Research on improved genetic algorithm in path optimization of aviation logistics distribution center

Abstract

At present, airlines cannot meet the requirements of increasing air cargo volume and cargo service quality by simply improving air transportation capacity. On the basis of increasing the construction of aviation logistics distribution centers, airlines also need to optimize the distribution route. By analyzing and comparing the application of genetic algorithm in the problem of distribution path optimization, this paper proposes a new adaptive genetic algorithm with adaptive mutation to improve the search ability in the local range, which has faster convergence speed than the general genetic algorithm. And solve the logistics node location model. The result proves that the algorithm is faster and more accurate, and can solve the transportation path problem in distribution.

Keywords

Improved genetic algorithm distribution center path optimization aviation logistics

1 Introduction

Air transportation is an important part of aviation logistics. The transportation route determines the time of freight transportation and the revenue of airlines. Freight services usually involve two issues, one is whether the capacity can meet the requirements; the second is the arrangement of the cargo transportation route. In fact, airlines can promise the time of delivery by providing flexible capacity arrangements and time-determined services. By analyzing the flight schedule and the capacity of each route, the airline selects the most suitable route to maximize revenue. Therefore, the problem of air cargo transportation route can be defined as: in order to meet the transportation requirements of the cargo owner, the airline can reasonably arrange its transportation capacity and search for the best transportation route based on the existing route network under the condition of meeting the actual transportation time requirement. The decision-making problem of maximizing transportation profit.

There are two main ways of transporting air cargo, one is carried by the passenger cabin, and the other is transported by all cargo planes. At present, there are about 50 full cargo planes in China, accounting for only 3% of the total aircraft. The transportation turnover and transportation volume of the cargo in the belly compartment accounted for 86% and 94% of the total freight volume, respectively, and only about 10% was all-cargo transportation. Most of China’s domestic freight transport is carried by the passenger cabin belly; the full cargo aircraft mainly serves international cargo transportation [1]. Therefore, the passenger cabin belly compartment is an important cargo resource, and it is necessary to analyze the cargo optimization problem in the case of passenger and cargo mixed cargo and full cargo aircraft transport.

The cargo transportation route is closely related to the transportation capacity, and the cargo capacity is related to factors such as aircraft type and aircraft scheduling. Therefore, this chapter discusses the optimization of the model assignment and freight route optimization in the case of passenger aircraft belly cargo loading and the optimization of freighter scheduling and freight routing in the case of full cargo aircraft transportation [2].

2 Model assignment and freight route optimization

The freight arrangement under the mixed cargo and passenger situation mainly depends on the capacity of the passenger aircraft, and the capacity depends on the model. At present, domestic airlines usually only consider passenger transportation, neglecting freight demand, and freight revenue is regarded as contribution income, which is not taken seriously. However, with the development of the economy, the volume of freight has increased rapidly, and the original model assignment scheme cannot meet the demand for freight, which often leads to delays in the inability of the cargo to be loaded into the aircraft. Frequent occurrences of this situation may result in the loss of freight revenue and even the loss of the company’s total revenue. If the development of freight is considered while the model is being assigned, the passenger cabin can also be used as an important resource, and scientific and rational use can improve the profit of the airline [3]. Therefore, it is necessary to use a scientific method to carry out model assignment, optimize the freight route, meet the passenger and cargo demand, and improve the total revenue of the airline while considering the passenger and cargo transportation demand.

Model assignment is a question of how to effectively arrange the different models of airlines to each voyage to maximize the company’s profits. Literature [4] proposed the following model by proposing a model assignment model for model assignment and freight route optimization.

The geographical location of the customer and the quantity of goods supplied by the goods are known and remain stable in the short term;

All customer goods are first sent to the only collection center and finally to the production line;

The number of aircraft is sufficient to meet the demand. All aircraft must start from the assembly center and eventually return to the assembly center. All aircraft travel at the same speed and are not allowed to be overloaded;

The manufacturer’s demand for the customer’s goods is only the earliest and latest delivery time requirement. The strict time window is determined by the customer and the manufacturer and the automobile goods after the line is determined;

Everything works normally during the picking process, and no abnormal events occur.

