Homo economicus to model human behavior is ethically doubtful and mathematically inconsistent

In many models in economics or business a dominantly self-interested homo economicus is assumed. Unfortunately (or fortunately), humans are in general not homines economici as e.g. the ultimatum game shows. This leads to the fact that all these models are at least doubtful. Moreover, economists started to set a quantitative value for the feeling of social justice, altruism, or envy and the like to execute utilitarian calculation. Besides being ethically doubtful, it delivers an explanation in hindsight with little predicting power. We use examples from game theory to show its arbitrariness. It is even possible that a stable Nash equilibrium can be calculated while it does not exist at all, due to the wide differences in human values. Finally, we show that assigned numbers for envy or altruism and the like do not build a field (in a mathematical sense). As there is no homomorphism to real numbers or a subset of it, any calculation is generally invalid or arbitrary. There is no (easy) way to fix the problem. One has to go back to ethical concepts like the categorical imperative or use at most semi quantitative approaches like considering knaves and knights. Mathematically one can only speculate whether e.g. surreal numbers can make ethics calculable.

INTRODUCTION
In economics and many business situations, it is assumed that humans tend to look for maximal economic wellbeing. The theoretical model behind this is called homo economicus. However, numerous experiments have shown that humans are not homines oeconomici. The best-known test is the ultimatum game. In it player A gets goods (e.g. € 10) and is allowed to share it with B within any ratio (from € 9 for A and € 1 for B to € 1 for A and € 9 for B). If B accepts it, each player gets the share determined by A. If B refuses it, both will get nothing. If at least B were a homo economicus, he should always accept. However, in reality reaching a certain level of injustice B will not accept any more because many people perceive an unequal share between two equal individuals as unfair or unjust. The exact level when an individual perceives a share as unfair varies not only from person to person but is also different over time and will depend on e.g. whether A as a person is known. This is already a first hint that justice cannot be quantified in a mathematical sense and that certain values (such as justice) might affect human behavior and lead to non-rational decisions in context of the homo economicus model.
Nonetheless, many decisions in business are made based on the model of homo economicus. In business administration, all decision models, optimization calculations, or variable salary models depending on performance are good examples. Let us assume two companies A and B facing a decision whether they start to invest abroad or not (Figure 1). If both companies do not invest abroad, they can both expect a profit of 4 each. In the same way, both will get a profit of 2 if both are investing abroad. A mathematician will immediately recognize that staying at home is not only maximizing profit, it is also the only stable Nash equilibrium. Any move from there will reduce the individual profit. Reality tells often a different story. Let us further assume going from (4,4) to (3,2) or (2,3) will increase the reputation of A or B, respectively. This leads to a deviation from the more profitable position due to certain character traits, such as envy of the other's reputation. Envy could obviously be so strong that A and B are willing to pay for it, ending up with half the profit in the end.
According to this example, the players' behaviour indicates that either they are acting irrationally or it is not explainable by the homo economicus model. One explanation would be an accounting for envy, altruism, and the like. In Figure 1 companies are willing to half their profits because of envy. If envy is worth half the profit, we have a number for envy. One may include it in the payoff matrix of Figure 1 leading to a stable Nash equilibrium at (2,2). In doing so one gets the result from reality though strict homines economici are assumed. This is often done in modern economics though it is extremely flawed. There are many other examples where one is misled if one assumes a homo economicus in situations like this.
In this article, we aim to show that this application of homo economicus bears inherent mathematical contradictions that question the logical consistency of the model. Since an inconsistent set of statements can be used to prove anything, including nonsense, absence from contradiction is indispensable for useful scientific theories. If our argumentation were successful, the homo economicus model would be methodological useless for scientific reflection and modelling of human behavior with moral values because of its inconsistency.
After a short review on the current understanding of the problem in section 2, we will give a philosophical perspective on how economics started to explain every human (economic) behavior by maximizing the individual profit in Section 3. After that, the main point of section 4 is to show that quantification of ethics is not working, is not only a philosophical remark. We can rigorously prove that it is impossible for mathematical reasons too. In Section 5 we conclude that a quantification of ethics does not make sense, neither from a philosophical nor mathematical point of view.

LITERATURE REVIEW
The model of homo economicus has been challenged with experiments like the ultimatum game, as stated above (Dilek & Kesgingöz, 2019). In a recently published article, Schreck, van Aaken and Homann (2020) have therefore defended homo economicus in the context of social dilemma situations against criticism, especially from the empirical side. The authors put forward an argument that sets the contradictory empirical observations on the homo economicus model analog to those on Galileo Galilei's law of motion: "Interpreting observed moral behavior as a falsification of the homo economicus model is like interpreting the slow fall of a feather as a falsification of Galilieo's laws of motion." (ibd.) Instead, the discrepancy between predicted behavior and actually observed behavior should be investigated. They suggest to see models of moral behavior that are based on social preferences not as alternatives but refinements of the homo economicus model. In their view, the purpose of these approaches is to contribute to a contingency theory of moral behavior, rather than falsifying and preplacing the homo economics model.
The argumentation of Schreck et al. with regard to the scientific-theoretical foundations seems convincing at first glance. However, the aim of our publication on this is not to repeat empirical counter-arguments against homo economicus, but to reflect on the model with regard to its methodological foundations and the mathematical assumptions made in the process for quantifying qualitative values.

THE PHILOSOPHICAL POINT OF VIEW
Before deep diving into mathematics, a philosophical perspective aims to broaden the understanding of the main problem here. Moral philosophy, or ethics, is the methodical reflection on human action and under what conditions it can be called "morally right" or "good". Various ethical theories answer this question differently. Utilitarianism is a form of ethics that, reduced to its essentials, states that an action is morally right precisely when it maximizes the aggregate total utility. Jeremy Bentham, founder of systematic utilitarianism, put this in the formula: "the greatest happiness for the greatest number" (Bentham, 1789). This principle states: "By the principle of utility is to be understood that principle which universally approves or disapproves of every action in proportion as it appears to have an inherent tendency to increase or diminish the happiness of the group whose interest is in question." (ibd.) The measurement of happiness is determined as follows: "Add up the values of all the pleasures on one side and all the sufferings on the other. If the side of pleasure predominates, the tendency of the action with regard to the interests of that individual person is good on the whole; if the side of suffering predominates, its tendency is bad on the whole." (ibd.) This definition of the utilitarian approach to ethics exposes a natural proximity to the homo economicus model. In order to maximize happiness, he describes what economists call a cost-benefit reasoning. Thereby, this normative theory could be used to open up an ethical perspective upon business related issues, since economic thinking is inherent.
But can values such as happiness, justice, or even human life be quantified and aggregated with arithmetic operators? Critics have challenged this approach (Taylor, 2017). They state that, moral values cannot be quantified, measured, or even calculated. To evaluate actions as morally "better" or "worse" in a quantitative sense seems to be a linguistic convention alone, but does not sustain ethical reflection.
An example illustrates very well what the problem is when qualitative values are quantified. In the 1960s, Japanese and German cars in the subcompact car class increasingly dominated the US car market. In 1968, Ford executives decided to launch the Pinto to challenge the competitors. A very tight schedule was to be followed to start with an early production of the new car in order to eliminate the foreign invaders from the market as quickly as possible.
Before Ford started producing their new car, they tested several prototypes. The crash-tests showed that the Pinto posed a serious danger to its occupants, as it easily caught fire when involved in a rear-end collision. Ford officials faced a decision. Should they stick to the current design and thus ensure that the ambitious schedule can be met, thereby putting the safety of its customers at risk? Or should they suspend the planned start of production and give the tank a redesign to make it safer which would result in the home market being dominated by foreign companies for another year? (Lineq, 2001) The evidence suggests that Ford relied, at least in part, on cost-benefit reasoning. Hereby, Ford concluded that, if they agreed on the technical upgrade of the tank design, the increased cost outweighed the benefit. Interesting is the calculation they made. They estimated the technical improvements to prevent gas tanks from leaking and prevent fire hazards to be $11 per vehicle in additional cost. They planned with a production of 12.5 million cars. This results in $137 million total additional cost if they would upgrade the gas tank. But what would be the benefit from that? On the benefit-side (if the gas tanks were replaced and the inmates life were saved), they estimated $49.5 million. They did the following calculation: They assumed society loses $200,725 every time a person is killed in a car accident. They broke down the cost in various positions such as direct and indirect productivity losses ($132,000 + $41,300), hospital ($700), insurance administration ($4,700), legal and court expenses ($3,000), employer losses ($1,000), property damage ($1,500), victim's pain and suffering ($10,000), funeral ($900), lost consumption (5,000$), miscellaneous accident costs ($200) and other ($425). They estimated 180 burn deaths ($200,000 per death), 180 serious injuries ($67,000 per injury), and 2,100 burned vehicles ($700 per vehicle) which leads to a total of $49.5 million. Since the cost of a new tank design ($137 million) clearly outweighs the benefit ($49.5 million), Ford's management decided to stick with the original design und not upgrade the Pinto's fuel tank, despite the results reported by its engineers. Between 1971 and 1978, the Pinto was responsible for a number of fire-related deaths. Ford puts the number at 23; its critics say the figure is closer to 500. For the details on the Pinto case, see Lineq (2001).
The simple example shows the problem that arises when values are quantified: their quantity remains arbitrary. Why do these individual positions determine exactly a loss of a human life? Why is only the suffering of the person directly involved considered, but not that of his or her relatives? Why is individual suffering and pain worth exactly $10,000 and not more or even less? Economists tend to neglect this normative foundation of their argumentation or if they do, it remains arbitrary.

