research article heuristic search for planning with...

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 963874 11 pageshttpdxdoiorg1011552013963874

Research ArticleHeuristic Search for Planning with Different ForcedGoal-Ordering Constraints

Jiangfeng Luo Weiming Zhang Jing Cui Cheng Zhu Jincai Huang and Zhong Liu

Science and Technology on Information Systems Engineering Laboratory National University of Defense TechnologyChangsha 410073 China

Correspondence should be addressed to Jiangfeng Luo nudtluojiangfenggmailcom

Received 3 May 2013 Accepted 11 June 2013

Academic Editors W-J Hwang S-S Liaw and S H Rubin

Copyright copy 2013 Jiangfeng Luo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Planning with forced goal-ordering (FGO) constraints has been proposed many times over the years but there are still majordifficulties in realizing these FGOs in plan generation In certain planning domains all the FGOs exist in the initial state Nomatterwhich approach is adopted to achieve a subgoal all the subgoals should be achieved in a given sequence from the initial stateOtherwise the planning may arrive at a deadlock For some other planning domains there is no FGO in the initial state HoweverFGO may occur during the planning process if certain subgoal is achieved by an inappropriate approach This paper contributesto illustrate that it is the excludable constraints among the goal achievement operations (GAO) of different subgoals that introducethe FGOs into the planning problem and planning with FGO is still a challenge for the heuristic search based planners Then anovel multistep forward search algorithm is proposed which can solve the planning problem with different FGOs efficiently

1 Introduction

A large majority of real-world problems have interferingsubgoals How to effectively plan for the interfering subgoalsespecially when there are forced goal-ordering (FGO) con-straints has been a long term focus As the Goal AgendaManager (GAM) [1] used in the FF planner [2] and theordered landmarks [3 4] introduced in the LAMA planner[5] quite a number of approaches have been proposed but theperformance results have scarcely improved This is becauseif any of the FGO constraints is violated forward search mayarrive at a deadlock from which there is no way to reach thegoal state However the proposed approaches such as GAMand landmark cannot detect all the deadlocks exactly andthe undiscovered deadlocks make a planning difficult In thiscase this paper proposes an approach that can automaticallyput right the planning process when it leads the search to adeadlock and significantly improve the planning efficiency

Many real-world problems as in military industrial avia-tion and space domains involve FGO constraints An exam-ple is a naval platform which has to counter many incomingmissiles with different weapons [6] Firing weapons at onemissile may interfere with the interception of others Thus it

can cause the naval platform to suffer from high probabilityof leaking if the missiles are countered in an incorrect orderAnother example is a scenario of a robot rescue [7] Eachrobot has a special ability such as survivor searchtransportcleaning barriers or medical distribution The rescue tasksshould be finished coordinately in constrained orderingswithrespect to a given environment Certain robots only careabout their own subgoals and achieving them too early mayresult in the failure of an entire military operation Addition-ally FGO constraint can be observed in a NASA scenario as adigger robot is allowed to dig the ground onmars only after aphotograph robot has taken a picture of the site [8] Alongwith the complex domain dependent constraints the FGOconstraint is one of the main challenges that a planner needsto overcome for the above problems

Next we first give the problem statement Some defi-nitions and properties are proposed to explain why FGOsoccur for a planning problem Then a novel forward searchalgorithm is proposed to solve the planningwith FGOs Basedon the evaluation it can be concluded that planning withFGOs is still a challenge for current automatical planners butour method can solve it efficiency

2 The Scientific World Journal

2 Problem Statement

Before introducing forced goal ordering (FGO) we first givethe description of a planning problem (119874 119868 119866) as given in [1]

In a planning problem definition 119874 is a finite set ofground actions with the STRIPS style (in this paper) For any119900 isin 119874 there is 119900 = (119901119903119890 119886119889119889 119889119890119897) where 119901119903119890 119886119889119889 and119889119890119897 are finite sets of ground atoms 119901119903119890 is the preconditionset under which the action is applicable 119886119889119889 and 119889119890119897 are theatoms added or deleted after the execution of the action 119868 and119866 are the finite sets of ground atoms and represent the initialand goal state of the problem For any given state 119904 and action119900 isin 119874 the result of applying 119900 to 119904 is

119903119890119904119906119897119905 (119904 119900) = 119904 cup 119886119889119889 (119900) 119889119890119897 (119900) if 119901119903119890 (119900) sube 119904119904 otherwise

(1)

For the action sequence 1199001 1199002 119900

119899 (119900119894isin 119874 1 ⩽ 119894 ⩽ 119899)

there is

119903119890119904119906119897119905 (119904 1199001 1199002 119900

119899)

= 119903119890119904119906119897119905 (119903119890119904119906119897119905 (119904 1199001 1199002 119900

119899minus1) 119900119899)

(2)

The planning problem is finding a sequence of actions120587 suchthat 119866 sube 119903119890119904119906119897119905(119868 120587) The set of action sequence is definedas Π119874 A state 119904 is reachable from the initial state if and onlyif exist120587 isin Π119874 st 119904 sube 119903119890119904119906119897119905(119868 120587) Similarly an atom 119891 isachievable from a certain state 119904 if and only if 119891 isin 119904 orexist120587 isin Π119874 st 119891 isin 119903119890119904119906119897119905(119904 120587) An action 119900 is applicable ina reachable state 119904 if and only if 119901119903119890(119900) sube 119904

Definition 1 (forced goal ordering (FGO) [1]) For the plan-ning problem (119874 119868 119866) let 119892 1198921015840 isin 119866 be the atomic goals Wesay that there is a forced ordering between119892 and1198921015840 written as1198921015840 ≺ 119892 if and only if for any state 119904(119892 not1198921015840) there is no plan120587 satisfying 1198921015840 isin 119903119890119904119906119897119905(119904(119892 not1198921015840) 120587) 119904(119892 not1198921015840) represents areachable state in which 119892 has just been achieved but 1198921015840 isfalse

In any given state an atomic goal remaining false meansthat the goal is not achieved Definition 1 illustrates thatforward search arrives at deadlock 119904(119892 not1198921015840) when there is1198921015840 ≺ 119892 and the atomic goal 119892 is achieved before 1198921015840 In some ofthe literatures [2 9] 119904(119892 not1198921015840) is called a dead-end state tooDuring the planning process forward search which violatesany of the FGOs may lead the planning to a deadlock fromwhich there is no way to the goal state

With respecting to the FGO there is goal ordering definedas reasonable goal ordering (RGO) written as 1198921015840≺

119903119892 There is

a reasonable ordering between 119892 and 1198921015840 if and only if forany reachable state 119904(119892 not1198921015840) there is no longer a plan thatcan achieve 1198921015840 from 119904(119892 not1198921015840) without deleting 119892 at leasttemporarily [1] So if there is 1198921015840≺

119903119892 and 119892 is achieved before

1198921015840 in order to get a plan solution the planning must firstdelete the achieved goal 119892 then to achieve 1198921015840 and last toachieve 119892 again In this paper as to focus the problem onFGO it is supposed that there is no goal deletion during theplanning process Each goal cannot be deleted once it hasbeen added

(3 1)

(0 3)(0 2)(0 1)

(1 3)(1 2)(1 1)

(2 3)(2 2)(2 1)

(3 3)(3 2)

Initial state

(3 1)

(0 3)(0 2)(0 1)

(1 3)(1 2)(1 1)

(2 3)(2 2)(2 1)

(3 3)(3 2) (3 1)

(0 3)(0 2)(0 1)

(1 3)(1 2)(1 1)

(2 3)(2 2)(2 1)

(3 3)(3 2)

goal state given state

Figure 1 Floortile domain in IPC 2011

The first planning domain for planning competition withFGOs is the Floortile proposed in the International PlanningCompetition (IPC) 2011 [10] During the competition noparticipating planner in the sequential satisficing track cansolve it well Figure 1 shows an example of the Floortiledomain in the IPC 2011 In the initial state the status ofall floor tiles is clear Floor tiles need to be painted blackand white while adjacent tiles should have different colorsRobots can only paint tiles that are in front (up) or behind(down) Moreover once a tile is painted a robot cannot standon it This particular configuration makes the domain veryhard to solve because of the existence of FGOs For examplesuppose a robot first selects the tile(21) to paint in whiteIn further planning steps the robot can only stand on atile(21) to paint the front tile(31) in black This process canbe achieved if and only if the atom (robot-at tile(21)) is trueHowever this atom is not true and cannot be added once thetile(21) has been paintedThe reason is that the atom (robot-at tile(21)) ismutually exclusivewith the atomic goal (paintedtile(21) white) and (painted tile(21) white) cannot be deletedonce it has been added Therefore painting tile (21) before(31) violates the FGO constraint in consequence causing thesearch to arrive at a deadlock

In this example there are many FGOs in the Floortileproblem and the robots should paint tiles obeying a correctsequence In the case of Figure 1 the FGOs are

(painted tile(31) black) ≺ (painted tile(21) white) ≺(painted tile(11) black)(painted tile(32) white) ≺ (painted tile(22) black) ≺(painted tile(12) white)(painted tile(33) black) ≺ (painted tile(23) white) ≺(painted tile(13) black)

In the above domain all the FGOs exist in the initialstate No matter which approach is adopted to achieve anatomic goal all the atomic goals should be achieved in agiven sequence starting from the initial state Otherwise theplanning may arrive at a deadlock However in some real-world planning problem there is no FGO in the initial stateFor any 119892 1198921015840 isin 119866 planning starting from the initial state tofirstly achieve 119892 or 1198921015840 would not lead to a deadlock Howeverthe planning may arrive at a given state 119904(not119892

