Proceedings of the 36th Chinese Control Conference July 26-28, 2017, Dalian, China
Optimal Schedule for Agricultural Machinery Using an Improved Immune-Tabu Search Algorithm Xinmeng Zhu1, Yongsheng Ding1*, Xin Cai1, Haishan Wang2, and Xiangfei Zhang3 1. Engineering Research Center of Digitized Textile & Apparel Technology, Ministry of Education, College of Information Sciences and Technology, Donghua University, Shanghai 201620, China 2. Shanghai Agricultural IOT Engineering Technology Research Center, Shanghai, China 3. Information Center, Shanghai Agricultural Committee. Shanghai 200335, China * E-mail:
[email protected]
Abstract: Considering the low efficiency and lack of intelligent dispatching decision of the agricultural machinery scheduling problem, an improved Immune-Tabu Search Algorithm (ITSA) based on the immune optimization algorithm is proposed. A new operator, named TSA, is designed through improvement on the generation of neighborhood solution based on the tabu search algorithm. At the beginning of the iterations, the algorithm makes use of the search results of TSA as the mutated antibodies, so as to improve the climbing performance of the algorithm and accelerate the convergence speed in the meantime. Then, random mutation and tabu search for parallel strategies are implemented to ensure the optimization of the result as well as shorten the optimization time. Compared with the immune optimization algorithm and tabu search algorithm, the simulation results show that ITSA can not only obtain a better search success rate, but also stabilize the results of the algorithm. Key Words: agricultural machinery, scheduling, immune optimization algorithm, tabu search, immune-tabu search algorithm
1
Introduction
Agricultural machinery scheduling is mainly based on evaluation of expert system and manual decision-making, but few automatic schedule system or tool. Thus there is still enormous potential upside for efficiency and accuracy of intelligent schedule to promote. Furthermore, in the process of promoting the precision of agricultural mechanization, agricultural machinery often needs to execute operations across regions due to the lack of concentrate distribution of fields, which also increases the difficulty of information management of agricultural machinery. Last but not least, existing agricultural monitoring and controlling systems are not fully adequate, which is defective in scheduling of machinery. It is necessary to generate a formulation approach of multiple machines among multiple fields by optimization algorithms taking the place of manual decision. There are few researches on agricultural machinery scheduling so far, mainly focusing on the genetic algorithm (GA) [1], the simulated annealing algorithm (SAA) [2] and the tabu search algorithm (TSA) [3] or the comprehensive application of various heuristic operators [4], which basically solve the agricultural machinery path planning and allocation issues. Bochtis [5] summarized the progress of agricultural management, not only did they do well preparation for agricultural intelligence automation for future stability operation, but also they provided a relative reference for the generalization and definition of the agricultural machinery scheduling problem.
*This work is supported in part by the National Natural Science Foundation of China (nos. 61473078, 61603090), Program for Changjiang Scholars from the Ministry of Education (2015-2019), Agricultural Project of the Shanghai Committee of Science and Technology (no. 16391902800), and Shanghai Science and Technology Promotion Project form Shanghai Municipal Agriculture Commission (no. 2016-1-5-12).
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In this paper, we optimize the dispatching decision of agricultural machinery, with consideration of the decreasing of costs and increasing of efficiency. Based on the analysis of the advantages and disadvantages of the immune optimization algorithm (IOA) and the TSA, an improved immune-tabu search algorithm (ITSA) is proposed, which improves the performance of the climbing performance of the immune optimization algorithm by using the tabu search algorithm. The IOA performs well in solving agricultural scheduling problem. However, at each iteration, there is no complete search in the neighborhood of antibodies due to the randomness of mutation and crossover. Thus, the proposed ITSA takes the result of random mutation by using the TSA as the mutated antibodies in the mutation stage, which guarantees the diversity of the population and accelerates the optimization process, so as to ensure the successful search of the optimal solution. According to simulation results, the feasibility and effectiveness of ITSA is verified through two testing cases. The remaining content of the paper is organized as follows. In Section II, the scheduling problem is presented. In Section III and Section IV, the proposed ITSA is developed based on IOA and TSA operators, respectively. Finally, Section V analyses and proves the performances of ITSA. A general conclusion of the work and the perspectives considered are given in the last section.