ACCOUNTING FOR ETHICS IN ECONOMICS
Considering ethics, morals, and many other values is necessary in economics. A good example is the theory of knaves (self-interest only) and knights (common-good oriented) as explained from Grand (2003). This theory suggests that not every human behavior is driven from the same motivation. Especially when it comes to salary, the output of a person's work depends not only from the amount of money she or he earns. Whether the person is a knave or a knight is equally important. For example, it has been applied to a new pay scale for German professors introduced in the beginning of this century (Grabinski, 2005). The main point was changing a system based on a fixed salary plus some growth due to gain of experience over time into a base salary plus some bonus depending on output quality bargained every year. In Figure 2, the dashed curve denoted by shows the typical behavior of a self-interested knave. Output and payment are about linearly related. The solid curve denoted by stands for a typical knight. For a rigorous theoretical discussion, please see Grand (2003). The points and state the payment and output quality of a typical knight or knave, respectively. As one clearly sees, a knight has a higher output with lower pay while it is vice versa with the knave. Besides it is very difficult to measure the output quality of professorial work, we have a much more severe problem. Professor are supposed to be dedicated to their work. This is nicely expressed in the German word Berufung used for the appointment of a professor. It is the same word as used for a priest being appointed by god. So a dedicated professor should have another motivation besides pay. However, treating a knight like a knave by trying to control him by varying the payment, will lead to a phasing out denoted by the line D in Figure 2. We end up with higher cost and lower quality due to assuming self-interest as the main trait.
In Figure 2 the quantitative deviation from home oeconomicus has been shown. Please note that this does not show how to calculate the behavior of Mother Teresa who was for sure an extreme knight. The whole consideration is semi quantitative only. In Figure 2 one can only show the (negative) effect at work. But the exact curves (especially ) can not be constructed. (Of course the gaps in Figure 2 are for sure exaggerated) Trying to construct the lines by e.g. differential equation would be mathematical nonsense as there is no homeomorphism from the space of dedication, curiosity, etc. to the ℝ . But this will be discussed in more detail in Section 5. The next example shows such misuse of numbers in a simple example. Now we come back to the game of Figure 1 in the introduction. Mathematically it is supposed to end in (4,4). Due to envy (2,2) is a typical end in reality. As stated envy makes A in point (3,2) feel better and B worse. So one may come up with the idea to include some envy parameters in Figure 1. In doing so we end up with Figure. The parameters to denote the envy. So if A has a profit of 3 but B a profit of 2, A feels a profit of 3 + and B is so annoyed as were its profit 2 − . With = = = = 1.5 the problem appears to be solved. The point (2,2) is a stable Nash equilibrium showing unfortunately the lowest total profit. It is like in the game prisoner's dilemma. The effect of envy even appears to be calculable. Every , > 1 and , > 0 will create a strong enough envy effect to end up in (2,2) while , ≤ 1 will lead to (4,4) as a Nash equilibrium. Nevertheless, Figure 3 gives an explanation in hindsight. In order to predict the future one has to know the envy parameters to beforehand. In order to do so, one needs a rigorous definition envy. This is maybe possible by certain hormone levels. Then we need a homomorphism from these parameters to ℝ. It will be impossible. If A envies B and C by a certain amount, will the hormone levels add up? Will this lead to simply added envy in our parameters to ? Furthermore, the amount of envy will not only depend on the difference of profit in Figure 1 but whether A likes or hates B. Mathematically spoken we have no transitivity.
One could also say that there are many more variables. But this makes it worse. We would need a homomorphism to ℝ for all these variables. Even in the very unlikely situation that one can find a homomorphism, one will probably end up in a chaotic system. This is very similar to a stock market. The price of a particular stock is determined by each individuum's willingness (for whatever reason) to buy or sell a particular stock at a particular price. Though this sounds like a simple predicting rule, it leads to an unpredictable chaotic stock market (Appel & Grabinski, 2011;Klinkova & Grabinski, 2017). So one might at least say that one can make a rough estimate. Due to chaos this rough estimate may be very rough. Having a very wide range for e.g. envy parameters and the like, creates another problem as has been shown for game theory in a very recent publication (Klinkova & Grabinski, 2021).
In Figure 4 a game has been introduced where each player has a choice of three strategies. It could be applied to two competitors trying to conquer the market with three strategies. In (Klinkova & Grabinski, 2021) the fun example of penalty shooting in soccer has been scrutinized. The following payoff matrix Figure 4 has been taken  from this (ibd.) publication where further details can be found. Estimating the from real championship games (The Economis, 2018), it has been calculated in (Klinkova & Grabinski, 2021) that jumping and shooting with 43.0 % probability to the left and with 37.5 % to the right (and 19.5 % else) would be the optimal strategy for player and keeper. As calculated in (Klinkova & Grabinski, 2021) it will lead to a probability for a goal of 79.9 %.
Even in soccer there are other soft variable besides the success probabilities . Player or keeper may have stress factors because their career is in danger. But as stated, the may also be success rates in investment strategies. So one may include envy like in Figure 3. As discussed above these envy parameters may have a broad distribution. Of course one can use the average only.
To see the point all in Figure 4 may keep their original value but and are broadly influenced by envy. So they may vary between 0 and 1. Figure 5 has been directly taken from Klinkova & Grabinski (2021). As explained there, only in the orange meshed area there exists a Nash equilibrium. (In the area not covered one had to shoot e.g. with over 100 % probability) Even if the average values (including envy) for and lie the orange meshed area, the entire calculation does not make any sense. One gets a result, but it is a weighed average of an existing solution and a non-existing one. It is even quite dangerous. One may get a result and of course one relies on it, though it has no meaning. Please note that calculating the area in Figure 5 is straight forward within this example. But this is due to the particular payoff matrix of Figure 4. To find the orange meshed area like in Figure 5 is for an arbitrary case would require fundamental research in e.g. topology, which has been explained in Klinkova & Grabinski (2021).

