Multiobjective Location Routing Problem considering Uncertain Data after Disasters

The relief distributions after large disasters play an important role for rescue works. After disasters there is a high degree of uncertainty, such as the demands of disaster points and the damage of paths. The demands of affected points and the velocities between two points on the paths are uncertain in this article, and the robust optimization method is applied to deal with the uncertain parameters.This paper proposes a nonlinear location routing problemwith half-time windows and with three objectives. The affected points can be visited more than one time. The goals are the total costs of the transportation, the satisfaction rates of disaster nodes, and the path transport capacities which are denoted by vehicle velocities. Finally, the genetic algorithm is applied to solve a number of numerical examples, and the results show that the genetic algorithm is very stable and effective for this problem.


Introduction
In recent years, man-made or natural disasters which caused huge casualties and economic losses occurred frequently in different regions and countries. Timely and effective rescue works are very important after disasters. The emergency logistics management mainly includes two aspects [1]: the facility location problem (FLP) and the vehicle routing problem (VRP). In fact, there is a close relationship between facility location problem and vehicle routing problem about emergency logistics, that is, location routing problem (LRP). Compared with the traditional transportation logistics, the facility location problem and vehicle routing problem about emergency logistics are more challenging and complex [2]. von Boventer [3] combined the facility location problem and the vehicle routing problem. Up to now, there have been abundant research results about LRP [4,5].
After disasters, there are a high degree of uncertainties in all aspects, such as the relief demands and the path transport capacities. Three major methods for dealing with the uncertain parameters are stochastic programming, fuzzy method, and robust optimization method. Ahmadi-Javid and Seddighi [6] studied a stochastic LRP problem. The materials which were provided by facilities are random variables. Under uncertain demands in the short time after disasters [7], large scale emergency scheduling problems were studied, including the selection of supply points, route selection, the decisions of the transport quantities, and the distribution methods, and the fuzzy set was applied to express the uncertain demands. Zare Mehrjerdi and Nadizadeh [8] studied the LRP problem with capacity restrictions of the vehicles and the distribution centers. Triangular fuzzy number represented the uncertainty demands. Lu [9] considered the robust weighted P-center model, and the travel time and demands of affected points were all uncertain. Wang et al. [10] proposed a dynamic time-space network model. It was a network flow model with multistages and multimaterials. Robust optimization method was applied to deal with the uncertain demand. Wang et al. [11] proposed a multiobjective location routing model with split delivery. It optimized the allocation of reliefs after earthquake. The objectives were to minimize the travel time, to minimize the total costs including the fixed costs of the distribution centers and the vehicles transportation costs and to maximize the path reliability, but the demands of disaster points and the probabilities of available arcs were certain. Koç et al. [12] studied the LRP problem with time windows. It was a certain problem with heterogeneous fleets and with spit delivery, and the objective was to minimize the total costs. It can be seen from the existing literatures that there are little researches on LRP problem with uncertain parameters which are solved by robust optimization. In this paper, the robust optimization method is used to deal with the uncertain demands and the uncertain velocities between two disaster points on the paths. In addition, the most scholars take the cost as the objective of LRP problem. Actually, the distances and the travel time between two points after disasters will be affected, and we will pay the price. Then, the distance and travel time can be understood as the travel cost. In this paper, the half-time windows constraints are quoted, so the time when the materials reach the demand points cannot be later than the specified time. Thus, the timeliness of emergency rescues is improved. After disasters, all kinds of materials are in short supplies, so it is very important to possess adequate relief supplies. Therefore, the minimization of the total distribution costs is one of the objectives, and the maximization of the worst path satisfaction rates is the second objective. After incidents, the transport network will be destroyed. In order to find a better path, the maximization of path transport capacities is the third objective in this paper.

The Problem Description
Generally, the distribution network of reliefs after disasters is described as a graph = ( , ). is a vertex set, and = { +1, +2, . . . , + } is a set of candidate distribution centers. We assume that there are no demands for a point in the set .
(2) Maximize the minimum material satisfaction rates of demand points.
(3) Maximize the transport capacities of the worst path.
(The transport capacities are represented by the velocities of the vehicle.) This paper has the following assumptions: (1) The disaster points and the candidate distribution centers are known and the capacities of the candidate distribution centers are large enough.
(2) The available arcs and distances between two points on the transport network are known.
(3) Because the reliefs are calculated by volume, different types of relief supplies can be regarded as a kind of material.
(4) The relief demands of the demand points are greater than or equal to the amounts of supplies because of the materials shortage after disasters.
(5) In this paper, we can only consider the disaster points which can be serviced by vehicles, and the disaster points which can be serviced by the special transportation methods (e.g., helicopters) are ignored.
The parameters and variables used in this paper are introduced in Notations.

Objective 1: Minimization of the Total Distribution
Cost-min 1 . In the location routing problem, it is necessary to determine the number and location of the distribution centers and to arrange the disaster points to the distribution centers, and the corresponding vehicle routing will be decided. Therefore, the total costs of the relief distributions include the fixed costs of distribution centers and the vehicle transport costs. (1)

Objective 2: Maximization of the Worst Path Satisfaction
Rates-max 2 . We hope that reliefs can reach the demand point timely and effectively after disasters, so we should consider the relief satisfaction rates of the demand points. We hope to maximize the relief satisfaction rates in affected areas, and the fairness of the relief distributions is taken into account. Therefore, the second objective is maximization of the worst satisfaction rates.

