Implementation of equity in resource allocation for regional earthquake risk mitigation using two-stage stochastic programming.

Risk Analysis, Vol. 35, No. 3, 2015

DOI: 10.1111/risa.12321

Implementation of Equity in Resource Allocation for Regional Earthquake Risk Mitigation Using Two-Stage Stochastic Programming Mohammad R. Zolfaghari∗ and Elnaz Peyghaleh

This article presents a new methodology to implement the concept of equity in regional earthquake risk mitigation programs using an optimization framework. It presents a framework that could be used by decisionmakers (government and authorities) to structure budget allocation strategy toward different seismic risk mitigation measures, i.e., structural retrofitting for different building structural types in different locations and planning horizons. A twostage stochastic model is developed here to seek optimal mitigation measures based on minimizing mitigation expenditures, reconstruction expenditures, and especially large losses in highly seismically active countries. To consider fairness in the distribution of financial resources among different groups of people, the equity concept is incorporated using constraints in model formulation. These constraints limit inequity to the user-defined level to achieve the equity-efficiency tradeoff in the decision-making process. To present practical application of the proposed model, it is applied to a pilot area in Tehran, the capital city of Iran. Building stocks, structural vulnerability functions, and regional seismic hazard characteristics are incorporated to compile a probabilistic seismic risk model for the pilot area. Results illustrate the variation of mitigation expenditures by location and structural type for buildings. These expenditures are sensitive to the amount of available budget and equity consideration for the constant risk aversion. Most significantly, equity is more easily achieved if the budget is unlimited. Conversely, increasing equity where the budget is limited decreases the efficiency. The risk-return tradeoff, equity-reconstruction expenditures tradeoff, and variation of per-capita expected earthquake loss in different income classes are also presented. KEY WORDS: Equity; optimization; resource allocation; seismic risk mitigation; Tehran

1. INTRODUCTION

to national gross domestic product (GDP). Such losses include damages to normal properties and infrastructures, business interruptions, and macroeconomic effects. Experiences from previous natural catastrophes also indicate that not all members of society experience natural disasters equally.(1) In fact, certain groups of people are more vulnerable to death and injury or economic loss. This is also the case for resilience as it might be hard for such groups to return to the situation before the disaster.(2) There are certain characteristics such as income, gender, race, ethnicity, age, and health status that may be used to categorize societies into different groups.

Natural catastrophes threaten human lives and properties, resulting in widespread social and economic disruptions. Although economic losses in the developing countries are of smaller size compared to those in the developed countries, they are still of significant importance when measured with regard Civil Engineering Department, K. N. Toosi University of Technology, Tehran, Iran, 15875-4416. ∗ Address correspondence to Mohammad R. Zolfaghari, Civil Engineering Department, K. N. Toosi University of Technology, Tehran, Iran 15875-4416; [email protected].

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C 2015 Society for Risk Analysis 0272-4332/15/0100-0434$22.00/1

Implementation of Equity in Resource Allocation Low-income and unemployed residents have less financial resources to withstand the impact of economic loss. This could be more significant for people in most developing countries where the proportion of population categorized as vulnerable groups outweighs the proportion in developed countries. There are various reasons for such vulnerability, namely: financial difficulties, migration from rural to urban areas, rapid unplanned development, and other socioeconomic factors. Natural catastrophe risk management measures have been improved in recent years to reduce the adverse effects of natural catastrophes. These measures include improving design codes, retrofitting of existing buildings and infrastructures, improving critical infrastructures, raising public awareness, urban planning toward emergency preparedness, using early warning systems for emergency response services, and insurance as a risk transfer mechanism. Disaster risk management incorporates measures to reduce adverse consequences and losses brought about by natural catastrophes (e.g., structural damage, human casualties, and damages to infrastructure, utility, and lifeline systems). However, concerns have always been expressed with regard to approaches used to distribute risk management funds, which, in most cases, are made regardless of the magnitude of risks and risk mitigation opportunities within a region and across different regions. Therefore, funds and national investments are not utilized as effectively and efficiently as expected. Another concern in the planning and performing risk management efforts is the issue of fairness among various groups suffering from natural catastrophes. For example, risk mitigation through structural retrofitting in some developing countries is an effective approach for buildings with certain structural integrity that are usually occupied by people of moderate to highincome level. Therefore, allocating retrofitting budget exclusively based on such criteria may distribute funds to groups of people with less financial vulnerability in contrast with those with low income and inexpensive assets. Thus, unfair financial resource allocation can increase social and financial inequality. Therefore, for sustained development in countries exposed to natural disasters, due attention should be paid to equity in the national mitigation and retrofitting planning as well as to efficiency. From the onset of proposal and implementation of risk management and mitigation plans, methods were also provided and introduced to assess the effectiveness and profitability (efficiency) of such

435 plans and also to compare the outcomes of their probable implementation. However, these methods should be similarly extended to assess issues such as equity and fairness of implemented risk mitigation measures. To this aim, this article develops a new framework for implementing the concept of equity in regional earthquake risk mitigation and discusses risk-equity tradeoff as a criterion for regional earthquake risk mitigation. To achieve this, a two-stage stochastic optimization program and its equivalent linear program are formulated to incorporate equity in financial resource allocation for regional earthquake risk mitigation strategies. In this article, the concept of equity is implemented as certain constraints in order to model fair financial resource allocation toward regional seismic risk management. To illustrate performance of the proposed model, a fullscale case study for residential buildings in Tehran, the capital city of Iran, is presented. The following questions are the main concerns addressed in this article using the results obtained from the model. (1) To what extent should mitigation in planning horizon be funded? (2) How should buildings be selected for mitigation? (3) How much should the reconstruction expenditures be for possible earthquake scenarios for a given mitigation decision? The model could also address the following questions. (1) Are the resources allocated in an equitable way and how does the inequity level affect the mitigation recommendations? (2) What is the effect of limitations in mitigation budget on the mitigation recommendations? (3) How does the financial risk aversion parameter affect the mitigation recommendations? There are more questions and ramifications to be pointed out here; however, they are likely to yield a more profound insight into some of the complexities involved in designing and prioritizing regional mitigation plans and strategies. In Section 3, a review of previous efforts and models used to measure equity in economics, public policy, transportation, waste and water management, and emergency management issues is presented. In Section 4, the framework and formulation of the two-stage stochastic optimization program are presented to incorporate equity into financial resource

436 allocation for regional earthquake risk mitigation strategies. Section 5 discusses the case study for Tehran and its results. 2. MODELS FOR FINDING OPTIMAL RISK MITIGATION MEASURES This section reviews different operational research methodologies and models proposed and used to find optimal natural disaster risk mitigation. These methods can help decisionmakers allocate available financial resources to the most beneficial measures among all available mitigation measures to maximize the benefit or minimize the related risks. Previous research related to resource allocation toward natural disaster risk management is categorized into four main approaches:(3) (1) Deterministic net present value (NPV) analysis or cost-benefit analysis;(4) (2) Stochastic NPV analysis;(5) (3) Multiattribute utility models;(6) and (4) Optimization models. Simple cost-benefit analyses incorporate a small number of predefined mitigation alternatives. Such methods can be used for a limited number of buildings and mitigation scenarios. In such cases, selection of optimal strategy can be made manually and comparison of alternative results can sometimes be time consuming and susceptible to human errors. Introduction of stochastic cost-benefit analysis provides facilities to account for earthquake uncertainties; however, due to the manual process for selection of optimal strategy, a limited number of earthquake scenarios could be used. Similarly, this is the case for the third method. In the deterministic optimization models, however, the concept of cost-benefit method is employed and NPV is used as an objective to compare many different alternatives. In this method, the mathematical tool known as optimization method is used to find the optimal measure automatically. In particular, a large set of mitigation alternatives in regional risk mitigation (including millions of alternatives) are compared automatically in order to maximize or minimize some stated objective(s) subject to certain constraints.(3,7) This method allows for more criteria to be included in the process, which could help the decisionmaker to ensure that the results are optimum with these criteria. Therefore, optimization models could enhance the process in contrast with the other three methods.

