Research ArticleTask Assignment for Multi-UAV under Severe Uncertainty byUsing Stochastic Multicriteria Acceptability Analysis

Xiaoxuan Hu Jing Cheng and He Luo

School of Management Hefei University of Technology Hefei 230009 China

Correspondence should be addressed to Xiaoxuan Hu xiaoxuanhuhfuteducn

Received 25 January 2015 Accepted 28 July 2015

Academic Editor Yakov Strelniker

Copyright copy 2015 Xiaoxuan Hu 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

This paper considers a task assignment problem for multiple unmanned aerial vehicles (UAVs)The UAVs are set to perform attacktasks on a collection of ground targets in a severe uncertain environmentTheUAVs have different attack capabilities and are locatedat different positions Each UAV should be assigned an attack task before the mission starts Due to uncertain information manycriteria values essential to task assignment were random or fuzzy and the weights of criteria were not precisely known In thisstudy a novel task assignment approach based on stochastic Multicriteria acceptability analysis (SMAA) method was proposed toaddress this problem The uncertainties in the criteria were analyzed and a task assignment procedure was designed The resultsof simulation experiments show that the proposed approach is useful for finding a satisfactory assignment under severe uncertaincircumstances

1 Introduction

Unmanned aerial vehicles (UAVs) are playing increasinglyimportant roles in military and civilian applications Overthe last decade UAVs were employed as intelligent units invarious missions in dangerous and complex environmentsWhen executing multiple missions UAVs form teams andare able to work cooperatively In this context themulti-UAVcooperative control and decisionmechanisms including taskassignment path planning and tactical decision makinghave received a great deal of attention

The purpose of multi-UAV task assignment is to allocatenecessary tasks to UAVs so as to maximize the overallperformance Here tasks involve reconnaissance searchattack and verification In applications wherein a groupof UAVs executed complex missions the result of taskassignment directly determined how well the missions wereperformedThis problem has generated many research inter-ests and a variety of approaches have been developed fordifferent scenarios Methods like linear programming [12] dynamic programming (DP) [3] and Markov decisionprocesses (MDP) [4] were employed For solving techniquesthe genetic algorithm [5 6] and tabu search algorithm [7]were used

Researchers have also considered task assignment underuncertain circumstances Choi et al [8] addressed single andmultiple assignment problems by presenting 2 decentralizedalgorithms the consensus-based auction algorithm (CBAA)and the consensus-based bundle algorithm (CBBA) These2 algorithms were independent of inconsistencies in situa-tional awareness and could respectively produce conflict-free solutions to the assignment problem Bertuccelli et al [9]extended the CBBA to resolve the heterogeneous UAVs real-time task assignment problem in uncertain environmentsThe first extension accounts for obstacle regions in order togenerate collision-free paths for UAVs The second exten-sion reduces sensitivity to sensor noise and minimizes thechurning behavior in flight paths Alighanbari and How [10]presented a robust filter-embedded task assignment (RFETA)algorithm to achieve the target assignment for UAVs inuncertain dynamic environments The algorithm used twotechniques one is proactive in order to hedge against theuncertainty while the other is reactive in order to limitchurning behavior by the vehicles Alighanbari et al [11]formulated a modification of the classical task assignmentunder noisy conditions They developed a noise rejectionalgorithm that reduced the effects of high frequency noise

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 249825 10 pageshttpdxdoiorg1011552015249825

2 Mathematical Problems in Engineering

on the planner This algorithm could be used to mitigate theeffect of noise in situational awareness on the solution

In this study we considered a task assignment problemfor multi-UAV that performed attack tasks on a collectionof ground targets in a severe uncertain environment Eachtarget had to be allocated to a UAV before the mission beganThe severe and uncertain environment refers to battlefieldconditions in whichmuch information vital to an assignmentis random fuzzy or unknown For example the location of atarget is vague within the task area Though the predetectedlocation provides a reference it may be not in accordancewith the real target location because of a sensor error or targetmovement the threat of a targetrsquos defense system to the UAVsis unclear the value of the target is fuzzy These conditionsare often encountered in real-world UAV applications Underthese conditions the common methods for task assignmentcannot be used and new methods that can deal with severeuncertainty are needed

