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 problem with 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.
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 [
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 [
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.
Generally, the distribution network of reliefs after disasters is described as a graph Minimize total costs including fixed costs and vehicle transportation costs. Maximize the minimum material satisfaction rates of demand points. 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: The disaster points and the candidate distribution centers are known and the capacities of the candidate distribution centers are large enough. The available arcs and distances between two points on the transport network are known. Because the reliefs are calculated by volume, different types of relief supplies can be regarded as a kind of material. The relief demands of the demand points are greater than or equal to the amounts of supplies because of the materials shortage after disasters. 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.
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.
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.
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. Let
The following formulas are the constraint conditions. In adition, the constraints conditions (4), (5), (6), (7), (8), (9), (16), (17), and (18) of the literature [
Formulas (
There are subloop elimination constraints in order to avoid the subloop. The constraints proposed by Dror et al. [
The travel velocities of vehicles on each arc are uncertain, so the interval of the vehicles travel velocity on
Because the uncertain demands and uncertain transport velocities are represented by continuous intervals, the scenario set is an infinite set. Therefore, it is very difficult to get the estimate values of The demands of point The transport velocities of arc on the path
With the above introduction of
Taking the second objective function as an example, the third objective function can be proved by the same method. Let
It can be seen that the main claim of Lemma
Before converting, since
The following theorem is obtained by Lemma
For any
Many practical problems need to optimize multiple objectives simultaneously. Sometimes these goals often compete with each other or contradict each other, so the definition of Pareto optimal solution is introduced.
The papers [
Generate initial population The variation population is obtained by the variation process, and trail population is obtained by the crossover operation. Combine the parent population Compute the objective values for each chromosome in Determine the values of Select the first NP individuals as the next generation population. Stop the procedure if the generation
Parameter setting is
The combination of the number of candidate distribution centers
The coordinates of each point are randomly generated in the plane, and the distances from point
Required number of vehicles.
Vehicle type | 10 points | 100 points | 1000 points |
---|---|---|---|
Large vehicle |
10 | 100 | 1000 |
Medium vehicle |
6 | 60 | 600 |
Small vehicle |
9 | 90 | 900 |
Vehicle parameters.
Vehicle type | Vehicle capacity |
General velocity |
Unit transportation cost |
---|---|---|---|
Large vehicle |
|
50 | 10.0 |
Medium vehicle |
|
30 | 3.1 |
Small vehicle |
|
20 | 1.7 |
Calculation results.
Examples | Total costs | Satisfaction rate (%) | Path velocity | |||
---|---|---|---|---|---|---|
Mean value of approximate Pareto fronts | Approximate Pareto solution | Mean value of approximate Pareto fronts | Approximate Pareto solution | Mean value of approximate Pareto fronts | Approximate Pareto solution | |
3-10 | 8150.1 | 5811 | 55.48 | 55.44 | 643.8333 | 1098 |
3-100 | 34625 | 25263 | 64.28 | 62.77 | 8644.1 | 8761 |
3-1000 | 158190 | 139560 | 59.35 | 59.80 | 89865 | 88913 |
6-100 | 42807 | 38976 | 60.57 | 64.49 | 8554 | 8575 |
6-1000 | 129560 | 116650 | 54.89 | 60.01 | 11145 | 12270 |
10-100 | 43869 | 48567 | 58.99 | 59.62 | 8097.4 | 8329 |
10-1000 | 137980 | 118960 | 58.67 | 61.13 | 87660 | 89437 |
Obtained by genetic algorithm (3-10).
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 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.
The number of candidate distribution centers
The number of disaster points
The number of vehicles
The fixed costs of distribution center
The velocity of arc
The demands of disaster point
The available quantities of reliefs on the transport network
The unit transportation cost of vehicle,
The load capacities of vehicle
The time of vehicle
The latest service time at point
The relief supply quantities transported by the vehicle
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research is supported by the Natural Science Foundation of China (Grant no. 71471007).