the EU. Then, the system works as follows:

• STEP 1. The World Health Organization (WHO) detects an epidemic that could develop into a worldwide pandemic. It issues an alarm to the rest of the world.

• STEP 2. The commissioner for the EU responsible for health care alerts the companies belonging to the system that they need to prepare for emergency production. This preparation includes the production of a relatively small initial stock. The alert applies to the chemical industry as well, that is, the suppliers and their suppliers.

• STEP 3. When the pandemic is detected within the EU, the commissioner orders the start of the emergency production. The states where the pandemic appeared start their own production as well. The commissioner has the right to ask states where the pandemic is not yet occurring to start production and distribute items to other member states.

• STEP 4. Distribution starts from the EU-level production. The reports of the member states form the basis for determining the daily quantities. These quantities are transported from the EU-level companies to the a priori determined centers of the member states. The member states must pay for the items obtained.

If the pandemic is detected in the EU, then the health authorities of the member state, the commissioner of the EU, and the WHO must cooperate directly.

4. Selection of Facility Location and Capacity

The aim of the model is to provide a calculation tool for the decision maker to select the facilities. An economic production network can be created in this way. The network can serve the whole EU by the proper cooperation of the member states.

Assume that the first round of the evaluation is over and that there is a known set of companies that have submitted acceptable applications. However, the final selection is still to be made. The locations and potential capacities of the companies are known. In the second round, the selection of the EU-level companies is carried out based on careful calculations. The demands are based on the population of the regions represented by centers in the model. Each center is assigned to a production facility such that its total demand can be obtained from that facility. The capacity of each production facility is at least as great as the total demand assigned to that facility. In this way, the population demands are distributed among production facilities according to population distributions and all demands are produced and provided to all the population.

A mathematical model, which minimizes the transportation cost, is discussed here.

The model uses the following data: the demand of the centers, the potential capacities of the accepted applicants, and the distances between the production facilities and the centers.

The model uses 2 types of variables: a 0–1 decision variable for each applicant representing its acceptance or nonacceptance, and the quantities transported from facilities to centers.

These parameters and variables are summarized in Table 2 as follows:

Table 2.

Parameters and variables

It is assumed that the system is feasible, that is, that the total capacity exceeds the total demand, which is defined according to the population distribution, that is,

$$\mathop \sum \limits_{i \in I} {K_i} \ge \mathop \sum \limits_{j \in J} {d_j}.$$

In the case of the equation, the only feasible solution is to establish all facilities. The objective function is to minimize the total transportation distance:

$$\min \mathop \sum \limits_{i \in I} \mathop \sum \limits_{j \in J} {c_{ij}}{d_j}{x_{ij}}$$

Every center must be assigned to exactly 1 facility:

$$\forall j \in J:\;\mathop \sum \limits_{i \in I} {x_{ij}} = 1.$$

A facility cannot deliver more than its capacity:

$$\forall i \in I:\;\mathop \sum \limits_{j \in J} {d_j}{x_{ij}} \le \;{K_i}{y_i}.$$

The number of accepted facilities is $p:$

$$\mathop \sum \limits_{i \in I} {y_i} = p.$$

The quantities transported are non-negative:

$$\forall i\varepsilon I:\;\forall j \in J:\;{x_{ij}} \ge 0.$$

The decision variable is either 0 or 1:

$$\forall i \in I:\;{y_i} = 0\;or\;1.$$

The problem is an integer programming problem. Instead of the non-negativity requirement of the x variables, it is also possible to claim that they must be 0 or 1. This means that a facility is completely responsible for the centers assigned to it. There is no significant difference from a computational point of view, as discussed in the next section.

5. Example and Numerical Results

An example is elaborated for the continental countries of the EU with the exception of Luxembourg. In most cases, the country is represented by its capital. Italy and Spain are each represented by 2 cities: Rome and Milan in Italy and Madrid and Barcelona in Spain. Hamburg, Berlin, and Munich represent Germany in the example. In these cases, the demand is distributed equally among the representative cities. Thus, the number of cities is 28. The basic data are listed in Table 3.

Table 3.

Countries and their cities

The demand is determined in a simple way. It is assumed that 2.5 billion masks must be produced and distributed in a time period of 20 days. This represents more than 5 masks per person. Hence, the daily production is 125 million pieces. This assumption takes into account the fact that the distribution takes time. The route of the distribution is as follows:

$$\begin{gathered} facility \to center\; \to other\;cities\;in\;the\;count \hfill \\ \quad\quad \to \left\{ {pharmacies,\;hospitals,\;etc.} \right\}. \hfill \\ \end{gathered}$$

The distances used in the calculation are surface distances from facilities to centers. The cost of this type of calculation appears on the EU level. Thus, the objective function contains this transportation. The costs of further transportation, that is, the transportation from the centers to the final customers, are covered by the member states. The unit of the transportation cost is 100,000 masks × km. It is supposed that the real transportation cost is proportional to this quantity.

Two different capacities of facilities have been investigated. The smaller one is a capacity of 25 million pieces per day, and the larger one is 50 million pieces per day. In the case of the larger capacity, there must be at least 3 facilities. In the case of the smaller one, 5 facilities are sufficient. Figure 2 shows that the total transportation cost of the EU-level distribution decreases exponentially with the number of facilities.

Figure 2.

The total cost versus the number of facilities. The capacity of each facility is 50 million mask per day (Total cost = $220142189{e^{ - 0,18689p}}$ , p: the number of facilities).

The structure of the system is presented in Table 4 and depends on the number of facilities. There is 1 structure if there are no more than 6 facilities. Another structure exists if there are between 7 and 10. It is not realistic to establish even more facilities on the EU level.

Table 4.

Locations of the facilities with different numbers of facilities

If the capacity of each facility is 25 million masks per day, then there must be at least 5 facilities. The results are identical with the exception of 5 facilities. Furthermore, there is no significant difference between the results obtained when the x variables are allowed to be continuous and the results obtained when they are binary. The obtained optimal solutions are exactly the same if the number of facilities is at least 4 in the case of a capacity of 50 million pieces and if the number of facilities is at least 6 in the case of a capacity of 25 million pieces.

The optimization problems were solved by Xpress solver on a laptop with 64 bits, 4G of memory, and an Intel Core I3-2310M processor operating at 2 × 2.10 GHz. If the x variables were claimed to be integers, then integer programming iterations were performed in the case of the larger capacity when the number of facilities was 3 or 4 and in the case of the smaller capacity when the number of facilities was 5. In any other case, the optimal solution was obtained immediately, that is, within 1 s.