1 not1198922 not1198923)

(1198921 1198922 1198923isin 119866) without FGO among 119892

1 1198922 and 119892

3 Now

The Scientific World Journal 3

Figure 2 Air defense of a naval group

if certain plan as 1205871is selected to achieve 119892

1while translating

the search to state 119904(1198921 not1198922 not1198923) then there is 119892

2≺ 1198923or

1198923≺ 1198922

Figure 2 shows an example for the air defense of a navalgroup (ADoNG) A naval group has some Surface to AirMissile (SAMs) and chaffs to intercept the incoming antishipmissiles The arrived antiship missiles at the same time issupposed to locate at a spherical surface above the navalgroup which can be transferred into a 119898 times 119899 rectangularplane as shown in Figure 2 In each rectangle there is anantiship missile while one ship of the naval group can firea SAM or chaff to intercept it However once a chaff isemployed to intercept an antishipmissile in a given rectanglethe rectangles in the up down left and right of the givenrectangle should be interfered by the chaff cloud If certainrectangle is interfered by the chaff cloud from one directionthe antiship missile in this interfered rectangle can only beintercepted by chaff because chaff cloud canprevent the radarfrom guiding the SAM interception Moreover if certainrectangle is interfered by the chaff cloud from more thanone direction the antiship missile in this rectangle cannot beintercepted because the radar of the naval group may losethe accurate position of the antiship missile The goal state isto intercept all the incoming missiles by the given SAMs andchaffs

For the planning of the air defense of a naval group thereis no FGO in the initial state Taking the problem shown inFigure 3 as an example there are 4 antiship missiles in a 2 times 2rectangular plane with 3 chaffs and 1 SAM Obviously in theinitial state every antishipmissile can be intercepted with thehighest priority However if the antiship missile in rectangle(2 1) is firstly intercepted by a chaff there are FGOs (1 2) ≺(1 1) and (1 2) ≺ (2 2) in the successor state and the antishipmissile in (1 2) must be intercepted by a SAM However ifthe antiship missile in rectangle (2 1) is firstly intercepted bya SAM there is no FGO for the remaining antiship missilesin (11) (12) and (22)

3 Related Works

Heuristic search planning (HSP) has become a dominantdomain independent paradigm over the last decade [5] TheHSP method first proposed by Bonet and Geffner [11] per-forms a forward search from an initial state to a goal state in a

(2 1)

1 2

1

2

(1 1)

(2 2)

(1 2)

(2 1)

1 2

1

2

(1 1)

(2 2)

(1 2)

Figure 3 Air defense of naval group

search graph This method employs some powerful heuristicestimators to guide the search in a fast forward mannertowards the goal state with the help of heuristics for choosinghelpful actions to extend the search closer to the goal stateIn the past great success has been achieved in heuristicsearch planning systems such as FF [2] FD [12] SGPlan[13] and LAMA[5] Researchers have also considered variousgoal interactions in multiple-goal achievement and detection[1 14] All of the above planners have their own approachesto deal with the goal orderings

Among the previous works the most relevant methods toour approach areGoal AgendaManager (GAM) and orderinglandmarks The concept of a Goal Agenda Manager (GAM)was proposed in [1] to detect the reasonable goal orderingsThe GAM is widely used in many planning systems suchas IPP and FF which can improve the performances of IPPand FF dramatically A GAM defines the order in which thesubgoals are achieved In the beginning of a search processa GAM is employed to check all the ordering relationships ofeach atomic goal pair Then the search divides the goal setinto many subsets so that the planner can achieve each ofthem in sequence

For each atomic goal pair as 119892 1198921015840 isin 119866 GAM uses 119865119892119863119892=

⋂119900isin119874 119892isin119886119889119889(119900)

119889119890119897(119900) and 119874lowast = 119874119892 119900 isin 119874 | 119901119903119890(119900) cap

119865119892

119863119892= 0 where 119874

119892= 119900 isin 119874 | 119892 notin 119889119890119897(119900) to calculate

whether there is an action sequence 119901 isin 119875119874lowast

satisfying1198921015840 isin 119903119890119904119906119897119905(119904(119892 not1198921015840) 119901) If there is not there exists anordering defined as 1198921015840 ≺

119903119892 Otherwise the GAM checks

whether 119892≺1199031198921015840 exists Therefore there are 1198752

|119866|atomic goal

pairs that need to be checkedA concept known as ldquolandmarkrdquo is defined to extend the

GAM on goal ordering at top-level atomic goals as well ascertain states known as landmarks during a planning process[3 4] For a planning problem (119874 119868 119866) an atom 119897 is calleda landmark if for any 119901 = 119900

1 1199002 119900

119899 isin 119875119874 and 119866 sube

119903119890119904119906119897119905(119868 119901) there is 119897 isin 119903119890119904119906119897119905(119868 1199001 119900

119894) (1 ⩽ 119894 ⩽

119899) A planning system that uses landmarks obtains all thelandmarks of the problem at the beginning of the planningprocess The planner then orders them heuristically It uses abacktracking method via a relaxed plan graph (RPG) [2] tofind the candidate landmarks and their orders For exampleall the atomic goals are landmarks For each atomic goal119892 the atoms in ⋂

forall119900isin119874 119892isin119886119889119889(119900)119901119903119890(119900) are treated as new

landmarks where there is relationship 119891≺119903119892 for each 119891 isin

⋂forall119900isin119874 119892isin119886119889119889(119900)

119901119903119890(119900) Then 119891 is treated as a new atomic


g11g12g13

g11g12g13

g22

g21g11g12g13

g22

g21

g33g32g31

(I O g11 g12 g13)

(s1 O g21 g22)

(s2 O g31 g32 g33)

s1 s2

s3

(s3 O middot middot middot )

Figure 4 The incremental planning process

goal The landmark-generation algorithm repeats the aboveprocess from the top level of the RPG to the lowest level

The thirdmethod to handle the goal ordering is the incre-mental planning process adopted by the planner SGPlan6 [1315] As shown in Figure 4 the incremental planning processtry to check part of the goal orderings in the initial stateSome of the checked atomic goals which can be achievedwith high priority are firstly handled Then the incrementalplanning process try to find out more other goal orderingsin current state and achieve part of atomic goals with highpriority The planning continues the above process until allthe atomic goals are achieved

4 Why FGOs Occur

Definition 2 (goal achievement operation (GAO)) For plan-ning problem (119874 119868 119866) 119900 isin 119874 is a GAO if exist119892 isin 119866 satisfying119892 isin 119886119889119889(119900) 119900 is written as 119900

119892

Definition 3 (available GAO) 119900119892is available in a given state

119904 if and only if 119901119903119890(119900119892) sube 119904 or exist120587 isin Π119874 st 119901119903119890(119900

119892) sube

119903119890119904119906119897119905(119904 120587)

For an unachieved atomic goal 119892 in 119904 an available GAO as119900119892represents a plan written as 120587

119900119892

while 119892 isin 119903119890119904119906119897119905(119904 120587119900119892

)So if 119901119903119890(119900

119892) sube 119904 there is 120587

119900119892

= 119900119892 Otherwise if 119901119903119890(119900

119892) sube

119903119890119904119906119897119905(119904 120587) there is 120587119900119892

= 120587 119900119892

Definition 4 (available GAO sequence) Suppose that 1198661015840 sube 119866while 119904(not1198661015840) is reachable and there is at least one availableGAO as 119900

119894isin 119874 (1 ⩽ 119894 ⩽ |1198661015840|) for each 119892

119894isin 1198661015840 in 119904(not1198661015840) (|1198661015840|

is the number of atomic goals contained in 1198661015840) These GAOscan be ranked as an available GAO sequence if and only if

(1) these GAOs are ranked in a correct sequence 1199001198951

1199001198952

119900119895|1198661015840|

(119895119894= 1 2 |1198661015840| 119895

119896= 119895119897if 119896 = 119897 1 ⩽ 119894

119896 119897 ⩽ |1198661015840|)(2) there are number of |1198661015840| corresponding plans 120587

1198951

1205871198952

120587119895|1198661015840|

such that

119901119903119890 (1199001198951

) sube 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

)

1198921198951

isin 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

1199001198951

) = 1199041

119901119903119890 (1199001198952

) sube 119903119890119904119906119897119905 (1199041 1205871198952

)

1198921198952

1198921198951

sube 119903119890119904119906119897119905 (1199041 1205871198952

1199001198952

) = 1199042

119901119903119890 (119900119895|1198661015840|

) sube 119903119890119904119906119897119905 (119904|1198661015840|minus1 120587119895|1198661015840|

)

1198661015840sube 119903119890119904119906119897119905 (119904

|1198661015840|minus1 119901119895|1198661015840|

119900119895|1198661015840|

)

(3)

Property 1 For the planning problem (119874 119868 119866) with FGOconstraints the reachable state 119904 is not a deadlock if and onlyif there is at least an available GAO sequence for the maxi-mum unachieved atomic goal set in 119904

Proof Based on Definition 4 suppose that the maximumunachieved atomic goal set in 119904 is 1198661015840 Since goal deletion isnot considered in this paper there is a plan

120587 = 1205871198951

1199001198951

1205871198952

1199001198952

120587119895|1198661015840|

119900119895|1198661015840|

(4)

st 119866 sube 119903119890119904119906119897119905(119904(not1198661015840) 120587) Therefore 119904(not1198661015840) is not a dead-lock Additionally if 119904 is not a deadlock there must be at leastone available GAO sequence in 119904

Definition 5 (excludable constraint of GAO) For a reachablestate 119904(not119892