MATHEMATICAL REQUIREMENTS TO CALCULATE ETHICS
From the examples from Section 4 it is clear what one needs. Objects like envy, greed, hate, altruism... must be measured in the first place. This could be extremely difficult, and mankind is far from being successful in this field. But even if done successfully one has to assign numbers to it in order to make any calculation. Consider the example of rating the most favored color for a new car. Firstly, one has to define the setup clearly. Does it mean a possible increase in price for a car being identical except for the color, the wellbeing while driving in such car or seeing such car. Going for the latter one may come up with detailed numbers. The color is uniquely defined by a wavelength (but only if it is a pure color, else one will have a distribution of wavelengths). The amount of happiness can be measured by hormone levels such as dopamine. Now we would have numbers which are scientifically sound. An average wavelength is a color in between. But will the dopamine level do the same? If the dopamine level (in whatever units) is 50 while watching a red car and 10 while watching a yellow car, does it mean it should be (50 + 10) 2 ⁄ = 30 when an orange car passes by? Instead of measuring dopamine levels one may ask people to rate colors on a scale from 1 to 5. This produces some arbitrariness but will show the same weakness.
Assigning numbers is often easy. But is an addition defined? How is it defined? It is a typical task for first year students of mathematics in linear algebra to find out whether some objects and a generalized addition (e.g. ⊕) will build a group. Unfortunately this is almost never done in business or economic research. From the examples of Section 4 the problems become obvious.
Dealing with numbers we need at least a reasonable defined addition and multiplication. Mathematically spoken the numbers must build a field. Probably everyone knows two fields: The real numbers ℝ and the rational numbers ℚ with ℚ ⊂ ℝ. As the first-year math students invent groups or fields with strange objects, one may say that one just needs creativity to define an addition to e.g. envy levels of Figure 3. Sadly, one can show that any (one-dimensional) field is homomorphic to ℝ or a subset of it. If objects build a field, it is just a systematic renaming of real numbers. So there is no chance overcoming the problems of Section 3 when using real numbers or a homomorphism of it.
Leaving real numbers is also no trivial option. The well-known two-dimensional field of ℂ has the drawback that it is not ordered. As there is no higher dimensional field and any two-dimensional field is isomorphic to ℂ, even the greatest creativity will not help. To see the point why order is essential let's go to the mathematical definition of a game: Given be a map with The are the possible strategies.
( 1 , . . . , ) ∈ 1 × 2 ×. . .× be a particular strategy bundle. If for any * ∈ holds, then ( 1 , . . . , ) is called Nash equilibrium. is the "profit" of a certain strategy. As one sees from inequality (2), one needs an ordered field. And of course everywhere in economics one wants e.g. to maximize a profit which demands order. Therefore mathematician always speak of real games in game theory.
Please note the problem of having no field is not limited to quantities like envy, greed, hate, altruism... In utility analysis one assigns e.g. points to potential profits when investing in A, B, or C. The profits itself or its estimates can be added. But the very point for assigning points is that such estimates are very hard to get. So the points are at most a monotonous function of the profit . One wants to maximize But instead one maximizes As is arbitrary and has the only restriction of being monotonous, the maximization in (3) and (4) will lead to arbitrarily different result depending on . (Management consultants might know it when they are using utility analysis to prove a best strategy desired by themselves or the CEO) Only if the were linear functions, utility analysis could work. But then one has to guess the weights accurately which is far from being easy. The falsification of utility analysis has been shown in Grabinski (2007) already. There almost no hardcore mathematics has been used.
The learning from this section and in some sense of the whole publication is that in order to add or multiply numbers, addition and multiplication must be defined first. The statement sounds trivial and indeed it is. However assigning a number to an object sounds trivial and arbitrary. As one learns in primary school how to add and multiply, people are doing it without thinking. Nobody would say e.g. one can add envy to hate. But as soon these values are quantified, an addition is performed. However it is by no means better then defining

CONCLUSIONS
We have shown that quantifying values like envy, greed, hate, altruism... may or may not be possible. But making calculations like finding optima with it, is almost ludicrous. Although, it appears to grow more and more common. We have also shown that there is no easy fix and maybe none at all. For the time being it is the best advice not using such tools. And even more importantly, not to rely on results created by such methods. Positively, a cost saving is possible by just not doing such calculations. We therefore conclude: The homo economicus model is neither a methodologically reasonable nor a normatively acceptable model for describing human behavior in business and economic contexts.
The point we made in this article may seem trivial from both a philosophical as well as from a mathematical point of view. However, the consequences for business administrations and economics are certainly not.
Leaving such evaluations to philosophers is one way out. For sure it will not please quite a few economists. But it will make economics more human and more scientific, as it is the basis of any science to find a proven answer. Peter Ulrich's Integrative Economic Ethics (Ulrich, 2008) is an approach for the use in economic context. This is based on the inviolable human dignity of every human being and should therefore not be forgotten in the context of economic action. According to Ulrich, a meaningful economy is one in which economists see the term "economic activity" as more than just acting for profit. Reasonable economic activity, in the sense of sensible and ethically correct, serves life, so to speak, since it integrates -in addition to survival through profit -the social and fulfilling aspect that society needs to flourish. The concept of knights and knaves (Grand, 2003) is another promising invention. Though it is hard to say who invented the concept, Richard Titmuss (1907 -1973) was for sure an essential contributor.
To find a rigorous calculation of ethical things is at least a very long way to go. For sure it will look very different from what people expect. Real numbers (ℝ) could not be involved. So the result could not be 42. On the other hand, a similar situation happened at the beginning of the 20 th century. Classical mechanics assigns any object a position and a velocity in order to describe it. These are all real numbers. Though classical mechanics is often an excellent approximation it is generally wrong. Fixing the problems is called quantum mechanics. Meanwhile it is easy to see (prove) that such theory is impossible to formulate with real number. One needs complex numbers ℂ. As stated above, complex numbers are not ordered. One cannot decide whether a number is bigger or smaller than another one. Therefore in quantum mechanics it is impossible to state e.g. whether an object is further to the left than another one. One can only speak of a certain probability of being further to the left.
Whether complex numbers can solve our problem here, can be answered with maybe. But it is a very big maybe. A may be better candidate are surreal numbers. In real numbers one says sometimes is arbitrarily close to . However, for every < there exists a with < < . Admittingly over simplified, surreal number are real numbers plus infinitesimal close number. With these numbers it would be impossible to find a as defined above. John Conway (1937 -2020) could prove that it is possible to build a group with them though no field. John Conway also worked on game theory though he generally considered games like go (圍棋) or chess. In these games there is of course also a finite number of strategies. But they are so big that even supercomputers are far from able to consider them all. Finding the best strategy needs something like creativity. John Conway observed some surprising connections between such games and surreal numbers (Conway, 1978). After Conway died of Covid-19 in 2020, research fortunately goes on (Henle, 2021).
Author Contributions: All the authors (ML, AK, GK, and MG) contributed to conceptualization, formal analysis, investigation, methodology, writing original draft, writing review and editing. All authors have read and agreed to the published form.
Funding: This research received no external funding.
Institutional Review Board Statement: Ethical review and approval were waived for this study because it does not involve any experiments with animals or humans.

Informed Consent Statement: This study does not involve any humans.
Data Availability Statement: This publication did not use any data not published within the paper or its references.


Exploring disaster impacts on adaptation actions in 549 cities worldwide

Whether disasters influence adaptation actions in cities is contested. Yet, the extant knowledge base primarily consists of single or small-N case studies, so there is no global overview of the evidence on disaster impacts and adaptation. Here, we use regression analysis to explore the effects of disaster frequency and severity on four adaptation action types in 549 cities. In countries with greater adaptive capacity, economic losses increase city-level actions targeting recently experienced disaster event types, as well as actions to strengthen general disaster preparedness. An increase in disaster frequency reduces actions targeting hazard types other than those that recently occurred, while human losses have few effects. Comparisons between cities across levels of adaptive capacity indicate a wealth effect. More affluent countries incur greater economic damages from disasters, but also have higher governance capacity, creating both incentives and opportunities for adaptation measures. While disaster frequency and severity had a limited impact on adaptation actions overall, results are sensitive to which disaster impacts, adaptation action types, and adaptive capacities are considered.