Zolfaghari and Peyghaleh Optimization models for earthquake mitigation decision-making processes were first performed by Shah et al.,(8) who presented integer programming with a budget constraint to maximize the NPV of earthquake mitigation investment for a single earthquake scenario. Dodo et al.(3) introduced a category of optimization models that centers on mitigation decisions and public decision-making perspectives. Such perspectives led to the formation of a very different modeling framework with a format that is much closer to regional loss estimation methodologies. Others contributing to these categories of modeling include Dodo et al.,(3,9) Davidson et al.,(2) Xu et al.,(10) Vaziri et al.,(7) Legg et al.,(11) and Motamed et al.(12) All these models (except Legg et al.(11) ) presented their work for earthquake risk mitigation decisions for part of a city or a region. In almost all these models, linear programs were developed to formulate regional earthquake risk mitigation resource allocation to perform for a pilot area. These models determined which buildings should be mitigated (in some cases which buildings should be reconstructed) to minimize expenditures such as total mitigation and expected postearthquake reconstruction expenditures. Incorporating uncertainty and the probabilistic nature of seismic events into the process led to the division of models into (1) deterministic programming and (2) stochastic programming (two-stage and multistage stochastic optimization model). From this aspect, all current models except Xu et al.(10) use deterministic programming and are based on the annual expected ground shaking. These models are founded on the assumption that a small loss (and thus mitigation benefit) occurs every year. Only Xu et al.(10) introduced stochastic programming and presented a model for a one-time period (one year) with the capability to model large variability in investment benefits. The model in Xu et al.(10) can simulate reality based on the assumption that there is no loss in most years and considerable loss occasionally. However, the stochastic optimization models still need further extension and improvement. Another shortcoming in these models, with regard to public perspectives, is ignoring equity in the decision-making process for resource allocation toward earthquake risk mitigation. As the budget for mitigation is obtained from the public purse, public policymakers could not rely on efficiency only, and the equity should be considered as well. Other potential enhancements to the current approaches could be made by introducing the following concepts in the process:

Implementation of Equity in Resource Allocation (1) other risk management and mitigation measures such as insurance, land-use planning, improving the preparedness, retrofitting of infrastructures, etc.; (2) other risks and losses such as business interruption loss, other socioeconomic losses, loss due to damaged infrastructures, etc.; (3) secondary hazards and multiple hazards; and (4) other factors such as migration, population growth, dynamic evolution of wealth, changes over time, etc. 3. MODELING EQUITY IN RESOURCE ALLOCATION The issue of equity and fairness entails further consideration in public policymaking, resource allocation planning, man-made and natural catastrophe risk management, and many other disciplines. In the context of providing public services, distributing public budget, managing public sector resources, and public policymaking in general, efficiency is not adequate and fairness is to be accounted for too. Savas(13) provided three criteria for procuring public goods and services: effectiveness, efficiency, and equity. Effectiveness is a measure to determine how helpful the public goods or services are when addressing the underlying needs. Efficiency means the cost of providing the goods or services with respect to the value of the produced output. Equity is a measure of the fairness as to how those goods and services are distributed. Moreover, equity considerations may be crucial to identify workable mitigation measures and attract different investors and stakeholders. In earthquake risk mitigation strategy making, if justification is made based on financial benefits out of structural retrofitting only, many stakeholders who do not gain such benefits may not be desirous to invest. Since in earthquake risk management the budget comes from public sources, if equity is considered, these stakeholders may be encouraged to invest if the death toll and financial risk are distributed equally among different groups of people. The concept of fairness and equity and efforts and models used to measure equity in economics and public policymaking have been reviewed in other articles. Davidson et al.(2) provided a state-of-theart framework to consider equity in deterministic optimization models. Johnson(14) made a review to include the issue of equity for allocation of public resources in various multidisciplinary researches, revealing the complexities in the equity concept. Marsh

437 and Schilling(15) also reviewed and discussed the alternative equity measures and how to model equity in the site selection process of public-sector facilities. The issue of economic fairness and equity has also been addressed in many other disciplines. Gopalan et al.(16) focused on creating a routing model for hazmat in which the total risk is spread evenly over a geographic area. Current and Ratick(17) addressed equity issues related to the joint optimization of the location of facilities that handle hazmat and the routing for transportation of hazmat to those facilities. ˜ et al.(18) and Alçada-Almeida Studies by Tralhao et al.(19) are among the efforts that consider the equity of the risk imposed on managing waste and (20) ´ locating waste facilities. Ward and Velazquez and Divakar et al.(21) used equity constraint in a water resource allocation optimization model. Yang(22) improved a model for resource allocation by considering equity in service operations (health-care services). Chanta et al.(23) presented the minimum P-envy location model for equitable distribution of emergency resources for emergency medical service systems. Green and Kolesar(24) discussed the successful applications of management science/operations research to improve police and firefighting resources. Horner and Widener(25) examined the issue of site selection of hurricane disaster relief facilities based on the tradeoff between equity and efficiency, using household income as a variable to measure populations’ socioeconomic differences. Shan and Zhuang(26) also studied equity issues in government’s financial resource allocation for risk mitigation against man-made disasters. They considered the tradeoff between equity and efficiency in homeland security resource allocation when facing a strategic attacker. Davidson et al.(2) developed a linear optimization model considering the Gini coefficient as an equity measure. The optimization process distributes resources for earthquake mitigation in a way that the amount of inequality per-capita loss does not pass a threshold.

4. GENERAL ASPECTS OF THE MODEL In this article, a new stochastic optimization process is developed to help the decision-making process for more appropriate distribution of funds in regional earthquake risk mitigation programs. The proposed model could be used in highly seismically active developing countries with limited financial resources

438 and widespread seismic damage where the issue of equity for resource allocation is usually neglected. The model is developed with a focus on regional earthquakes from public-sector perspective, but the methodology is likely to be adapted to other hazards and risk management perspectives. This model is presented as an improvement upon the previous efforts made by Xu et al.(10) In the proposed model, a two-stage stochastic optimization model is developed for the whole planning horizon, which is an improvement upon the previous model in which only a one-time period was considered. The stochastic program developed in this article can take into account the probabilistic earthquakes of various magnitude and probability by explicitly modeling the variability in earthquake loss during the whole planning horizon instead of the annual probabilities used in previous models. Both the proposed model and the Xu et al.(10) model assume that all unit mitigation and reconstruction costs are stationary (do not change over time) and also include a risk-return tradeoff in order to avoid the possibility of experiencing losses exceeding some allowable threshold value. The details about modeling this tradeoff are explained extensively by Xu et al.(10) The allowable threshold value represents a limit below which earthquake loss is manageable considering the regional capacity. The allowable threshold value can also be defined as equivalent to limited financial resources for postearthquake reconstruction. Further on in this article, the concept of equity is incorporated in the stochastic optimization model similar to the study by Davidson et al.,(2) which modeled equity in deterministic optimization. The challenges and modeling specifications are described in the following section. In addition, in this model, the limitation in budget available for risk mitigation, which is a common difficulty in developing countries, is considered. Also, as there are still many nonengineered structures in these countries, in this model the set of possible mitigation alternatives is expanded to allow changes in structural type for nonengineered structures, such as adobe buildings, as well as upgrading of particular structural types to higher seismic design levels. The proposed model here deals with direct losses related to structural damages only in terms of repair or reconstruction costs. By changing the input data, however, the model in the current formulation could also consider other indirect losses as a result of damage to buildings. The model also focuses on structural upgrading policies for groups of buildings and replacements as mitigation alternatives. Other

Zolfaghari and Peyghaleh mitigation alternatives such as land-use planning, insurance, improving critical infrastructures, or emergency response services could be taken into account too. Buildings are grouped into categories based on their local administrative unit locations, structural types (e.g., masonry, steel), occupancy types (e.g., residential, hospital), and seismic design levels. Floor surface area of buildings of a given structural and occupancy type, located in a specified local administrative unit that undergoes upgrading process is used as a decision variable to find optimal mitigation alternatives. From a computational perspective, modeling the decision variables as continuous variables (square meters of floor area) instead of integers (e.g., number of buildings) simplifies the optimization and is more appropriate given the quality and quantity of data available. The upgrading could mean either improvement from one seismic design level to another or from one structural type to a more seismically resistant type. The new model also allows damaged buildings to be reconstructed to any specified seismic design level and structural type. It also allows the decisionmaker to restrict mitigation and reconstruction decisions. The proposed formulation can cope with any geographical resolution (different scales for local administrative units), while the choice of geographical resolution depends on data availability, desired accuracy, and computational demands. 4.1. Implementation of Equity in Risk Mitigation Decisions The overall strategy flowchart for incorporating equity in an optimization model for resource allocation toward risk mitigation measures is summarized in Fig. 1. It shows key challenges that must be addressed to incorporate the concept of equity into regional earthquake risk mitigation decisions. Some of these challenges have been already addressed by others.(2) The framework is as follows. (1) Selecting the mathematic method to formulate equity in the optimization process. For example, equity can be considered as a constraint or as an objective. If modeled as an objective, it could be formulated in an objective function in multi-objective optimization formulations or, instead, it can be added to an existing objective function. (2) Selecting an index to measure operationalized equity. A large number of metrics such as the Gini coefficient, Theil index, Atkinson index,