Objective 3: Maximization of the Worst Path Transport
Capacities-max 3 . The original paths after disasters are affected more or less. At this time, we take into account the path transport capacities, and the worst path transport capacities are maximized in this paper. The sum of velocities indicates the path transport capacities. The vehicle velocities are uncertain.
≥ 0, ∀ ∈ , ∈ , (8) Formulas (4) and (5) are the time window constraints. is a great positive number. This article requires that the sum of travel time and the service time is not bigger than the set time. Constraint (6) shows that the total amounts of reliefs delivered from the distribution centers to the disaster points shall not exceed the total amounts of available reliefs. Constraint (7) ensures that the relief volume transported to the affected areas by vehicle cannot exceed the load capacities of vehicles. Constraints (8)-(9) ensure that the decision variables are 0-1 nonnegative variables.
There are subloop elimination constraints in order to avoid the subloop. The constraints proposed by Dror et al. [13] are applied in this paper. Let indicate the outgoing degree of point : ∈ is called a scenario. In order to facilitate the description of the problem, 2 , 3 are changed equally.

The Solution Method of This Model
Given a scenario, problems (12) and (13) are certain. In certain problem the objective function 2 is We need to define the maximum velocities sum of path ( ) under the scheme and the scenario . .
The objective function 3 is expressed by formula (17) under the scheme and the scenario .
The robust costs of a scheme can be represented by the following problems: The second objective and the third objective of the above problem can be expressed as follows: Formula (21) implies minimizing the maximum of the robust deviations.

Robust Cost
where * ( ) and * ( ) are optimal schemes under the scenarios and .
Proof. Taking the second objective function as an example, the third objective function can be proved by the same method. Let * ( ) be the optimal solution of problem (14), and ( , * ( )) is the optimal solution of problem (19).
Main Claim. ( , * ( )) is the optimal solution of problem (19), so is the worst scenario under scheme , and * ( ) is the optimal scheme of 2 ( ). Then the following formulas can be obtained: It can be seen that the main claim of Lemma 1 can be proved converting to . We can finish this conversion by the following steps.
Step 1. The upper bound of point substitutes ( ).
The following theorem is obtained by Lemma 1, which greatly simplifies formulas (19) and (20). The papers [14][15][16][17] are the common methods to solve multiobjective optimization problems. In this paper, genetic algorithm [17] is applied to solve the uncertain problem with half-time windows. The specific processes are as follows.

Theorem 2. For any ∈ ,
(1) Initial Population. According to the characteristics of the LRP problem, each chromosome includes three substrings.
We can calculate three objective function values through decomposing [11]. Choose the worst scenario according to (2) Mutation Operation. Variation vectors can be obtained through mutating. In order to avoid being trapped in local optimum, this paper introduces the inversion sequence variation method for each substring of chromosome [18]. It randomly chooses two notes within a chromosome and then reverses their contents. The following two formulas are the examples. Parent is [5 6|2 3 7 8 9|1 4], and child is [5 6|9 8 7 3 2|1 4] after reversing.
(3) Crossover Operation. Crossover operation is applied to obtain the trail vector after mutation operator. According to the literature [19], the two points crossover method is used in 3 , and single point crossover method is applied to produce 1 , 2 . 1 , 2 , and 3 which constitute the trail vector .
(4) Selection Operation. The weighted method [20] is applied to sort individuals in population. 1 is selected randomly on interval (0.1, 0.3], and 3 is obtained randomly on interval (0.3, 0.5]. Because the satisfaction rate is less than 1, it is easy to lose with the total costs and the total path velocities, so we select on interval [500, 1000] randomly. Then, we let 2 = × ( is the number of disaster points).

(5) Steps of Genetic Algorithm
(S1) Generate initial population 0 of size NP randomly. (S2) The variation population is obtained by the variation process, and trail population is obtained by the crossover operation.

Numerical Experiment
Parameter setting is NP = 5 × ( + ); mutation probability is = 0.7; crossover probability is = 0.7.   Table 1. Let the time when the disaster happened be the zero time. The last time (hour) for each demand point to be served is randomly generated on the interval (0, 100]. Table 2 gives the parameters of the vehicles. Table 3 shows the results of the calculation, and the approximate Pareto front is shown in Figure 1 (3-10).

Conclusion and Prospect
In this paper, a multiobjective nonlinear location routing model with half-time windows is proposed. The arrival time of the reliefs to the demand points cannot be later than the specified time, so the timeliness of emergency reliefs is enhanced. The affected points can be visited more than one time in this article. After the disasters, all kinds of materials are in short supplies, so it is very important to have adequate relief supplies. Therefore, the minimization of the total distribution costs is one of the objectives, and the maximization of the worst path satisfaction rates is the second objective. After incidents, the transport network will be influenced. In order to find a better path, the maximization of path transport capacities is the third objective in this paper. After disasters situations are very complex with a high degree of uncertainties, such as demands, transportation time, and path through velocities. Therefore, this paper assumes that the demands of the disaster nodes and the transportation velocities of the 6 Discrete Dynamics in Nature and Society   available path are uncertain, and the robust optimization is applied to deal with the uncertainty. The genetic algorithm is applied to solve a number of numerical examples; the results show that the algorithm is very stable and effective for this problem. Finally, the method of solving the problem can also apply simulated annealing algorithm or nondominated sorting genetic algorithm.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.