Using Family Competition Genetic Algorithm in Pickup and Delivery Problem with Time Window
Constraints
Wan-rong Jih Cheng-Yen Kao Jane Yung-jen Hsu
Department of Computer Science and Information Engineering National Taiwan University
Taipei 106, Taiwan
jih@agents.csie.ntu.edu.tw cykao@csie.ntu.edu.tw yjhsu@csie.ntu.edu.tw
Abstract— In this paper, we propose a novel research scheme to solve the single vehicle pickup and delivery prob- lem (PDPTW) with time window constraints. The fam- ily competition genetic algorithm (FCGA) is a modern ap- proach that has been successfully applied to solve the travel- ing salesman problem. We illustrate the family competition GA and give the experimental results that show the FCGA is a brilliant algorithm for solving the single vehicle PDPTW.
Genetic algorithms (GA) have been successful applied to solve the combinatorial computation problems. The fam- ily competition will improve the achievements for obtaining optimal solutions and the probability to hit the feasible solu- tions. By comparing FCGA with traditional GA, this excel- lent approach does not need enormous resources. Applying FCGA to single vehicle PDPTW, it succeeded in finding fea- sible solutions for all problems and obtained efficient results in our experimentation.
I. Introduction
The pickup and delivery problem (PDP) is a NP-hard problem[7] and has been extensively studied in computer science, management and operational research. Conse- quently, the domain space of this problem grows exponen- tially. Under such time explosion, exact algorithms can only achieve an optimal solution with in a very small prob- lem size. Therefore, the majority of the related researches apply heuristic algorithms to solve the pickup and delivery problems.
Many practical pickup and delivery problems had been applied to transportation of goods, robotics and the job- shop scheduling[1]. This problem is similar to Traveling Salesman Problems (TSP), Dial-a-Ride Problems (DARP) and the Vehicle Routing Problems (VRP). Except for its augmented constraints. For the detail discussion of pickup and delivery problems, readers might refer to the survey paper by Savelsbergh[10].
Genetic algorithms (GA) have been applied to solve the TSP[4],[8] and VRP[12],[2],[5],[6] successfully. Given this strength in routing related problems, we present a scheme that employs family competition genetic algorithms (FCGA) to solve the single vehicle pickup and delivery problem with time window constraints (PDPTW). The al- gorithms for single vehicle PDPTW are used as subroutines in multiple vehicles cases. The approaches for solving sin- gle vehicle PDPTW would be the principal algorithms on
finding solutions of the multiple vehicles problems.
The organization of this paper is as follows. In the next section, we will introduce the single vehicle PDPTW. The chromosome representation and the fitness function of the FCGA will be given in Section . An outline of FCGA will show in section IV. The related operators of FCGA that will be used to solve the single vehicle PDPTW are described in section IV. Section V and section VI depict the experimental results and conclusions, respectively.
II. The Pickup and Delivery Problem In the single vehicle pickup and delivery problem (PDP), a route must be constructed in order to satisfy transporta- tion requests. Each transportation request specifies the weight of the load to be transported, a location where it is to be picked up and another location where it is to be delivered. Suppose that the beginning of a route is a depot where the vehicle departs from it. By starting from the depot, the vehicle travels through all the locations where the transportation requests specified. After the vehicle has visited all the transportation requests, the vehicle will park at one of the delivery location.
Pickup and delivery problems with time constraints (PDPTW) are strictly harder than the fundamental prob- lems. In addition to the intrinsic precedence constraints and the vehicle capacity constraints, time constraints com- plicate the problem significantly. A time window is associ- ated with every pickup location and the delivery location, which indicates the time interval that the corresponding location is available for service. The vehicle should visit every location and satisfy the given service time interval of the locations. If the vehicle arrives early, the vehicle has to wait at the arrived location until it is open.
A. Single vehicle PDPTW
Suppose that N = {1, . . . , n} is a set of transportation requests, where n is the number of transportation requests.