Implementation of Equity in Resource Allocation

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Fig. 1. Overall strategy flowchart for incorporating equity in optimization process for resource allocation toward risk mitigation measures.

and coefficient of variation have already been used, corresponding to different equity principles. The right selection of equity index varies by case and no single index is best in all circumstances. Understanding the characteristics of each index and associated socioeconomic values expressed by each can help the decisionmaker decide which index to use. (3) Determining the strategy to select groups of people or facilities that should be examined with respect to equity. People or facilities could be categorized by income class, location, access to important facilities (e.g., hospitals), or some other factors such as age, gender, etc.(2) (4) Choosing equity principles, such as horizontal and vertical equity. As described by Davidson et al.,(2) horizontal equity states that people or similar facilities should be treated equally (e.g., as in sales tax). Vertical equity includes the notion that different people or facilities may need to be treated differently proportional to some other measures (e.g., as in the graduated income tax). (5) Deciding on the stage to which the equity principle could be applied; examples are the initial allocation, final outcome, or process.(27) In the case of earthquake risk, equity could be based on how risk mitigation funds are spent (initial allocation), in what conditions the final risk or

expected losses are after risk mitigation efforts (outcome), or by which process risk mitigation efforts are allocated.(2) (6) Selecting the value that should be distributed equitably among facilities or people. For example, in the case of outcome-based models for seismic risk, such values could represent earthquake losses (e.g., reconstruction expenditures), number or monetary value of damaged buildings, business interruption losses, or number or monetary value of human fatalities. (7) Clarifying exactly what kind of average, percentage, or proportion of selected value in step 6 should be distributed equitably among facilities or people. For instance, an outcomebased criterion could focus on average perperson earthquake loss, percentage reduction in per-capita expected loss, or average pergroup earthquake loss.(2) Fig. 1 also shows selected alternatives for each of these factors. The concept of equity in this article is formulated as a constraint in the optimization model. The Gini coefficient has been used to measure equity in the expected per-capita earthquake loss (reconstruction expenditures) across different income classes. However, it is assumed that income classes are based on the location of occupants, i.e., group of people who live in the same geographic location have the same income class. Therefore, the

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equity model is set horizontally and outcome based, that is, different groups of people are treated equally. It means that the optimization model distributes the reconstruction expenditures equally among different groups of people (e.g., low income vs. high income or people living in different districts). The aim is to make different groups undergo the same reconstruction expenditures (loss). With such an assumption, the characteristics of risks (i.e., exposure taxonomy and structural vulnerability) also affect the reconstruction expenditures and thus are included in the model. Alternatively, if the strategy was initial based to allocate mitigation expenditures equally among different groups of people, the impact of earthquake risk and reconstruction expenditures would be ignored. Taking other alternatives shown in Fig. 1 could lead to a new model. The Gini coefficient is often described in reference to the Lorenz curve,(28) but for brevity, the formula of Yitzhaki(29) (Equation (1)) is used. Here, pg and ph are the population in income class g and h, respectively. Note that φ g is per-capita earthquake loss (reconstruction expenditures) for income class g, and φ¯ is average per-capita loss (reconstruction expenditures) across the entire population. Zero value for the Gini coefficient represents complete equity among different groups and Gini = 1 represents complete inequity: N N

Gini =

pg ph ϕg − ϕh

g=1 h=1

2

N

.

2 pg

(1)

ϕ¯

g=1

4.2. Two-Stage Stochastic Optimization Modeling Approach Conceptually, the two-stage stochastic program and its equivalent linear program are formulated to represent the evolving condition of the study area’s building inventory, described in terms of total floor area of each structural and occupancy type in each local administrative unit (Fig. 2). The model is called two-stage as in the first stage it initiates with the decisions on the choice of buildings to mitigate and also their mitigation approach, and in the second stage, it calculates the reconstruction expenditures caused by each potential earthquake scenario, given the selected mitigation decisions and also assuming the damage is reconstructed as the user specifies.

Following in this section, the constraints and the objective function for the two-stage stochastic programming and associated formulation are presented. Equations (2)–(14) represent the constraints while Equation (15) shows the objective function. Mitigation: Let Xicjk be the initial floor area (m2 ) of buildings of structural type i, occupancy type j, located in local administrative unit k; and designed to seismic design level c. To simplify the notation, we use m to represent buildings of occupancy type j located in local administrative unit k; therefore, Xicjk c can be replaced by Xim . The model has to first decide which buildings should be mitigated and how. ic be the floor area of buildings (m2 ) of strucLet Zimc tural type i, building class m, designed to seismic design level c that are mitigated to structural type i’ and seismic design level c’. It is assumed that we can mitigate by changing the seismic design level c and/or the structural type i. The initial floor area is either mitigated (if c’ > c or i’i) or not (if c = c’ and i’ = i). The mitigation decisions are represented by Equation (2). c = Xim

i ,c

ic Zimc

∀i, m, c

(2)

Mitigation Restrictions: Certain restrictions are imposed on the allowable mitigation alternatives. Equation (3) states that we cannot mitigate to a lower seismic design level; however, mitigation to the same seismic design level is acceptable. Equations (4) and (5) show that buildings cannot be mitigated to any structural type i’ that belongs to the set of undesirable structural types Z (e.g., adobe) or to any seismic design level c’ that belongs to the set of undesirable seismic design levels Z, respectively. In Equation (6), we assume the building inventory is partitioned into N mutually exclusive structural type subsets [n , where n(1, . . . ,N)], and that change in structural type as a mitigation choice can be implemented between buildings within set n only.

∀i, m, i , c > c

(3)

∀i, m, c, c , i ∈ Z

(4)

∀i, m, c, i , c ∈ Z

(5)

∀m, c, c , i ∈ n , i ∈ n

(6)

ic Zimc =0 ic Zimc =0 ic =0 Zimc ic Zimc =0

Mitigation Budget: It is assumed that there is a maximum budget G to be spent on mitigation ic be the unit cost of mitigation for a stage. Let Fimc


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Fig. 2. Evolution of building inventory with model floor area variables.

building of structural type i, building class m, and seismic design level c to structural type i’ and seismic design level c’. The decision of how to allocate the available budget to mitigation is represented by Equation (7).

ic ic Fimc Zimc ≤ G

(7)

i,m,c,i ,c

Risk of Especially Large Loss: The desire to guard against scenarios that would impose unacceptably large reconstruction expenditures for each earthquake scenario l is expressed with Equation (8). It calculates β l , the reconstruction expenditures in excess of the allowable loss B for each potential earthquake scenario l. B is the allowable level of loss (reconstruction expenditures) and can be defined as the losses that are considered to be manageable. The first term on the left side is the reconstruction expenditures if earthquake l occurs and based on sedic is the lected mitigation decisions. In this term Rm unit construction cost of building with structural type i’, building class m, and seismic design level c’, which is damaged to damage state d. Note that all restrictions on reconstruction are reflected by these coefficients. All seismically undesirable structural types for reconstruction options and seismic design levels that are unacceptable reconstruction options are bedic valing defined and considered by the user in Rm ldic ues. am is the proportion of buildings of structural type i’, building class m, and seismic design level c’

that will be damaged to damage state d if earthquake l happens. i c di c ldi c Rm am Zimc − β l ≤ B ∀l m,i ,c ,d i,c (8) Equity: The model measures equity among households based on median income levels. For each potential mitigation investment plan, the expected per-capita reconstruction expenditures for each local administrative unit are considered. We assume all people living in a specific building class m (i.e., people living in a specific administrative unit k and occupancy level j) have the same income and would suffer the same earthquake losses. We use an outcome-based horizontal equity principle measured by the Gini coefficient. Equations (9) and (10) represent the Gini coefficient as two linear constraints. Equation (11) calculates the per-capita loss in income class g. With these constraints, the optimization process will only consider mitigation plans that attain a threshold level of equity − pg ph ϕg − pg ph ϕh − + gh + gh = 0

g

ϕg =

− (+ gh + gh ) ≤ Q

pg

g

h>g

1 l P pg l

m∈O(g),i ,c ,d

∀g∀h > g (9)

pg ϕg

(10)

g

di c ldi c Rm am

ic Zimc

∀g

i,c

(11)

442


Pl is the probability of the earthquake scenario l occurring during the planning horizon, defined by a magnitude and an epicentral location. A simulated probabilistic earthquake catalogue can represent these events. Q is a user-defined value (0 Q 1) representing level of acceptable inequity, where complete equity is denoted by Q = 0. Larger values of Q imply more tolerance for inequity. Note that gh represents the difference between the per-capita expected earthquake loss for income classes g and h, − where + gh and gh represent positive and negative values for such differences, respectively. Note that O(g) contains a set of buildings of class m for which their local administrative unit k is categorized as an income class g. Nonnegativity Requirements: The following nonnegativity constraints must also be held for the decision variables:

∀ i, m, c, i , c ,

(12)

β ≥0

∀ l,

(13)

∀ g, h.