Coding Adaptation Actions and Natural Hazard Events by Geographic Region
In order to ensure spatial proximity between the locations of the natural hazard events in the EM-DAT dataset and the adaptation actions in the CDP dataset, we reviewed the geographic information provided and determined the first-order administrative unitthe largest sub-national territorial unit within a countrywhere these events occurred. Given the CDP adaptation actions dataset provides both the country and city names, the geocoding process was a straightforward matching exercise using a global list of first-order administrative units 1 . The EM-DAT dataset provides information containing country names and a non-uniform, subjective description of the subnational areas impacted by a given natural hazard event. In this case, the process involved manually reviewing the location descriptions to determine the first-order administrative unit(s).

Coding Adaptation Actions by Hazard Type
To link the EM-DAT and CDP datasets, we coded each unique adaptation action in the CDP dataset according to the nine EM-DAT hazard types: drought, earthquake, extreme temperature, flood, landslide, mass movement, storm, volcanic activity, and wildfire. In this step, we only considered primary first-order effects. For example, while wildfires often coincide with droughts and heatwaves, we only consider wildfire-related adaptation actions to be those directly related to wildfire events. However, it is possible for one action to be directly related to multiple hazard types, or even to allhazard types. Codes are binary; either a specific adaptation action is interpreted as being directly related to any given hazard event type (coded as '1') or it is coded as being unrelated to that specific event type ('0'). We include the full table of coded CDP adaptation actions below (Supplementary  Table B1). We also include the results from an internal inter-coder reliability test to verify the coding scheme (Supplementary Figure B1). Lastly, in instances where actions were described in languages other than English, we used Google Translate to convert the texts into English prior to coding (action descriptions in Supplementary Table B1 are presented in the original language, as they appear in the CDP dataset).

Inter-coder reliability test of adaptation action type categorization
In order to verify the results of the coding scheme in Supplementary Table A1, we conducted an internal inter-coder reliability test. In the first step, JH coded all adaptation actions according to the four types. Next, JH selected 20 adaptation actions from the CDP dataset (see the bold-faced adaptation actions in Table A1 above). While all individual actions were identified at random using a random number generator in the R software environment, the process was "semi-random" to ensure a representative sample of the different types of adaptation actions. In the second step, JH provided the adaptation action descriptions and detailed coding instructions, and DN, MM, and CP coded the 20 actions according to the types of hazard events each action directly addresses. In the last step, we calculated the level of agreement among the three coders. There was, on average, over 70% agreement between all four coders ( Figure A1) concerning the categorization of adaptation actions. We then discussed all issues of non-agreement, and the final coding for each action of non-agreement was determined by applying codes agreed by two of the three coders).

Supplementary Figure 2:
Results from internal inter-coder reliability test for twenty semi-randomly selected adaptation actions.

Section 3: Dependent, Independent, and Control Variables
In this section, we provide the full descriptions of all variables investigated in the multiple regression analyses, including the data sources for each variable (Table B1). We provide additional discussion of the adaptive capacity variables, including the relevant theoretical foundations and justification for selecting these specific variables for inclusion in the analysis.

Variable descriptions
Supplementary The total urban population of a given city.

CDP Time lag
The average number of years between natural hazard events and adaptation actions taken by a given city.
CDP/EM-DAT *For some models (Supplementary Tables 1b-5b, 11b, 14b, and D16e-D16h), economic damages were normalized using Gross Domestic Product (GDP) per capita. Disaster frequency was normalized by the total number of disasters that occurred within the country over the time period of the study, while affected population and fatalities were normalized by city population.

Adaptive Capacity Control Variables
We included several relevant measures as controls for determinants of adaptive capacity, which is an aggregation of properties shaping the propensity or ability of any governing system to undertake adaptation actions. Due to the lack of comparable global data to determine the adaptive capacity of cities and local governments, we rely on national-level indicators of the preconditions for adaptation actions. Hence, the caveat is that these indicators may overlook information concerning local variation. The selection of measures is based on three premises.
First is the analytical separation of adaptation actions from the adaptive capacity of a system. In this study, adaptation actions refer to measures undertaken by a city to alleviate the adverse impacts of climate change and/or enhance resilience to natural hazard events. A number of system attributes are important preconditions that shape the ability of cities to formulate, initiate, and implement these adaptation actions.
Second, adaptation actions are defined here relatively narrowly as measures resulting from deliberate decisions by policy actors, including politicians, bureaucrats, and other stakeholders, to change the existing order. Therefore, for this study, we assume that only those system attributes that directly affect decisions concerning adaptation actions are relevant to consider in the models. These measures include different system attributes that support the collective ability of policy actors to translate the experience of hazard events into adaptation actions. Other community features commonly associated with adaptive capacity, for instance, various social relationships and the ability of individuals to selforganize and innovate, are crucial elements of adaptive capacity, more generally but not relevant factors conditioning policy-making associated with adaptation actions.
Third, the literature identifies several important determinants of adaptive capacity related to attributes of individuals and the internal dynamics of public organizations, which in turn may enable or constrain policy-making concerning adaptation actions. Examples include institutional memory, the ability of policy actors to learn and change assumptions, and the vision and entrepreneurial skills of leaders. These attributes, however, are elusive, and to our knowledge, currently, no data exist to measure these across a wider set of cases. Hence there is a lack of data to enable comparisons of these attributes in cities around the world.
Following these premises, we set out from core dimensions identified by established frameworks of adaptive capacity and searched available datasets with global reach for the best possible proxy measures for each theme, respectively. To identify relevant data, we consulted two datasets: Varieties of Democracy (V-Dem) 1 and the Quality of Government (QoG) 2 . The selection of indicators has also been guided by the scope of each dataset, respectively, where some indicators were deemed relevant but excluded because data were only available for a limited number of the 58 countries included in the analysis. For example, the QoG dataset (p. 565) entails indicators of the quality of democracy, which recognize the essential role of democratic participation and oversight to learning and adaptation. Although these indicators would be relevant for the purpose of this study, data for these indicators were only available for 41 countries.
Political stability, including, e.g., the absence of civil conflict and the functioning of democracy 3,4 , is depicted as a prerequisite for adaptation. Some studies 5 define political stability as a proxy measure indicating a willingness to invest in adaptation. Other work 6 emphasizes the importance of stable political institutions that provide predictability in support of collective action. Political stability may also interact with other factors associated with adaptive capacity. For example, a decrease in political stability may trigger consequent problems related to economic conditions, which, in turn, can constrain adaptation initiatives 7 . The current study measures political stability by an aggregated index of 'state legitimacy' derived from the Fragile States Index (https://fragilestatesindex.org/indicators/p1/) accessed via the Quality of Government dataset, combining state performance in five areas: public confidence in the political process, political opposition (peaceful demonstrations and riots), transparency (evidence and considerations of corruption), openness and fairness of the political process (political rights, representative government, leadership transition, free elections), and political violence (political assassinations, armed insurgents, and terrorism). For this study, we recoded the Fragile States Index indicator, which originally measures political stability on a scale 10-0 where higher scores indicate less stability, so that higher scores indicate greater stability.
Studies suggest that adaptive capacity is conditioned by national development since financial resources are needed to support the implementation of adaptation policy measures. In addition, various other resources (political, human, legal, and technological) condition institutional effectiveness, including the ability of institutions to change norms and rules 3,6,[8][9][10] . Given that adaptive capacity is related to economic well-being, assessments of adaptive capacity use Gross National Income (GNI) as one indicator to demonstrate wealth 11 . On this basis, the current study includes GNI (data derived from the World Bank) as one proxy measure of adaptive capacity. It should be noted that GNI is indirectly included in one of the regression models (Supplementary Table 21b), where we used Gross Domestic Product (GDP) per capita to normalize economic losses of disasters. Therefore, GNI was removed as a control from this model.
Another dimension of adaptive capacity considered in this work concerns meritocracy, specifically knowledge of individuals representing state bureaucracy. Knowledge is generally defined both as a determinant and indicator of the adaptive capacity of individuals, communities, organizations, and countries 4,12 . Given our interest in adaptation actions undertaken by cities as government entities, the study focuses on the role of knowledge in supporting policy-making. Specifically, we focus on the capacity of individuals to interpret information regarding current and future local environmental conditions as the basis for making policy decisions about adaptation actions 10 . To this end, we use a generic indicator of 'meritocracy' (derived from the V-Dem dataset), capturing whether appointment decisions within state administration are based on personal and political connections versus skill and merit. Thus, the assumption is that appointment decisions based on skill and merit help sustaining a cadre of bureaucrats and other policy professionals with the capacity to make informed decisions based on experience. The scale ranges 0-4, with higher values indicating greater reliance on skill and merit.
Stakeholder diversity, defined here as the participation and influence of multiple interests and actors in policy-making, is another determinant of adaptive capacity. Ensuring broad participation of actors representing organizations at different levels and sectors, including non-state actors, is a way to give room for multiple opinions, beliefs, and problem definitions and to widen the range of options in policy-making 6 . Such variety provides access to different types of knowledge 13 that reduce uncertainty and support learning and the development and implementation of adaptation measures 14 . To account for stakeholder diversity, we relied on information concerning the 'range of consultation' (derived from the V-Dem dataset), which measures the width of consultations at elite levels when important policy changes are considered. This scale ranges from 0 (indicating no consultation where a leader or small group make all decisions on their own) to 5 (decisions are based on consultations involving actors from across the political spectrum and sectors of society).
Finally, the study includes a measure of local government power as a means to control for whether adaptation actions are more frequent in cities located in countries with elected local governments that are able to operate independently from unelected local actors (with the exception of judicial bodies). The Intergovernmental Panel on Climate Change recognizes the importance of empowering local communities to take action for reducing vulnerability and strengthening resilience, including responsibility and decision-making to improve preparedness through post-disaster assessments 15 . While the literature 5 emphasizes the multifaceted nature of local-level adaptation (spanning broad sets of community characteristics and interactions), we focused exclusively on local government independence from the involvement of unelected local actors. Hence, the assumption is that greater independence facilitates adaptation actions, partially by avoiding or bypassing political conflict and power-struggles 16,17 . We include a measure of local government power (derived from the V-Dem dataset) ranging from 0 (countries have no elected local governments) to 1 (elected local governments operate without restrictions).