1 not1198922) there are available GAO 119900

1198921

for 1198921and

1199001198922

for 1198922 There is an excludable constraint in 119904(not119892

1 not1198922)

written as 1199001198921

999424999426999456 1199001198922

if and only if 1199001198922

is unavailable in state119903119890119904119906119897119905(119904(not119892

1 not1198922) 1205871199001198921

)

Definition 6 (excludable GAO set) In a reachable state119904(not1198661015840) (1198661015840 sube 119866) the excludable GAO set for the availableGAO 119900

119892(119892 isin 1198661015840) is written as 119874

119900119892

(119904(not1198661015840)) if and only if (1)for any 119900

119892999424999426999456 1199001198921015840 (1198921015840 isin 1198661015840 1198921015840 = 119892) there is 119900

1198921015840 isin 119874119900119892

(119904(not1198661015840))(2) for any 119900

1198921015840 isin 119874

119900119892

(119904(not1198661015840)) there is 119900119892999424999426999456 1199001198921015840 (3) for any

1199001198921015840 notin 119874119900119892

(119904(not1198661015840)) there is not 119900119892999424999426999456 1199001198921015840

Definition 7 (equivalent GAO) In a reachable state119904(not119892) (119892 isin 119866) 1199001

119892and 1199002

119892are two different available GAOs

for 119892 1199001119892and 1199002119892are equivalent in 119904(not119892) written as 1199001

119892cong 1199002119892 if

and only if 1198741199001

119892

(119904(not119892)) = 1198741199002

119892

(119904(not119892))

Definition 8 (equivalent State) Reachable states 1199041and 1199042are

equivalent written as 1199041cong 1199042 if and only if for any 120587 isin Π119900

there is119866 sube 119903119890119904119906119897119905(1199041 120587) there must be 119866 sube 119903119890119904119906119897119905(119904

2 120587) and

vice versaIn a reachable state 119904(not119892) for an available GAO 119900

119892with

119901119903119890(119900119892) sube 119904(not119892) there may be two different action sequences

1205871and 120587

2satisfying 119901119903119890(119900

119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

1) and

119901119903119890(119900119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

2) However 119903119890119904119906119897119905(119904(not119892) 120587

1)

might not be equivalent with 119903119890119904119906119897119905(119904(not119892) 1205872) In this case

the atomic goal 119892 achieved by the same GAO 119900119892may lead the

search to two different nonequivalent states Planning withthis feature may increase the search space dramatically whenhandling the planning with FGOs Therefore with respectto the above case we can define two different GAOs as


1199001205871119892

and 1199001205872119892

to replace 119900119892to ensure that in a given state

achieving an atomic goal by the same GAO should lead thesearch to the equivalent state Namely for a given reachablestate 119904(not119892) and an available GAO 119900

119892 for any 120587

1 1205872isin Π119874

if there are 119901119903119890(119900119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

1) and 119901119903119890(119900

119892) sube

119903119890119904119906119897119905(119904(not119892) 1205872) there must be 119903119890119904119906119897119905(119904(not119892) 120587

1 119900119892) cong

119903119890119904119906119897119905(119904(not119892) 1205872 119900119892)

Property 2 For a reachable sate 119904(not1198661015840) (1198661015840 sube 119866 119892 isin

1198661015840) and two available GAOs 1199001119892and 1199002

119892 there are 119904

1=

119903119890119904119906119897119905(119904(not1198661015840) 1205871199001

119892

) and 1199042= 119903119890119904119906119897119905(119904(not119866

1015840) 1205871199002

119892

) If both 1199041and

1199042are not deadlock and there is 1199001

119892cong 1199002119892 theremust be 119904

1cong 1199042

Proof Suppose that 1198741198661015840 is the set of all possible available

GAO sequences contained in 119904(not1198661015840) 11987411198661015840119892

is the set of allpossible available GAO sequences contained in 119904

1 and 1198742

1198661015840119892

is the set of all possible available GAO sequences containedin 1199042 As 1199001

119892and 1199002119892have the same excludable GAO set there

must be 1198741198661015840 11987411198661015840119892= 1198741198661015840 11987421198661015840119892 It can be inferred that

11987411198661015840119892= 11987421198661015840119892 As 119904

1and 1199042have the same set of possible

available GAO sequences it can be declared that for anyavailable GAO sequence contained in 119904

1(1199042) this available

GAO sequence must be available in 1199042(1199041) So there is 119904

1cong

1199042

Property 3 For the planning problem (119874 119868 119866) 119904(not119892) is areachable state 1199001

119892and 1199002


of 119892 in 119904(not119892) while 1199001119892cong 1199002119892 Starting from 119904(not119892) if selecting

1199001119892to achieve 119892 leads the planning to state 119904

1and selecting

1199002119892leads the planning to state 119904

2 then for any not119892

1 not1198922isin 119904119894

(119894 = 1 2) if there is 1198921≺ 1198922in 1199041(1199042) there must be 119892

1≺ 1198922

in 1199042(1199041)

Proof Based on the Property 2 it can be inferred that 1199041and

1199042have the same set of possible available GAO sequences So

if there is 1198921≺ 1198922in 1199041 there is no available GAO sequence

contained in 1199041which can achieve 119892

2before 119892

1 As 119904

1and

1199042have the same set of possible available GAO sequences so

starting from 1199042 1198922cannot be achieved before 119892

1too So there

is 1198921≺ 1198922in 1199042 Obviously by the same way it can be inferred

that if there is 1198921≺ 1198922in 1199042 there must be 119892

1≺ 1198922in 1199041

Property 3 illustrates that during the planning processselecting the equivalent GAO to achieve an atomic goalintroduces the same possible FGOs into the planning

Definition 9 (independent goal set) For the planning prob-lem (119874 119868 119866) the goal set119866 can be divided into 119896 independentgoal sets written as 119866

1 1198662 119866

119870 (1 ⩽ 119896) while 119866

1cup

1198662cup 119866

119896= 119866 and 119866

119894cap 119866119895= 0 (for all 1 ⩽ 119894 119895 ⩽ 119896) 119866

119894

and 119866119895are called independent with each other if and only if

for any 119892119894isin 119866119894and 119892

119895isin 119866119895 there is no excludable constraint

between the GAO of 119892119894and 119892

119895in each reachable state

Based on the above discussion it can be concluded thatFGOs occur just because there are excludable constraintsamong the GAOs of different atomic goals Take the instance

shown in Figure 1 as a example In the initial state the avail-able GAO for atomic goal (painted tile(31) black) is (paint-uprobot1 tile(31) tile(21)) The available GAOs for atomic goal(painted tile(21) white) are (paint-up robot1 tile(21) tile(11))and (paint-down robot1 tile(21) tile(31)) as there are

(paint-up robot1 tile(21) tile(11)) 999424999426999456 (paint-up robot1tile(31) tile(21))(paint-down robot1 tile(21) tile(31)) 999424999426999456 (paint-uprobot1 tile(31) tile(21))So there is (painted tile(31) black) ≺ (painted tile(21)white) in the initial state

Furthermore for the planning which has no FGO in theinitial state the excludable constraints among the GAOs mayintroduce FGOs into the planning process As the exampleshown in Figure 3 in the initial state each antiship missilecan be intercepted by a SAM or chaff However as there are

(Chaff-Intercept (21)) 999424999426999456 (SAM-Intercept (11))(Chaff-Intercept (21)) 999424999426999456 (SAM-Intercept (22))the FGOs (intercept (12)) ≺ (intercept (11)) and(intercept (12)) ≺ (intercept (22)) occur

Obviously for the planning problem (119874 119868 119866) if for all119892119894 119892119895isin 119866 (119894 = 119895) while 119892

119894is independent with 119892

119895as

Definition 9 described no FGO may occur during the plan-ning process

Next a forward search algorithm is proposed based onthe above discussion to solve the planning problem with dif-ferent FGOs

5 A Novel Forward Planning Algorithm

During the planning process selecting aGAOwith the biggerexcludable GAO set to achieve an atomic goal has the higherprobability to introduce FGOs into the planning Generallyfor the same atomic goal a search algorithm prefers to selectthe GAOwith the smaller excludable GAO set first to achieveit However for some planning problem keeping to selectthe GAO with smaller excludable GAO set first may causecertain operation resource excessively consumed In this casethe later planning process can only select the GAO with thebigger excludable GAO set to achieve each atomic goalThenplanning may lead to a deadlock as the FGOs introduced bythe excludable GAOsTherefore with respect to the planningalgorithm proposed in this paper the atomic goal as 119892

1with

the fewest number of available GAO is firstly selected to beachieved by an available GAO with the biggest excludableGAO setThen calculate the number of availableGAO for theremaining unachieved atomic goals The atomic goal whosenumber of available GAO is decreased after the achievementof1198921 is selected to be achievedwith high priority Continuing

the above process if planning arrives at a state which containsan unachieved atomic goal without available GAO move theachievement sequence of this atomic goal ahead and findan available GAO with the smallest excludable GAO set toachieve it while ensuring that all the prior achieved goals can


also be achieved by the prior selectedGAOs (or the equivalentGAOs)

The excludable GAO set based forward search algorithmEx MsFS (multi-step forward search) for the planning withFGOs is displayed in Algorithm 1 In the initial state 119904 theplanning selects an atomic goal with the fewest number ofavailable GAO to be achieved first (step 04) With respect tothe selected atomic goal as 119892 the available GAO as 119900