2.1 Model description

N represents the total number of customers; i, j represents a single customer, i, j = (0, 1, . . . N); k represents the aircraft number, k = (1, 2 . . . , K); c_ij represents the transportation distance from i to j; α represents the cost of the unit transportation distance; β represents the fixed cost of the unit aircraft; Indicates the maximum load capacity of the aircraft; d_i represents the customer’s supply; t_ij represents the time of the aircraft from i to j, where i ≠ j; f_i represents the service time required for the aircraft to complete the mission; w_i represents the necessary arrival of the aircraft to the customer’s point i in advance Waiting time; $T_{0}^{k}$ is the departure time of aircraft k; $T_{R}^{k}$ is the time required for aircraft to return [5].

2.2 Objective function and constraints

The total cost is minimal to improve customer service levels and create more customer value. The total cost cycle is divided into three parts: the cost of the aircraft, the fixed cost of the aircraft and the opportunity cost lead time and the time delay of the shortage cost. Among them, the fixed cost of the aircraft is labor and management fees, which can limit the number of aircraft and the number of employees [6]. $\begin{matrix} MinF = \sum_{k = 1}^{K} \sum_{i = 1}^{N} \sum_{j = 1}^{N} α C_{ij} X_{ijk} \\ + \sum_{i = 1}^{K} \sum_{j = 1}^{N} β X_{0 ik} + \sum_{k = 1}^{K} \sum_{i = 1}^{N} P_{i} (s_{i}) \end{matrix}$ (1)

The meaning of the constraints is as follows:

The number of aircraft departing from the assembly center shall not exceed K;

Each vehicle departs from the assembly center and eventually returns to the collection center;

Each customer happens to be visited by an aircraft once;

The quantity of goods loaded per vehicle does not exceed the aircraft’s load limit;

The flow is conserved, that is, after the aircraft arrives at a customer, it must leave the customer;

The departure time and return time of the aircraft are within the specified time;

$s . t . {\begin{matrix} T_{0}^{k} + \sum_{i = 0}^{N} \sum_{j = 0}^{N} x_{ijk} (t_{ij} + f_{i} + w_{i}) \leq T_{R}^{k}, k \in {1, 2, . . . K} \\ \sum_{k = 0}^{K} \sum_{i = 0}^{N} x_{ijk} (s_{i} + f_{i} + w_{i} + t_{ij}) \leq s_{j}, j \in {1, 2, . . . N} \\ x_{ijk} \in {0, 1}, i, j \in {1, 2, . . . N}, k \in {1, 2, . . . K} \\ P_{i} (s_{i}) = p max (e_{i} - s_{i}, 0) + q max (s_{i} - l_{i}, 0) \end{matrix}$ (2)

The time constraint of the aircraft arriving from i to j;

Integer constraint, which limits x_ijk to only 0 or 1;

A penalty function cost expression that is advanced or delayed.

2.3 Detailed punishment function

The cycle picking path optimization is a typical just-in-time supply logistics network transportation system, in which the restrictions of the unloading crossing and the continuous production of the main engine have strict requirements on time. In order to minimize the total cost of the implementation process when designing the pick-up route, time-effect costs must be taken into account. The cost function expression is as follows [7]: $P_{i} (s_{i}) = {\begin{matrix} p (e_{i} - s_{i}) \begin{matrix} s_{i} < e_{i} \end{matrix} \\ 0 \begin{matrix} \begin{matrix} \end{matrix} & \begin{matrix} e_{i} \leq s_{i} \leq l_{i} \end{matrix} \end{matrix} \\ q (s_{i} - l_{i}) \begin{matrix} s_{i} {> l}_{i} \end{matrix} \end{matrix}$ (3)

The above formula can be expressed as: $P_{i} (s_{i}) = p max (e_{i} - s_{i}, 0) + q max (s_{i} - l_{i}, 0)$ (4)

e_i represents the starting point of customer i’s allowed service time window;

l_i represents the end of the service time window allowed by customer i;

s_i represents the time when the aircraft arrived at the customer’s i;

p represents the opportunity cost of the waiting unit time that occurs when the aircraft arrives at the customer in advance;

q represents the unit time penalty value of the aircraft arriving at the customer after the time window.

3 Improved genetic algorithm for calculating distribution center path optimization

The freight path is closely related to the freight capacity [8]. If the airline adjusts the model, it will also lead to the adjustment of the freight route. Especially in the case of the need for transshipment, the freight route will undergo complex changes due to changes in factors such as randomness, flight schedule, and freight demand. If the airlines are based on the quarterly flight plan and transportation resources, and comprehensively consider the passenger and cargo demand of the transportation market, the model assignment will not only meet the needs of aviation logistics development, but also positively help the airline’s total revenue.