To address this problem we propose a novel task assign-ment method based on the stochastic multicriteria accept-ability analysis (SMAA) SMAA is amethod used for stochas-ticmulticriteria decision problems It ranks alternatives basedon multiple criteria values that are random variables and canalso address problemswithoutweight information Lahdelmaet al [12] gave the first version of the SMAA method andlater studies presented a series of extended versions includingSMAA-2 [13] SMAA-3 [14] SMAA-O [15] SMAA-A [16]SMAA-TRI [17] and SMAA-P [18] Here we have employedSMAA-2 to handle battlefield uncertainty The aim of thisstudy is to offer a suitable and reliable method for taskassignment under severe uncertain circumstances

The rest of this paper is organized as follows Section 2defines the problem Section 3 analyses the criteria essentialto task assignment Section 4 provides the task assignmentprocedure In Section 5 the results of simulation experimentsare given and analyzed Finally the conclusion is presented inSection 6

2 Problem Description

Theproblem is to assign the attack tasks of 119873119879geographically

dispersed ground targets to 119873119880UAVs Each UAV is equipped

with an air-to-ground missile and can attack one targetduring one flight Each UAV is assigned only one target andeach target should only be allocated to a UAV (ie 119873

119880=

119873119879) The task assignment problem needs to be solved before

the mission starts The objective of task assignment is tomaximize the total reward one example of an assignment isshown in Figure 1 In this example 3 targets were assigned to3 UAVs one target for each UAV

The major symbols used for the problem are listed inSymbols

To make an assignment we consider 3 criteria 119877119860 119862119865

and 119862119871 Their values are computed through the following 6

subcriteria (1) the value of the target (2) the value of theUAV (3) the distance between the UAV and the target (4)

the flying cost per unit distance of UAV (5) the probability

UAV

UAV

UAV

Target

Target

Target

Figure 1 Example of a task assignment 3 UAVs and 3 targets

of UAV loss and (6) the probability of killing target The 3criteria are given by the following expressions

119877119860

=

119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119901119905

119894119895V119905119895

119862119865

=

119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119889119894119895

119888119906

119894

119862119871

=

119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119901119906

119894119895V119906119894

(1)

The common model for task assignment is given as follows

max (1205961119877119860

minus 1205962119862119865

minus 1205963119862119871)

= max119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

(1205961119901119905

119894119895V119905119895

minus 1205962119889119894119895

119888119906

119894minus 1205963119901

119906

119894119895V119906119894)

(2)

st119873119880

sum

119894=1119909119894119895

= 1 (3)

119873119879

sum

119895=1119909119894119895

= 1 (4)

3sum

119896=1120596119896

= 1 (5)

119909119894119895

= 0 1 (119894 = 1 2 119873119880

119895 = 1 2 119873119879

) (6)

120596119896

isin [0 1] (119896 = 1 2 3) (7)

Objective function (2) aims to maximize the total taskreward which is the difference between the expected attackreward and the cost120596

119896(119896 = 1 2 3) are weights given bymil-

itary commanders or experts Constraints (3) and (4) ensurethat there is a one-to-one relationship between the UAV andthe target Constraint (5) ensures that theweights are normal-ized Constraints (6) and (7) give the range of the variables

Mathematical Problems in Engineering 3

The task assignment problem can be settled by solvingthe above model when all values of the parameters are givenHowever in complex battlefield scenarios much informationis severely uncertain Uncertainty increases many difficultiesfor task assignment In these conditions we need newmethods to handle uncertainty

3 Criteria Analysis

In this section we will provide an analysis of the criteriaand give methods to quantitatively describe the involveduncertainties

31 The Value of the Target The value of a target evaluatedby military commanders or experts determines whetherthe target is worth attacking For ease of calculation in thispaper the lower and upper bounds of the value of targethave been set to be 0 and 100 A higher number indicates ahigher value However it is difficult for the commanders orexperts to provide a precise value They typically prefer touse linguistic variables such as ldquovery highrdquo and ldquofairly lowrdquoThese linguistic variables are fuzzy descriptors that cannotbe used in task assignment models To solve this problemwe have defined a fuzzy linguistic set 119878 based on the work in[19 20]