119892with

the biggest excludable GAO set is selected with the highestpriority to achieve 119892 (step 08) Then in the successor state1199041015840 after 119892 achieved (step 09) there exist the following twocases In case one there is an unachieved atomic goal as 1198921015840which has no availableGAO in 1199041015840 In this case 1199041015840 is a deadlockAll possible available GAOs for 1198921015840 in 1199041015840 are excluded by theGAOswhich have been adopted to achieve the earlier selectedatomic goals contained in 119871 So the achievement sequenceof 1198921015840 should be moved ahead (step 12) The detailed movingalgorithm is lately discussed in Algorithm 2 For case two(step 20) if all unachieved goals in 1199041015840 have available GAOthe atomic goals which have fewer available GAOs in 1199041015840 thanthat in 119904 should be selected to be achieved with high prioritystarting from 119904

1015840 by the depth-first search strategy (steps 2307)

The algorithm119872119900V119890 119886ℎ119890119886119889 used in step 12 of Algorithm 1is displayed in Algorithm 2 For the step 11 of Algorithm 1when there is an atomic goal as 119892 having no available GAOin current state it means that certain achieved goals stored inlist 119871 (step 09 of Algorithm 1) should not be achieved before119892 So the achievement sequence of 119892 should be tried to moveahead Suppose there are 119899119906119898 elements contained in 119871 Eachelement is written as (1199041015840 119900

119892 119904) whichmeans that selecting the

available GAO 119900119892in current state 119904 to achieve goal 119892 transfers

the state to the successor 1199041015840 The atomic goals contained inthe list 119871 are achieved by the sequence from the head to theend The algorithm tries to set 119892 as the 119894th (1 ⩽ 119894 lt 119899119906119898)goal to be achieved (step 02) The atomic goals stored in 119871from the location index 1 to 119894 minus 1 are still achieved by thepreviously selected GAOs (steps 04ndash06) Then it is the turnto select an available GAO to achieve 119892 As to ensure thatall the atomic goals stored in list 119871 from the location index119894 to 119899119906119898 can still be achieved by its previously selected GAOsor the equivalent GAOs of the previously selected GAOsthe algorithm chooses an available GAO with the smallestexcludable GAO set to achieve 119892 (step 09ndash17) Now thealgorithm starts to check whether the atomic goals stored inlist 119871 from the location index 119894 to 119899119906119898 can still be achievedby its previously selected GAOs or whose equivalent GAOs(steps 21ndash35) If it is it means that with respect to the atomicgoals in 119871 119892 can be taken as the 119894th goal to be achieved Sothe algorithm returns 119905119903119906119890 (steps 31ndash33) Otherwise move 119892ahead as the (119894minus1)th goal to be achieved If achieving 119892 beforeall the atomic goals in list 119871 still cannot ensure that the atomicgoals stored in list 119871 can be achieved by their previouslyselected GAOs or whose equivalent GAOs the Move aheadalgorithm returns 119891119886119897119904119890 (step 36)

Based on Algorithms 1 and 2 and the Definition 9 it canbe inferred that for each do while loop (steps 06ndash24) of thealgorithm Ex MsFS the achieved goals contained in each

list 119871 defined in step 03 of Algorithm 1 come from the sameindependent goal set During the depth-first search processall the atomic goals which are related with each selectedgoal (in step 04 of Algorithm 1) are stored in each list 119876Obviously the atomic goals in each list 119876 are from the sameindependent goal set For themoving ahead process displayedin Algorithm 2 the requirement that all the achieved goalsin each list 119871 can still be achieved by their previouslyselected GAO or their equivalent GAO after the achievementsequence of certain atomic goal is moved ahead is to ensurethat with respect to each list 119871 the elements having containedin list 119876 would not be changed during the moving aheadprocess The reason is that the equivalent GAO makes anatomic goal lose the same number of available GAOs So thedepth-first process could not be inferred by themoving aheadprocess

Figure 5 gives an example of the air defense for a navalgroup to illustrate the search process of Ex MsFS In theinitial state there are 6 antiship missiles There are 5 chaffsand a SAM that can be used to intercept all the antishipmissiles while the antiship missile in rectangle (21) mustbe intercepted by a chaff As there is only one availableGAO (119862ℎ119886119891119891 119868119899119905119890119903(2 1)) for the missile in (21) Ex MsFSselects the missile in (21) to firstly intercept by a chaff andtransfers the state to 119904

1(step 04 of Algorithm 1) Element

(1199041 119862ℎ119886119891119891 119868119899119905119890119903(2 1) 119868) is pushed into list 119871 (steps 08-09

of Algorithm 1) When the missile in (21) is intercepted bya chaff the rectangles (11) and (22) are interfered by thechaff cloud coming from rectangle (21) So the missiles in(11) and (22) cannot be intercepted by SAMThe number ofavailableGAOs formissiles in (11) and (22) in 119904

1is fewer than

that of in 119868 Therefore missiles in (11) and (22) are pushedback into list 119876 (steps 22-23 of Algorithm 1) and assignedthe higher priority to be intercepted during in the depth-firstprocess Suppose that missile in (22) is firstly selected to beintercepted by a chaff in state 119904

1(step 07) Planning leads to

1199042and element (119904

2 119862ℎ119886119891119891 119868119899119905119890119903(2 2) 119904

1) is pushed back into

list 119871 (step 09 of Figure 4) At the same time missiles in (12)and (23) are pushed back into list119876 (step 22 of Algorithm 1)

Continuing the above process element (1199043 119862ℎ119886119891119891 119868119899119905119890119903(2

3) 1199042) is pushed back into list 119871 andmissiles in (13) is pushed

back into list 119876 Now the planning arrives at state 1199043and

list 119876 pops back the missile in (13) to intercept As rectangle(13) is interfered by the chaff cloud from rectangle (23) themissile in (31) can only be intercepted by a chaff In this caserectangle (23) should be interfered by the chaff cloud fromrectangles (22) and (13) as the state 119904

4shows So there is no

available GAO for the missile in (13) and the interceptionsequence of the missile in (13) needs to be moved ahead(steps 11-12 of Algorithm 1)

Now the move ahead algorithm tries to intercept themissile in (12) before that of in (13) It means to interceptthe missile in (12) using (119862ℎ119886119891119891 119868119899119905119890119903(1 2)) starting fromstate 119904

3 This selection may lead the search to state 119904

5 in

which missiles in (11) and (13) cannot be intercepted Sothe intercept sequence of missile in (12) should be furthermoved ahead Themove ahead algorithm tries to intercept itbefore themissile in (23) which leads the planning to state 119904

6

Obviously the missile in (11) cannot be intercepted starting


Input planning problem (119874 119868 119866)Output plan 120587 with 119866 sube 119903119890119904119906119897119905(119868 119901)01 119904 = 119868 1198661015840 = 0 120587 = 002 do03 119876 = 0 119871 = 004 select not119892 isin 119904 with the fewest number of available GAO05 119876push back((119904 119892)) 119905119890119898

119904larr 119904

06 do07 119904 larr 119876pop back()119904 119892 larr 119876pop back()11989208 select an available GAO 119900

119892with the biggest excludable GAO set

09 1199041015840larr 119903119890119904119906119897119905(119904 120587

119900119892

) 119871push back((1199041015840 119900 119904))10 for each not1198921015840 isin 119904101584011 if 1198921015840 does not have available GAO in 119904101584012 if Move ahead(119871 1198921015840 1198711015840)==true13 119871clear() 119871 larr 119871

1015840 1199041015840 larr 119871back()119904101584014 else15 if 1198661015840contain(1198921015840)== true16 return FAILURE17 else18 119876clear() 119871clear() 1198661015840insert(1198921015840)19 119876push back((119905119890119898 119904 1198921015840)) go to step 0520 for each not1198921015840 isin 119904101584021 if Available GAOs of 1198921015840 in 1199041015840 is fewer than that of in 11990422 if 119876contain((1199041015840 1198921015840))==false23 119876push back((1199041015840 1198921015840))24 while(119876 = 0)25 for 119894 = 0 119871size()-126 120587push back(119871[119894]119900)27 119904 larr 119904

1015840 119871clear()28 while(119866 sube 119904)29 return120587

Algorithm 1 Ex MsFS algorithm

from 1199046 So the missile in (12) should be intercepted before

the missile in (22) It means intercepting the missile in (12)starting from 119904

1 Now the available GAO (119878119860119872 119868119899119905119890119903(1 2))

with the smaller excludableGAOset is selected Furthermoreall the missiles in (22) (23) and (13) can still be interceptedby its previously selected GAOs After the moving ahead pro-cess planning arrives at state 119904

8 from which the goal state

can be arrived after the missile in (11) is intercepted by thelast one chaff

6 Discussion of the Problem and Algorithm

This section proposes some properties about the planningwith FGOs and the search algorithm Ex MsFS

Property 4 The complexity for solving the planning withFGOs all of which exist in the initial state and are irrelevantwith the approach for each atomic goal to be achieved is P119899

119899

where 119899 = |119866|

Proof As all the FGOs exist in the initial state and are irrele-vant to the approach for each atomic goal to be achieved thesearch can arrive at the goal state if and only if all the atomicgoals are achieved by a correct sequence It is a completepermutation problem So the complexity is P119899

119899

Property 5 Thecomplexity for solving the planning problemin which there is no FGO in the initial state but FGOs wouldoccur during the planning process if and only if certain goal isachieved by an inappropriate approach is up to P119899

119899sdot119870119899 where

119899 = |119866| and119870 is the average number of approach that can beadopted to achieve each atomic goal

Proof In this case all the atomic goals should be achieved in acorrect sequence and each atomic goal should be achieved byan appropriate approach However as there is no FGO in theinitial state not all the atomic goals need to take part in thecomplete permutation So the complexity is up to P119899