To accomplish each transportation request i ∈ N , the ve- hicle should service the pickup location i+before the de- livery location i−. Accordingly, V+ = {i+ | i ∈ N } and V− = {i− | i ∈ N } are the set of pickup locations and delivery locations respectively. Let G = (V, A) be a di- Proceedings of the 2002 IEEE
International Symposium on Intelligent Control Vancouver, Canada • October 27-30, 2002
rected graph where V = {0} ∪ V+∪ V− is a set of lo- cations, which includes an initial depot 0 and the spec- ified location sets of the n transportation requests. Set A = {(r, s) | r 6= s, r, s ∈ V } is a set of arcs. To measure the arrival time of each step, a given non-negative travel time drs is assigned to each arc (r, s) ∈ A. Each trans- portation request i is associated with a positive number qi; it indicates the load of the transportation request. A ve- hicle should not be allowed to exceed the vehicle capacity Q.
In addition to the precedence constraints and the capac- ity constraints, PDPTW must also satisfy the time win- dow constraints. For every transportation request i, are required to pick up the goods within the time interval [ai+, bi+], and transport the goods to its delivery location i−on the time interval [ai−, bi−].
The time windows [ai+, bi+] and [ai−, bi−] are the avail- able time interval of the pickup location i+ and delivery location i−, respectively. If the vehicle arrives in location r ∈ V+∪ V−, and its arrival time tr is earlier than the lower bound of the allowed arrival time ar, it has to wait.
That is, the departure time of a vehicle at location r will be equal to max{ar, tr}.
Every transportation request i ∈ N is subject to ai+ ≤ bi+, ai− ≤ bi− and bi−− ai+ ≥ di+i−. The first two con- straints declare the upper bound and lower bound of the time windows; the value of upper bound should higher than its lower bound. Constraint bi−− ai+≥ di+i−corresponds to formula ai++ di+i−≤ bi−. It means the time windows for every transportation request i ∈ N should be admis- sible. Given a transportation request i ∈ N , if a vehicle departs from a pickup location i+at time ai+, and its im- mediate destination is location i−, which is the delivery location of the transportation request i. The vehicle will arrive location i− at time ai++ di+i−, where di+i− is the traveling cost between location i+ and location i−. Ac- cordingly, value of ai++ di+i− should less than the upper bound of time window bi−.
B. Goal
Initially, we assume that the vehicle is parked on an ini- tial depot 0. When the transportation request i ∈ N is assigned, the weight qi of this transportation request i is known a priori. The value of travel time drsis pre-defined, where r, s ∈ V . Each transportation request i is associ- ated with a pickup location i+ and a delivery location i−, besides, each location is associated with a time window.
Our goal is to find a vehicle route starting from an initial depot, finishing all transportation requests, and end at one of the delivery locations. According to the path, both the total traveling time and the total waiting time of the vehicle are minimized. Certainly, the path must be a feasible route that satisfies the capacity and the time window constraints of every transportation requests. Moreover, the feasible routes should satisfy the precedence constraints, that is, for every transportation request i ∈ N , the vehicle should visit its pickup location i+ prior to the delivery location i−.
III. Construction of Genetic Algorithm Genetic algorithms are search algorithms based on the mechanics of natural selection and natural genetics. The
“fittest” string structures are recombined with a structured yet randomized information exchange to form such a search algorithm. Genetic algorithm is a superior approach to solve the NP-hard problems[7].
To solve a problem by the genetic algorithm, we should define the representation of each chromosome first. Ac- cording to the chromosome representation, an evaluation function will be defined. Conventionally, in genetic algo- rithm, the evaluation function is also named fitness func- tion. The chromosome representation and fitness function play important roles in the design of genetic algorithm.
Most parents selection methods choose individuals accord- ing to the their fitness value The recombination operators should be decided while the chromosome representation has been given.
A. Chromosome Representation
Typically, the representation for a route is a sequence of the location labels, to list the successional labels that a vehicle is visited. Suppose that the transportation re- quests N = {1, 2, 3}, a route 0 → 3+→ 1+→ 1−→ 2+→ 2− → 3− the corresponding chromosome representation will be (0 3+1+1−2+2−3−). This representation is in- tuitively by following the traveling sequence in order, and it is easy to understand. The permutation representation is simple and easy to manipulate. If the transportation requests N = {1, . . . , n}, by simply counting the pickup and delivery locations of every transportation request and a initial depot, the length of every chromosome will be 2n + 1. Consequently, for solving the order-based prob- lems, the permutation representation is the most common and popular representation.
B. Fitness function
Solutions to the single vehicle pickup and delivery prob- lems with time window constraints are subject to the fol- lowing constraints[3].