2
Problem Description and Formulation
2.1 State of the problem The problem of agricultural machinery schedule involves the allocation of heterogeneous agricultural machinery and dispersed fields. Heterogeneous means that the service each machine can provide and each field needs are both different. The objective function of this problem is to minimize the total distance traveled of all agricultural machineries, and
keep the variance of distance traveled by each machine as small as possible at the same time. Consider an undirected graph G (S , E) where S ={0,1,2, ,H,H+1} ,H,H+1 is the set of nodes (0: node of depot, 1~H: nodes of H fields waiting for service, H+1: node of depot) and E {(i, j ) | i, j Sфi z j} is a set of arcs. An arc
The sequence of fields always begins and ends at the depot, namely field 0 and H+1. Sh z St , h z t , h, t {1, 2, , H } (6) Each field is visited only once. The model described in Eqs. (1-6) is based on the model described in [3]. Table 1: Schedule Parameters and Input Parameters
(i, j ) E is said to be the edge joining any two nodes. For each arc (i, j E ) , there is a corresponding distance d ij , so a symmetric distance matrix D
Variable
(dij ) can be defined which
satisfies the triangle inequality theorem, where d ij is the
1) Each field is ought to be served until the end and each machine should be utilized. 2) The service capability that each machine provides and service time of each field are proportional. 3) Taking it into account that inter-field travelling speed of each machine is the same, the travel distance is proportional to the corresponding time of each machine. 4) Considering the objective function takes time as the dimension. 5) The serving sequence of each machine always begins at the agricultural machine depot, by way of several fields, and finally comes back to the depot.
M
H
H
¦
H
M
m 1
H
¦
i 0 j zi, j 0
H
(1)
h
The objective function is to minimize the weighted sum of total distance traveled of all agricultural machineries and the variance of distance traveled by each machine. M
N
N
¦¦ ¦
M
xijm
m 1 i 0 j zi , j 0
N
¦¦ x
jm
M
(2)
m 1 j 1
For i 0 ˈthe number of machines providing the depot with service is the total number of machines. tij dij / v, i, j {0,1, , H }, },ii z j (3) The travel time between the two fields i and j equals to the distance d ij divided by the constant speed v . N
Dm
N
¦ ¦
xijm dij , m {1, 2,
, M}
H)
Each field is labelled 1 to H, with the depot labelled as 0 and H+1
0
Agricultural machine depot
d ij
Distance between the two fields i and j
v
Constant speed of machines
Mpm
Travel time of the machine between the two fields i and j Service which field i need, the corresponding value of the depot is 0 Service efficiency of machine m
Dm
Total distances which machine m travels
dij / v )
w
r
xijm
Weight of the total travel distances of all machines Weight of the variance of the travel distances of each machine 1 if machine m travels from field i to field j , otherwise 0 1 if machine m serves field i , otherwise 0
An Improved Immune-Tabu Search Algorithm
3.1 Immune optimization algorithm
d ij xijm / v
d ij xijm / v ¦ Svi ymi / Mpm ))
Total number of fields to be served
Svi
3
m 1 i 0 j zi, j 0
r * V (¦
H
ymi
The scheduling of agricultural machinery creates a plan for each individual machine executing the same service at dispersed fields. The plan decides when and where the machines have to execute the service. A number of schedule parameters and input parameters are required for the procedures involving the number of available machineries, location information of fields, and so on (see Table 1). The scheduling problem is formulated as follows: min( w * ¦¦
Total number of available machines
tij ( tij
2.2 Scheduling
Objectivefunction : f
M
Si , (i 1,
distance between the two fields i and j [6, 7]. An agricultural machinery scheduling model requires the following assumptions;
Description
(4)
i 0 j 0, j z i
The total travel distance of machine m equals to the sum of distances between any two fields served by machine m . S0 SH 1 (5)
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In the immune system, lymphocytes are responsible for recognizing invading antigens and producing antibodies to eliminate antigens. Moreover, the immune system is immune to itself, which can inhibit the production of excessive antibodies, and then return to normal state. Through this mechanism, the immune system can deal with almost infinite number of antigens [8]. The IOA evolves based on the mechanism of the immune system, which eliminates the antibodies differing greatly from the optimal solution by selection, crossover and mutation. This algorithm has strong search ability to find the global optimal solution with great probabilities. Many traditional optimization methods start from the single point of the search space and then determine the next point by some transformation rules. Those point-to-point search methods would be caught in the local optimal solution easily. What’s more, the IOA introduces the self-regulation, the characteristics of memory as well as the diversity of the immune system. Especially its evolutionary dealing with the antibodies, which can prevent the optimization process from the prone to "premature", showing better global search ability [9]. On the other hand, the local climbing performance of the IOA is a little weak, the global searching speed of which is proved to be slower due to random mutation.