119878 = 1199040 1199041 1199042 1199043 1199044 1199045 1199046 1199047 1199048

= extremely high very high high fairly high medium fairly low low very low extremely low

(8)

Each linguistic variable in 119878 matches a value intervalextremely high = [90 100] very high = [80 90] high =

[70 80] fairly high = [55 70] medium = [45 55]fairly low = [30 45] low = [20 30] very low = [10 20] andextremely low = [0 10] Using the fuzzy set the commandersor experts can use a linguistic variable to describe the value ofa target and then the variable is transformed to a numericalinterval

32 The Value of the UAV When a UAV attacks a target theUAV may also be at risk of being attacked Therefore theUAVrsquos value is an important factor that we need to considerBy contrasting it with the value of target the value ofUAV canbe accurately obtained and can be represented by a precisenumber

33 The Distance between the UAV and the Target Thedistance between the UAV and the target is an importantcriterion for task assignment To minimize the flying costsa UAV tends to select the nearest target when other variablesare equal

The distance between the UAV 119906119894and target 119905

119895is calcu-

lated by using the Euclidean distance as shown in

119889119894119895

= radic(119909119906

119894minus 119909119905

119895)2

+ (119910119906

119894minus 119910119905

119895)2 (9)

In battlefield circumstances the coordinate (119909119905

119895 119910119905

119895) of

target 119905119895is not a certain value The information we have is

the detected position (119909119905

119895

1015840

119910119905

119895

1015840

) of 119905119895 However there is usually

a deviation between a targetrsquos detected position and its realposition

Normally the real position (119909119905

119895 119910119905

119895) follows a two-

dimensional Gaussian distribution whose expectation is(119909119905

119895

1015840

119910119905

119895

1015840

) as shown in

119891 (119909119905

119895 119910119905

119895) =

1

212058712059011205902radic1 minus 1205882

sdot 119890minus(12(1minus1205882))[(119909119905

119895minus119909119905

119895

1015840

)2120590

21minus2120588((119909

119905

119895minus119909119905

119895

1015840

)1205901)((119910119905

119895minus119910119905

119895

1015840

)21205902)+(119910

119905

119895minus119910119905

119895

1015840

)2120590

22 ]

(10)

Because 119909119905

119895and 119910

119905

119895are independent we let 120588 = 0 and we

assume 1205901

= 1205902

= 1 Then (10) can be simplified as

119891 (119909119905

119895 119910119905

119895) =

12120587

119890minus(12)[(119909119905

119895minus119909119905

119895

1015840

)2+(119910119905

119895minus119910119905

119895

1015840

)2] (11)

In (11) the probability distribution of a targetrsquos realposition within the task area is shown in Figure 2 Theprobability is 0159 that a targetrsquos real position is in accordancewith the detected position

The position can be accurately detected only when theUAV approaches the target However at the time the taskassignment ismade theUAVonly has uncertain information

34 The Flying Cost per Unit Distance of UAV This criterionas well as the distance between the UAV and the targetdetermines the flying cost of a mission For a UAV the valueof this criterion is fixed

35The Probability of UAVLoss Ahigh value target is usuallyprotected by a defense system that is equipped with surface-to-air missiles When performing the attack task the UAValso faces the possibility of being destroyed Washburn andKress [21] provided the probability of 119906

119894loss when attacking

target 119905119895as

119901119906

119894119895=

exp(minus(119877 minus 1198770)

2

21198872) 119903

min119895

le 119877 le 119903max119895

0 119877 gt 119903max119895

or 119877 lt 119903min119895

(12)

where 119877 denotes the real-time distance between 119905119895and 119906119894and

1198770and 119887 are parameters calculated using

1198770 =

(119903min119895

+ 119903max119895

)

2

119887 = radic2 (119903max119895

minus 119903min119895

)

(13)

Equation (12) indicates that the probability of 119906119894loss is

0 when being out of the range of a surface-to-air missile

4 Mathematical Problems in Engineering

02

015

01

005

0

(xt998400

j yt998400

j )