(14)

ic ≥0 Zimc l

+ − gh , gh

≥0

Objective Function: The goal is to minimize the total expenditures of risk mitigation (first term), reconstruction expenditures as a result of building structural damages (second term) and the risk of especially large losses (reconstruction expenditures) exceeding the allowable level B weighted by a userspecified weight κ (third term). Here, κ represents risk aversion parameter. i c i c di c ldi c Fimc Zimc + Pl Rm am Min i,m,c,i ,c

l

m,i ,c

ic Zimc +κ

i,c

d

Pl β l

(15)

l

The model delivers the recommended mitigation decisions on how to allocate the budget among ic mitigation alternatives Zimc and the resulting reconstruction expenditures for each possible earthquake scenario are defined as: di c ldi c ic Rm am Zimc . (16) m,i ,c

d

i,c

The resulting Gini coefficient can be calculated from the output as well as the value of loss over the “allowable loss threshold” (Pl β l ). In formulating the two-stage stochastic program in this article, it is also assumed that the full mitigation budget is available in advance. As the sequence order of the retrofitting is not determined within the

process, this assumption doesn’t allow the recommended mitigation strategies to be prioritized. However, it allows us to recommend what would make sense to retrofit at some point during that whole time period. The model does not assign a specific time for performing the reconstruction either. Therefore, like Vaziri(10) and Dodo,(9) all unit reconstruction costs are considered stationary, i.e., with no discounting rate. In reality, in a seismologically active region, a variety of small to large earthquakes could happen during the planning horizon. In effect, within the planning horizon one single earthquake (hereinafter referred to as single-earthquake) or several earthquakes (hereinafter referred to as multi-earthquakes) might happen although the latter is more unlikely to occur in a short planning horizon. In this article, the possibility of experiencing only one single-earthquake (scenario l) with probability Pl is taken into account. According to the model formulation, this earthquake could be any of the simulated events in the earthquakes catalogue. In other words, we do not consider the possibility of multi-earthquakes during the planning horizon. The planning horizon is the time duration for which the budget for mitigation measures is invested and planned for the earthquakes. This assumption signifies that the focus of this model is on larger earthquakes. If multi-earthquakes within the planning horizon are considered, minor changes need to be implemented in the formulation and earthquake index l. Different values can be considered for the planning horizon that depend on the seismicity of the area, mitigation budget, and view of risk mitigation planning and risk manager. For other user-defined parameters, sensitivity analyses over a range of values are likely to provide more insight. 5. CASE STUDY As an example for cities in the developing countries with a long history of destructive earthquakes, an area within Iran is selected as the case study. There have been many large-scale earthquakes in Iran, such as Manjil Earthquake in 1990 and Bam Earthquake in 2003, causing widespread human casualties and economic damages. Like other nations, the lessons from past Iranian earthquakes have created significant motivation toward the implementation of earthquake risk reduction and mitigation plans and activities in Iran.(30) However,

Implementation of Equity in Resource Allocation despite all these activities,(4,31–33) there are still many weaknesses, such as limited financial resources, unfair financial resource allocation for both public and private buildings as well as lifelines and emergency facilities,(4) and the absence of optimized plans for financial resource allocation, undermining the efficiency of mitigation measures in Iran. In this article, the built environment data for Tehran, the capital and political and economic center of Iran, are used to conduct the case study. The Greater Tehran Area is located at the foot slope area of the Alborz Mountains, in an area of high seismic potential with many peculiar active faults. Urban growth has been rapidly progressing in Tehran. Tehran’s population has boomed in recent decades, growing from 1 to 7 million from 1950 to 2000,(34) with a current population of more than 8 million. Although several studies have been carried out in relation to earthquake disaster management issues for Tehran in recent years, the city is sprawling in the absence of proper disaster prevention systems against potential earthquakes. 5.1. Input Data and Assumptions for Pilot Study Required input data and variables for the twostage stochastic optimization model are defined here. The model relies on probabilistic loss estimation, which in turn requires earthquake hazard and risk analyses in order to assess seismic hazard and associated seismic damage to buildings. Building inventory for residential buildings in Tehran is collected from census study. Tehran is divided into 3,173 census zones, 114 communes, and 22 district units. For each census zone, number of buildings, sum of building areas by structural type, and population count are provided. To reduce computation run time in this study, data provided for census zones are aggregated by 114 commune units, i.e., k = 114. Statistical data on population and building inventory are extracted from data collected in a micro-zoning study of the Greater Tehran Area conducted by JICA.(4) The database refers to residential occupancy type only (j = 1); therefore, there are m = k = 114 groups of buildings located in k = 114 administrative units. The building inventory database is grouped into 13 structural types (i). Table I illustrates the mapping between building taxonomies used here and those defined by JICA(4) and HAZUS(35) in order to use their fragility curves. Three damage states of slight, moderate, and heavy/collapse are considered here (d = 3). For the heavy damage and collapse state,

443 fragility curves presented by JICA,(4) which are prepared based on the characteristics of building structural types in Iran, are used. For the other two damage states, fragility curves proposed by HAZUS(35) are utilized since there are no reliable fragility curves for buildings in Iran related to these two damage states. As the current state of building inventory in Tehran cannot provide one-to-one mapping with HAZUS taxonomy, some aggregations are performed to map building with fragility curves. The optimization process requires preprocessed ldic , i.e., the proportion of building estimation of am area in building class m in structural type i’ and designed to seismic code c’ that enters damage state d if earthquake l happens. To calculate these values, it is necessary to prepare a loss estimation platform. The basic component of any seismic loss estimation is a representation of seismic hazard. To address the objective of minimizing losses or maximizing benefits, one or even a few scenarios are not adequate. This is due to the fact that an optimal investment strategy for a specific earthquake may not necessarily result in optimal cases for all possible earthquakes or it may even result in a completely ineffective strategy for other earthquakes.(3) This is why the stochastic programing, which in turn requires full probabilistic earthquake hazard assessment, is introduced in this article. To represent seismic hazard and probabilistic earthquakes in this article, the regional probabilistic hazard analysis framework proposed by Han and Davidson(37) is performed. Monte Carlo simulation (MCS) is used to generate a synthetically generated earthquake catalogue that represents 10,000 years of events with some 84,000 earthquakes of magnitude 4.6–7.55 distributed in a 200 km radius from Tehran city.(38) Consequently, taking into account a threshold peak ground acceleration (PGA) of 0.01 g for all 114 communes, many of these simulated earthquakes are filtered out, resulting in a reduced set of some 11,660 earthquakes. Ground motion attenuation relationships of Ramazi,(39) Ambraseys and Bommer,(40) and Sarma and Srbulov(41) with equal weights are used to estimate mean PGA values and hazard curves for 114 centroid points representing 114 city communes. Using this catalog, the full probabilistic seismic hazard is estimated for the city. However, the number of simulated events providing full MCS for the city is too big to be used in the optimization process. To reduce the selection, the “optimization-based probabilistic scenarios (OPS) selection” method is performed to reduce