The model assignment problem is to assign different aircraft models to the designed scheduled flights according to the capacity, operating cost and potential benefits of different models. As mentioned above, the constraints of the general model assignment model include flight coverage constraints, aircraft. Flow balance constraints and aircraft quantity constraints. If the OD demand of the goods is considered when the model is built, and the route of the freight is optimized according to the capacity of the transportation, the passenger and cargo transportation demand can be satisfied at the same time. Based on the improved genetic algorithm model, this chapter adds the constraints of freight route optimization, and establishes an integrated optimization model of model assignment and freight route with the maximum objective of the airline’s total return. Since the established model problem belongs to the NP-Hard problem, the heuristic algorithm with improved genetic algorithm is proposed to solve the model. The genetic algorithm steps are designed as shown in Fig. 1.

Fig. 1

Improved genetic algorithm flow.

3.1 Coding and decoding

Cyclic pick-up is a problem of aircraft path with time window. It is a sequence-based combinatorial optimization problem. In order to reduce the generation of invalid solutions, natural numbers are used for chromosomes, i.e., ordinal coding. For example, the individual code (0532071690840) for 9 customers served by 3 cars can be interpreted as: sub-path 1 is 0 → 5 →3 → 2 →0, sub-path 2 is 0 → 7 →1 → 6 →9 → 0, and sub-path 3 is 0 → 8 →4 → 0. The subpaths are ordered, and the subpaths are unordered. Since the number of aircraft is uncertain in advance, 0 is not added to the chromosome structure as the path-separated collection center, but the accesses in all paths are directly encoded into one chromosome, and the above three sub-paths are encoded as E. The decoding operation is the inverse of the encoding operation, that is, the process of mapping the coding vector of the chromosome into a feasible solution that satisfies all constraints.

3.2 Initial group

Since the chromosomes are integer-encoded, that is, each individual (denoted as i) is a full array of natural numbers from 1 to n, the initial population is generated at any time to generate popsize such individuals as the initial population, where n is the number of client nodes, and popsize is Population size.

3.3 Selection operator

Use roulette to select operators. Suppose a chromosome i has a fitness of f_i and a population size of M. The probability that the individual i is selected is P_si: $P_{si} = \frac{F_{i}}{\sum_{i = 1}^{M} F_{i}}, I = 1, 2, . . ., M$ (5)

3.4 Crossover operator

PMX-like is adopted because of the natural number coding method of the customer arrangement. The PMX-like method is different from the traditional crossover operator. Instead of directly exchanging the cross-section of the chromosome, it first adds the cross-segment gene to the first gene of the other chromosome, and then removes the same as the cross-segment gene in the original individual part one by one. Genes to get the individual after the cross. The operation process is as follows:

Randomly select an intersection area among the parent individuals, such as two parent individuals and the intersection area are selected as: A = ″256|8437|19″, B = ″359|4178|26″, where ″∥″ represents the intersection area;

Adding the intersection of B to the front of A, the intersection of A is added to the front of B, and two intermediate individuals are obtained: A′ = ″4178|256843719|″, B′ = ″8437|359417826|″;

In A′ and B′, the same genes as the crossover region are deleted in turn from the crossover region, and the final two individuals are: A₁ = ″843759126″, B₁ = ″417825639″.

3.5 Mutation operator

The invariant mutation operator is used to carry out the mutation operation. The specific process is to randomly select two mutation points of a chromosome and invert the mutation region to obtain a new individual. In the process of evolution, inversion mutation can effectively adjust the individuals in the population, prevent the premature convergence problem, and improve the global optimization performance of genetic operations. The operation process is as follows:

Randomly generate a body temp = ″9258735461″ for two variants, such as 2 and 3, i.e., where ″∥″ represents the region of variation. The new region temp₁ = ″9|37852|5461″ is obtained by inserting the variant region gene into the original position in reverse order.

