Robust Mission Planning based on Nested Genetic Algorithm

Luohao Tang, Cheng Zhu, Weiming Zhang, and Zhong Liu

Abstract—Military mission planning involves resource allocation and task scheduling and it is a complex combination optimization problem. Traditionally, this problem is solved to get a static solution (plan) with minimal makespan. However, there are lots of uncertainties in the military operations, such as disruptions of actions or unexpected increases of task duration, a static plan is prone to be invalid. This paper focus on generating robust plan which can absorb some deviations. Time-slack based robustness measure is used to qualify the robustness, based on which a nested Genetic Algorithm is proposed to resolve the robust mission planning problem. At last, an illustrative mission instance is presented, three plans with different robustness are tested through simulations and the results provide strong evidence of the benefits of this method.

I. INTRODUCTION

ILITARY mission planning is to answer question that “Who when and where to accomplish

which tasks”. Traditionally it is reformulated as a resource allocation and the task scheduling problem in [1] [2]. In the mission planning model proposed by G. M. Levchuk [2], a set of platforms (clustering of resources) should be allocated to a logical ordered tasks under the consideration of resource temporal and space constraints. As it is a complex combination optimization problem, a heuristicbased Multi dimensional Dynamic List Scheduling (MDLS) algorithm is proposed to get a static solution (plan) with minimal makespan. However, as the environment of military operations is full of uncertainties, a static solution always becomes invalid in practice due to unforeseen increases of task durations, disruption of some actions and so on. As a result, we need robust plans to reduce the side-effect of uncertainties.

But what does the “robust” mean? and how can we qualify the robustness of a plan? Recently, Bernard Roy gives a detailed survey on robustness in operational research [3].

Manuscript received August 2, 2011. This work was supported by the National Science Foundation of China under Grant No. 91024006, No. 71031007, No. 71001105, No. 71071157.

Luohao Tang is with the Science and Technology on Information Systems Engineering Laboratory, National University of Defense Technology, Changsha, China. (phone: 86-0731-84576612; fax: 86-0731-84576612; e-mail: tlh_nutshell@126.com).

Cheng Zhu is with the Science and Technology on Information Systems Engineering Laboratory, National University of Defense Technology,Changsha, China. (e-mail: chzhu3@gmail.com).

Weiming Zhang is with the Science and Technology on Information Systems Engineering Laboratory, National University of Defense Technology, Changsha, China. (e-mail: weimingzhang@nudt.edu.cn).

Zhong Liu is with the Science and Technology on Information Systems Engineering Laboratory, National University of Defense Technology,Changsha, China. (e-mail: philipsliu@263.com).

Slightly abusing the notion of [4], in our opinions, a solution (plan) is robust if all of tasks can be finished before a given deadline despite some small unexpected perturbations such as tasks last longer or start later than expected. As the deadline is of great importance in military operations, this robustness is eagerly pursued by decision makers.

We use the time-slack based robustness measure to qualify the robustness, which is used widely in the field of project scheduling. For instance, He´di Chtourou [5] and Olivier Lambrechts [6] use slack based method to generate robust Resource-Constrained Project Scheduling (RCPS). However, the mission planning problem is more complex than the RCPS and there is little research concerning with robust mission planning.

This paper proposes a nested GA (Genetic Algorithm) method to generate robust plan. The mission planning problem is decomposed into two sub problems: platform allocation and task scheduling, the former is to determine which platforms should be allocated to each task and the latter is to determine the schedule intervals of tasks. They are resolved by the external and internal GAs respectively.

The rest of this paper is organized as follows: the robust mission planning problem is introduced in section 2, and the nested GA is described in detail in section 3. In section 4, the experiment is presented which demonstrates the effectiveness of the algorithm. Section 5 gives the final conclusion of this paper and the future work.

II. PROBLEM STATEMENT According to [2], the mission planning is to allocate

proper platforms to the tasks and determine the scheduling of tasks, which involves two kinds of entities: Taskset and PlatformSet.

Taskset 1{ , , }Nt t, }Nt, N is a set of logical ordered tasks. Every task it has three attributes: id is its duration, ( , )

i it tx y

is its geographical location, 1[ , , ]i iLr r ]iL,ri, is its resource requirement vector, where ijr represents the quantity of resource jr required by it .