The immune system produces different antibodies by recognizing the genotype. The degree of match between the antigen and the antibody can be described by fitness. Thus, if the genotype of the antibody produced by any cell line is the same, the fitness value indicates that the antibodies are identical. When it comes to the problem of agricultural machinery schedule, the fitness between the antigen, antibody, antigen and antibody corresponds to the objective function, the optimal solution and the matching degree of the solution and objective function respectively.
mH M ; [m1 , m2 , mH ] , where m1 m2 The lower bound constraint vector of scheduling vector is as follows: mH 1 . [m1 , m2 , mH ] ,where m1 m2 Initialize all the unites Initialize the antibodies Antibody diversity evaluation Record the current optimal fitness value, calculate and record the average fitness value
3.2 Tabu search algorithm The main idea of TSA is as follows: starting from an initial solution, choosing a series of specific search direction (or called movement) as a temptation, TSA chooses to implement the movement which can reduce the objective function value the most. To avoid to fall into the best local solution, TSA adopts a flexible "memory" technology, which records and selects the optimized process, so as to guide the direction of the next search, which is the establishment of the tabu list. The tabu list limits a movement’s availability to be utilized for a period of time. A movement, so long as it is in the tabu list, cannot be implemented in the current iteration, so that the revisit of algorithm can be avoided since it has visited the solution during several recent iterations, which helps algorithm to be prevented from converging to local optimal solution[10]. However, TSA has stronger dependence on the initial solution. A good initial solution can help TSA to find a good solution in the solution space, while poor initial solution can reduce the convergence rate of TSA [11]. In general, while solving a particular problem, it is better to use other optimization algorithms to generate an initial solution of high quality. Beginning with that solution, TSA can improve the quality and efficiency of searching [12]. There is no doubt that the greed of TSA will rob the algorithm of global optimization capability.