(1) Initially, a single vehicle parks on the initial depot 0, and it will visit all the specified locations exactly once.
(2) Every location i+∈ V+should be visited before its delivery location i− ∈ V−. Such consideration is called precedence constraints.
(3) The capacity constraints represents the total load of a vehicle cannot exceed its capacity Q.
(4) Suppose that a vehicle arrives in the location i ∈ V+∪ V− at time ti, the criterion ti ≤ bi must be satisfied. In general, we named this constraint the time window constraint.
(5) The vehicle routes will be the open paths, ending at any one of the delivery locations.
Given a chromosome, which represents a route S, the corresponding fitness function is defined in equation 1.
Φ(S) = ftravelcost(S) + fpenalty(S) (1) The first part of function Φ(S) represents the total travel time for a vehicle to complete the route S. The value of ftravelcost(S) will includes the waiting time while the vehi- cle arrives a location early.
The second part of Equation (1) is the penalty function of route S. A route might violate the constraints during the exploration of genetic algorithm. Therefore, a chromosome might represent an infeasible route. Function fpenalty(S) defines the punishments for violating the constraints. If the vehicle arrives a location and it is overloading, penalization for violating the vehicle capacity will be given. Besides, penalties for delay of arriving a location will be adopted.
Suppose that a vehicle arrives in location r at time tr. If the arrival time tr< br, there is a penalty for the vehicle arrives late. Symbol br stands for the upper bound of the time window [ar, br], which is associated with location r. The intrinsic precedence constraints should be strictly observed.
Given a chromosome, if the corresponding route S is a feasible route, the value of function fpenalty(S) will equal to zero. Otherwise, for an infeasible route, there are pe- nalizations, the value of penalty function certainly larger than zero. A feasible route denotes a route that might have waiting time but satisfy the precedence constraint, capacity constraint and time window constraint. If a route violates any constraints of the PDPTW, it is an invalid route, or namely the infeasible route.
Suppose that route Si is an infeasible route and route Sf is feasible. According to the definition of Φ(S), the value of Φ(Si) will much larger than that of the Φ(Sf).
Consequently, we will try to find a feasible route S and the value of Φ(S) is the minimum. The goal of the single vehicle PDPTW is to minimize Φ(S) apparently.
C. Crossover
We will consider four crossover operators. The order crossover (OX) is an order-based crossover[8] and uni- form order-based crossover (UOX) is the position-based crossover[11], both operators had been adapted to TSP during the past years. Merge cross #1 (MX1) and merge cross #2 (MX2) invented by Blanton[2] for solving VRP with time window constraints. The first two operators are traditional crossover operators and the last two oper- ators use a global precedence vector to be the guidance of crossover.
Many studies had given exhaustive research on the first two traditional crossover operators. Therefore, in this pa- per, we will not describe the two traditional crossover op- erators. In regard to the MX1 and MX2, a primitive spec- ification will be given on the following paragraph.
The key concept of merge crossover operators is based on a global precedence, an independent antecedence among the genes of the populations, that is, each gene in the chromosome has a precedence relation to every other gene.
From the characteristics of constraints of the single vehi- cle PDPTW, a global precedence relation probably exists among genes. The global precedence vector is formed by such relationship and it could be the offspring-generating guidance.
In single-vehicle PDPTW, except the initial depot 0, ev- ery gene is either a delivery point or a pickup point and has an associated time window. We might utilize the time windows in order of precedence.
The operations of MX1 and MX2 are based on the global precedence vector. Operator MX1 produces a child that is close to the order of the global precedence. After perform operator MX2, will produce an offspring that the genes with lower priority will be moving to the end of the chro- mosome. For detail information related to merge crossover, readers might refer to the publications[2] of Blanton and Wainwright.
D. Mutation
In general, the mutation is worked with a single chro- mosome. A chromosome will be created by applying the mutation operator, and it will substitute the new chromo- some for its original one. We will consider two mutation operators that can be applied to the single vehicle PDPTW problem.
The first mutation will select two genes randomly, and their positions are interchanged. This operator will create a new route, which has four different edges from its original route. The second mutation operator choose two cut sites randomly, and the order of the sub-route specified by the genes is inverted. Such mutation operator is the 2-opt move in TSP; the difference between the new and the original route is two edges.