Figure 2 Probability distribution of target position

within the range of the missile the probability approximatesa Gaussian distribution

To calculate 119901119906

119894119895 we reasonably assume the following (1)

the defense system will attack a UAV at the same time theUAV attacks the target (2) a UAV always attacks a targetat the UAVrsquos best striking distance which depends on theUAVrsquos onboard weapon Different weapons have their ownbest strike distances

Based on the above assumptions we substitute 119877 in (12)with 119889

str119894 119889

str119894

typically is an interval such as [20 km 25 km]Consequently 119901

119906

119894119895is also an interval like [050 065] and so

forth

36 The Probability of Killing Target As mentioned abovethe UAV always attacks the target at the UAVrsquos best strikingdistance Washburn and Kress [21] presented the probability119901119905

119894119895of UAV 119906

119894killing target 119905

119895as

119901119905

119894119895= 1minus (05)

(119903dam2119894cepdam

2119894)

(14)

where cep represents the circular error probability thedefinition of which is the following If the probability thatthe weaponrsquos two-dimensional shooting error falls within thecircle is 05 the radius of circle will be called the circular errorprobability

4 Assignment Model

The SMAA-2 method was employed to solve the taskassignment problem The SMAA-2 method can handle mul-tiattribute decision making problems in which uncertaincriteria values and uncertain weights exist simultaneouslyThe SMAA-2 method has been applied in many aspects ofthe real world Kangas et al [22] used the SMAA-2 methodfor strategic forest planning Hokkanen et al [23] used theSMAA-2 method in a technology competition for cleaningpolluted soil in Helsinki Menou et al [24] gave a decisionsupport for centralizing cargo at a Moroccan airport hubusing the SMAA-2 method Rahman et al [25] used theSMAA-2 method to evaluate the choices for sustainablerural electrification in developing countries Pesola et al [26]

applied the method on alternatives for remote monitoringsystems of municipal buildings

41 The SMAA-2 Method Considering 119898 alternatives 119860 =

1199091 1199092 119909119898

and 119899 evaluation criteria 120596 is a weightvector for the decision maker (DM) to express his subjectivepreference 120596

119896represents the weight of criterion 119896 satisfying

sum119899

119896=1 120596119896

= 1 A weight distribution with density function119891(120596) is used to represent the DMrsquos partially known orunknown preference The weight distribution is in the setof feasible weights 119882 defined as 119882 = 120596 isin 119877

119899 120596 ge

0 and sum119899

119896=1 120596119896

= 1 120585119894119896represents the uncertain or imprecise

value of criterion 119896 for alternative 119909119894 Similarly a joint

probability distribution with density function 119891(120585) in thespace 119883 is used to represent the uncertain or imprecisecriteria values The utility of alternative 119909

119894is defined as

119880119894

= 119880 (120585119894 120596) =

119899

sum

119896=1120596119896119880119896

(120585119894119896

) (15)

where 119880119896(120585119894119896

) is the utility function of criterion 119896The SMAA-2 method defines a ranking function

rank(120585119894 120596) which is given by the following to represent the

rank of alternative 119909119894

rank (120585119894 120596) = 1+

119898

sum

ℎ=1120588 (119880 (120585

ℎ 120596) gt 119880 (120585

119894 120596)) (16)

where 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) is a judgment function If

119880(120585ℎ 120596) gt 119880(120585

119894 120596) then 120588(119880(120585

ℎ 120596) gt 119880(120585

119894 120596)) = 1

otherwise 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) = 0

The SMAA-2 method is based on analyzing the sets offavorable rank weights 119882

119903

119894(120585) Alternative 119909

119894always obtains

rank 119903 for arbitrary weight 120596 isin 119882119903

119894(120585) 119882

119903

119894(120585) is defined as

119882119903

119894(120585) = 120596 isin 119882 rank (120585

119894 120596) = 119903 (17)

The SMAA-2 method presents 3 important analysisindices All the indices are based on properties of thesestochastic sets The first index is the rank acceptability index119887119903

119894 defined as the expected volume of 119882

119903

119894(120585)