3.6 Fitness function

In genetic algorithm, the greater the individual’s fitness, the better the individual performance, and the VRPTW is the minimum combinatorial optimization problem. The goal is to minimize the total cost of distribution, that is, the objective function value is the smallest, so the objective function is transformed into fitness. function: $f_{i} = \frac{1}{z_{i}}, i \in {1, 2, . . ., popsize}$ (6)

In the above formula, z_i is the objective function value corresponding to the i chromosome in the population, reflecting the total distribution cost corresponding to the staining of the i; f_i is the fitness corresponding to the i chromosome, and its value determines the probability of the offspring of the chromosome [9].

3.7 Termination of evolution

Here, the evolutionary algebra is determined to be 100 as the termination rule, that is, whether the algebra of the evolution is the required algebra N, and if so, the evolution is stopped, and the distribution path set corresponding to the best performing chromosome is selected as the optimal solution output of the problem sought. Conversely, continue to perform evolutionary operations. The process of substituting into MATLAB is shown in Fig. 2.

Fig. 2

MATLAB computing interface diagram.

The genetic algorithm is programmed by MATLAB, and the visual operation interface is designed. The operator can change the corresponding parameters according to the actual situation and get the preliminary route. The specific operation steps are as follows: Then, double-click to open the bottom of the visual operation main interface diagram to see the file of the given data. MAT, you can get the parameter setting interface. There are 16 variables that can be changed in the parameter setting interface diagram. The operator can change the following 16 variables according to the actual situation. The specific meanings of the variables are shown in Table 1.

Table 1

Variable meaning

Parameter	Meaning
Q	Aircraft capacity limit
Alpha	Aircraft operating unit distance cost
Begin time	Earliest service time
Beta	Fixed cost of aircraft
Supplier name	client’s name
Coord	Customer coordinates
Crossp	Cross probability
SupplierNet	number of clients
Demand	Customer demand
End time	Latest service time
Maxgen	Maximum genetic algebra
Mutp	Mutation probability
P	The unit time cost of waiting for the aircraft to arrive early
Pop size	Population size
q	The unit time penalty value of the aircraft arriving later than the time window
serve time	service hours

Finally, run the car routing file to get a relatively good aircraft path map, run it multiple times, and then compare and select the optimal initial line with the lowest cost.

4 Case analysis

4.1 The analysis process of the Case 1

It is known that an airline has a total of six passenger aircraft types and one cargo aircraft type. Its one-week flight plan includes 1,600 flights, including 1,400 passenger flights and 200 cargo flights. In order to analyze the effectiveness of the optimization model and algorithm, the paper randomly selected some flights to conduct research, and generated five sets of data. The data is shown in Table 2.

Table 2
Study related data

No. Passenger segment Freight segment Passenger model Freight model Freight volume (piece)

1 60 0 6 0 6235

2 100 0 6 0 9860

3 500 200 6 1 28830

4 1400 200 6 1 56526

5 1400 200 6 1 63500

No.	Passenger segment	Freight segment	Passenger model	Freight model	Freight volume (piece)
1	60	0	6	0	6235
2	100	0	6	0	9860
3	500	200	6	1	28830
4	1400	200	6	1	56526
5	1400	200	6	1	63500

In order to effectively reduce the amount of calculation, it is assumed that there is only one hub airport in the airline route network. All cargo can only be transferred at the hub airport, and the cost of cargo transshipment refers to the charging standard of the domestic hub airport. Passenger and freight transportation costs and benefits refer to the airline’s average cost, average fare and average freight rate [10]. The airline model parameters are shown in Table 3. The cargo capacity of the passenger aircraft is the maximum carrying capacity of the passenger cabin. Since the cargo capacity of the passenger compartment is closely related to the passenger volume, if the passengers carry more luggage, the cargo volume will be reduced. In the calculation, the average passenger load rate of the set flight is 75%, and the weight of each passenger carrying the luggage is 10 kg.

Table 3

Model parameter

	Model	Cargo capacity (kg)	Number of available seats	Maximum load (kg)
Passenger aircraft	8737–500	4404	136	14379
model	8737–700	6299	138	16649
	8737–800	4250	170	17000
	8757	9267	200	24267
	A319	7540	124	16840
	A320	12020	152	23270
Freighter model	8747–200E	1000000		1000000

According to the model and algorithm, ILOG/PLEX9.0 is used to solve the main problem and the problem. The main problem is the 0-1 type planning problem, which is a difficult problem. It adopts the branch and bound method and solves it with the depth-first search principle. 1%. The problem is solved by the simplex method, and the error δ is set at 0.1%. The five sets of data were calculated separately, and the results are shown in Table 4.

