A Multiobjective Allocation Model for Emergency Resources That Balance Efficiency and Fairness YanyanWang and

Efficiency and fairness are two important goals of disaster rescue. However, the existing models usually unilaterally consider the efficiency or fairness of resource allocation. Based on this, a multiobjective emergency resource allocation model that can balance efficiency and fairness is proposed. The object of the proposed model is to minimize the total allocating costs of resources and the total losses caused by insufficient resources. Then the particle swarm optimization is applied to solve the model. Finally, a computational example is conducted based on the emergency relief resource allocation after Ya’an earthquake in China to verify the applicability of the proposed model.


Introduction
Major sudden disaster often affects many areas and causes a lot of casualties and a great loss of property [1,2].For example, in 2008, more than 100 thousand square kilometers were seriously damaged by the Wenchuan earthquake on May 12, resulting in a number of seriously affected disaster areas, including Wenchuan, Beichuan, Qingchuan, Mianzhu, and other hard-hit areas [3].After a disaster, a large demand for emergency resources, such as food, drugs, and other supplies, is inevitably generated at the multiple affected sites [4].At this time, how to allocate emergency resources will directly affect the rescue effect.Efficiency and fairness are two important but conflicting goals in emergency rescue [5].The most efficient allocation scheme is not necessarily the fairest, and the fair allocation scheme may result in very high allocating costs.Therefore, appropriate trade-off between efficiency and fairness may be needed in the context of different disasters.Based on this, it is very necessary to construct a resource allocation model that considers both efficiency and fairness to meet the demands of all affected sites.
The goals of most existing models for emergency resource allocation mainly reflect the efficiency, such as the lowest total allocation costs.Equi et al. [6] proposed a product delivery model that minimizes the total costs, including loading costs, transportation costs, unloading costs, and storage costs.Viswanath and Peeta [7] developed low-cost routes that cover the maximum population while satisfying budget constraints for earthquake rescue.Zhan and Liu [8] presented a model that aims to minimize both the transportation costs and the rescue time.Barbarosoglu et al. [9] proposed a mathematical model for minimizing the cost of different helicopters in a disaster relief operation.Haghani et al. [10,11] analyzed the multicommodity allocation models that aim to minimize the sum of the vehicular flow costs and the transfer costs.Bai [12] used the minimum total costs of allocation as one of the objectives to consider the issue of emergency resource allocation under integrated uncertainty.Yang et al. [13] constructed a bilevel programming model of resource allocation for sudden disaster with the minimum allocation cost as the goal of the upper level.However, in the actual allocation of resources, the low-cost allocation plan is not necessarily the fairest and often leads to inaccurate allocation of resources among disaster locations.
With the continuous development of research, the issue of fairness has gradually been considered in the resource allocation process.Mandell [14] put the Gini coefficient as the fairness index to build the resource allocation model of public service facilities system.Wang et al. [15] constructed a fair allocation model for emergency resource based on the double-layer decision-making method.Pang et al. [16] set a fair coefficient within (0,1) to ensure the fairness of resource allocation.Chen and Fu [17] proposed an emergency resource allocation model with the minimum total weighted envy value as the fairness goal and the proportional fairness as constraints.Ye et al. [18] developed an equilibrium decision model for earthquake rescue based on demand information update that aims to realize a fair allocation to all disaster locations.Zhan et al. [19] presented a fair decision-making model for resource allocation by introducing Bayesian demand updating method, but the dynamic change characteristics of supply were not considered.
From the perspective of stage of emergency resource allocation, the above researches mainly focused on single-stage resource allocation.There are few related studies although the issue of multistage resource allocation has gradually received more emphasis.For example, Linet [20] constructed a multistage emergency resource allocation model with the goal of minimizing transportation time, but this model did not consider the issue of fairness.Feng et al. [5] proposed multiobjective optimization model of the emergency resource allocation with multicycle and multi-item; it deals with the allocation of resources between a single supply site and multiple disaster sites, but in the actual rescue process, the resource reserves of a single allocation center are usually limited, and it is not possible to meet the needs of multiple disaster sites at the same time.Thus, the study of resource allocation between multiple supply sites and multiple affected sites may be more in line with the actual situation of rescue.
Through the analysis of the above literatures we found that most of the existing researches usually unilaterally take the efficiency or fairness as one of the goals of model for the resource allocation and the issue of balancing efficiency and fairness of resource allocation is rarely studied.However, efficiency and fairness are important goals and should be considered in the process of resource allocation.
On the basis of existing studies, this paper constructs a multiobjective allocation model for emergency resources, which aims to balance the efficiency (the total allocating costs) and fairness (the total losses caused by insufficient resources) of resource allocation.Then, a case study for the Ya'an earthquake in China is applied to verify the effectiveness and feasibility of the proposed model.
The main contributions of this paper can be summarized as follows: (1) The proposed model can consider both efficiency and fairness and can provide support for resource allocation decisions in different emergency contexts (2) The unmet proportion of required resources is used to quantify the fairness, which will facilitate the fair allocation of emergency resources among multiple affected sites during large-scale disasters The rest of this paper is organized as follows: Section 2 describes the general problem and assumptions.Section 3 proposes the mathematical model.Section 4 presents the solution algorithm.In Section 5, a computational example is given to illustrate the applicability and potential advantages of the proposed model.Finally, conclusions are drawn in Section 6.