Section 4: Full Regression Results for All Types of Adaptation Actions
Tables 4-8: bivariate regression models of each type of adaptation action with respect to absolute hazard frequency and severity measures     Tables 9-13: bivariate regression models of each type of adaptation action with respect to normalized hazard frequency and severity measures          Tables 19-23 contain bivariate regression models of each type of adaptation actions with respect to adaptive capacity control variables     Tables 24-26 contain multiple regression models of each type of adaptation action with respect to absolute, normalized, and baseline hazard frequency and severity measures, respectively, and do not include any additional control variables.   Tables 27-28 contain multiple regression models of each type of adaptation action with respect to adaptive capacity control variables  Tables 29-31 contain raw, normalized, and baseline multiple regression models of each type of adaptation action with respect to frequency, severity, and relevant control variables.     Tables 33-36 contain multiple regression models of each type of adaptation action with respect to disaster frequency and severity measures, individually, and all control variables.      Table 35: Adaptation action types with respect to affected population and controls.

Dependent variable:
All Specific Expansive General Other (1)   Tables 37-44 contain multiple regression models of each type of adaptation action with respect to normalized disaster frequency and severity measures, individually, and all control variables.   Table 38: Adaptation action types with respect to normalized disaster damages and controls.

Dependent variable:
All Specific Expansive General Other (1)  Table 39: Adaptation action types with respect to normalized affected population and controls.

Dependent variable:
All Specific Expansive General Other (1)      Tables 45-56 contain multiple regression models of each type of adaptation action with respect to disaster frequency and severity measures, individually, and all control variables.            Note: * p<0.1; ** p<0.05; *** p<0.01


A Survey of Location-Allocation of Points of Dispensing During Public Health Emergencies

Public health emergencies such as disease outbreaks and bioterrorism attacks require immediate response to ensure the safety and well-being of the affected community and prevent the further spread of infection. The standard method to increase the efficiency of mass dispensing during health emergencies is to create emergency points called points of dispensing (PODs). PODs are sites for distributing medical services such as vaccines or drugs to the affected population within a specific time constraint. These PODs need to be sited in optimal locations and have people (demand points) assigned to them simultaneously; this is known as the location-allocation problem. PODs may need to be selected to serve the entire population (full allocation) or different priority or needs groups (partial allocation). Several previous studies have focused on location problems in different application domains, including healthcare. However, some of these studies focused on healthcare facility location problems without specifying location-allocation problems or the exact domain. This study presents a survey of the PODs location-allocation problem during public health emergencies. This survey aims to review and analyse the existing models for PODs location-allocation during public health emergencies based on full and partial demand points allocation. Moreover, it compares existing models based on their key features, strengths, and limitations. The challenges and future research directions for PODs location-allocation models are also discussed. The results of this survey demonstrated a necessity to develop a variety of techniques to analyse, define and meet the demand of particular groups. It also proved essential that models be developed for different countries, including accounting for variations in population size and density. Moreover, the model constraints, such as those relating to time or prioritizing certain groups, need to be considered in the solution. Finally, additional comparative studies are required to clarify which methods or models are adequate based on predefined criteria.

INTRODUCTION
Public health emergencies such as disease outbreaks and bioterror attacks demand an immediate response to save people's lives and to prevent the further spread of infection. The Centers for Disease Control and Prevention (CDC) has created a new approach and organizational unit to meet the challenges in responding to bioterror attacks and other large and complex health emergencies, whether naturally occurring, intended, or casual, such as anthrax attacks (1). Furthermore, being able to respond to a range of future emergencies means that the CDC has to expand its capabilities and responses in terms of scale and speed to cover two global trends: faster and more frequent international travel and the increasing global population (2).
The standard method to increase the efficiency of mass dispensing during health emergencies is to create emergency points, called points of dispensing (PODs) (3). PODs are locations for dispensing medical services, such as vaccines or drugs, to a large number of people, while meeting a specific time constraint (4). PODs are necessary to prevent people who are not sick from becoming infected (5). There are two types of PODs, opened and closed (6). While opened PODs are located at public sites such as schools, closed PODs are operated by a partner organization while the operation continues during the emergency (6).
Poor decision-making while determining the location of healthcare facilities such as PODs results in negative outcomes. These adverse outcomes are not limited to cost and service but also increased infection numbers and deaths due to difficulty in accessing the service (5). A POD need to be sited in a location and, simultaneously, have people assigned to the located facilities; this is known as the location-allocation problem (5). Locationallocation models are used to locate optimal facility locations and allocate demand points to each facility (7). Location-allocation models are operated by planners, who usually determine the number of facilities required (7). The model then locates the facilities before the demand points are allocated to the nearest facility based on a measure such as shortest travel distance or time (7), as shown in Figure 1. Facilities may need to selected to serve the the entire population (full demand points allocation) or populations at high risk, or with different priorities or needs (partial demand points allocation).