Table 4

Passenger aircraft belly cargo loading results

No.	Calculation time (seconds)	Number of iterations	Error rate (%)	Objective function value (yuan)
1	2.1	4	0.02	2622750
2	3	3	0.09	4291500
3	14.2	6	0.07	18387000
4	105.7	12	0.08	47853900
5	120.8	8	0.09	53931300

Through 5 sets of data, it is found that the algorithm can complete the calculation in a short time, and the error rate is within the specified range. Comparing the data of Group 4 and Group 5, the total revenue of passengers and goods can increase by 12.7% when the freight volume increases by 12.3%. Therefore, in consideration of the demand for freight, this method can effectively improve the total passenger and cargo revenue of the airline. At present, the domestic airline’s route network is mainly a passenger route network. The passenger aircraft belly cabin is still an important part of air cargo capacity, which plays a certain role in the development of air cargo. Therefore, it is of practical significance to conduct research on the assignment of models in consideration of freight revenue. Of course, the existence and development of modern air cargo as an accessory of air passenger transport can no longer meet the competition of the modern air cargo market. Its development direction should be that air cargo enterprises have strong all-cargo capacity, scientifically select and analyze target markets, and rationally allocate routes. Capacity layout, vigorously integrate existing network resources, build a corresponding air cargo network, and improve the efficiency of aviation operations [11 –16].

4.2 The analysis process of the Case 2

Through research, collect relevant data of a certain airline and each airport in China, including the unit operating cost of the cargo flight, the flight time, the flight distance and flight time of each airport, and the ground handling cost of the cargo at each airport (including fixed costs and Variable cost), the time of the aircraft at the airport, and the transit time of the goods at the airport. There are three BOEING747-200E aircraft, each with a maximum payload of 100 tons. In 18 cities, 30 OD cargoes are transported. The classification criteria L, and L of the route are set to be 1,500 km and 3000 km respectively, that is, short-haul routes of less than 1,500 km and long-haul routes of more than 3,000 km. The maximum transfer time for setting the multiple transfer method (Scheme c) is 1800 seconds, and the improvement range of the solution is 5%, that is, if the lower limit of the solution is exceeded 5%, the operation is stopped. Using the optimization software ILOG/CPLEX9.0, according to the optimization model and algorithm, the results of the five schemes in one week are calculated separately, as shown in Table 5.

Table 5
All cargo aircraft operation results

Program Self-transport One transfer Multiple transfer Mixed solution Improved hybrid solution

Number of aircraft 3 3 3 3 3

Flight frequency (week) 28 28 28 28 28

Average loading rate 61.5% 66.4% 68.7% 68.1% 68.5%

Transfer rate 0 6.7% 9.5% 9.4% 9.4%

Optimal value (yuan) 20664 000 22310400 22881 600 22848 230 23016 270

Calculation time (seconds) 3.98 14.36 1800 346.2 98.5

Program	Self-transport	One transfer	Multiple transfer	Mixed solution	Improved hybrid solution
Number of aircraft	3	3	3	3	3
Flight frequency (week)	28	28	28	28	28
Average loading rate	61.5%	66.4%	68.7%	68.1%	68.5%
Transfer rate	0	6.7%	9.5%	9.4%	9.4%
Optimal value (yuan)	20664 000	22310400	22881 600	22848 230	23016 270
Calculation time (seconds)	3.98	14.36	1800	346.2	98.5

According to the calculation results, although the direct transportation method can improve the efficiency of cargo transportation, it cannot improve the utilization rate of the aircraft, and thus the profit is also the least. A single transfer can increase aircraft loading rate and profit to a certain extent, but is inferior to multiple transfer and mixed transportation options. In this paper, the longest calculation time for multiple transshipment is 1800 seconds, and the optimal value is 22881600. The improved hybrid scheme can get the optimal solution in a short time, and the benefit is 5 more than the multiple transshipment scheme. %. Figure 3 is a schematic diagram of the aircraft route. The three patterns represent the flight routes of the three aircraft at 18 airports.

Fig. 3

MATLAB computing interface diagram.