PlatformSet 1{ , , }Kp p }Kp, is a set of platforms which provides resources for the tasks. Each platform ip has two attributes: moving velocity iv and resource capacities vector

1[ , , ]i iLR R ]iL, Rii, , where ijR represents the quantity of resource

jr provided by ip . A mission can be depicted by a space-time graph as in

Fig. 1 [1], in which , the Taskset contains 10 logical ordered tasks located in different positions, and each task needs some platforms to provide the necessary resource, for example 1p and 2p are allocated to 1t . The solution to the

M

Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011

978-1-61284-375-9/11/$26.00 @2011 IEEE 45

mission planning is an optimal platform-task allocation and task scheduling, which satisfies resource temporal and space constraints.

Fig.1. Space-time graph of a mission scenario The robust mission planning problem can be formulated

as a mixed integer programming model.

1

1 0

1

max. .

,1; 1

0,1;1

( )

1,;1

N N

ijm jimj j

iim

iji i ijm ij j ij

m

N

RMs t

x x i N m K

x i N m Kdis

s d x a D s a Dv

i j N m Ks D

�

� �

�

� � � �� � ���������

� � � � � ����������������������

� � � � � ��������������

� � � � �������������������������������� �

� �

) j ijD s a D) j ij ������������s a Ds) j ij

1 0

(4)

,1;1K N

jim ml ilm j

x R r i N l L� �

���������� ����������������������������������������������������� � � � � ����������� ,ml il ,1R r 1ml ilr ,1il

There are two types of variables: the routing variables {0,1}ijmx � representing whether platform mp serves task

jt after accomplishing it , the scheduling variables is R��

{0} representing the start time of it . Parameter ijarepresents precedence constraint, 1ija � if it precedes jt ,otherwise 0ija � , ijdis represents the Euclidean distance between it and jt ,and D is the deadline before which all tasks should be finished. Two dummy tasks 0t and 1Nt � are used to represent the first and last task.

Constraints (1) and (2) define the routing network of platforms. Constraint (3) represents that a task can start only after all its predecessors have been finished and all platforms it needs have arrived. Constraint (4) represents the deadline constraint. Constraint (5) represents the resource constraint that the platforms can provide enough resources to each task.

The objective is to maximize the robustness measure RM, in this paper the RM is defined as:

1

N

i ii

RM LST SET�

� �� (6)

where iLST and iSET represent the latest and earliest start time of it respectively. Other RM can also be used such as those defined in [5].

The mission planning problem combines both 0-1variables and continuous variables and it is a NP-hard problem [2].

III. ROBUST MISSION PLANNING BASED ON NESTED GA

We propose a nested GA-based method to solve the above mission planning problem. In fact, the problem can be decomposed to two sub problems: platform allocation and task scheduling, which can be solved by the external and internal GA respectively.

We use the external chromosomes to encode the platform-task allocations and the internal chromosomes to encode the task scheduling. First, the external GA is invoked to generated feasible platform-task allocations. Once the allocation is given, the problem is transferred to a variant of job shop with setup time scheduling, and the internal GA is invoked to solve it, i.e. the internal GA is to find the best schedule for the given platform-task allocation represented by the external chromosome and return the RM(corresponding to fitness) to it, which is essentially the evaluation step of external GA. When the external GA stops, the optimal solution can be decoded from corresponding external and internal chromosomes. The framework of nested GA is depicted in Fig.2.

Fig.2. The framework of nested GA

A. External GA synthesis The external GA is used to find the candidate

PlatformSet for the tasks, which corresponds to a 0-1 integer program and it is obviously NP-hard.

Using ( )iP t to represent the candidate PlatformSet

allocated to task it , it must satisfy two conditions: 1. 1 l� �

( ),

i

ml ilm P t

L w R�

� � � , 2.( ) /

( ), . . .i

i ml ilP t p

p P t l s t w R� � �� �� , whe-

re condition 1 means that ( )iP t can provide enough resources to it , condition 2 means that none of platform in

( )iP t is redundant. The external GA is designed as following:

Encoding: The external chromosome is represented as a 0-1 array with length N K� , where N is the number of tasks and K is the number of platforms. The array is defined as:

1,[( 1)* ( 1)]0

j iif p is allocated to tAllocate i K jotherwise

����

� � � �

,1 ;1 (5)

(3)

(1)

(2)

46

To each task it , the array encodes a candidate platform set ( )iP t allocated to it. The chromosome encoding is depicted in Fig.3.