Motivation
In March 2020, the World Health Organization (WHO) declared the novel coronavirus disease (COVID-19) a global pandemic (8). During the COVID-19 pandemic, public health authorities required an immediate response plan to access the affected population and ensure they have easy access to the healthcare services and supplies (such as water or food) they need. According to the CDC, distributing food via delivery may be an ideal approach to inhibiting the spread of COVID-19 (9). However, in the case of healthcare services where a physical examination is required, patients need to be able to access physical locations (face to face) to receive services (10).
Given the nature of the fast-spreading of COVID-19, public health authorities across the world needed to respond immediately to ensure the protection of the public, stop the infection from spreading further and provide mass prophylaxis. In the early days of this pandemic, locations needed to be assigned and set up for COVID-19 testing. Then, when COVID-19 vaccines were approved, health facilities and other sites or PODs were needed to provide fair access to the vaccine.
These PODs differ from traditional healthcare facilities, since they involve the provision of easy and fair access for all demand points (5) and have a role in providing medical supplies to protect the public during public health emergencies. However, setting up PODs needs to be done on a large scale and within specific timeframes (4). Thus, locating PODs and allocating the demand points to the designated PODs is a critical and challenging task.
PODs may need to be located to serve the entire population (full allocation) or those who have difficulty accessing PODs and those with different priorities or needs (partial allocation). This survey provides taxonomy-based demand points allocation as full and partial demand points allocation. Which may help improve location-allocation models for efficient PODs assignment while ensuring fair access for all populations or addressing different populations' needs.

Contributions
This survey focuses on location-allocation problems for PODs during public health emergencies. Previous surveys have focused on location problems in different application domains, including the location-allocation of general healthcare facilities (11,12). Location problems "intend to determine the optimum locations for a set of facilities by minimizing or maximizing some objectives for satisfying the existing and/or projected demand with respect to a set of constraints in some given space (11,13)", while location-allocation problems intend to locate facilities and simultaneously allocate demand points to facilities (5). Surveys, such as Ahmadi-Javid et al. (5) and Afshari and Peng (14), have focused on the locations of healthcare facilities without a focus on the location-allocation problems or specific applications such as PODs during public health emergencies. To the best of our knowledge, there exist no studies surveying PODs locationallocation problems during public health emergencies in the form of allocation decisions (full and partial). Thus, the contributions of this paper are as follows: • Providing a taxonomy for PODs location-allocation problems during public health emergencies based on demand allocation (full and partial); • Comparing existing solutions based on their key features, strengths, and limitations; • Discussing the challenges and issues associated with the current solutions for PODs location-allocation during public health emergencies and providing future directions.
The rest of this survey is structured as follows: Section 2 introduces the background of the components of locationallocation problems, location-allocation models and locationallocation solutions. Section 3 presents survey methodology while, Section 4 presents existing solutions to PODs locationallocation during public health emergencies. Section 5 then discusses the challenges of, and potential future research direction for, location-allocation solutions for PODs during public health emergencies. Finally, Section 6 concludes the study.

Components of Location-Allocation Problems
There are three essential components for location-allocation problems: facilities, locations, and customers (15,16). Facilities reflect various objects with spatial positions to optimize interaction with preexisting objects; an example of a facility is a school or hospital. Locations are a set of candidate points where facilities can be positioned or located. Finally, 'customers' is a general term to indicate any objects interacting with facilities. Customers can be users demanding a service from the facility, for example communities requesting a public service in a rural area. The modeling and analysis of interactions between facilities and customers often require the knowledge of their spatial  X j 1 if a facility is located at candidate location j, otherwise 0.
Y ij 1 demand point i is assigned to facility at the candidate location j, otherwise 0.
positions (16). In the remainder, we refer to customers as demand points.

Location-Allocation Models
The location problem can be discrete, continuous (5,17) or network (16,18). Continuous location problems locate the facility anywhere in a spatial region, while discrete location problems limit facility locations to a set of prespecified candidate points which may include the locations of the demand points (5,16). Network models implement graph theory to model location problem (16), and for the purpose of composing nodes and links (18). Facilities and demand points can be located only on links or nodes, and any travel between them should be within the network (18). We focus on the discrete case for several reasons. It is highly flexible in terms of including various economic and geographical features in the models (16). It also provides more natural design results when free land is available and requires locating a new facility at a prespecified place (16). More importantly, using a discrete set of candidate locations is the preferred approach for the majority of location problems and appears frequently in healthcare applications (5). There are two categories of discrete location-allocation problems which are coveringbased and median-based problems (5). Table 1 explains the mathematical notations and input parameters used in their mathematical formulations.

Covering-Based Problems
In covering-based problems, each demand point should be withing a particular distance or time range from the facility that serves them. P-centre location problems (PCLPs) are covering-based problems that aim to minimize the maximum travel distances or times between the selected facilities and the demand points. The decision variables of the PCLPs are presented in Table 2, and the formulation is as follows (5,18):
In the formulations listed above, (1) illustrates the first objective of PCLPs, namely, to minimize the maximum travel time or distance between the demand points and the closest facilities. The constraints are listed from (2) to (8), where (2) requires assigning each demand point to only one facility. Constraint (3) limits the number of established facilities. Constraint (4) states the maximum travel time or distance, which means that W must be greater than the distance between any demand point and its serving facility. Constraint (5) is the facility that should be opened to serve demand points. Finally, constraints (6), (7) and (8) are the domain constraints. Note that (18) considers W a decision variable, while (5) consider it an auxiliary variable.

Median-Based Problems
In this set of location-allocation problems, facilities are located at the candidate locations to minimize the weighted average distance cost among each demand point and the facility to which they are assigned. Median-based problems are divided into pmedian location problems (PMLPs) and fixed-charge facility location problems (FCLPs) (5). PMLP aims to locate the facilities while minimizing the weighted travel time or distance. The decision variables of the PMLPs are presented in Table 3, and the formulation is as follows (5,16,18): Variable Description X j 1 if a facility is located at candidate location j, otherwise 0.
Y ij 1 demand point i is assigned to facility at the candidates locations j, otherwise 0.

Formulation of PMLPs
In the formulations listed above, (9) objective of PMLPs minimizing the total demand-weighted travel time or distance between every demand point and their closest facility. The constraints are listed from (10) to (14), where (10) limits the assignment of each demand point to a single facility. Constraint (11) limits the number of established facilities. Constraint (12) requires a facility to be opened in order to serve demand points. Finally, constraints (13) and (14) are integrality constraints. FCLPs attempt to minimize total costs in terms of traveling and setting up the facilities. The decision variables of the FCLPs are presented in Table 4, and the formulation is as follows (5,18).
Formulation of FCLPs Frontiers in Public Health | www.frontiersin.org In the formulations listed above, (15) illustrates the objective of FCLPs, namely, to minimize the total cost of demand-weighted distance and facility opening. The constraints are listed from (16) to (19), where (16) requires assigning each demand point to a single facility. Constraint (17) requires opening a facility that serve demand points. Finally, constraints (18) and (19) are integrality constraints.

Location-Allocation Solutions
There are three solutions approaches for the location-allocation problems (15): Exact solutions, heuristic methods, and metaheuristic methods. Exact solutions (15,19,20), such as branch and bound algorithm, find the optimal solution through systematically examining a large subset of all possible feasible combinatorial set of facility location and demand allocation solutions. On the other hand, heuristic methods (15,19,20), such as greedy assignment, find the near-optimal solution by examining only a limited subset of the potential combinations. Finally, metaheuristic methods such as genetic algorithm (GA) (15,21), provide a general framework and a higher-level procedure to design heuristic algorithms which are more powerful than standard heuristic methods.