119887119903

119894= int119883

119891 (120585) int119882119903

119894(120585)

119891 (120596) 119889120596 119889120585 (18)

It is computed as a multidimensional integral over thecriteria distributions and the favorable rank weights repre-senting the acceptability or the probability of alternative 119909

119894

rank 119903The rank acceptability index is a real number in [0 1]

and is generally expressed as a percent Normally the moreacceptable alternatives should have higher acceptabilities forthe best ranks

The second index is the central weight vector 120596119888

119894 the

best single weight vector representation of a hypotheticalDM supporting alternative 119909

119894to rank first It is computed

as a double integral over the criteria distributions and thefavorable first rank weights 119882

1119894

(120585)

120596119888

119894=

int119883

119891 (120585) int119882

1119894(120585)

119891 (120596) 120596 119889120596 119889120585

1198871119894

(19)

Mathematical Problems in Engineering 5

The central weight vector 120596119888

119894is the expected center of

gravity of the favorable first rank weights 1198821119894

(120585) It can helpthe DM understand what preferences support the differentalternatives Moreover the central weight vectors are used tocompute the confidence factor

The third index is the confidence factor 119901119888

119894 representing

the probability for the alternative 119909119894ranking first if the central

weight vector is chosen It is computed as an integral over thecriteria distributions

119901119888

119894= int120585rank(120585119894120596119888119894 )=1

119891 (120585) 119889120585 (20)

The confidence factor can also be used to judge whetherthe criteria value is accurate enough to distinguish thealternatives when the central weight vector is used Providingany weight vector can also calculate the correspondingconfidence factor in a similar way

Comparing the alternatives according to their rankacceptabilities can be seen as a ldquosecond-orderrdquo multicriteriadecision problem [13] The SMAA-2 method additionallydefines the holistic acceptability index to provide ameasure ofthe overall acceptability of each alternative It is representedas a weighted sum of the rank acceptabilities

119886ℎ

119894=

119898

sum

119903=1120572119903119887119903

119894 (21)

where 120572119903are referred to as metaweights (or rank weights) A

complete priority order between the metaweights should bewell defined Lahdelma and Salminen [13] gave three possiblechoices linear weights 120572

119903= (119898 minus 119903)(119898 minus 1) inverse weights

120572119903

= 1119903 and centroid weights 120572119903

= sum119898

119894=1199031119894 sum

119898

119894=1 1119894 Aftercomparison they preferred using centroid weights

In practice accurately calculating these indices requirescomplex computation processes In order to reduce the com-putational complexity Tervonen and Lahdelma [27] gave anapproximate computation method by using the Monte Carlotechnique Even when dealing with large-scale problemsthis method can quickly solve them Furthermore Tervonen[28] presented JSMAA open source software for SMAAcomputations

42 The Decision Model To solve the problem of assigningthe attack tasks on119873

119879targets to119873

119880UAVs we decompose the

problem into 119873119879parts based on the number of targets For

target 119905119895

(119895 = 1 2 119873119879

) let the 119873119880UAVs be alternatives

and use 119887119903

119894119895 119886ℎ

119894119895 120596119888

119894119895 and 119901

119888

119894119895to respectively represent the

rank acceptability index holistic acceptability index centralweight vector and confidence factor of alternative 119906

119894 These

indices are calculated by using the SMAA-2 method On thebasis of the indices the DMs make the assignment decisions

In this study the DMs have weight intervals of thecriteria and the widths of the intervals are small The mainconsideration of the DMs is the overall acceptability Sothe holistic acceptability index is the most suitable index

Step 1 get all the criteria information and the a priori weightinformation and define a utility function for each all criteria

Step 4 use the Monte Carlo simulation method to calculateYes

No

Step 6 input the indices into the task assignment model to getthe optimal assignment scheme

the indices brij wcij

let j = 1Step 2

if j le NTStep 3

do j = j + 1Step 5

pcij and ahij

Figure 3 The steps of task assignment

for making decisions Then the task assignment model inSection 2 is modified as

max119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119886ℎ

119894119895(22)

st119873119880

sum

119894=1119909119894119895

= 1

119873119879

sum

119895=1119909119894119895

= 1

119909119894119895

= 0 1

(119894 = 1 2 119873119880

119895 = 1 2 119873119879

)