444

Zolfaghari and Peyghaleh Table I. Building Structural Type Variants

Category

Structural Type (Precode) Name in HAZUS for Slight and Moderate Damage

Category Name in JICA for Heavily Damage

Name

Structure Type Steel low rise Steel medium rise Steel high rise Concrete low rise Concrete medium rise Concrete high rise Half-frame low rise Half-frame medium rise Half-frame high rise Masonry low rise Masonry medium rise All wood frame Sun-dried brick

Low Medium High Low Medium

Mean of S1L and S2L * Mean of S1M and S2M Mean of S1H and S2H Mean of C1L and C2L Mean of C1M and C2M

BT-2 BT-3 BT-3 BT-5 BT-6

SLR SMR SHR CLR CMR

High Low Medium

Mean of C1H and C2H Mean of S5L and C3L Mean of S5M and C3M

BT-4 BT-1 BT-1

CHR HLR HMR

High Low Medium

Mean of S5M and C3M URMM URMM

BT-1 BT-8 BT-8

HHR MLR MMR

Low Low

W2 For slight: G. Ghodrati Amiri,(36) curve; for moderate: mean of slight curve and heavily and collapse curves

BT-7 BT-9

W SDB

the set of probabilistic earthquake scenarios. OPS was first introduced by Vaziri et al.(42) This method incorporates the effects of all possible events and their occurrence rate in a loss estimation process for a region. In addition, this method captures spatial correlation better than the probabilistic seismic hazard method (PSHA).(43) Moreover, the number of scenarios provided by this method is small enough to make optimization formulation computationally feasible. In this method, a relatively small set of all possible earthquake scenarios is selected and their annual occurrence probabilities are adjusted so that they could represent, as closely as possible, the full PSHA results for each 114 communes.(43) In the OPS method, a mixed-integer linear optimization model is used to minimize the sum of errors over all 114 communes (called control points in Vaziri et al.(43) ) and return periods r, between points on the “full” hazard curves and the corresponding points on hazard curves developed by the reduced set. As criteria for selection of earthquake scenarios in the OPS process, PGAs with return periods of 250, 475, 1,000, and 2,475 years and for all 114 city communes are used as reference points. The OPS approach is performed many times to reach a minimum number of simulated events that satisfy desired accuracy as well as computational runtime. In the end, a reduced set of events containing 62 earthquakes with magnitude of 4.6–6.75 and in close vicinity of

Tehran are selected and their annual occurrence probabilities are adjusted using the OPS method. More details about mixed-integer programing and the formulation of objective function and constraints are presented in Han and Davidson.(37) To calculate the exceedance probability for earthquakes within the planning horizon, Poisson distribution is used (Equation (17)). In this equation nyl is the rate of occurrence of earthquake (eql ). P(at least one eql in 20 years) =1 − exp(−20nyl ) (17) In this case study, the planning horizon is assumed to be 20 years, during which only one earthquake from all possible scenarios will occur. This is a vital assumption to make the model computationally manageable. However, in reality, there is the possibility of more than one earthquake taking place during this horizon. In other words, in this study, the focus is on larger earthquakes with potential damage only. Historical seismicity within 150 km vicinity of Tehran shows on average one earthquake of magnitude 6 and above in every 20 years.(44) Considering a shorter planning horizon (for example, 10 years) leads to removing the larger events from the process and results in short-term planning with more attention paid to smaller earthquakes. On the other hand, considering a longer planning horizon makes the assumption of one earthquake during the planning horizon unacceptable. In addition, from the risk

386 386 386 – – 332 332 332 – – 332 332 332 – – 299 288 288 161 – 233 227 227 122 – 166 166 166 83 83 216 205 205 120 – 150 144 144 81 – 83 83 83 42 42 133 122 122 78 – 67 61 61 39 – SLR SMR, SHR CLR, CMR, CHR HLR, HMR, HHR MLR, MMR, W, SDB

Mitigated level 2 Mitigated level 1 Not mitigated Mitigated level 2 Mitigated level 1 Not mitigated Mitigated level 2 Mitigated level 1 Not mitigated Mitigated level 2 Mitigated level 1 Structure Type

URC ($) for Heavily Damage to URC ($) for Moderate Damage to URC ($) for Slight Damage to UMC ($) to

mitigation planning point of view at a regional level, very long-term plans do not have practical justification as they have a very long payback period. Moreover, the development plan in Iran is performed with a 20-year vision plan that consists of four five-year development plans. The next component of seismic loss estimation ldic values is done by estimafor calculation of am tion of PGAs from the reduced set of simulated earthquakes for all city communes here. Seismic damage is assigned to each group of buildings based on estimated PGA and associated damage ratios estimated from building fragility curves. Fragility curves deliver cumulative probability for meeting a given damage state against PGA. However, the probability can be used as the damage ratios of buildings entering each damage state when exposed to a certain PGA since the decision variables in this article are the summation of areas by building groups and, therefore, a continuous variable. For each structural type, three seismic design levels are considered: not mitigated, mitigated level 1, and mitigated level 2. Based on engineering judgments and according to Vaziri et al.,(7) the goal of mitigation level 2 is assumed to achieve the life safety performance level, defined in retrofit guidelines in Monograph No. 360,(45) which is similar to FEMA 356.(46) Based on such assumptions and knowledge of existing buildings in Tehran, the effect of mitigation is defined as a rightward shift in the fragility curve, i.e., increasing PGA by 200% to reach the same damage ratios. Similarly, for mitigation level 1, collapse prevention performance level is achieved with 150% shift in fragility curves. The associated unit mitigation cost F, and the unit reconstruction costs R, are estimated, as shown in Table II. As three different damage states are considered in this case study, the reconstruction expenditures (R values) are presented for different damage states. It is assumed that sun-dried brick (SDB) and masonry buildings (masonry low rise [MLR] and masonry medium rise [MMR]) are among unacceptably weak structural types not appropriate for mitigation or reconstruction measures when they are under heavy damage or collapse conditions. MLR and MMR are the types of unreinforced masonry buildings that are structurally weak and obsolete. These are unreinforced buildings made of stone and cement and, in fact, construction of these buildings has been forbidden in Iran even in rural areas during the last 25 years. This is also the same for all wooden structural types; therefore,

445

Table II. The Unit Mitigation Cost (UMC) and Unit Reconstruction Cost (URC) Estimated by Authors Based on Statistic Data(47,48) and Vaziri et al.(7)


446 mitigation for such buildings means reconstruction to new building types and also to new mitigated levels in the same category. The cost for changing structural type of building in the mitigation stage is equal to the cost of reconstruction of a new structure. Structural types are divided into three categories: low rise, mid rise, and high rise. It is assumed that demolishing a building from one category and reconstructing it as mitigation or reconstruction decision to a structural type from other categories is not an option. In practice, under some circumstances, the decisionmakers might also prefer to demolish an old and low-quality structure and replace it with a new structural type as a mitigation decision in order to decrease the regional vulnerability. The cost for changing structural type in the stage of mitigation is equal to the cost of reconstruction of a new structure. As shown in Table II, mitigating SLR to SLR mitigated level 2 costs US$ 133 while the reconstruction to SLR mitigation level 2 costs US$ 365. Moreover, it is possible to reconstruct buildings to different types of design levels. All buildings (not mitigated and mitigated) that enter slight and moderate damage states can be reconstructed to their primary situation. Therefore, the damaged mitigated structures should also be reconstructed to their previous mitigated levels. The reconstruction costs in columns 5, 6, 8, 9, 11, and 12 in Table II are reconstruction costs for such damaged mitigated buildings. In this model, all restrictions on recondic . We struction are reflected by the coefficientsRm assume that reconstruction costs for nonmitigated structures are equal to the costs for mitigating to level 1 structures. This is to show that the reconstruction of buildings with nonmitigated design level, when destroyed completely, is not possible and nonmitigated heavily damaged or collapsed structures should be reconstructed to structures with mitigated level 1 because the codes have been changed during previous years. In addition, if a wood, SDB, masonry buildings (MLR and MMR), and half-frame buildings (half-frame low rise [HLR], half-frame medium rise, half-frame high rise) are damaged heavily or collapsed, they have to be reconstructed to the steel and concrete structural types. In this study, in order to measure and incorporate equity in the optimization model, annual income is used to categorize different groups of people. However, in the absence of reliable and detailed information on income, people are grouped based on their geographic locations as an indicator of their