SURVEY METHODOLOGY
This survey focuses on PODs location-allocation during public health emergencies. Thus, two stages were involved in achieving the objectives of the study: examining the results of previous relevant surveys and reviews, and carrying out a literature review.
Examining previous surveys helps to understand the topic and identify gaps in survey and review articles. The literature review stage helps find, present and compare research papers proposing a solution for PODs location-allocation during public health emergencies. This stage also analyzes the solution's strengths, limitations and key features to identify particular issues and challenges to guide future directions. This survey focuses on solutions for location-allocation of PODs during public health emergencies; a more detailed explanation and discussion will be presented and provided for these solutions in Section 4. A hybrid search strategy was adopted using systematic and non-systematic approaches to obtain relevant research papers. In non-systematic approaches, the papers were retrieved through backward and forward citations or snowballing. The databases considered in the search were Web of Science, Scopus, and Science Direct. We also search Google Scholar, a web search engine that indexes a full text or metadata of scholarly literature in almost any area. The existing research papers in PODs location-allocation were retrieved using different keywords, namely location-allocation, points of dispensing, mass vaccination, mass dispensing, emergency medical services. All papers had to be in the field of emergency healthcare. While to find existing survey papers about PODs location-allocation, terms such as survey, review and healthcare facility location were used in addition to the previous search keywords. The title and abstract of each paper were evaluated to consider the relevant papers. The search is limited to papers written in English. A detailed overview of the survey methodology is presented in Figure 2.
Most recent and relevant survey and review articles in a range between 2012 and 2021 were examined to obtain an overview of the locations of healthcare facilities or healthcare facility locationallocation in emergency situations. Basar et al. (12) provided a taxonomy for the locations of emergency service stations; the taxonomy covers the type emergency, such as a fire or pandemic disease, as well as use of different objectives, assumptions, constraints, modeling choices, and solutions techniques. Afshari and Peng (14) examined challenges based on the current needs of decision-makers, and methods for locating healthcare facilities to guarantee an optimal solution. They classified literature based on resolved challenges, methods and solutions for facility locations. They also examined the measures used to evaluate the studies in terms of their objective functions and constraints. Ahmadi-Javid et al. (5) presented a paper for both emergency and nonemergency healthcare facility location that included PODs, but presented only three relevant research papers.
Turkoglu and Genevois (11) presented a comparative survey for service facility location problems in different application fields, including healthcare. First, they defined several key features for location problems, such as the number of facilities or objectives. They then defined descriptive dimensions such as application fields, and used these 19 characteristics to present existing solutions. The solutions were categorised based on the application fields and these characteristics were summarized for each field. Moreover, they compared the solutions in different application fields based on these characteristics.
Finally, in the location-allocation problem, Gwalani et al. (22) evaluated and compared four heuristic algorithms to solve the p-median problem in different terms, namely objective function value, time, and stability. Also, they considered the effect of scale (number of sources and destinations) and spatial distribution of destination locations on the performance of these algorithms. Finally, they applied the evaluation in synthetic and real datasets, the latter including resource distribution during bio-emergencies.

LOCATION-ALLOCATION MODELS FOR PODS DURING PUBLIC HEALTH EMERGENCIES
This section presents the existing research papers for locationallocation models for PODs during public health emergencies. We focused on retrieving the research papers between 2017 and 2021. As a result, there were (2) and (2) research papers from 2017 and 2018, respectively, while in 2020 and 2021, there are (1) and (4) research papers. The existing location-allocation methods for PODs, as shown in Figure 3, are divided based on the allocation of demand points as full demand points allocation and partial demand points allocation.

Full Demand Points Allocation
Full demand points allocation presents the solutions for locating the facilities regardless of the type of demand points. In other  words, it provides equal access to PODs for all populations, as shown in Figure 4. Risanger et al. (23) suggested a model to select testing sites for COVID-19 using pharmacies to overcome the gap in coverage in the US in terms of the number of individuals who wish to perform testing at their nearest site and reduce the distance required to travel. They adopted an optimization model (24), which was used to optimize pharmacy-based distribution of antiviral drugs during the H1N1 pandemic. The objective is to maximize the number of people probably who can travel to limited testing sites (pharmacies). The optimization model is based on mixed-integer linear programming (MILP), they used mathematical optimization such as JuMP or CPLEX to solve it (25,26), and implementation of solution is freely available via (25). The model used geographic region zip code and data with the population willing to travel as the input (24). The zip codes (mail delivery address) were converted to ZIP Code Tabulation Areas (ZCTAs). ZCTAs are used by the US Census Bureau to help obtain such information as counts, centroid coordinates, and latitudes and longitudes for each ZCTA. The pharmacy datasets were obtained from InfoGroup, Amazon Web Services, and Esri's national COVID-19 database. The results estimated that 94% of the population could access pharmacies, with coverage exceeding 80% and 90% for 47 and 12 states, respectively. However, the capacity of pharmacies is not considered, which may make it difficult to manage matching testing capacity with the testing demand.
In another study, Deng et al. (27) aimed to minimize the number of emergency medical service (EMS) facilities that need to be added to current hospital networks to cover 90% of the population within 15 min of travel time. The population and facility data were obtained from LandScan (28), the Health Bureau of Sichuan Province and the Health Bureau of Chengdu (not available to the public). The proposed  solution used Nearest-Neighbor to find the nearest EMS using geographical information systems (GIS), with the shortest one used. To solve the location set covering formulation, a GA was then used. The results demonstrated that 55 of new EMS facilities were able to cover 90% of the population within 15 min. It was also found that to access an EMS facility, the weighted median shortest travel time were reduced by 14.57%. However, in some cases the road network may be affected by traffic jams, leading to an increase in access time.
Devi et al. (29) suggested a model based on MILP for temporary testing laboratories, aiming to minimize the total cost and the travel time from demand points to the laboratories. The overall cost includes fixed, operating, traveling and capacity underutilization costs. They applied the model to Holmberg et al. (30) and found underutilization of capacity. Based on that, they added a third decision variable to capture the capacity needs. The model was applied to a case study in Maharashtra, India, and the data are included in the paper. The results demonstrated that when there are 27 temporary testing laboratories, the total cost for the first objective is 6.08E + 11, while the second objective achieves 30.681 min. However, it would prefer to use ward-village level data to ensure that the centers area located near people.