IV. Family Competition GA
The family competition genetic algorithm (FCGA) is based on genetic algorithm (GA) with adding the concept of families. For every population, each individual owns its family. To maintain the constant size of a population, only the champion at a family survived.
The main concept of the family competition genetic al- gorithms (FCGA)[13] is similar to the rule of the natural evolution; the best solution is the only survivor from its family. Given a population t, a family contains u individu- als and these individuals are the recombination of a family father Fit, where i = 1, . . . , m and m is the population size.
Every individual Iit in population t has its turn to be the family father Fit.
In any generation t, each individual Iit in the current population Pt owns its family Cit. The individual Fit is the family father of family Cit and the size of the family is u. Accordingly, a generation t will produce u × m new individuals, where m is the population size.
Algorithm 1 elaborates the procedure of family competi- tion genetic algorithm. The selection process for crossover operators is to select two parents; one is the family father Fit and the other is the alternative parent Atj. Alternative
Algorithm 1 The procedure of FCGA 1: t = 0;
2: Initialize population: Pt← {I1t, I2t, . . . , Imt};
3: Evaluation: Pt← {Φ(I1t), Φ(I2t), . . . , Φ(Imt)};
4: repeat 5: T ← ∅;
6: for i = 1 to | Pt| do 7: Family father: Fit← Iit; 8: Family: Cit← ∅;
9: for j = 1 to u do
10: Selection: alternative parent Atj∈ Pt; 11: Reproduction: cj← Om(Oc(Fit, Atj));
12: Evaluation: Φ(cj);
13: Cit← Cit∪ {cj};
14: end for
15: T ← T ∪ best(Cit);
16: end for
17: New population Pt+1: Pt+1← Pt∪ T ; 18: t = t + 1;
19: until the termination condition is reached 20: Output the solutions;
parent Atjis randomly selected from the population Ptand is distinct from the family father Fit.
The procedure of reproduction will perform the recombi- nation operations. The reproduction process might execute the crossover Ocand then perform the mutation Omor sim- ply execute the crossover. The family father Fitgenerates a new offspring cj by using the recombination operators with a specific probability. The crossover rate is a fixed real number between 0 and 1, that is, analogous to the traditional genetic algorithm, using a fixed rate to perform crossover. In our FCGA, the occasion to execute the mu- tation depends on the similarity between the parents and its offspring. If the offspring is similar to one of its par- ents, perform the mutation. Accordingly, our FCGA will perform mutation while the offspring and one of its parents represents the same route. Such adaptive strategy of exe- cuting mutation supports the route diversity and prevents the search space be bound in local optimal solutions. It also precludes the algorithm from premature convergence.
In Algorithm 1, the evaluation function shown in the line 3 and 12 has been depicted in the previous section.
The definition of function Φ(S) is consistent in this paper.
The fitness function of the family competition genetic algo- rithms is function Φ(cj), where cjis a chromosome, which represents a route for the single vehicle pickup and delivery problems.
After the end of the inner for loop, the family Citcontains u individuals. Generally, most genetic algorithms support the fixed population size in the same run. to preserve the legitimately constant population, the individual who owns the best fitness value will be chosen from the family Cit. Given a family C, the function best(C) will return an in- dividual cj, which has the minimum fitness value Φ(cj).
Accordingly, the new population size of T will be identical to the size of population Pt. Population Ptholds m indi-
viduals and the population T will hold the same individuals while the program executes the command after the end of the outer for loop. The individuals of the new population Pt+1 are selected from the population Pt and population T .
Algorithm 1 will repeat on executing the new popula- tion generation procedure until the termination condition is reached. Finally, output result will equal to the best solution in the current population.
V. Experimental design and results An algorithm for generating test sets had been devel- oped, for obtaining the authentic datasets. The exper- imental results are use the datasets which generated by Algorithm 2.
A. Experimental design
Up to present, there is no suitable test sets for PDPTW.
The datasets for PDPTW should concern with the prece- dence constraint, capacity constraint and time window con- straint. The test data for TSP definitely not suit to the PDPTW because the datasets are not associated with any of the above constraints. As for the VRPTW instances, there are no precedence constraints. Therefore, we invent an algorithm to create PDPTW datasets for the experi- ments, shows in Algorithm 2.