Form the parent group, and then update the memory unit Selection Crossover Mutation(TSA) No Termination?? Yes Output put
Fig. 1: The overall procedures of the ITSA
Fig. 2: The working style of the agricultural machines
3.3 The immune-tabu search algorithm (ii) Initialize the antibodies: A ( N n) u H dimensional matrix is generated by random value according to the upper and lower bound constraint of the scheduling vector, namely the antibody population. Each vector X k ,G ( x1k ,G , xk2,G ,..., xkH,G ) in the antibody population
We use the TSA operator as a mutation operator in the mutation stage of the IOA, and propose an improved Immune-Tabu Search Algorithm (ITSA). The overall procedures of ITSA are illustrated in Fig. 1. (i) Initialize all the units: Firstly, each arrangement of individual schedule is denoted as the scheduling vector X ( x1 , x 2 ,..., x H ) , where H is the total number of fields to be served. The value of each bit of the vector x i is the number of service machine at the field i (to illustrate how the agricultural machine works, see Fig. 2, where M1~M4 represents four machines, respectively).Then, load the scheduling and input parameters, as well as the necessary parameters of ITSA, such as the antibody population size N, memory capacity n, the current iteration number G, the maximum number of iterations MAXGEN, the threshold of the stepwise iteration IG, the diversity evaluation parameter ps, the crossover probability pc and the mutation probability pm; The upper bound constraint vector of scheduling vector is as follows:
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represents an antibody, where k is the number of the antibody in the antibody population and G is the number of the current iteration. (iii) Antibody diversity evaluation: In each iteration, each antibody would be calculated for diversity evaluation. The diversity evaluation is composed of the two indicators, as follows: The fitness of antibodies is: M
f
H
min( w * ¦¦
H
¦
d ij xijm / v
m 1 i 0 j zi , j 0
M
H
r * V (¦ m 1
H
¦
i 0 j zi, j 0
H
d ij xijm / v ¦ Svi ymi / Mpm ))
(7)
h
which is also the objective function of the problem. The expected reproductive probability is calculated based on the parameter ps:
exc( X k ,G )
f ( X k ,G )
¦f
* ps
con( X k ,G )
¦ con
*(1 ps)
(8)
By the diversity evaluation above, antibodies with high concentration and low fitness would be inhibited. In addition, con( X k ,G ) is said to be the concentration of each antibody in the antibody population as well as ps to be the diversity evaluation parameter. (iv) Record the current optimal fitness value, calculate and record the average fitness value. (v) Form the parent group, and then update the memory unit: Select N antibodies from the N + n antibodies after step (iii), which are ranked in descending order according to the expected reproductive probability. The first n antibodies are extracted to constitute an antibody memory library. Those N antibodies selected are formed to be a parent antibody group, and the last n antibodies are eliminated. (vi) Immunization operations: Special immunization operations, including selection, crossover, mutation operations, are used to generate the subsequent population: Selection: According to the expected reproductive probability of antibodies, roulette method is used to select antibodies from the parent group. In this operation, N rounds of roulette are operated to obtain N antibodies which are naturally selected, that is the population with high reproductive probability. Crossover: Two antibodies are randomly selected from the antibodies selected by the selection procedure described above. Then selecting two gene bits within the range of antibody length H, crossing all gene bits in the two position ranges. After N cycles, the crossed population is obtained. Mutation (TSA): Firstly, the TSA is utilized to be a mutation operator on improving the climbing performance of the IOA until the current iteration number G reaches the threshold of the stepwise iteration IG, by taking the best local solution of the TSA as the mutated antibody. The antibody to be mutated is accelerated to find the optimal solution in the neighborhood by TSA instead of randomly mutated. After those iterations, in order to ensure the algorithm to find the optimal solution, TSA mutation is utilized on those antibodies whose fitness ranking last forty percent to find neighborhood optimization. Only when the solutions found by the TSA are superior to the current antibodies, they would be taken as the antibodies mutated. (vii) Termination: After getting optimal solutions by immunization operations of step (vi), the algorithm stops further iteration when it reaches the stopping criteria. Otherwise, the n antibodies stored in the memory library would be added into the antibody population consisting of the N antibodies produced by the immunization operations to generate the parent population of the next iteration, then the process proceeds to step (iii) to perform diversity evaluation. The key parameters and operators of ITSA are as follows. The coding and decoding methods: Taking the number of agricultural machineries m as the hex value to encode the antibody and constructing the antibody structure with the length of H. Each antibody represents a feasible scheduling result, where the i-th gene bit of the antibody represents the machine number serving the field i.