1( )P t ( )NP t

Allocate 1 0 0 1 Fig.3. Chromosome encoding of the platform-task allocation

Evaluation: To calculate the fitness of a external chromosome, we should find the best task scheduling with highest RM for the platform allocation encoded in it, then the RM is returned as the fitness of this chromosome. As there are many feasible schedules corresponding to a given platform allocation, it is also NP-hard to find the best one, so the internal GA is used to solve it. These will be described in details in next section.

Selection: We use the roulette wheel method to select chromosomes based on their fitness (corresponding to the RM).

Crossover: For two selected parents, we first randomly choose a crossover point [2, 1]n N� � , and interchange their genes in this point ,which means the first child solution consists of the first ( 1)n K� � genes from the first parent and ( 1)N n K� � � genes from the second parent, and vice versa for the second child. The crossover operator is with a high probability e

cp . Mutation: The mutation operator is applied to every

chromosome with a small probability emp . To a selected

chromosome, we randomly choose a number [1, ]n N� , and randomly generate a new candidate platform set ( )nP t for task nt .

B. Internal GA synthesis Once the platforms allocation is given by the external

GA, the internal GA should be invoked to find the best schedule with highest RM. If we view the platform as a machine, and the moving time of platform from one task to anther as the setup time, then the task scheduling problem is transferred to variant of a job shop scheduling with setup time, which is NP-hard .

Encoding: We use the task list representation to encode a schedule. The array TaskList contains N tasks according to their scheduling sequence, i.e. TaskList[i] will be the ith task chosen to be scheduled. As there is precedence between tasks, a task must be scheduled after all its predecessors and before all its successors.

Fig.4. depicts two feasible task lists generated for the mission scenario in Fig.1.

10 1 2 4 6 8 3 5 7 9 1 10 2 4 3 6 7 5 8 9

Fig.4. Two feasible chromosomes examples The pseudo code to randomly generate a feasible

scheduling sequence is depicted in Fig.5.

Randomly generate a task scheduling sequence

Ready= � ; TaskList[0]=t0;

For(i=1 to N) Ready= Ready { tasks whose predecessors have all

already been in TaskList };Randomly select a task tk in Ready; TaskList[i]= tk ;Ready=Ready- tk;

End ForFig.5. Pseudo code of generating a feasible task list randomly

Evaluation: The evaluation of internal chromosome is a process of decoding, which utilizes platform allocation encoded in external chromosomes and task sequence encoded in internal chromosomes to determine precedence and interval between takes, based on which the earliest and latest start time of tasks can be computed by the CPM (Critical Path Method), thus the RM can be got. The process of chromosome evaluation is shown in Fig.6.

Double Evaluate (ExternalChromosome, InternalChromosome)

// T(pi): tasks allocated to the platform pi; TD(k,l): transition //interval between task k and task l; dis(k,l): Euclidean //distance between task k and l; Vpi: moving velocity of pi

1).For (P1 to PM) 1.1).Determine T(pi) according to the allocation

information encoded in ExternalChromosome; 1.2).Determine the relative orders of tasks in T(pi)

according to the scheduling information encoded in InternalChromosome;

1.3).Update TD(k,l)=max{ TD(k,l),dis(k,l)/vpi}, where k,l T(pi) and k,l are two consecutive tasks on the routing of Pi

End For 2).Construct a temporal network based on the task sequence,task durations and the intervals TD(i,j),(i,j) N×N;3).If the earliest finish time of the last task exceeds the deadline,RM=0, goto 6); 4).Calculate the earliest and latest start time of tasks by CPM. 5).RM=∑(LSTi-ESTi) 6).Return RM;

Fig.6. Calculation of the RM

Selection: As in the external GA, the roulette wheel method is used here to select chromosomes based on their fitness.