(23)

In this model objective function (22) aims to maximizethe holistic acceptability of the assignment schemes Tocalculate the holistic acceptability the DMs select centroidweights 120572

119903= sum119898

119894=1199031119894 sum

119898

119894=1 1119894The steps of task assignment for multiple UAVs under

severe uncertainty are given in Figure 3In step 1 linear utility functions are used Let 120585

119894119896(119896 =

1 2 3) respectively represent the values of criteria 119877119860 119862119865

and 119862119871for alternative 119906

119894 Their values can be calculated using

(1) 119877119860is an income-type criterion 119862

119865and 119862

119871are cost-type

criteriaTheir utility functions can respectively be defined as

119880119896

(120585119894119896

) =120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 1

119880119896

(120585119894119896

) = minus120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 2 3

(24)

where 120585min119894119896

= min1le119894le119898120585119894119896and 120585

max119894119896

= max1le119894le119898120585119894119896The effects

of the utility functions are to normalize the criteria valuesBased on the work of [27] the detailed procedure of Step

4 is given in Algorithm 1

6 Mathematical Problems in Engineering

(1) Initialize the data(2) For 119897 = 1 to 119868

119861do 119868

119861is the number of iterations

(3) Randomly generate a weight vector 120596 = (1205961 1205962 1205963) based on the prior weight information(4) Randomly generate a set of sub-criteria values (119901119905

119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(5) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(6) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(7) Calculate 119880119894for each 119906

119894using (15)

(8) Sort 119906119894according to the size of 119880

119894 getting ℎ

119894119895119897 ℎ119894119895119897

represents the sorting of 119906119894in iteration 119897

(9) If 119906119894rank the first then 120596

119888

119894119895= 120596119888

119894119895+ 120596

(10) End for(11) For 119894 = 1 to 119873

119880do

(12) For 119903 = 1 to 119873119880do

(13) ℎ119903

119894119895= sum119868119861

119897=1(ℎ119894119895119897

= 1199031 0) ℎ119903

119894119895is the total number of times that 119906

119894obtains rank 119903

(14) 119887119903

119894119895= ℎ119903

119894119895119868119861

(15) End for(16) Calculate 119886

ℎ

119894119895according to (21)

(17) End for(18) For 119894 = 1 to 119873

119880do

(19) 120596119888

119894119895= 120596119888

119894119895ℎ

1119894119895

(20) End for(21) For 119897 = 1 to 119868

119875do 119868

119875is the number of iterations

(22) Randomly generate a set of sub-criteria values (119901119905119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(23) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(24) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(25) For 119894 = 1 to 119873119880do

(26) 120596 = 120596119888

119894119895

(27) Calculate 119880119894for each 119906

119894using (15)

(28) Sort 119906119894according to the size of 119880

119894

(29) If 119906119894rank the first then 119901

119888

119894119895= 119901119888

119894119895+ 1

(30) End for(31) End for(32) For 119894 = 1 to 119873

119880do

(33) 119901119888

119894119895= 119901119888

119894119895119868119875

(34) End for

Algorithm 1 Calculating the indices for target 119905119895

Table 1 Attribute values of the UAVs

UAV (119909119906

119894 119910119906

119894) 119903

dam119894

cepdam119894

V119906119894

119889str119894

119888119906

119894

1199061

(755 760) 20 12 80 [500 600] 141199062

(750 720) 15 8 75 [505 530] 151199063

(785 775) 15 10 90 [500 550] 131199064

(770 740) 13 8 85 [520 540] 14

5 Simulation Experiments

The proposed task assignment method was tested by sim-ulation experiments In the experiments the task area wasrepresented by 1000 lowast 1000 grid which was populated by 4UAVs and 4 targets as seen in Figure 4The experiments wererun on a computer with an Intel Core 2 Duo E7500 293GHzprocessor and 2GB RAM

The parameters used for the simulations are summarizedin Tables 1 and 2

In these simulations the DMrsquos preference has beendivided into 2 categories as follows