Zolfaghari and Peyghaleh income. Therefore, it is assumed that people living in the same district have the same income level. Based on the 2005 Iran’s National Report,(49) Iran spends 2.5% of its annual budget on disaster reduction and mitigation efforts. The total national 2005 budget is about 1,600 trillion Rials.(50) Assuming half of this budget is allocated to earthquake risk mitigation, and taking into account Tehran’s contribution to national GDP, which is around 26%,(50) the annual budget for risk mitigation in Tehran is approximately US$600 million. Therefore, for a planning horizon of 20 years, the whole available budget for reconstruction and mitigation is around US$12 billion. For the base case analysis (displayed in Fig. 3) the summation of B value (representing allowable level of loss [reconstruction expenditures]) and mitigation budget (G value) is considered equal to the whole budget that the government allocates to earthquake mitigation and reconstruction measures during 20 years. According to the Iranian Budget Law, one-fifth of the total national budget considered for mitigation and reconstruction expenditures should be allocated to mitigation.(51) Therefore, it is assumed that 20% of the whole available budget calculated in this way has to be dedicated to mitigation measures. The remaining 80% of the whole available budget is presumably assigned to reconstruction measures and as an allowable level of loss. 5.2. Analyses and Results In order to perform the stochastic optimization model, its equivalent linear model is designed and coded in IBM ILOG CPLEX Optimization Studio software and solved using CPLEX.(52) The model has been compiled many times, taking into account variation in the input parameters and assumptions, in order to investigate the sensitivity and variation of the results in relation to the input data uncertainty and in order to understand the effect of different parameters on the selected mitigation strategy. The optimization process is performed for a combination of five alternative budget limitations (G = US$ 1.2, 2.4, 3.5, 5.2, billion), seven alternative maximum acceptable inequity levels (Q = 0, 0.05, 0.1, 0.15, 0.2, 0.25, 1), and 24 selected values for financial risk-aversion parameter (κ = 0, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 3.00, 4.00, 5.00, 6.00, 7.00, 8.00, 9.00, 10.00, 20.00, 30.00, 40.00, 50.00, 60.00, 70.00, 80.00), altogether 840 different scenarios. The


447

Fig. 3. Mitigation expenditures (million US$) by commune.

model, once parameterized with the data for the case study, has 1,200,000 decision variables and 2,400,000 constraints. The analysis Case-1 (Fig. 3), which is also named “base case,” refers to a case where equity is not considered (Q = 1). “Case-3” is another main case similar to the base case but where equity is considered (Q = 0.05). The value of B is considered US$9.6 billion in these cases. The variation in other userdefined parameters results in further analyses, which are presented and compared with these two main analyses. These comparisons illustrate how such variation could impact the final result and could help the selection of suitable mitigation strategies. In fact, these enable the users to check if their limitations have any untoward and significant impacts on the results, in which case the user can consequently select the results of other case analyses as the recommended mitigation strategies.

5.2.1. Mitigation Recommendations There are several factors controlling the process of selecting mitigation strategy (retrofitting or

changing structural type). These factors include severity and frequency of seismic hazard, initial building taxonomy and associated structural vulnerability, available mitigation alternatives and improvements, and costs associated with such alternatives. In addition, selecting the mitigation strategy depends on the budget available for mitigation and the level of risk that is accepted by the decisionmaker. Fig. 3, for example, illustrates the mitigation expenditures by geographic locations for four optimization analyses, with and without equity constraints and with limited and unlimited budget. In cases where equity is not considered (Q = 1), expenditures show high sensitivity to variation in budget limitation value (G). There are also higher mitigation expenditures in central and eastern parts of Tehran compared to western communes, which may be due to higher PGA values and/or more densely built environment in those locations. For cases where equity is included (Q = 0.05), the recommended mitigation expenditures go mostly to the northern half and northwestern Tehran and also small districts in the central part of the city. These simple examples show the substantial effect of considering equity in

MLR to CLR c-2 c-3 0 0 0 0 0 0 0.8 6 SBD to CLR c-2 c-3 0 0 13 0 18.9 0 2.4 4.8 SMR to SMR c-2 c-3 0 0 23.4 0 37.4 0 17.2 18.2 SLR to SLR c-2 c-3 0 0 0.6 0 3.9 0 2.6 2.4 CHR to CHR c-2 c-3 0 0 0.2 0 0.07 0 0.1 0.06 CMR to CMR c-2 c-3 23 0 13 0 28.9 0 3.7 11.9

Percentage of mitigation expenditures to buildings structural taxonomy (%)

2.4 8.2 2.4 11 base case (Case-1) Case-2 Case-3 Case-4

(billion US$) Case analyses

Mitigation expenditures

HLR to HLR c-2 c-3 75 0 48 0 6.5 0 24.5 3.3

CLR to CLR c-2 c-3 2 0 1.5 0 4.4 0 0.6 1.5

7% 1.2% 34% 5% 0.1% 6.5% 7.5% 45% Prevalence of buildings

0.9%

HLR Initial Buildings Taxonomy

CLR

CMR

CHR

SLR

SMR

SDB

MLR


Table III. Initial Prevalence of Buildings Structural Type, Mitigation Expenditures, and Percentage of Mitigation Expenditures Allocated to Each Building Structural Type According to the Mitigation Strategy for All Four Analyses Cases

448

decision-making process. They also show how sensitive the distribution of recommended mitigation expenditures is to the amount of the available budget. Table III demonstrates the initial building taxonomies and prevalence and percentage of mitigation expenditures allocated to each building structural type according to the mitigation strategy for all four cases. Fig. 4 shows the recommended total mitigation expenditures by initial and mitigated structural types for the base case (Case-1) and other three case analyses. Case-1 and Case-2 in Fig. 4 show distribution of mitigation expenditures by initial and mitigated structural types where no equity is taken into account. The same results are shown for Cases 3 and 4 but with equity effect. Inclusion of equity (Cases 3 and 4) results in more involvement of other structural types in the mitigation process. In the base case (Case-1), the result suggests that most of the budget expenditures go to mitigation of HLR (brick and steel) buildings. HLR and SMR buildings make a big proportion of buildings in Tehran although according to the fragility curves proposed by JICA,(4) they are two of the least vulnerable structural types within their respective subgroups of low and mid rise buildings. Mid and lowrise reinforced concrete (concrete medium rise [CMR] and concrete low rise [CLR]) should be mitigated next, respectively. Reinforced concrete is the next most common structural type in Tehran, which is characterized with higher vulnerability and, therefore, it is an appealing target for mitigation measure. The choice of structural types to mitigate seems to be driven largely by their relative prevalence (HLR) and vulnerability (CMR and CLR). As the available budget increases in Case-2, the mitigation expenditures increase and the fraction of these three structural types in mitigation expenditures decrease (Table III). Moreover, other building structural types like steel structures (SLR and SMR), highrise reinforced concrete (CHR), and the most vulnerable type, i.e., SDB, are chosen to be mitigated. Except for SDB structural type, other types are recommended to be mitigated to the same structural type and to the first level of mitigation (c-2). Results also show that SDB structural type is replaced by CLR-mitigated level 1 as a mitigation decision since SDB structural type is not allowed for mitigation. The cost of replacing SDB with CLR-mitigated level 1 is considered equal to the cost of reconstruction of CLR-mitigated level 1. These results indicate that as the budget increases, more expensive mitigation alternatives, like reconstruction of SDB buildings, can be justified (Table III).


449

Fig. 4. Mitigation expenditures (billion US$) by initial and mitigated structural type.

Masonry buildings (MLR) have the same mitigation cost as the SDB buildings, but with less vulnerability. This is the reason why this structural type is not mitigated in Case-2 analyses compared to SDB. In Case-3, equity is considered with limiting the inequity level (Q) to 0.05 compared to the base case. Although the values of total mitigation expenditures are the same for these two case analyses, they have totally different distribution patterns of the recommended mitigation expenditures among various structural types. Compared to the base case, (Case-1) in which only three structural types were recommended for mitigation, in Case-3 all types of structures except masonry buildings are recommended for mitigation. In this case, the result suggests that most expenditure goes to mitigation of SMR, CMR, and SDB buildings. As stated before, SMR and CMR make a big proportion of buildings in Tehran and are also characterized by high vulnerability. SDB structural type, known as the most vulnerable type, is also recommended for mitigation. Except for SDB structural type, other types are recommended to be mitigated to the same structural type and to the first level of mitigation (from not mitigated to mitigated level 1). Equity constraint controls the selection of structural types to be mitigated. In this case, mitigation recommendation seems to be allocated to more vulnerable buildings like CMR and SDB buildings.