Crossover: In this step, we use the precedence set crossover method in [7].We use the example in Fig.1 to illustrate how it works. First, a task T9 is selected randomly, all the predecessors and successors (not necessarily immediate) of T9 combining with itself forms a set {1,2,3,5,7,9},and we call it the RelatedSet. In the son (daughter) chromosome, tasks in RelatedSet should inherit their relative orders from the father (mother) and tasks not in RelatedSet should inherit their relative orders from the mother (father).As depicted in Fig.7, the son preserves the relative orders of {1,2,3,5,7,9} as the father and the relative orders of {10,4,6,8} as the mother. For more details about precedence set crossover, see [7].

47

father 10 1 2 4 6 8 3 5 7 9mother 1 10 2 4 3 6 7 5 8 9

son 1 10 2 4 3 6 5 7 8 9daughter 10 1 2 4 6 8 3 7 5 9

Fig.7. Example of precedence set crossover

Mutation: we randomly choose a internal chromosome with a probability of i

mp , and select one or more tasks randomly in its task list, then we put each selected task into a new randomly selected position which must be higher all of its predecessors and lower than all of its successors,through which we get a difference task sequences. For example we randomly select two tasks {5,6}, then T5 can only swap position with T7 and T6 may swap position with T3, after that a new chromosome is generated, see Fig.8.

Before 10 1 2 4 6 8 3 5 7 9 After 10 1 2 4 3 6 6 7 5 9

Fig.8. Example of mutation

IV. EXPERIMENT To empirically test the robust planning method proposed

above. We use a military planning instance from [2] The instance contains 10 tasks, 20 platforms, and 8 resources types. The initial precedence between tasks is depicted in Fig.1, and the parameters such as moving velocity of platforms, resource capacity/requirement vector and locations of tasks etc. are the same as in [2]. The deadline is set to 100. We use the algorithm to find robust solutions for this instance, and what’s more, we want to check whether the solutions are really robust.

The parameters of external GA is set as: the constant size of population ExPopSize=30, the maximal generation ExMaxGen=100, the crossover probability 0.8e

cp � , the mutation probability 0.1c

mp � .The parameters of internal GA is set as: InPopSize=20, InMaxGen=10, 0.85i

cp � ,0.15i

mp � . In the experiments, 20 runs were done to theinstance, and the means and standard deviations of fitness (RM) are depicted in Fig.9.

0

50

100

150

200

250

300

350

400

450

500

0 10 20 30 40 50 60 70 80 90 100

Fig.9. Mean and standard deviation of fitness, 10 runs to the instance From Fig.9, we can see the optimal solution has obvious

higher RM than the initial solutions and the nested GA can find optimal solution in less than 100 generation, which

indicates the algorithm can converge to the optimal solution fast.Fig.9 also shows that the standard deviation becomes smaller as the generation increase, which demonstrates the high quality of the algorithm.

To test whether the solutions with high RM is really robust, next we compare three solutions with different fitness through Monte Carlo simulations. These solutions are depicted in Table 1 and Table 2, where Table 1 presents the platform-task allocation and Table 2 presents the RMand task scheduling sequence of the solutions. Letter “T”represents task and “P” represents platform.

Table 1 Task-platform allocation in three solutions Solution -1 Solution -2 Solution -3

T1 P{3,10,19} P{1,10,19} P{9,11,12,20} T2 P{1,7,18} P{9,12,13,18} P{10,11,13,19} T3 P{10,12,13,20} P{8,9,18} P{10,11,13,18} T4 P{1} P{1} P{10,12} T5 P{3} P{3} P{1} T6 P{8,14,19} P{4,10,19} P{4,8,9} T7 P{4,7} P{4,9,11,13} P{10, 14,18} T8 P{5,9,19} P{10,19,20} P{5,7,19} T9 P{17,18,20} P{5,17,18} P{17,18,20} T10 P{4,17} P{4,17} P{9,14,15,16,20}

Table 2 RM and task scheduling sequence of three solutions RM Task Scheduling Sequence

Solution-1 442.113 T{10,1,2,4,3,5,6,7,8,9} Solution-2 327.619 T{1,10,2,3,6,5,4,8,7,9} Solution-3 199.595 T{1,10,2,3,6,4,8,7,5,9}

The simulation process is as following: we randomly pick some tasks with a probability p and increase their durations with q, for instance, if p=0.5 and q=0.3, 50% tasks will be picked randomly and their duration be increased with 30%.Then we check whether the deadline is violated following the schedule represented by the solution, this process is repeated 5000 times in every simulation. We record the number of that the deadline is exceeded, noted as NV, and calculate the ratio of deadline violation ijf =NV /5000,(j=1 ),which represents the statistic probabilityof the scheduling failing to meet the deadline according to solution i in the jth simulation.