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The TSA mutation operator˖The solution’s neighboring solution is generated using three improved movements [3] .The first movement switches the order of two fields, to create a new sequence for the fields to be served, Eq. (9). Movement1: S0 , Sk , Sk 1 , SH 1 o S0 , Sk 1 , Sk , SH 1 (9) k 1, , H 1 The second movement increases the number of machines available at a field by one, unless in doing so would exceed the maximum machines available, Eq.(10). Movement2: Ui o Ui 1 i 1, H ; Ui 1 d M (10) The third movement decreases the number of machines available at a field by one unless in doing so would result in zero, Eq. (11). Movement3: Ui o Ui 1 i 1, H ; Ui 1 t 1 (11) In the formulations above, H is the total number of fields, S0 and SH 1 are the start point and end point of the path of each agricultural machinery respectively, namely the depot of machinery. The machines serving the field i is U i , and the total number of available machinery is M. Movement1, Eq. (9), generates H*(H-1)/2 neighboring solutions, while Movements 2 and 3, Eqs. (10) and (11) resp., each generate a maximum of H solutions. This results in a maximum of H*(H-1)/2+2H solutions in a local neighborhood of solutions. All of the local solutions would be calculated. The best local solution, of those which are not tabu, is then chosen as the movement to be made. The best global solution produced after all the iterations of TSA would be chosen as the mutated antibody. Route refine of machines: When it comes to the calculation of the fitness function, the actual traveling path between any two fields is taken into account, by which, the distance between any two fields is optimized to be the sum of distance of the two right-angled sides of the right-angled triangle corresponding to the line connecting the fields centers[13]. Furthermore, the de-interleaving process of the traveling path is also considered [14].
4
Simulation Results
In this paper, two examples are optimized by the IOA and the improved ITSA, respectively. Some stochastic components of the location of fields are added referring to [3]. 4.1 Set of parameters The location and service required of each field as well as service that each machine can provide per hour are summarized in Tables 2, 3, 4, 5. This problem was simulated on MATLAB2016a platform with the parameters of IOA and ITSA as follows: The size of antibody population N=100, the capacity of memory library n=15, the size of excellent antibodies s=5,the maximum number of iterations MAXGEN=100, the threshold of the stepwise iteration IG=10, the diversity evaluation parameter ps=0.95, the crossover probability pc =0.9 and the mutation probability pm=0.7, tabu list length T = 3, solution number of tabu Ti = 10, the maximum iteration number of tabu g = 20. The
parameters of the basic immune algorithm is the same to the ITSA. Table 2: Service rate of machines for test1 Number of Machine
1
2
3
4
Machine_perhour
3.38
2.93
2.64
3.33
Table 3: Service rate of machines for test2 Number of Machine
1
2
3
4
Machine_perhour
3.29
2.76
3.62
3.07
Table 4: Parameters of test1 x-coordinate of field location
y-coordinate of field location
Service
29.4743
19.5502
15.02
44.2759
22.0332
12.05
92.7854
8.7500
16.71
95.0064
14.2389
10.09
76.2527
52.2289
13.03
52.4704
99.6853
17.8
20.2699
83.4859
13.4
14.8598
81.0007
15.7
12.1471
79.2958
13.6
28.1811
65.1118
12.35
Fig. 3: Convergence process of ITSA and IOA for Test1
Fig. 4: Convergence process of ITSA and IOA for Test2
Table 5: Parameters of test2
Table 6: Results of IOA for two tests
x-coordinate of field location 1.5403
y-coordinate of field location 18.8955
16.8990
18.3511
14.33
35.3159
29.6321
13.46
45.0924
8.1126
64.9115
Service 14.79
Tests
Test1
Test2
fitness
times
fitness
times
1
0.6016
6
1.2328
6
2
0.6846
2
1.2677
1
13.31
3
0.7531
1
1.2698
1
36.8485
13.34
4
0.7902
1
1.2808
1
73.1722
62.5619
12.92
5
-
-
1.3120
1
82.1194
74.4693
13.53
Sum
-
10
-
10
64.7746
78.0227
13.53
Average running time
4.3024
68.6775
14.45
23.4780
54.7009
14.38
All fields are dispersed in a square area of 100 * 100 square meters and the depot is located at coordinates (50, 50).