119899sdot119870119899

For the Floortile problem the solving complexity is P119899119899

while the air defense planning problem for a naval group isup to P119899

119899sdot 119870119899

Property 6 For a solvable problem with FGOs the Ex MsFSalgorithm is sufficient to returning a plan solution if themove ahead algorithm returns 119905119903119906119890 (step 12 of Algorithm 1)for each time it is called

Proof If the move ahead algorithm returns 119905119903119906119890 (step 12 ofAlgorithm 1) for each time that it is called it means the searchprocess which violates certain FGO constraints has been put


Input Achieved goal sequence list 119871 Goal 119892 whose achievementsequence needs to be moved aheadOutput New goal achievement sequence list 1198711015840

01 bool 119904119906119888119888119890119904119904 119898119900V119890119889=false 119899119906119898 larr 119871size()02 for 119894 = 119899119906119898103 1198711015840 larr 004 for (119896 = 1 119896 ⩽ 119894 minus 1 119896 + +)05 119904119905119886119905119890 119900119901 119901119886119894119903 larr 119871[119896]06 1198711015840push back(119904119905119886119905119890 119900119901 119901119886119894119903)07 119904

1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898

119892(1199041) is the available GAO

set of 119892 in 1199041 For forall119900 1199001015840 isin 119874119899

119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)

The GAO in set 1198741198991119892(1199041) has the smaller excludable GAO

set than that of in 1198741198992119892(1199041) if there is 1 ⩽ 119899

1lt 1198992⩽ 119898

10 for 119897 =111989811 select an available GAO 1199001015840

119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)

13 if (existnot11989210158401015840 isin 1199042 11989210158401015840 has no available GAO in 119904

2)

14 continue15 else16 ℎ119886V119890 119900119901=true17 break18 if (ℎ119886V119890 119900119901==true)19 1198711015840push back((119904

2 1199001015840119892 1199041))

20 bool 119894119904 119887119903119890119886119896 =false21 for 119895 = 119894 11989911990611989822 119900 larr 119871[119895]11990023 if (119900 or its equivalent GAO 1199001015840 is available in 119904

2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))

28 else29 119894119904 119887119903119890119886119896=true30 break31 if (119894119904 119887119903119890119886119896==false)32 119904119906119888119888119890119904119904 119898119900V119890119889 =true33 return 119904119906119888119888119890119904119904 119898119900V11989011988934 else35 continue36 return 119904119906119888119888119890119904119904 119898119900V119890119889

Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)

right So the search process can arrive at the goal state andreturn a plan solution

For some planning problem the search may arrive atcertain state from which no matter which atomic goal isfirstly selected to achieve the search should arrive at a dead-lock In this case the move ahead algorithm always returns119891119886119897119904119890 and the Ex MsFS algorithm cannot return the plansolution Therefore further works need to be done to extendthe Ex MsFS algorithm to a more general case

Property 7 Evaluated by the Relaxed Graph [2] Ex MsFSalgorithm is an enforce hill-climbing algorithmwhich climbsmultiple steps each time

Proof Suppose119867(119904) is the distance calculated by the RelaxedPlan Graph between the reachable state 119904 and the goal state119866 where119867(119866) = 0 In the estimator of Relaxed Graph

119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)

where 119892119894isin 119866 (1 ⩽ 119894 ⩽ 119899) is the unachieved atomic goal

in 119904 and ℎ119892119894

(119904) is the number of actions in the relaxedgraph to achieve the atomic goal 119892

119894starting from 119904 Set 119874

119904

contains actions shared by different atomic goals during theirachievements in the relaxed graph where |119900| is the frequencyof 119900 that has been shared For each atomic goal as 119892

119894selected

in step 04 of Algorithm 1 the Ex MsFS algorithm selects an


s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception

cloud from one direction

from more than one direction

((s1 Chaff inter(2 1)) I) ((s2 Chaff inter(2 2)) s1)

((s3 Chaff inter(2 3)) s2) ((s4 Chaff inter(1 3)) s3)


((s7 Chaff inter(1 2)) s1)

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

interfered by the chaff

interfered by the chaff cloud

Figure 5 The example of air defense for a naval group

available GAO and generates a plan to achieve 119892119894 which

transfers the search from the current state 119904 to the successorstate 1199041015840 As action deletion is not considered in the relaxedgraph there are ℎ

119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)

Suppose that119892119894has 119896 independent actions to other atomic

goals in the Relaxed Graph starting from 119904 The value ofsum119874119904

(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894

(119904) minus 119896 Therefore for eachselected atomic goal there is 119867(119904) minus 119867(1199041015840) = 119896 The search

direction of the Ex MsFS algorithm is along the direction ofthe enforced hill climbing by 119896 steps at a time toward a goalstate

7 Evaluation

This paper evaluates the proposed algorithm Ex MsFS bycomparing it with the following planning systems

(1) FF planning system [2] in which the GAM heuristicis used for the detection of goal ordering

(2) SGPlan6 [13] which adopts the incremental planningprocess to handle the goal orderings


(3) LAMA 2008 planning system [5] which adopts thehill-climbing strategy where the preferred actionsof each step are selected by the FF and landmarkheuristics Moreover the orderings of (disjunction)landmarks are calculated based on the domain tran-sition graph and the causal graph

FF SGPLan6 and LAMA 2008 won the 1st prize ofthe satisficing planning track in IPC 2001 2006 and 2008respectively In addition the LAMA 2008 planning systemconsists of translating processing and searching modulesThe translating and processingmodules are used to constructsome structured data based on which searching moduleis employed for forward search In this paper in order tocompare the Ex MsFS with the previous approaches we usethe Ex MsFS algorithm to replace the search modules of theLama 2008 planning system

The competition domains are the Floortile proposed inIPC 2011 and the air defense of naval group problem intro-duced in Section 1 For an air defense of naval group problemthere are119898times 119899 antiship missiles while |(119898 times 119899)3| SAMs and119898times119899minus|(119898times119899)3| chaffs can be used As this paper focuses onthe efficiency when to solve a planning problem with FGOsthe quality of the planning solution is not considered So weevaluate each planner by the scale of planning problems itcan solve The experiments are implemented in the Mac OSX with 24GHz Intel Core 2 Duo and 2GB 1067MHz DDR3memory Each instance fails if the running time is more than1200 seconds

Figure 6 shows the running time curves of the fourdifferent search approaches on the Floortile domain Thereare 20 instances while LAMA 2008 can only solve 3 instancesand FF can only solve 4 instances However SGPlan6 andthe Ex MsFS algorithm can solve all of the instances More-over in this domain SGPlan6 can solve each instance withthe fewer time cost than Ex MsFS The reason is that theincremental planning process is very suitable in handling theFGOs of the Floortile domain Taking the instance displayedin Figure 1 as an example the incremental planning processfirst paints the upest tiles as (31) (32) and (33) Then itpaints the tiles (21) (22) and (23) At last it paints thetiles (11) (12) and (13) This process satisfies the FGOconstraints

For the air defense of naval group problem SGPlan6can only solve only 1 out 0f the 17 instances while FF andLAMA 2008 can solve 10 instances Moreover as there aretime constraints during the air defense process of the navalgroup in real world if we require that each instance shouldbe solved within 20 seconds FF can solve only 7 instancesand LAMA 2008 can only solve 6 instances Howeverthe Ex MsFS approach can solve all of the 17 instancesand return the solution plan for each instance within 10seconds

Based on Figure 7 it can be inferred that air defense ofnaval group problem is a new challenge for many currentplanners and Ex MsFS can solve the planning problems withdifferent FGOs efficiently

1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)

Figure 6 Planning results of the Floortile

1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS

Figure 7 Planning results of air defense of naval group problem

8 Conclusions and Future Works

This paper contributes to introduce a new planning domainwith FGOs for which all the current related planners do notperform well when solving it Then a new search algorithmis proposed which can solve the planning problem withdifferent FGOs efficiently

For future works the Ex MsFS should be extended to amore general case to solve more real-world problems withFGOs Also machine learning approach [16 17] can beemployed to obtain more informed about FGOsThe learnedknowledge can improve the efficiency of the search processMoreover a human-computer method is considered forplanning with FGO constraints As some FGO constraintsmay easily be inferred by humans and other FGO constraintscan be learned easily by computers a human-computermethod can be employed in amixture of automatic and hand-crafted control rules [18] In addition if two different domainshave similar subgoals interaction the knowledge of deadlockchecking leaned from one domain can be considered trans-ferrable to other domains [19]

Acknowledgment

Thanks are due to the NSF of China which supported theresearch with Grants nos 71001105 91024006 and 61273322


References

[1] J Koehler and J Hoffmann ldquoOn reasonable and forced goalorderings and their use in an agenda-driven planning algo-rithmrdquo Journal of Artificial Intelligence Research vol 12 pp 339ndash386 2000

[2] J Hoffmann and B Nebel ldquoThe FF planning system fast plangeneration through heuristic searchrdquo Journal of Artificial Intel-ligence Research vol 14 pp 253ndash302 2001

[3] J Hoffmann J Porteous and L Sebastia ldquoOrdered landmarksin planningrdquo Journal of Artificial Intelligence Research vol 22pp 215ndash278 2004


[5] S Richter andMWestphal ldquoThe LAMA planner guiding cost-based anytime planning with landmarksrdquo Journal of ArtificialIntelligence Research vol 39 pp 127ndash177 2010

[6] A Benaskeur E Bosse andD Blodgett ldquoCombat resource allo-cation planning in naval engagementsrdquo Tech Rep ValcartierCanada 2007