In a similar way, as the available budget increases in Case-4, the mitigation expenditures increase and the fraction of these structural types in mitigation expenditures changes, as shown in Table III. In addition, more expensive mitigation strategies like retrofitting MLR buildings are recommended as more budgets are available. Also, the mitigation expenditures are allocated to more prevalent structural types instead of more vulnerable ones, compared to previous analyses. Comparing this case to Case-2, in which equity is not considered, the mitigation expenditures increase by 35%. Moreover, with more budget available to use, the strategy of mitigation changes in a way that a fraction of buildings are mitigated to the higher seismic design level (mitigated level 2 [c-3] instead of mitigated level 1). 5.2.2. Sensitivity to Risk-Aversion Parameter We can examine the tradeoff between risk and return on regional earthquake mitigation investment by comparing optimization results for many values of the risk aversion weight κ. The results are plotted in Figs. 5(a) and (b) for the analyses with given B and G for Q = 1 and Q = 0.05, respectively. Fig. 5(a) shows the variation of recommended mitigation expenditures, expected reconstruction expenditures (over all 62 earthquakes), and expected total

450


Fig. 5. Return-risk relationships for cases in which there is no limitation for mitigation budget, B = US$9.6 billion. (a) Q = 1. (b) Q = 0.05. Each point on these curves corresponds to an analysis with a specified κ, with κ increasing from right to left. Note that κ takes 24 various values as explained in Section 5.2.

expenditures (mitigation plus reconstruction) versus the average excessive reconstruction expenditures (E = l Pl β l ) as a measure of risk. Each point on these curves corresponds to an analysis with a given κ, with κ increasing from right to left. The optimal solutions for all κ values from 40 to 80 are the same; resulting as single points on these graphs. It is likely that once the decisionmakers’ risk aversion reaches the threshold of κ = 40, mitigation costs more, and the incremental cost of further reducing the probability of a large loss becomes too high, which is not efficient any more. Therefore, even if the decisionmakers become more risk averse, the recommended mitigation strategy will remain the

same. This is because the main objective of the optimization model is to minimize all costs, including mitigation and reconstruction costs, as well as excessive loss. As κ decreases, the mitigation expenditures decrease, which causes the average excessive reconstruction expenditures to increase. The reason is that for a small value of κ, more excessive reconstruction expenditures are allowed and, therefore, less mitigation is required. On the other hand, the total reconstruction expenditures rise as κ decreases. The sum effect is that total expenditures decrease with increasing allowable risk. The optimal solution results for all κ from 0 to 2 are similar as the risk-aversion

Implementation of Equity in Resource Allocation parameter is too small to motivate implementing of mitigation. Hence, the mitigation expenditures are equal to zero for such values of κ. Fig. 5(b) shows the same result but for Q = 0.05. The same trend can be observed here, too; however, smaller variations of the average excessive reconstruction expenditures versus κ are seen compared to those in Fig. 5(a). In addition, for small values of κ, the mitigation expenditures are greater than zero in order to satisfy the equity constraints. Also, the mitigation expenditures in medium values of κ (κ = 2–10) are considerably greater compared to those in Fig. 5(a) to allow more mitigation to satisfy the required equity as expected. Moreover, there is a noticeable increase in the reconstruction and total expenditures compared to those in Fig. 5(a). This demonstrates existence of a tradeoff between equity and efficiency in allocation of mitigation expenditures. Figs. 6(a)–(d) show the variation of the average excessive reconstruction expenditures, the recommended mitigation expenditures, expected reconstruction expenditures, and expected total expenditures versus the risk-aversion weight κ for the analysis cases with given B, five alternative budget limitations explained in Section 5.2, and where the equity is not considered (Q = 1). Each point on these curves corresponds to a case analysis with a given κ value. No differences are seen for results from cases with limited and unlimited budget for small values of κ, which is indicative of a situation in which recommended mitigation expenditures are smaller than the budget limitation values. However, as κ increases, the mitigation expenditure increases (Fig. 6b) until it reaches the budget limitation value. Therefore, further increases in mitigation expenditures are restricted by the limited budget, as shown in Fig. 6(b). Similar effects are projected for the average excessive reconstruction expenditures, expected reconstruction, and expected total expenditures as shown in Figs. 6(a), (c), and (d). In other words, as the mitigation expenditures reach the budget limitation, further increases for κ value cause no decrease in the reconstruction expenditures and the average excessive reconstruction expenditures. The curve representing no budget limitation follows the same pattern of those with limited budget for small values of κ. Figs. 7(a)–(d) show the same results as Fig. 6 except that in the former, equity is considered (Q = 0.05). In all cases where the budget is limited, except where G = US$ 5.2 billion, even for small values of κ, mitigation expenditures are restricted by the limited

451 budget, as shown in Fig. 7(b). This is because of the equity constraint, which tries to meet the level of considered equity. For the case where G = US$ 5.2 billion, no differences are seen in the results from cases with limited and unlimited budget for small values of κ, which is indicative of the situation in which recommended mitigation expenditures are smaller than budget limitation values. However, as κ increases, the mitigation expenditure increases (Fig. 7b) until it reaches the budget limitation value. Therefore, further increase in mitigation expenditures is restricted by the limited budget, as shown in Fig. 7(b). Similar effects are projected for the average excessive reconstruction expenditures, expected reconstruction, and expected total expenditures as shown in Figs. 7(a), (c), and (d). In other words, as the mitigation expenditures reach the budget limitation, further increases for κ value cause no decrease in the reconstruction expenditures and the average excessive reconstruction expenditures. In addition, in all cases where the budget is limited, decrease of budget limitation value and restriction of the mitigation recommendation by that limited budget lead to increase of other expenditures, which can be observed as a leftward shift in the related graphs. The curves representing no budget limitation follow the same pattern of the ones in Fig. 6. Comparing Fig. 7 to Fig. 6 demonstrates that incorporating equity in the decision-making process when the budget is limited leads to an extreme increase of the average excessive reconstruction expenditures, expected reconstructions, and expected total expenditures, which result in a significant decrease in efficiency. 5.2.3. Sensitivity to Inequity Level The tradeoff between expected reconstruction expenditures and level of inequity achieved on regional earthquake mitigation investment can be examined by comparing optimization results for many values of inequity level Q. Fig. 8(a) displays the risk-equity tradeoff for the cases with given G, κ, and B. The chart indicates that as inequity decreases (as measured by the Gini coefficient), the expected reconstruction expenditures and total loss (summation of mitigation expenditures, reconstruction expenditures, excessive loss) increase. It can also be seen that as inequity decreases, the mitigation expenditures increase to allow more mitigation expenditures to satisfy the required equity as expected. However, as the budget is limited, further reduction in inequity does not change the mitigation

452


Fig. 6. Variation of expenditures versus risk aversion parameter κ for analyses with B = US$9.6 billion, Q = 1, and five alternative budget limitations as explained in Section 5.2. (a) The average excessive reconstruction expenditures, (b) recommended mitigation expenditures, (c) expected reconstruction expenditures (over all 62 earthquakes), (d) expected total expenditures.

expenditure values. As κ is very small in this case (κ = 0.25), the excessive loss has a small value compared to the reconstruction expenditures. In this case, if the inequity constraint is ignored, there will be no need to sustain mitigation expenditures. Fig. 8(b) shows the same results for the cases with greater κ (κ = 6) and when no limitation for the budget is considered. As κ has a larger value in this case, higher excessive loss, mitigation expenditures, and, consequently, total loss are observed compared to those shown in Fig. 8(a). In a similar fashion, further decrease in inequity causes the expected reconstruction expendi-

tures, excessive loss, and total loss to increase. In addition, it can be seen that as inequity decreases, the mitigation expenditures show a mild increase until inequity reaches the value related to Q = 0.05. More decrease in inequity causes a dramatic decrease in mitigation expenditures and, as a result, an intensive increase in other expenditures. Extreme constraint for implementing equity seems to cause the acceptance of more damage to satisfy equity and thus less required mitigation, which is not beneficial any more. In this case, even if we do not want to satisfy equity (for example, Q > 0.25), we have to pay the costs of


453

Fig. 7. Variation of expenditures versus risk aversion parameter κ for analyses with B = US$9.6 billion, Q = 0.05, and five alternative budget limitations as explained in Section 5.2. (a) The average excessive reconstruction expenditures, (b) recommended mitigation expenditures, (c) expected reconstruction expenditures (over all 62 earthquakes), (d) expected total expenditures.