4.2 Results of two examples Fig. 3 and Fig. 4 show the convergence process of these two algorithms for the two examples. First of all, the IOA cannot always find the optimal solution. For ITSA, it has obvious convergence tendency as well as the ability of converging to the optimal solution in earlier iteration. Taking the results of the IOA into analysis individually: the IOA has a high probability to find the optimal solution. In Test1, the average solution of IOA is 0.6522, which is only 8.41% higher than the optimal solution 0.6016. The average solution of IOA in the second test is 1.2527, only 1.32% greater than the optimal solution 1.2328. However, IOA has a disadvantage in searching the local solution around each antibody while expanding the global search range.
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17.16s
17.3s
Table 7 shows that the probability of TSA to find the optimal solution is a little small due to the influence of the random initial solution. Since the running time of TSA is extremely less, it is theoretically feasible to utilize TSA as the mutation operator in order to enhance the performance of local search. As the ITSA algorithm converges earlier, some parameters of the ITSA algorithm are adjusted, such as the size of antibody population N = 50, the capacity of the memory library n = 10, the number of excellent antibodies s = 3, the maximum number of iterations MAXGEN=40, in order to ensure the efficiency of algorithm, as well as to reduce iteration algebra. As we can see from Table 8, ITSA algorithm runs ten times with finding the optimal solution each time. It is proved that ITSA has the advantage of high precision and fast convergence speed. The results show that the introduction of the TSA into the IOA can help to find the optimal solution more effectively, making the algorithm more efficient.
Taking those three results into comprehensive comparisons (see Table 9), ITSA has obvious advantage in improving the efficiency of solving and shortening iteration time. Known ITSA in global search and local search has more strong searching ability, it is better to be utilized for optimization problems. Table 7: Results of the tabu search algorithm for two tests Tests
Test1
Test2
References
fitness
times
fitness
times
1
0.6016
3
1.2328
3
2
0.7531
1
1.2677
1
3
1.0085
3
1.3120
3
4
1.5804
1
1.7157
2
5
1.6122
2
1.7712
1
Sum
-
10
-
10
Average running time
0.21s
[1]
[2]
[3]
0.22s
Table 8: Results of ITSA for two tests Test1
Tests 1
[4]
Test2
fitness
times
fitness
times
0.6016
10
1.2328
10
Average running time
49s
[5]
45s
[6]
Table 9: Results of three algorithms for two tests
Test1
Test2
5
In actual agricultural production process, as the frequent occurrence of the dynamic event, the traditional scheduling algorithm cannot adapt to the various changing of actors in the agricultural production environment. In order to utilize the available resources effectively and improve the efficiency of agricultural production, in future we aim to design dynamic and multi-operational agricultural machinery scheduling algorithm.
Algorithm
IMMUNE
TSA
ITSA
Success rate of search
60%
30%
100%
Average iteration amount of the optimal solution
48.8
10
15.2
Average running time
17.16
0.21
49
Success rate of search
IMMUNE
TSA
ITSA
Average running time
60%
30%
100%
Average iteration amount of the optimal solution
47.3
8.9
16.7
Average running time
17.3
0.22
45
[7]
[8]
[9]
Conclusion and Future Work
[10]
In this paper, we have proposed an improved ITSA for agricultural machinery scheduling problem. Because of the characteristics of self-regulation, memory and diversity, the IOA has large search range as well as better global search ability, but it has the disadvantage of incomplete searching in local search. While the TSA has strong ability of local search, it can help to find the optimal solution effectively. Meanwhile, in the calculation of the fitness value of the problem, the traveling path of agricultural machine is refined according to the actual situation. Finally, the ITSA is applied to two cases. The results are compared with the IOA and the TSA, which proves the feasibility and effectiveness of the proposed algorithm. It can be seen that ITSA improves the search efficiency at the cost of increasing local search time. However, for static scheduling problem, this time consumption can be acceptable. But it is actually worth considering how to shorten the time costs by local search.
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[11]
[12]
[13]
[14]
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