Algorithm 2 Create test data for single vehicle PDPTW Require: Give the number of task n, and width is the
width of time window 1: N = {1, · · · , n}
2: for i ∈ N do
3: Generate locations i+and i−randomly;
4: Generate the demand qirandomly;
5: end for
6: Evaluate the traveling time drs, r, s ∈ V ;
7: Randomly generate a route which satisifies the prece- dence contraint;
8: Let tibe the arrival time of location i ∈ V ;
9: Suppose that the average traveling time between the locations is AverageT ime;
10: for i ∈ V+∪ V−do
11: ai = ti− (random(width) × AverageT ime;
12: bi = ti+ (random(width) × AverageT ime;
13: end for
14: [ai, bi] is the time window for location i ∈ V ;
B. Experimental results
A dynamic programming approach proposed that can achieves the optimal route and this dynamic programming had been elaborated on the publications of Psaraftis[9].
Use the dynamic programming to solve the single vehicle PDPTW, the execution time will explode, as the prob- lem size increase. Furthermore, the dynamic programming method needs gigantic computer resources to execute and store the intermediate results. Therefore, most researcher
could not attain to optimal solutions while the task size is large, neither we are.
In our experiments, every test case had been performed crossover with 3 different crossover rate, said 45%, 60%
and 75%. Moreover, given a specified crossover operator and the crossover rate, the experimentation would associate with a mutation operator. We apply the 2-point mutation and the 2-opt mutation operators to our experimentations.
After determined the associated parameters of genetic al- gorithm, each case will be executed 30 runs.
Table I demonstrates the percentage above the optimal value by using the order-based crossover (OX) and merge crossover (MX1). Columns 2 to 4 represents the relative error of the best solutions to the optimum; if an item with value 0 means it is equal to the optimal solution. Symbol
’x’ represents that no feasible solution had been found. The last column given the optimal solutions that obtained by using the Psaraftis[9] approach. Optimal values in paren- theses stand for the best solution in our experimental re- sults.
TABLE I
The best solution results (percentage of relative error) – OX & MX1
relative error(%)
task OX MX1 optimal
size GA FCGA GA FCGA value
10 0.344 0.000 0.000 0.000 872
20 x 7.685 0.000 0.000 2030
30 x x 0.000 0.189 3713
40 x x 0.068 0.068 4386
50 x x 0.017 0.017 (5752)
60 x x 0.000 0.000 (5658)
70 x x 0.429 0.249 (7221)
80 x x 0.688 0.764 (7849)
90 x x 0.046 0.313 (8618)
100 x x 0.151 0.000 (10600)
Obviously, the order-based crossover (OX) is not suit for single vehicle PDPTW. Even the small problems OX still cannot get the feasible solutions. The merge crossover #1 (MX1) could not always obtain the optimal solution.
Though FCGA improve the quality of the most solutions, for problem size 30 and 80, the fitness value is increased.
Figure 1 shown the best and average fitness value while the problem size is 80. The solid line represents the aver- age fitness values that achieved by GA; it has the highest solution cost in every generation. Symbol FCGA(avg) in- dicates the average fitness values of FCGA. Figure 1 shows the average fitness values of FCGA are lower than that of GA, and both the two algorithms obtain the similar results of the best fitness values.
Table II shows the results of the UOX and MX2. Table I and Table II are refer to the same optimal values. Values in parentheses are the number of best value found under the trials. By applying the position-based crossover (UOX), the solution qualities of FCGA have significant improved.
The FCGA might not promote the solution qualities while the transportation requests are less than 50. For larger
0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000
0 50 100 150 200 250
Fitness value
Generation
The average fitness value of 80 tasks (MX1, 60% crossover rate, 2-point mutation, 30 runs) GA(avg) FCGA(avg) GA(best) FCGA(best)
Fig. 1. Fitness values of MX1, 80 transportation requests
problem, FCGA achieve better results than GA.