Fig. 8. Expenditures versus inequity tradeoff for: (a) cases with G = US$ 2.4 billion, B = US$ 9.6 billion, κ = 0.25 and (b) cases with unlimited mitigation budget, B = US$ 9.6 billion, κ = 6.

454 mitigation measures to reduce the risks. To accomplish an equity of approximately 95%, which is related to Q = 0.05, we just need to pay 35% more. Fig. 9 demonstrates per-capita expected earthquake loss by income classes and various levels of inequity for cases with unlimited mitigation budget and fixed к and B. For a complete equity, i.e., Q = 0, there is a uniform per-capita expected earthquake loss for different income classes. Predictably, as Q increases (i.e., moves toward inequity) the scatter of percapita expected earthquake loss in different income classes gets wider. For the complete inequity, i.e., Q = 1, highest scatter for the per-capita expected earthquake loss is observed. Extreme constraints for implementing equity (when the level of equity is made too tight [Q = 0]) cause the model to allow for more damage to meet the requirements for the equity. In effect, these constraints force the value of per-capita expected earthquake reconstruction expenditures to be equal for each income group (here, Tehran’s 22 districts); therefore, less mitigation and greater per-capita expected earthquake reconstruction expenditures will be entailed. 6. DISCUSSION ON APPLICATION AND DRAWBACKS In this article, the proposed two-stage stochastic model and its application to a case study for the city of Tehran is presented. The base case (Case-1) and Case-3 are considered as the main cases with the same predefined limited budget, the same allowable loss and risk-aversion parameter, yet with different predefined inequity levels (Q = 1 and Q = 0.05, respectively). The results of the base case (Case-1) are proposed as recommended mitigation (retrofitting) strategies if the user ignores the effect of equity; otherwise, results from Case-3 should be taken as the proposed strategies where effect of equity is implemented. To demonstrate the sensitivity of the results to the user-defined input data, different values for budget limit, risk-aversion parameter and inequity level, and the variation in mitigation alternatives and retrofitting decisions are investigated and compared with the two main analyses. This enables the users to check if their limitations have any untoward and significant impacts on the results, in which case the user can consequently select the results of other case analyses as the recommended mitigation strategies. Due to the probabilistic nature of the proposed model, many sources of uncertainties are incorporated and combined to find the optimal solution.

Zolfaghari and Peyghaleh Therefore, disaggregating the results to show the effect of each regional or generic factor calls for further efforts that are beyond the scope of this study. Nevertheless, the model shows strong sensitivity to the regional building taxonomy and retrofitting measures, retrofitting cost, regional seismic hazard, budget limitation, equity consideration, and other regionspecific factors. Moreover, the pattern of equity variation and its tradeoff against reconstruction expenditures, risk of large reconstruction expenditures, and mitigation expenditures seems to be generic. The method and similar results shown for the pilot study presented in this article can be used by different organizations and users depending on their responsibilities for planning, implementing, and managing mitigation efforts in developing countries. For example, policymakers, decisionmakers, and risk managers at city levels could use such tools to select effective strategies and to prioritize budget allocation for these strategies. In effect, the model was developed stressing a regional (e.g., metropolitan area) public-sector perspective, such as governmental organizations and other authorities; however, the method could be adjusted to other hazards and risk management perspectives (e.g., insurance industry). Using two-stage stochastic programing provides a unique opportunity to take the probability of all possible earthquakes and the variation of their consequences into account in the decision-making process for resource allocation toward earthquake risk mitigation. The mitigation measures here include retrofitting and reconstructing of buildings only and no infrastructures are considered. Moreover, the model only considers the effect of retrofitting of building seismic performance. Therefore, other impacts following building retrofitting measures such as migration and wealth increase are disregarded. In addition, only financial losses due to structural damage are considered in this article. In order to calculate the earthquake financial loss, only direct repair and construction costs are considered. However, the model in the current formulation can also consider other indirect losses as a result of damage to buildings, such as business interruption losses, by changing the input data (R value). Other financial losses due to human causalities and infrastructure damages are disregarded in order to prevent the model from becoming unwieldy and on account of computational efficiency considerations. Incorporating multiple socioeconomic factors and their uncertainties into the process increases the processing time significantly. It should be mentioned


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Fig. 9. Per-capita expected earthquake loss (reconstruction expenditures) by income class g (district) for different values of inequity for the cases with unlimited mitigation budget, B = US$ 9.6 billion, κ = 6, and various Q values.

that the model in its current status for the case study in this article has 1,200,000 decision variables and 2,400,000 constraints. Increasing these numbers would be risky and could result in the process ending up with no solution. However, the presented twostage stochastic optimization model may not capture all the complexities of the social, political, economic, and cultural contexts in which resource allocation decisions are actually made. It is only a beginning step on the long way of ongoing efforts to consider equity in optimization models for resource allocation toward risk mitigation measures. These models can be improved further to incorporate other mitigation measures involving wider socioeconomic risks and other factors such as migration, age, gender, culture, employment, dynamic evolution of wealth, etc.

7. CONCLUSIONS Financial resource allocation and its importance in natural catastrophes risk management are discussed in this article. This article also highlights the importance of issues such as effectiveness, efficiency, and fairness of financing among the main objectives for the allocation of public resources. Specifically, for public financial resource allocation as to pre- and

postnatural catastrophes mitigation measures, fairness plays a key role because various groups of people may suffer from earthquake impacts unequally. In this article, an optimization process is employed as a tool to seek optimal mitigation measures for resource allocation. A two-stage stochastic model is developed at a regional level with the objective of minimizing adverse effects of potential earthquakes in the future and expenditures of many plausible risk mitigation measures with special emphasis on the risk-equity tradeoff. To achieve this, several aspects and challenges of considering equity and related formulations in stochastic programming are also expanded. Application of optimization methods in this article has led to comparing large numbers of various risk mitigation strategies for financial resource allocation and consideration of joint effects as for different objectives such as risk of excessive reconstruction expenditures and mitigation expenditures and also effects of constraints like budget limitation and equity. The results of the case studies show that for the constant risk-aversion, the mitigation strategies and the distribution of recommended mitigation expenditures are sensitive to the amount of available budget and equity. For small values of the risk-aversion parameter, if the equity is ignored, there will be no

456 need to sustain mitigation expenditures. As inequity decreases (as measured by the Gini coefficient), the expected reconstruction expenditures and total loss increase. It can also be seen that as inequity decreases, the mitigation expenditures increase to allow more mitigation expenditures to satisfy the required equity as expected. This confirms the existence of an expenditure-equity (risk-equity) tradeoff. However, as the budget is limited, further reduction in inequity does not change the mitigation expenditure values. In such condition, although inclusion of equity cannot change the total allocated mitigation budget, inclusion of equity causes the pattern of the recommended mitigation expenditures to be changed and allocated mostly to the northern half and northwestern parts of Tehran instead of the center. This also results in allocating the recommended mitigation expenditures mostly to more vulnerable buildings. Inclusion of equity also results in more involvement of other structural types in mitigation process. It should also be mentioned that incorporating equity in case the budget is limited will increase the reconstruction, excessive reconstruction, and total expenditures and will decrease the efficiency. However, inclusion of equity in conjunction with an unlimited mitigation budget raises the recommended mitigation, reconstruction, excessive reconstruction, and total expenditures. Equity implementation also reduces the sensitivity of these expenditures to the variation of the risk-aversion value. However, this is true only when the mitigation budget is unlimited. Generally, equity is more easily achieved if the budget is unlimited. Limitation of mitigation budget bounds the recommended mitigation expenditures to the limited budget for the moderate and large values of the riskaversion parameter. Therefore, it limits the number of buildings that can be mitigated and decreases the varieties of structural types that are recommended for mitigation. It also results in the selection of less expensive and more frequent building taxonomies for mitigation. The results also show that regardless of equity level, as the decisionmaker’s riskaversion increases, the mitigation expenditures increase, but the expected reconstruction expenditures decrease. However, as risk-aversion gets larger, successive reductions in the risk come at higher costs. As expected, there is also a clear tradeoff between the probability of a large loss and the average total expenditures.

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