[7] T Morimoto How to Develop a RoboCupRescue AgentRoboCupRescue Technical Committee 2000

[8] A Beynier and A-I Mouaddib ldquoSolving efficiently Decentral-ized MDPs with temporal and resource constraintsrdquo Autono-mous Agents and Multi-Agent Systems vol 23 no 3 pp 486ndash539 2011

[9] J Hoffmann ldquoWhere ldquoignoring delete listsrdquo works local searchtopology in planning benchmarksrdquo Journal of Artificial Intelli-gence Research vol 24 pp 685ndash758 2005

[10] T De la Rosa Floortile Domain httpwwwplginfuc3mes2011

[11] B Bonet and H Geffner ldquoHSP heuristic search plannerrdquo inAIPS Planning Competition Pittsburgh 1998

[12] M Helmert ldquoThe fast downward planning systemrdquo Journal ofArtificial Intelligence Research vol 26 pp 191ndash246 2006

[13] Y Chen B W Wah and C-W Hsu ldquoTemporal planningusing subgoal partitioning and resolution in SGPlanrdquo Journalof Artificial Intelligence Research vol 26 pp 323ndash369 2006

[14] D H Hu and Q Yang ldquoCIGAR concurrent and interleavinggoal and activity recognitionrdquo in Proceedings of the 23rd AAAIConference on Artificial Intelligence and the 20th InnovativeApplications of Artificial Intelligence Conference (AAAI rsquo08) pp1363ndash1368 July 2008

[15] C-W Hsu Y Chen and B W Wah ldquoSubgoal ordering andgranularity control for incremental planningrdquo InternationalJournal on Artificial Intelligence Tools vol 16 no 4 pp 707ndash7232007

[16] QYangKWu andY Jiang ldquoLearning actionmodels fromplanexamples using weighted MAX-SATrdquo Artificial Intelligence vol171 no 2-3 pp 107ndash143 2007

[17] J Yin X Chai and Q Yang ldquoHigh-level goal recognition in awireless LANrdquo in Proceedings of the 19th National Conference onArtificial Intelligence (AAAI rsquo04) pp 578ndash583 July 2004

[18] F Bacchus and F Kabanza ldquoUsing temporal logics to expresssearch control knowledge for planningrdquo Artificial Intelligencevol 116 no 1-2 pp 123ndash191 2000

[19] D H Hu and Q Yang ldquoTransfer learning for activity recog-nition via sensor mappingrdquo in Proceedings of the 22nd Inter-national Joint Conference on Artificial Intelligence (IJCAI rsquo11)Barcelona Spain 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



2 Problem Statement

Before introducing forced goal ordering (FGO) we first givethe description of a planning problem (119874 119868 119866) as given in [1]

In a planning problem definition 119874 is a finite set ofground actions with the STRIPS style (in this paper) For any119900 isin 119874 there is 119900 = (119901119903119890 119886119889119889 119889119890119897) where 119901119903119890 119886119889119889 and119889119890119897 are finite sets of ground atoms 119901119903119890 is the preconditionset under which the action is applicable 119886119889119889 and 119889119890119897 are theatoms added or deleted after the execution of the action 119868 and119866 are the finite sets of ground atoms and represent the initialand goal state of the problem For any given state 119904 and action119900 isin 119874 the result of applying 119900 to 119904 is

119903119890119904119906119897119905 (119904 119900) = 119904 cup 119886119889119889 (119900) 119889119890119897 (119900) if 119901119903119890 (119900) sube 119904119904 otherwise

(1)

For the action sequence 1199001 1199002 119900

119899 (119900119894isin 119874 1 ⩽ 119894 ⩽ 119899)

there is

119903119890119904119906119897119905 (119904 1199001 1199002 119900

119899)

= 119903119890119904119906119897119905 (119903119890119904119906119897119905 (119904 1199001 1199002 119900

119899minus1) 119900119899)

(2)

The planning problem is finding a sequence of actions120587 suchthat 119866 sube 119903119890119904119906119897119905(119868 120587) The set of action sequence is definedas Π119874 A state 119904 is reachable from the initial state if and onlyif exist120587 isin Π119874 st 119904 sube 119903119890119904119906119897119905(119868 120587) Similarly an atom 119891 isachievable from a certain state 119904 if and only if 119891 isin 119904 orexist120587 isin Π119874 st 119891 isin 119903119890119904119906119897119905(119904 120587) An action 119900 is applicable ina reachable state 119904 if and only if 119901119903119890(119900) sube 119904

Definition 1 (forced goal ordering (FGO) [1]) For the plan-ning problem (119874 119868 119866) let 119892 1198921015840 isin 119866 be the atomic goals Wesay that there is a forced ordering between119892 and1198921015840 written as1198921015840 ≺ 119892 if and only if for any state 119904(119892 not1198921015840) there is no plan120587 satisfying 1198921015840 isin 119903119890119904119906119897119905(119904(119892 not1198921015840) 120587) 119904(119892 not1198921015840) represents areachable state in which 119892 has just been achieved but 1198921015840 isfalse

In any given state an atomic goal remaining false meansthat the goal is not achieved Definition 1 illustrates thatforward search arrives at deadlock 119904(119892 not1198921015840) when there is1198921015840 ≺ 119892 and the atomic goal 119892 is achieved before 1198921015840 In some ofthe literatures [2 9] 119904(119892 not1198921015840) is called a dead-end state tooDuring the planning process forward search which violatesany of the FGOs may lead the planning to a deadlock fromwhich there is no way to the goal state

With respecting to the FGO there is goal ordering definedas reasonable goal ordering (RGO) written as 1198921015840≺

119903119892 There is

a reasonable ordering between 119892 and 1198921015840 if and only if forany reachable state 119904(119892 not1198921015840) there is no longer a plan thatcan achieve 1198921015840 from 119904(119892 not1198921015840) without deleting 119892 at leasttemporarily [1] So if there is 1198921015840≺

119903119892 and 119892 is achieved before

1198921015840 in order to get a plan solution the planning must firstdelete the achieved goal 119892 then to achieve 1198921015840 and last toachieve 119892 again In this paper as to focus the problem onFGO it is supposed that there is no goal deletion during theplanning process Each goal cannot be deleted once it hasbeen added

(3 1)

(0 3)(0 2)(0 1)

(1 3)(1 2)(1 1)

(2 3)(2 2)(2 1)

(3 3)(3 2)

Initial state

(3 1)

(0 3)(0 2)(0 1)

(1 3)(1 2)(1 1)

(2 3)(2 2)(2 1)

(3 3)(3 2) (3 1)

(0 3)(0 2)(0 1)

(1 3)(1 2)(1 1)

(2 3)(2 2)(2 1)

(3 3)(3 2)

goal state given state

Figure 1 Floortile domain in IPC 2011

The first planning domain for planning competition withFGOs is the Floortile proposed in the International PlanningCompetition (IPC) 2011 [10] During the competition noparticipating planner in the sequential satisficing track cansolve it well Figure 1 shows an example of the Floortiledomain in the IPC 2011 In the initial state the status ofall floor tiles is clear Floor tiles need to be painted blackand white while adjacent tiles should have different colorsRobots can only paint tiles that are in front (up) or behind(down) Moreover once a tile is painted a robot cannot standon it This particular configuration makes the domain veryhard to solve because of the existence of FGOs For examplesuppose a robot first selects the tile(21) to paint in whiteIn further planning steps the robot can only stand on atile(21) to paint the front tile(31) in black This process canbe achieved if and only if the atom (robot-at tile(21)) is trueHowever this atom is not true and cannot be added once thetile(21) has been paintedThe reason is that the atom (robot-at tile(21)) ismutually exclusivewith the atomic goal (paintedtile(21) white) and (painted tile(21) white) cannot be deletedonce it has been added Therefore painting tile (21) before(31) violates the FGO constraint in consequence causing thesearch to arrive at a deadlock

In this example there are many FGOs in the Floortileproblem and the robots should paint tiles obeying a correctsequence In the case of Figure 1 the FGOs are

(painted tile(31) black) ≺ (painted tile(21) white) ≺(painted tile(11) black)(painted tile(32) white) ≺ (painted tile(22) black) ≺(painted tile(12) white)(painted tile(33) black) ≺ (painted tile(23) white) ≺(painted tile(13) black)

In the above domain all the FGOs exist in the initialstate No matter which approach is adopted to achieve anatomic goal all the atomic goals should be achieved in agiven sequence starting from the initial state Otherwise theplanning may arrive at a deadlock However in some real-world planning problem there is no FGO in the initial stateFor any 119892 1198921015840 isin 119866 planning starting from the initial state tofirstly achieve 119892 or 1198921015840 would not lead to a deadlock Howeverthe planning may arrive at a given state 119904(not119892

1 not1198922 not1198923)

(1198921 1198922 1198923isin 119866) without FGO among 119892

1 1198922 and 119892

3 Now




1while translating


2≺ 1198923or

1198923≺ 1198922



3 Related Works


(2 1)

1 2

1

2

(1 1)

(2 2)

(1 2)

(2 1)

1 2

1

2

(1 1)

(2 2)

(1 2)





⋂119900isin119874 119892isin119886119889119889(119900)


119865119892

119863119892= 0 where 119874






|119866|atomic goal



1 1199002 119900

119899 isin 119875119874 and 119866 sube

119903119890119904119906119897119905(119868 119901) there is 119897 isin 119903119890119904119906119897119905(119868 1199001 119900

119894) (1 ⩽ 119894 ⩽




⋂forall119900isin119874 119892isin119886119889119889(119900)



g11g12g13

g11g12g13

g22

g21g11g12g13

g22

g21

g33g32g31

(I O g11 g12 g13)