The General Problem Description and Assumptions
2.1.Problem Description.The multiobjective emergency resource allocation discussed in this paper is about dealing with the multiple supply sites, multiple affected sites, and multistage allocation for large-scale sudden disasters; the decision makers should consider both efficiency and fairness of resource allocation at each stage to meet the needs of all affected sites as soon as possible with the least cost, as shown in Figure 1.

Assumptions.
Combined with the actual situation of emergency resource allocation, the following assumptions are made: (1) The emergency response center has an advanced information platform that can timely grasp and update the related disaster information at every stage, such as the number of affected sites, road conditions, casualties, resource demands, and available resources (2) All types of emergency resources are irrelevant items; that is, whether a certain type of resource arrives at the affected area has no effect on the allocation of other types of resources (3) The impact of secondary disasters on resource allocation is not considered in this paper

The Multiobjective Allocation Model for
Emergency Resources where the parameter  is used to show the importance placed on large losses.Then the objective functions can be written as the following: The first objective function ( 2) is to minimize the total allocating costs and stands for "efficiency" in process of relief resources allocation.The second objective function ( 3) is to minimize the total losses caused by insufficient resources and represents "fairness" of relief resources allocation.

Constraints. The emergency resource allocation model formulated should satisfy the following six constraints.
A The total amount of resources that can be allocated should not exceed the sum of the inventory and the new supply.Here is the mathematical expression of the following constraint equation ( 4);  ,−1 is the inventory of resources at supply sites at the end of stage  − 1: B The total amount of resource allocated to all affected sites should not exceed the sum of the current demand and the unmet proportion remaining from previous stages.It is given by (5);  ,−1 is the unmet proportion of required resources at affected sites at the end of stage  − 1: C The fixed cost should be paid as long as the supply site allocates resources to the affected site.This constraint limits the capacity for transportation and also enforces the fixed charge, as shown in D The binary variable must be equal to 0 or 1, as demonstrated in E The variables for the amount of allocation, inventory, and unmet proportion of required resource must be nonnegative, as shown in ≥ 0, ∀ ∈ ,  ∈ ,  ∈  (10)

Solution Algorithm
4.1.The Basis for the Selection of Algorithm.The model proposed in this paper contains many parameters which are dynamically changing.The traditional optimization methods (such as gradient descent, Newton's method and quasi-Newton methods, and conjugate gradient) are limited in the process of solving the model [22].Therefore, we need to explore the adaptive intelligent optimization method.Currently, the intelligent algorithms for solving optimization problems mainly include genetic algorithm (GA), ant colony algorithm (ACO), and particle swarm optimization (PSO).This paper selects the particle swarm optimization algorithm to solve the model.The reasons are as follows: Compared with GA, the advantage of PSO is that the rule is simpler.It does not have the "Crossover" and "Mutation" operations of GA and is easy to implement.There are not many parameters that need to be adjusted, and all particles can converge to the optimal solution quickly [23].Compared with ACO, the advantage of PSO is that it is quick to approximate the optimal solution, the solution speed is fast, and the parameters of the system can be effectively optimized [24].Meanwhile, resource allocation under emergency situations is often subjected to severe time pressure, which places stringent requirements on the efficiency of solving the model.The chosen algorithm should be able to quickly solve for dynamic changes of parameters, while the accuracy requirements of the solution may be relatively loose.Therefore, the selection of the algorithm for solving the model should follow the principles of fast convergence, less resource usage, and good robustness.Combined with the characteristics of the model proposed in this paper, PSO can meet the need for solving this kind of problem.It can avoid complex genetic operations like evolutionary algorithms and is simple, easy to implement, and fast to converge.Therefore, this paper selects the PSO to solve the proposed model.