TABLE II
The best solution results (percentage of relative error and hit ratio) – MX1 & MX2
relative error%(optimum/trials)
task UOX MX2
size GA FCGA GA FCGA
10 0.000(18/30) 0.000(27/30) 0.000(30/30) 0.000(30/30) 20 0.000(22/30) 0.000(30/30) 0.000(21/30) 0.000(11/30) 30 0.000(5/30) 0.000(20/30) 0.000(24/30) 0.000(7/30) 40 0.000(3/30) 0.000(23/30) 0.000(28/30) 0.000(12/30) 50 0.000(1/30) 0.000(17/30) 0.000(3/30) 0.017(24/30) 60 0.000(3/30) 0.000(5/30) 0.000(29/30) 0.000(30/30) 70 0.000(2/30) 0.000(19/30) 0.000(2/30) 0.000(2/30) 80 0.000(1/30) 0.000(4/30) 0.013(1/30) 0.000(1/30) 90 x 0.000(5/30) 0.000(1/30) 0.000(1/30) 100 0.000(1/30) 0.000(5/30) 0.000(10/30) 0.000(26/30)
Using UOX by the FCGA, the numbers of finding the best solution are increased substantially. For size is smaller than 50 request pairs, MX2 could not improve the best so- lution hit ratios. Since the best fitness value of size 50 could not improved by MX2, Figure 2 will show the best and average fitness values of MX2. Solid line is the aver- age fitness values of GA, which converge to near the best solutions. However, the average fitness values of FCGA went prematurely. MX2 moves the least preferred gene to near the end of the route, this might cause the search ex- ploration converged early while the problem size has not adequate spaces for the algorithms to search through.
Figure 3 depicts the average CPU time of FCGA and GA. The unit of the CPU time is second. Though the family competition will take more steps during the repro- duction procedure of genetic algorithm, the execution time does not increase substantially. Study both the Table II and Figure 3, UOX obtained the best solutions by FCGA, whereas it required much time to achieve the solutions. The average execution time of MX1 is less than that of MX2, however, the solution qualities of MX2 are better than that of MX1.
0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07 3e+07
0 50 100 150 200 250 300 350
Fitness value
Generation
The average fitness value of 50 tasks (MX2, 60% crossover rate, 2-opt mutation, 30 runs) GA(avg) FCGA(avg) GA(best) FCGA(best)
Fig. 2. Fitness value of MX2, 50 transportation requests
0 500 1000 1500 2000
0 20 40 60 80 100
CPU time(sec)
Task size The average CPU execution time, 180 runs UOX(FCGA)
UOX(GA) MX2(FCGA) MX2(GA) MX1(FCGA) MX1(GA)
Fig. 3. A comparison of CPU time (UOX, MX1 and MX2)
In real-world applications, users should not be waiting too long for service. A feasible and near-optimal solution will be allowed to serve customers, whereas efficient and reliable response should be satisfied. Table III describes the percentage of the feasible solution hit ratio. FCGA improved the probability to obtain feasible solutions in all the three crossover operators. Under execution time con- sideration, MX1 and MX2 might suit for the real-time ap- plications, by referring to Figure 3.
VI. Conclusion
The FCGA is a modern approach for solving the single vehicle pickup and delivery problems with time windows constraints. Every recombination operator has its partic- ular characters but not all the reombination operators are suitable for solving the single vehicle PDPTW. We should exploit the experiment results to expand expertise. Un- derstanding which operators are admissible operators for solving our problems will improve the probability of ob- taining optimal solution.
The majority of genetic algorithms can obtain the opti- mal or near-optimal solutions. By comparing the solution cost of FCGA with that of traditional GA, in most cases, FCGA always improve the solution qualities of GA.
TABLE III
Percentage of feasible solution found, 180 runs
feasible/trials(%)
task UOX MX1 MX2
size GA FCGA GA FCGA GA FCGA
10 100.00 100.00 100.00 100.00 100.00 100.00 20 98.89 100.00 100.00 100.00 100.00 100.00 30 70.00 93.33 100.00 97.78 98.89 98.33 40 45.00 90.00 100.00 100.00 99.44 96.11 50 8.33 46.67 100.00 100.00 99.44 84.44 60 5.00 15.56 98.33 100.00 96.67 87.22 70 7.22 47.22 100.00 100.00 97.22 95.00 80 11.67 11.67 21.67 55.00 61.11 80.56
90 x 8.89 61.11 96.11 71.67 87.22
100 1.11 13.33 7.78 26.67 25.00 63.89
We will continue to design the affiliated programs and recombination operators, so that we can accomplish the scheme and improve it. Furthermore, we hope that appli- cations of this novel technique to the other practical science will achieve the similar outstanding results.
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