(s1 O g21 g22)

(s2 O g31 g32 g33)

s1 s2

s3





4 Why FGOs Occur


119892



119892) sube

119903119890119904119906119897119905(119904 120587)


119900119892

while 119892 isin 119903119890119904119906119897119905(119904 120587119900119892

)So if 119901119903119890(119900

119892) sube 119904 there is 120587

119900119892

= 119900119892 Otherwise if 119901119903119890(119900

119892) sube

119903119890119904119906119897119905(119904 120587) there is 120587119900119892

= 120587 119900119892


119894isin 119874 (1 ⩽ 119894 ⩽ |1198661015840|) for each 119892

119894isin 1198661015840 in 119904(not1198661015840) (|1198661015840|



1199001198952

119900119895|1198661015840|

(119895119894= 1 2 |1198661015840| 119895

119896= 119895119897if 119896 = 119897 1 ⩽ 119894


1198951

1205871198952

120587119895|1198661015840|

such that

119901119903119890 (1199001198951

) sube 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

)

1198921198951

isin 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

1199001198951

) = 1199041

119901119903119890 (1199001198952

) sube 119903119890119904119906119897119905 (1199041 1205871198952

)

1198921198952

1198921198951

sube 119903119890119904119906119897119905 (1199041 1205871198952

1199001198952

) = 1199042

119901119903119890 (119900119895|1198661015840|

) sube 119903119890119904119906119897119905 (119904|1198661015840|minus1 120587119895|1198661015840|

)

1198661015840sube 119903119890119904119906119897119905 (119904

|1198661015840|minus1 119901119895|1198661015840|

119900119895|1198661015840|

)

(3)



120587 = 1205871198951

1199001198951

1205871198952

1199001198952

120587119895|1198661015840|

119900119895|1198661015840|

(4)




1198921

for 1198921and

1199001198922


1 not1198922)

written as 1199001198921

999424999426999456 1199001198922

if and only if 1199001198922


1 not1198922) 1205871199001198921

)



119900119892


119892999424999426999456 1199001198921015840 (1198921015840 isin 1198661015840 1198921015840 = 119892) there is 119900

1198921015840 isin 119874119900119892

(119904(not1198661015840))(2) for any 119900

1198921015840 isin 119874

119900119892


1199001198921015840 notin 119874119900119892

(119904(not1198661015840)) there is not 119900119892999424999426999456 1199001198921015840


119892and 1199002



119892cong 1199002119892 if

and only if 1198741199001

119892

(119904(not119892)) = 1198741199002

119892

(119904(not119892))




2 120587) and


119892with


1205871and 120587

2satisfying 119901119903119890(119900

119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

1) and

119901119903119890(119900119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

2) However 119903119890119904119906119897119905(119904(not119892) 120587

1)





1199001205871119892

and 1199001205872119892



119892 for any 120587

1 1205872isin Π119874


1) and 119901119903119890(119900

119892) sube


1 119900119892) cong

119903119890119904119906119897119905(119904(not119892) 1205872 119900119892)



119892 there are 119904

1=

119903119890119904119906119897119905(119904(not1198661015840) 1205871199001

119892

) and 1199042= 119903119890119904119906119897119905(119904(not119866

1015840) 1205871199002

119892




1cong 1199042




1 and 1198742

1198661015840119892




11987411198661015840119892= 11987421198661015840119892 As 119904





1cong

1199042


119892and 1199002




1and selecting



1 not1198922isin 119904119894


1≺ 1198922

in 1199042(1199041)





2before 119892

1 As 119904

1and



1too So there



1≺ 1198922in 1199041



1 1198662 119866

119870 (1 ⩽ 119896) while 119866

1cup

1198662cup 119866

119896= 119866 and 119866

119894cap 119866119895= 0 (for all 1 ⩽ 119894 119895 ⩽ 119896) 119866

119894


for any 119892119894isin 119866119894and 119892











119895as





1with






119892with













1is fewer than



1199042and element (119904

2 119862ℎ119886119891119891 119868119899119905119890119903(2 2) 119904











5 in


6




119904larr 119904



09 1199041015840larr 119903119890119904119906119897119905(119904 120587

119900119892














119899

where 119899 = |119866|


119899


119899sdot119870119899 where



119899sdot119870119899



119899sdot 119870119899






1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898



119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)



1lt 1198992⩽ 119898


119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)


2)


2 1199001015840119892 1199041))


2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))


Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)





119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)


in 119904 and ℎ119892119894



119904


119894selected



s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1while translating


2≺ 1198923or

1198923≺ 1198922



3 Related Works


(2 1)

1 2

1

2

(1 1)

(2 2)

(1 2)

(2 1)

1 2

1

2

(1 1)

(2 2)

(1 2)





⋂119900isin119874 119892isin119886119889119889(119900)


119865119892

119863119892= 0 where 119874






|119866|atomic goal



1 1199002 119900

119899 isin 119875119874 and 119866 sube

119903119890119904119906119897119905(119868 119901) there is 119897 isin 119903119890119904119906119897119905(119868 1199001 119900

119894) (1 ⩽ 119894 ⩽




⋂forall119900isin119874 119892isin119886119889119889(119900)



g11g12g13

g11g12g13

g22

g21g11g12g13

g22

g21

g33g32g31

(I O g11 g12 g13)

(s1 O g21 g22)

(s2 O g31 g32 g33)

s1 s2

s3





4 Why FGOs Occur


119892



119892) sube

119903119890119904119906119897119905(119904 120587)


119900119892

while 119892 isin 119903119890119904119906119897119905(119904 120587119900119892

)So if 119901119903119890(119900

119892) sube 119904 there is 120587

119900119892

= 119900119892 Otherwise if 119901119903119890(119900

119892) sube

119903119890119904119906119897119905(119904 120587) there is 120587119900119892

= 120587 119900119892


119894isin 119874 (1 ⩽ 119894 ⩽ |1198661015840|) for each 119892

119894isin 1198661015840 in 119904(not1198661015840) (|1198661015840|



1199001198952

119900119895|1198661015840|

(119895119894= 1 2 |1198661015840| 119895

119896= 119895119897if 119896 = 119897 1 ⩽ 119894


1198951

1205871198952

120587119895|1198661015840|

such that

119901119903119890 (1199001198951

) sube 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

)

1198921198951

isin 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

1199001198951

) = 1199041

119901119903119890 (1199001198952

) sube 119903119890119904119906119897119905 (1199041 1205871198952

)

1198921198952

1198921198951

sube 119903119890119904119906119897119905 (1199041 1205871198952

1199001198952

) = 1199042

119901119903119890 (119900119895|1198661015840|

) sube 119903119890119904119906119897119905 (119904|1198661015840|minus1 120587119895|1198661015840|

)

1198661015840sube 119903119890119904119906119897119905 (119904

|1198661015840|minus1 119901119895|1198661015840|

119900119895|1198661015840|

)

(3)



120587 = 1205871198951

1199001198951

1205871198952

1199001198952

120587119895|1198661015840|

119900119895|1198661015840|

(4)




1198921

for 1198921and

1199001198922


1 not1198922)

written as 1199001198921

999424999426999456 1199001198922

if and only if 1199001198922


1 not1198922) 1205871199001198921

)



119900119892


119892999424999426999456 1199001198921015840 (1198921015840 isin 1198661015840 1198921015840 = 119892) there is 119900

1198921015840 isin 119874119900119892

(119904(not1198661015840))(2) for any 119900

1198921015840 isin 119874

119900119892


1199001198921015840 notin 119874119900119892

(119904(not1198661015840)) there is not 119900119892999424999426999456 1199001198921015840


119892and 1199002



119892cong 1199002119892 if

and only if 1198741199001

119892

(119904(not119892)) = 1198741199002

119892

(119904(not119892))




2 120587) and


119892with


1205871and 120587

2satisfying 119901119903119890(119900

119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

1) and

119901119903119890(119900119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

2) However 119903119890119904119906119897119905(119904(not119892) 120587

1)





1199001205871119892

and 1199001205872119892



119892 for any 120587

1 1205872isin Π119874


1) and 119901119903119890(119900

119892) sube


1 119900119892) cong

119903119890119904119906119897119905(119904(not119892) 1205872 119900119892)



119892 there are 119904

1=

119903119890119904119906119897119905(119904(not1198661015840) 1205871199001

119892

) and 1199042= 119903119890119904119906119897119905(119904(not119866

1015840) 1205871199002

119892




1cong 1199042




1 and 1198742

1198661015840119892




11987411198661015840119892= 11987421198661015840119892 As 119904





1cong

1199042


119892and 1199002




1and selecting



1 not1198922isin 119904119894


1≺ 1198922

in 1199042(1199041)





2before 119892

1 As 119904

1and



1too So there



1≺ 1198922in 1199041



1 1198662 119866

119870 (1 ⩽ 119896) while 119866

1cup

1198662cup 119866

119896= 119866 and 119866

119894cap 119866119895= 0 (for all 1 ⩽ 119894 119895 ⩽ 119896) 119866

119894


for any 119892119894isin 119866119894and 119892











119895as





1with






119892with













1is fewer than



1199042and element (119904

2 119862ℎ119886119891119891 119868119899119905119890119903(2 2) 119904











5 in


6




119904larr 119904



09 1199041015840larr 119903119890119904119906119897119905(119904 120587

119900119892














119899

where 119899 = |119866|


119899


119899sdot119870119899 where



119899sdot119870119899



119899sdot 119870119899






1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898



119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)



1lt 1198992⩽ 119898


119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)


2)


2 1199001015840119892 1199041))