The PSO Algorithm. PSO is a new evolutionary algorithm
proposed by Kennedy and Eberhart, which was inspired by the bird foraging behavior [25].It uses a simple velocitydisplacement model to achieve population-based global search guided by fitness function information, while its memory function can track the search path and dynamically adjust the search strategy [26].The mathematical description of standard PSO is as follows: there are  particles in a dimensional space, the position of each particle  is denoted as X i =[x i1 , x i2 ,..., x iD ], and each position corresponds to a fitness function  −1 () related to the optimization objective function.The velocity of each particle  is denoted as the best position that the particle  has experienced is denoted as Pbest i =[p i1 , p i2 ,..., p iD ], and the best position that the population has experienced is denoted as   = [ 1 ,  2 , ...,   ].The velocity and position updating formula for particle  in the D-dimensional space are as follows: where  is the dimension of the particle position,  ∈ {1, 2, . .The detailed algorithmic steps are described as follows.
Step 5. Update the velocity   and position   of each particle according to formula ( 11) and ( 12).
Step 6.If the algorithm attains the maximum iterations, turn to Step 7; otherwise turn to Step 2.
Step 7. Stop optimization and output the result.

Computational Example
In this section, we provide a computational example to illustrate the applicability of the proposed model in solving the problem of emergency resources allocation for realistic large-scale disasters.Here, take the Ya'an earthquake as a case.A magnitude 7.0 earthquake suddenly occurred at 8:02:46 AM, China Standard Time, on April 20, 2013, in Lushan County of Ya'an city, Sichuan Province.The maximum intensity was 9. A total of 18,682 square kilometers was affected, 196 lost their lives, 11,470 were injured, and 21 were still missing [27].Lushan County (LS), Yucheng District (YC), Baoxing County (BX), Tianquan County (TQ), and Yingjing County (YJ) were seriously affected in this earthquake.The general disaster information on these affected sites is shown in Table 1.Chengdu (CD) and Ziyang (ZY) were selected as supply sites, and the tents and blankets are selected as the required emergency resources.We take one day as a stage, to analyze the emergency resources allocation in the first 5 days after the large-scale disaster.The relevant data for computational example were chosen using a combination of real and hypothetical data, since some disaster data could not be obtained through official reports.The estimated demand for resources at each affected site and the supply at each supply site are shown in Tables 2 and 3, respectively.Tables 4 and 5, respectively, show the fixed costs of transportation and variable costs of allocating resources.This computational example was solved using MATLAB R2016a on a computer with an Intel(R) Core(TM)1.90GHz processor with 16.0 GB of RAM.The parameters of PSO are set as follows: particle size is 50, largest number of iterations is 600, =0.9, and c 1 =c 2 =2.The allocation results and unmet  proportion of required resources when =-0.007 are shown in Table 6.
From Table 6, the model allocates resources to every affected site at each stage, and the unmet proportion of required resources at all affected sites are equal at each stage.
The unmet proportion of resources at the previous stage can be replenished in the next stage until all their demands are satisfied (e.g., all demands of every affected site can be fully satisfied until the fifth stage).When demand is significantly greater than supply, the fair allocation can still be guaranteed.When demand is less than supply, the demands of all affected sites are fully satisfied.Obviously, the proposed model can avoid large shortage and huge losses caused by the insufficient resources at any one affected site, which is beneficial to the fair allocation of resources.
In the actual material allocation process, when the fixed transportation costs are high enough, the high transportation costs may exceed the consideration of fairness.To prove this, we now give the following example (all fixed costs increased by 100 times), also assuming =-0.007.The allocation results and unmet proportion of resources are shown in Table 7.
Table 7 shows unequal unmet proportion due to the higher fixed costs of transportation, and the model does not allocate resources to every affected site at each stage.For example, YJ and TQ do not receive any required resources in the second and third stage, respectively.However, in the next stage, they can receive more resources.This can save the costs  of transportation.Thus, the proposed model can effectively trade off efficiency versus fairness and has better flexibility and applicability in emergency resource allocation.In order to verify the validity of this PSO algorithm, this paper employs MATLAB programming to directly solve the above computational example mentioned (the same data in Tables 2-5).The solutions obtained by the PSO algorithm and direct programming are 43,726,000 and 44,108,000, respectively.The corresponding computing time is 56 seconds and 439 seconds, respectively.Among them, the solution obtained by PSO algorithm is the average value of 20 tests.It can be seen that the PSO algorithm obtains a better objective value, which can reduce the total losses caused by resources shortage and the total costs of allocation.Moreover, the computing time of PSO is only 1/8 of direct programming, indicating that the PSO algorithm can improve the timeliness of resource allocation and save time for emergency rescue.