Partial Demand Points Allocation
Partial demand points allocation considers a partial group of the population assigned to PODs, as shown in  optimization model, secondary optimization model, and postprocess. The primary optimization model aims to produce an optimal coverage rate to achieve equal access for each locationpriority group pair. The secondary optimization model then reduces the type of vaccine for each priority group and ensures geographic equity by providing similar proportions of vaccine allocation among regions. The final step in the model is the postprocessing to obtain integer values for vaccine dose allocations; both primary and secondary optimization model ignores the integrality requirements. The model used priority group data (32) and healthcare locations where the vaccine was available using a tool designed for Texas (33). The results showed that with a small, reserved amount of vaccine (6.8%), the model satisfied the requirement of equal access, with 61.1% coverage of the priority population. However, the number of doses required to achieve immunity may reduce the chance of ensuring fair access for all communities.
Gao et al. (34) proposed two models for the temporary emergency center location-allocation such as M1 and M2. M1 aims to minimize travel time, whereas M2 aims to minimize the rate of mortality risk. Then, GA and modified fuzzy c-means (MFCM) (GA-MFCM) were developed. MFCM was modified to introduce the initial value for mortality risk. Whereas, GA was used to optimize the center of each cluster, and MFCM to group the data. They applied the proposed model in the form of a real dataset from Portland (America) post-earthquake emergencies (34). The results show the M1 model achieves less travel time than the M2 model, and M2 minimizes the total rate of mortality risk for patients. However, the injury severity level at the threshold of death is stated as a deterministic value.
In another study, Hudgeons (35) compared the use of the opened PODs only against using the Home Health Agency (HHA) as closed PODs and examined the impact on maximizing throughput (people's access to PODs) and minimizing resource allocation (staff). The suggested solution consists of replicating the process of services in PODs without HHA and identifying the necessary HHA nurse process. In replicating PODs, RealOpt simulation modeling software was used to show the patient's state traveling to, entering and leaving the PODs. The process of the HHA nurse was then defined as in-home and in-transit activities. For in-home activities, Monte-Carlo simulation was used to estimate the nursing processes required from entering to leaving the patient's home. For the transit modeling, traveling salesman heuristics were used to find the shortest path from the current location to the following location. Data from the nursing faculty at the University of Arkansas (not available to the public), the zip code for the HHA, and the squared area of each zip code (36) were used. The results showed that including the HHA as closed PODs increased throughput by 2.4% and required 16 fewer members of staff. Moreover, the solution targets the vulnerable group but is not based on a real-life case as such it does not specify who was involved and where they were located.
Memari et al. (37) suggested a solution to allocate injured people to temporary emergency stations, with four priorities group from high to low priority. They proposed a new fuzzy dynamic location-allocation with multi-medical servers M/M/c queue model. The model has bi-objectives to minimize the cost to construct temporary emergency stations and death rate and minimize the travel time and queue's response time. Furthermore, a fuzzy number is used to tackle treatment demand, travel time, and arrival treatment rate uncertainty. They used the augmented ε-constraint method to validate the model and a non-dominated sorting genetic algorithm (NSGA-II) to solve the problem; they applied the model in synthetic data and real case from Iran, Tehran (no information about the data). The results demonstrated that the model was able to achieve a good result to find the solutions. However, in some cases the road network may be destroyed due to disaster, leading to need for vehicle routing.
Li et al. (38) suggested a model that could achieve three objectives for locating vaccine sites to provide preventive vaccination to a specific group, in this case children. The three objectives were to minimize average travel distance to demand points, maximize the number of fully open vaccine stations, and minimize total costs, such as the fixed cost of opening the vaccine station or the cost of medical staff. A mixed-integer nonlinear programming (MINLP) model was used, which was divided into two stages to reduce the computational burden. Firstly, minimizing average travel distance is considered the main objective, and this objective and other objectives are solved independently using MILP; secondly, the other two objectives are considered constraints. The model was applied to Nanshan CDC in Shenzhen, China; the data for address and areas of vaccination stations are obtained from CDC. Vaccination stations, demand points and other data are available in (39). The results demonstrated that opening 50 vaccine sites involves a trade-off between three objectives, which helps CDC's decisionmaking. However, moving the demand point from one location to another may involve the need to re-assign people to a new and closer vaccine station.
Emu et al. (40) suggested the priority in conjunction with distance-based vaccine distribution model (PD-VDM) based on constraint satisfaction programming (CSP) to optimize the  vaccine distribution, aiming to maximize distribution for priority groups. In this case, they considered age and minimized average travel distance for individuals to get vaccinated. The PD-VDM model is initialized by using k-medoid to select the number of distribution centers (DCs) and then the priority groups are assigned by minimizing the average distance. They performed two experiments based on the randomly generated data and real data obtained from Chennai, India. Additionally, they compared their model with other models using different factors, such as none or one of the distance and priority factors. They found that, their proposed model (using randomly generated datasets) was able to vaccinate the highest priority groups at 100% while reducing the average travel distance by more than 40%. However, in the real case, the model could vaccinate less than 90% of the three highest priority groups while reducing the average travel distance by more than 70%, which made the proposed model better than others. However, number of doses required to achieve immunity may reduce the chance of fair access for all communities and other groups needs. Table 5 summarizes current work on location-allocation for PODs during public health emergencies in terms of reference, methodology, objective function, constraints, decision variables, type of the location-allocation model, datasets, and case study region (when applicable). The methodology is the solution for the proposed model. The objective function of the numerical value that could be maximized or minimized (42). Constraints are defined as the variable's possible values, while decision variables are the set of values that need to be defined to solve the problem (42). The location-allocation model is as described in Section 2.2. Datasets refer to the data used in the proposed model, while case study refers to whether the model is applied to a specific location, such as a particular city. Table 6 presents a comparison of current work on locationallocation for PODs based on their advantages and limitations during public health emergencies. Table 7 compares studies based on a range of key features: locating facilities, identifying number of facilities, the number of new facilities, type of facility, density, and full or partial demand points allocation. In terms of locating facilities, the physical locations for PODs are selected, while the identifying number of facilities defines the number of PODs needed to accomplish the desired objective. Meanwhile, the number of new facilities identifies the number of new PODs needed in addition to the current set of facilities to accomplish the desired objective. The type of facility defines the place allocated to provide the service, such as a pharmacy. Finally, in terms of density, it is stated if the proposed solution covers different population densities.

CHALLENGES AND FUTURE RESEARCH DIRECTIONS
Section 4 presented different solutions currently proposed for the location-allocation of PODs during public health emergencies. However, certain issues and limitations remain that must be addressed in future studies. This section discusses these challenges and suggests future research directions for PODs location-allocation during public health emergencies.

• Analysis of demand points
Demand for services in PODs will vary from country to country and from one kind of emergency event to another. For example, the COVID-19 vaccine is not appropriate for children, which may change the allocation decision. Existing solutions either assume that specific groups (38) are involved, -It would prefer to use ward-village level data to ensure that the centers area located near people.  or select them based on priorities in a specific type of emergency event (31). Thus, any analysis of the desired group (such as a priority or a vulnerable group) is very challenging and needs more attention due to the varied types of emergency (such as a pandemic or tsunami) and the needs of different groups of people (for example the elderly, children, or pregnant women). More importantly, the point or location of demands needs to be considered, along with whether it is easy to access the PODs or a new POD needs to be set up.
• Covering different countries and population Population size and density differ from one country to another; current solutions focus either on China (27,38) or the US (23,31). A few cases from other countries such as Iran and India (37,40). A wider variety of solutions is needed to cover populations of different sizes and densities in other countries and cities in addition to China or the US. Doing so may help decision-makers to locate and utilize PODs according to the country's needs and to allocate them to populations of different sizes and densities.

• Model constraints
PODs are locations for dispensing medical services, such as vaccines or drugs, to a large number of people while meeting a specific time constraint (4). As shown in Table 5, PODs location-allocation problem considers variety constraints such as budget or number of the fully open facility.
Therefore, PODs location-allocation problem needs to consider other types of constraints related to PODs location-allocation problems, such as solving the problem with the required timeframe to ensure the coverage of either entire population or partial population. Also, the partial group such as vulnerable or priority groups can be considered a constraint for the model.

• Evaluation and comparative studies
To address the location-allocation problem, Gwalani et al. (22) evaluated and compared four heuristic algorithms to resolve the p-median problem as applied to different terminologies, namely objective function value, time, and stability. However, there is a dearth of comparative studies evaluating methods utilizing several criteria, involving determining which methods are most applicable to maximize population coverage for populations of different sizes and densities. This will be a very interesting research direction for researchers seeking to provide data to evaluate and compare models and methods efficiently.

CONCLUSION
Public health emergencies such as disease outbreaks need PODs to dispense medical services such as vaccines to a large number of people within a specific period of time. PODs need to be situated in an optimal location and have demand points assigned to them simultaneously; this is known as the location-allocation problem. PODs may need to be selected to serve the entire population (full allocation) or different priority or needs groups (partial allocation). This paper presents a survey of PODs location-allocation models during public health emergencies and provides a taxonomy according to full and partial demand points allocation, and discusses a range of studies that have employed this taxonomy, comparing the advantages, limitations and key features of each one.
The survey concluded that in order to improve PODs location-allocation models during public health emergencies, there is a need to develop various techniques to analyze and define the demand of partial groups and provide the desired coverage to those demand points. Efforts to propose models to cover the needs of different countries, including variation in population size and density, are urgently needed. Furthermore, model constraints, such as time or priority groups need to be considered in the solution. Moreover, additional comparative studies are required to clarify which methods or models are adequate based on predefined criteria. Finally, this work has discussed the current challenges associated with existing techniques and recommended future research areas in PODs location-allocation models.