5 Conclusion

Since air cargo can be transported by passenger aircraft belly or full cargo aircraft, this chapter analyzes the cargo transportation route problem in two different situations. For the cargo compartment of the passenger aircraft, the cargo transportation route is affected by the passenger aircraft model and flight plan. Therefore, the passenger aircraft type assignment and the freight route optimization problem are In combination for this conducted research, and the aircraft total revenue is maximized as the goal. The assignment and freight route integration optimization model, and the improved genetic algorithm to solve the problem. For the whole freighter transportation, an integrated optimization model of aircraft scheduling and freight routing is constructed. The cargo transportation route is divided into four different situations: direct transportation, primary transportation, multiple transportation and mixed transportation [12, 13]. A heuristic algorithm for improving hybrid transportation is proposed. The effectiveness of the algorithm can be compared by calculation. Good solution to the problem of optimization of freight routing for medium-sized airlines.

Footnotes

Acknowledgments

This work was supported by the Safety Capacity Building Project of China Civil Aviation Administration under Grant No. AS-SA2015/21 and Scientific Research Innovation Plan of Jiangsu Province under Grant No.KYLX15_0312.

References

Cunbing

L.I.

, Xie

and Yang

, Low carbon-oriented route optimization of tobacco logistics via double-layer genetic algorithm, Tobacco Science & Technology 47(2) (2018), 15–18.

Ahmad

, Razali

S.F.M.

, Mohamed

Z.S.

and El-Shafie

, The application of artificial bee colony and gravitational search algorithm in reservoir optimization, Water Resources Management 30(7) (2016), 2497–2516.

Cui

L.X.

, Joint optimization of production planning and supplier selection incorporating customer flexibility: An improved genetic approach, Journal of Intelligent Manufacturing 27(5) (2016), 1017–1035.

Xiao

, He

, Hu

and S.O. Aeronautics, A new method of optimizing bolts layout in connector based on improved genetic algorithm, Journal of Northwestern Polytechnical University 35(3) (2017), 414–421.

Baumgarten

, Malina

and Lange

, The impact of hubbing concentration on flight delays within airline networks: An empirical analysis of the us domestic market, Transportation Research Part E 66(1) (2014), 103–114.

Paul

, Daniel

, Ahmad

and Rho

, Cooperative cognitive intelligence for internet of vehicles, IEEE Systems Journal 11(3) (2017), 1249–1258.

Ramapriya

, An Improved Approach of Text Steganography in Application with Rotational Symmetry, International Journal of Innovative Research in Computer and Communication Engineering (IJIRCCE) 5(7), 2017.

Banomyong

, Thai

V.V.

and Yuen

K.F.

, Assessing the national logistics system of vietnam, Asian Journal of Shipping & Logistics 31(1) (2015), 21–58.

Kanmani

and Jayabalan

, Improved Load Balancing Algorithm for Reducing Idle Iteration in Virtual Machine on Hadoop Distributed File System (HDFS), International Journal of Innovative Research in Computer and Communication Engineering (IJIRCCE) 5(6), 2017.

10.

Cen

and Zhuang

, Two-stage security screening strategies in the face of strategic applicants, congestions and Screening errors, Annals of Operations Research 258(2) (2017), 237–262.

11.

Zhen

, Zhang

, Sun

and Tian

, Fault detection and diagnosis method for batch process based on elm-based fault feature phase identification, Neural Computing & Applications 27(1) (2016), 167–173.

12.

Sirisha

, Arun Jyothi

and Parimala

K.S.

, Twitter, Facebook and Google Data Analysis Using Open Source Hadoop Technologies Hive, International Journal of Innovative Research in Computer and Communication Engineering (IJIRCCE) 5(5), 2017.

13.

Trautrims

, Defee

and Farris

, Preparing business students for workplace reality – using global virtual teams in logistics and scm education, International Journal of Logistics Management 27(3) (2016), 886–907.

14.

Sheng

, Li

Z.C.

, Xiao

Y.B.

and Fu

, Slot auction in an airport network with demand uncertainty, Transportation Research Part E 82(1) (2015), 79–100.

15.

Balasubramaniam

, Paul

, Hong

W.H.

, Seo

H.C.

and Kim

, Comparative Analysis of Intelligent Transportation Systems for Sustainable Environment in Smart Cities, Sustainability 9(7) (2017), 1120.

16.

Alfred Daniel

S.R.