Conclusions
This paper is motivated by the need to solve the allocation problem of emergency resource in large-scale disaster.We propose a multiobjective emergency resource allocation model and the particle swarm optimization algorithm is applied.Then the case study for the Ya'an earthquake in China verifies the effectiveness and feasibility of the proposed model.This study can provide enlightenment on the emergency resource allocation decisions for emergency decisionmakers: (1) In emergency resource allocation decision, efficiency and fairness are important goals of disaster relief.Decision-makers must realize a balance between efficiency and fairness to meet the material needs of all disaster-stricken areas as soon as possible with the least cost (2) In the case of resource shortage in the initial stage of disaster relief, the primary objective should be to reduce the losses caused by insufficient resources at each affected site, ensuring the fairness of resources allocation.When resources continue to be supplied, the cost of allocation should be considered (3) With higher costs of transportation, decisionmakers can choose to centralize allocation or allocate resources once every two stages to save the transportation costs However, the allocation process of emergency resources is affected by many factors, such as the response time, urgency of resources, and disaster severity.Thus, future research on resource allocation model should comprehensively consider the above constraints.

Figure 1 :
Figure 1: The diagram of emergency resource allocation.
For the sake of simplicity, we adopt the notations displayed as follows.Sets: : set of affected sites,  ∈ ,  = {/1, 2, ..., }, where  is the total number of affected sites : fixed cost of transportation for using  ∈  to  ∈  at stage  ∈   : variables cost of allocating resource  ∈  from  ∈  to  ∈  at stage  ∈   : maximum total amount of resource that can go from  ∈  to  ∈  at stage  ∈ Variables.There are one binary variable and three continuous variables for this model:  : binary variable indicating whether  ∈  allocates resource to  ∈  at stage  ∈  or not   : amount of  ∈  that allocates from  ∈  to  ∈  at stage  ∈   : inventory of resource  ∈  in  ∈  at the end of stage  ∈   : unmet proportion of required resource  ∈  in  ∈  at the end of stage  ∈ ; it can be measured by   =   − ∑ ∈   .This paper proposes to use a (convex) utility function   [⋅]: + →  + to measure the penalty (or loss) for the insufficient resources  ∈  in affected site  ∈  at stage  ∈ . [ 3.1.Notations.: set of supply sites,  ∈ ,  = {/1, 2, . . ., }, where  is the total number of supply sites : set of types of emergency resources,  ∈ ,  = {/1, 2, . . ., }, where  is the total number of types of resources : set of stages of rescue,  ∈ ,  = {/1, 2, . . ., }, where  is the maximum number of stages Parameters:   : demand for resource  ∈  in  ∈  at stage  ∈   : new supply of resource  ∈  in  ∈  at stage  ∈ particles leaving the search space, the movement of particles is appropriately limited by setting the velocity interval [ min ,  max ] and the position range [ min ,  max ].
. , , . . ., };  is the inertial weight factor;  − 1 is the present iteration;  1 and  2 are positive acceleration constants; and  1 and  2 are random numbers uniformly allocated within [0, 1].Moreover, in order to reduce the possibility of

Table 1 :
General information on the affected sites.

Table 2 :
The resource demand for the affected sites.

Table 3 :
The resource supply of the supply sites.

Table 4 :
The fixed costs of transportation from supply sites to affected sites (yuan).

Table 5 :
The variable costs of allocating resource from supply sites to affected sites (yuan).

Table 6 :
The allocation results and unmet proportion of emergency resources.

Table 7 :
The allocation results and unmet proportion of resources with higher fixed costs of transportation.