Totally we run 10 simulations, and the average ratio of deadline violation for each solution can be calculated as:

10

1/10,( 1,2,3)i ij

jF f i

�

� �� (7)

Obviously, to every solution i, Fi approximates its probability of failing to meet the deadline when the durations increase, so the solution with smaller Fi is better than those with bigger Fi, and it can absorb more disruptions.

Fig. 10 depicts the Fi of the three solutions under three settings (p=0.5,q=0.8), (p=0.75,q=0.8) and (p=0.9,q=0.8).From the figure, we can clearly see that solution with higher RM is much more robust than those with lower RM. For example, when 50% tasks last longer 80%, the probability of meeting the deadline is near 100% for solution 1, but less than 10% for solution 3, which provide strong evidence of the benefit of a robust solution.

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Fig.10. Fi of three solutions , 10 runs to the instance

More statistical Fi is presented in Table 3, from which we can also see the solution with higher RM can better absorb disruptions than those with lower RM under all parameter settings, so it’s of great importance to find the most robust solution, which also demonstrates the significance of the robust planning method.

Table 3 Fi of the three solutions under different parameter settings q=0.3 q=0.5 q=0.8

(F1,F2,F3 ) (F1,F2,F3 ) (F1,F2,F3 ) p=0.5 (0, 0,0.186 ) (0, 0.087 0.648 ) (0.041, 0.428, 0.908 )

p=0.75 (0, 0, 0.567 ) (0,0.238,0.970 ) (0.36, 0.868, 1.0 ) p=0.9 (0,0, 0.864 ) (0, 0.64, 0.996 ) (0.728 0.988,1.0 )

V. CONCLUTIONS Military mission planning involves resource allocation

and task scheduling and it is a NP-hard combination optimization problem. Most traditional methods focus on getting a static schedule with a minimal makespan. In contrast to that, this paper takes the uncertainties into account and proposed a robust mission planning algorithm based on nested GA. Experiments have been done on an military mission scenario, the result shows the algorithm can converge to a optimal solution fast. What’s more, the advantage of a robust solution is demonstrated through Monte Carlo simulations, which provides strong evidence of significance of the robust planning method.

As the instance in the experiment is of small size, the efficiency and effectiveness of this algorithm should be tested on problems of bigger size, especially those from the real data.

REFERENCES

[1] L. Belfares, W. Klibi, L. Nassiro, and A. Guitouni, “Multi-objectives Tabu Search based algorithm for progressive resource allocation”, European Journal of Operational Research, vol. 177, P. 1779 , 2007.

[2] G.M. Levchuk, Y. N. Levchuk, J. Luo, K.R. Pattipati, and D.L. Kleinman, “Normative Design of Organizations - Part I: Mission Planning”, in IEEE Transactions on Systems, Man, and Cybernetics,vol. 32,no. 3, pp. 346-359, 2002.

[3] B. Roy, “ Robustness in operational research and decision aiding: A multi-faceted issue”. European Journal of Operational Research ,vol. 200, no.3, pp. 629-638, 2010.

[4] S. Ali, A. A. Maciejewski, H. J. Siegel, and J. K. Kim, “Measuring the robustness of a resource allocation,” IEEE Transactions on Parallel and Distributed Systems, vol. 15, no. 7, pp. 630–641,Jul. 2004.

[5] H.D. Chtourou, and M. Haouari, “A two stage priority rule based algorithm for robust resource constrained project scheduling”.Computers & Industrial Engineering,vol. 55, pp. 183–194, 2008.

[6] O. Lambrechts, E. Demeulemeester, and W. Herroelen, “Time slack-based techniques for robust project scheduling subject to resource uncertainty scheduling subject to resource uncertainty”.Annals of Operations Research., vol. 186, pp.443-464, 2011.

[7] J. Alcaraz and C. Maroto, “A Robust Genetic Algorithm for Resource Allocation in Project Scheduling”. Annals of Operations Research 102, pp. 83–109, 2001.

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