2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))


Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)





119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)


in 119904 and ℎ119892119894



119904


119894selected



s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




g11g12g13

g11g12g13

g22

g21g11g12g13

g22

g21

g33g32g31

(I O g11 g12 g13)

(s1 O g21 g22)

(s2 O g31 g32 g33)

s1 s2

s3





4 Why FGOs Occur


119892



119892) sube

119903119890119904119906119897119905(119904 120587)


119900119892

while 119892 isin 119903119890119904119906119897119905(119904 120587119900119892

)So if 119901119903119890(119900

119892) sube 119904 there is 120587

119900119892

= 119900119892 Otherwise if 119901119903119890(119900

119892) sube

119903119890119904119906119897119905(119904 120587) there is 120587119900119892

= 120587 119900119892


119894isin 119874 (1 ⩽ 119894 ⩽ |1198661015840|) for each 119892

119894isin 1198661015840 in 119904(not1198661015840) (|1198661015840|



1199001198952

119900119895|1198661015840|

(119895119894= 1 2 |1198661015840| 119895

119896= 119895119897if 119896 = 119897 1 ⩽ 119894


1198951

1205871198952

120587119895|1198661015840|

such that

119901119903119890 (1199001198951

) sube 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

)

1198921198951

isin 119903119890119904119906119897119905 (119904 (not1198661015840) 1205871198951

1199001198951

) = 1199041

119901119903119890 (1199001198952

) sube 119903119890119904119906119897119905 (1199041 1205871198952

)

1198921198952

1198921198951

sube 119903119890119904119906119897119905 (1199041 1205871198952

1199001198952

) = 1199042

119901119903119890 (119900119895|1198661015840|

) sube 119903119890119904119906119897119905 (119904|1198661015840|minus1 120587119895|1198661015840|

)

1198661015840sube 119903119890119904119906119897119905 (119904

|1198661015840|minus1 119901119895|1198661015840|

119900119895|1198661015840|

)

(3)



120587 = 1205871198951

1199001198951

1205871198952

1199001198952

120587119895|1198661015840|

119900119895|1198661015840|

(4)




1198921

for 1198921and

1199001198922


1 not1198922)

written as 1199001198921

999424999426999456 1199001198922

if and only if 1199001198922


1 not1198922) 1205871199001198921

)



119900119892


119892999424999426999456 1199001198921015840 (1198921015840 isin 1198661015840 1198921015840 = 119892) there is 119900

1198921015840 isin 119874119900119892

(119904(not1198661015840))(2) for any 119900

1198921015840 isin 119874

119900119892


1199001198921015840 notin 119874119900119892

(119904(not1198661015840)) there is not 119900119892999424999426999456 1199001198921015840


119892and 1199002



119892cong 1199002119892 if

and only if 1198741199001

119892

(119904(not119892)) = 1198741199002

119892

(119904(not119892))




2 120587) and


119892with


1205871and 120587

2satisfying 119901119903119890(119900

119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

1) and

119901119903119890(119900119892) sube 119903119890119904119906119897119905(119904(not119892) 120587

2) However 119903119890119904119906119897119905(119904(not119892) 120587

1)





1199001205871119892

and 1199001205872119892



119892 for any 120587

1 1205872isin Π119874


1) and 119901119903119890(119900

119892) sube


1 119900119892) cong

119903119890119904119906119897119905(119904(not119892) 1205872 119900119892)



119892 there are 119904

1=

119903119890119904119906119897119905(119904(not1198661015840) 1205871199001

119892

) and 1199042= 119903119890119904119906119897119905(119904(not119866

1015840) 1205871199002

119892




1cong 1199042




1 and 1198742

1198661015840119892




11987411198661015840119892= 11987421198661015840119892 As 119904





1cong

1199042


119892and 1199002




1and selecting



1 not1198922isin 119904119894


1≺ 1198922

in 1199042(1199041)





2before 119892

1 As 119904

1and



1too So there



1≺ 1198922in 1199041



1 1198662 119866

119870 (1 ⩽ 119896) while 119866

1cup

1198662cup 119866

119896= 119866 and 119866

119894cap 119866119895= 0 (for all 1 ⩽ 119894 119895 ⩽ 119896) 119866

119894


for any 119892119894isin 119866119894and 119892











119895as





1with






119892with













1is fewer than



1199042and element (119904

2 119862ℎ119886119891119891 119868119899119905119890119903(2 2) 119904











5 in


6




119904larr 119904



09 1199041015840larr 119903119890119904119906119897119905(119904 120587

119900119892














119899

where 119899 = |119866|


119899


119899sdot119870119899 where



119899sdot119870119899



119899sdot 119870119899






1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898



119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)



1lt 1198992⩽ 119898


119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)


2)


2 1199001015840119892 1199041))


2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))


Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)





119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)


in 119904 and ℎ119892119894



119904


119894selected



s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1199001205871119892

and 1199001205872119892



119892 for any 120587

1 1205872isin Π119874


1) and 119901119903119890(119900

119892) sube


1 119900119892) cong

119903119890119904119906119897119905(119904(not119892) 1205872 119900119892)



119892 there are 119904

1=

119903119890119904119906119897119905(119904(not1198661015840) 1205871199001

119892

) and 1199042= 119903119890119904119906119897119905(119904(not119866

1015840) 1205871199002

119892




1cong 1199042




1 and 1198742

1198661015840119892




11987411198661015840119892= 11987421198661015840119892 As 119904





1cong

1199042


119892and 1199002




1and selecting



1 not1198922isin 119904119894


1≺ 1198922

in 1199042(1199041)





2before 119892

1 As 119904

1and



1too So there



1≺ 1198922in 1199041



1 1198662 119866

119870 (1 ⩽ 119896) while 119866

1cup

1198662cup 119866

119896= 119866 and 119866

119894cap 119866119895= 0 (for all 1 ⩽ 119894 119895 ⩽ 119896) 119866

119894


for any 119892119894isin 119866119894and 119892











119895as





1with






119892with













1is fewer than



1199042and element (119904

2 119862ℎ119886119891119891 119868119899119905119890119903(2 2) 119904











5 in


6




119904larr 119904



09 1199041015840larr 119903119890119904119906119897119905(119904 120587

119900119892














119899

where 119899 = |119866|


119899


119899sdot119870119899 where



119899sdot119870119899



119899sdot 119870119899






1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898



119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)



1lt 1198992⩽ 119898


119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)


2)


2 1199001015840119892 1199041))


2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))


Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)





119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)


in 119904 and ℎ119892119894



119904


119894selected



s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119892with













1is fewer than



1199042and element (119904

2 119862ℎ119886119891119891 119868119899119905119890119903(2 2) 119904











5 in


6




119904larr 119904



09 1199041015840larr 119903119890119904119906119897119905(119904 120587

119900119892














119899

where 119899 = |119866|


119899


119899sdot119870119899 where



119899sdot119870119899



119899sdot 119870119899






1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898



119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)



1lt 1198992⩽ 119898


119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)


2)


2 1199001015840119892 1199041))


2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))


Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)





119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)


in 119904 and ℎ119892119894



119904


119894selected



s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119904larr 119904



09 1199041015840larr 119903119890119904119906119897119905(119904 120587

119900119892














119899

where 119899 = |119866|


119899


119899sdot119870119899 where



119899sdot119870119899



119899sdot 119870119899






1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898



119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)



1lt 1198992⩽ 119898


119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)


2)


2 1199001015840119892 1199041))


2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))


Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)





119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)


in 119904 and ℎ119892119894



119904


119894selected



s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






1larr 119871[119894]119904

08 bool ℎ119886V119890 119900119901=false09 119874

119892(1199041) = 119874

1

119892(1199041) 1198742

119892(1199041) 119874

119898



119892(1199041) there is 119900 cong 1199001015840(1 ⩽ 119899 ⩽ 119898)



1lt 1198992⩽ 119898


119892from 119874119897

119892(1199041)

12 1199042= 119903119890119904119906119897119905(119904

1 1205871199001015840

119892

)


2)


2 1199001015840119892 1199041))


2)

24 119905119890119898 119900 larr 119900 or 119905119890119898 119900 larr 119900101584025 1199041015840 = 119904

2

26 1199042= 119903119890119904119906119897119905(1199041015840 120587

119905119890119898 119900)

27 1198711015840push back((1199042 119905119890119898 119900 1199041015840))


Algorithm 2119872119900V119890 119886ℎ119890119886119889(119871 119892 1198711015840)





119867(119904) = ℎ1198921

(119904) + ℎ1198922

(119904) + sdot sdot sdot + ℎ119892119899

(119904) minus sum119874119904

(|119900| minus 1) (5)


in 119904 and ℎ119892119894



119904


119894selected



s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




s1 s2

s3 s4

s5 s6

s7 s8

Chaff interception

SAM interception







1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3

1

2

1 2

(2 1)

(1 1)

(2 2)

(1 2)

(2 3)

(1 3)

3






119892119894

(1199041015840) = 0 and

119867(119904) minus 119867 (1199041015840) = ℎ119892119894

(119904) minus (sum119874119904

(|119900| minus 1) minussum1198741199041015840

(|119900| minus 1))

(6)



(|119900| minus 1) minus sum1198741199041015840

(|119900| minus 1) is ℎ119892119894



7 Evaluation











1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










1000

100

10

1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

FFLAMA

SGPlan 6Ex MsFS

(s)


1000

100

10

1

01

001

(s)

Problem size

22 23 24 25 33 34 35 36 44 45 46 55 56 57 66 67 77

FFLAMA

SGPlan 6Ex MsFS





Acknowledgment



References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




References



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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