Target t1

Target t2

Target t3

Target t4

UAVu1

UAVu2

UAVu3

UAVu4

Figure 4 Task area

(1) The primary purpose of the mission is to kill allthe targets The DM considers criterion 119877

119860to be the most

Mathematical Problems in Engineering 7

12

34

1

23

4

0

20

40

60

80

100

Rank

Target 1

Alternative (UAV)

Acce

ptab

ility

Figure 5 Rank acceptability indices 1198871ndash1198874 for target 119905

1

Table 2 Attribute values of the targets

Target (119909119905

119895

1015840

119910119905

119895

1015840

) V119905119895

119903max119895

119903min119895

1199051

(80 95) (90 100) 600 3001199052

(70 64) (80 90) 800 3501199053

(52 80) (90 100) 600 4001199054

(58 68) (90 100) 700 400

Table 3 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 04 01 025120596max 055 02 04

important 119862119871to be the second most important and 119862

119865to

be the third most important The weight intervals are givenin Table 3

Using the Monte Carlo technique the results are shownin Table 4 The rank acceptability indices are presented inFigures 5ndash8

Using (22)-(23) the final results of task assignment arelisted in Table 5 where 1 indicates that the target was assignedto the UAV and 0 otherwise

(2) The primary purpose of the mission was to kill all thetargets in the case of ensuring that the UAVs underwent zerodamage as far as possible The DM considers criterion 119862

119871to

be the most important 119877119860to be the second most important

and 119862119865to be the third most important The weight intervals

are given in Table 6 The results are shown in Table 7The final results of task assignment are listed in Table 8

6 Conclusion

This paper presented a task assignment method for multipleUAVs under severe uncertainty conditions in which the

Target 2

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 6 Rank acceptability indices 1198871ndash1198874 for target 119905

2

Target 3

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 7 Rank acceptability indices 1198871ndash1198874 for target 119905

3

values of the criteria essential to task assignment wererandom fuzzy or unknown Taking advantage of the SMAA-2 method this paper established the solving model and thesolution process

In the simulations we selected different weight infor-mation for 2 simulation experiments In the experimentsthere was little difference in the central weight vector foreach assignment so we only needed to select alternativeson the basis of the holistic acceptability index However ifthe central weight vectors vary widely the selection shouldinstead be based on the holistic acceptability index thecentral weight vector and the confidence factor

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

on the planner This algorithm could be used to mitigate theeffect of noise in situational awareness on the solution

In this study we considered a task assignment problemfor multi-UAV that performed attack tasks on a collectionof ground targets in a severe uncertain environment Eachtarget had to be allocated to a UAV before the mission beganThe severe and uncertain environment refers to battlefieldconditions in whichmuch information vital to an assignmentis random fuzzy or unknown For example the location of atarget is vague within the task area Though the predetectedlocation provides a reference it may be not in accordancewith the real target location because of a sensor error or targetmovement the threat of a targetrsquos defense system to the UAVsis unclear the value of the target is fuzzy These conditionsare often encountered in real-world UAV applications Underthese conditions the common methods for task assignmentcannot be used and new methods that can deal with severeuncertainty are needed

To address this problem we propose a novel task assign-ment method based on the stochastic multicriteria accept-ability analysis (SMAA) SMAA is amethod used for stochas-ticmulticriteria decision problems It ranks alternatives basedon multiple criteria values that are random variables and canalso address problemswithoutweight information Lahdelmaet al [12] gave the first version of the SMAA method andlater studies presented a series of extended versions includingSMAA-2 [13] SMAA-3 [14] SMAA-O [15] SMAA-A [16]SMAA-TRI [17] and SMAA-P [18] Here we have employedSMAA-2 to handle battlefield uncertainty The aim of thisstudy is to offer a suitable and reliable method for taskassignment under severe uncertain circumstances

The rest of this paper is organized as follows Section 2defines the problem Section 3 analyses the criteria essentialto task assignment Section 4 provides the task assignmentprocedure In Section 5 the results of simulation experimentsare given and analyzed Finally the conclusion is presented inSection 6

2 Problem Description

Theproblem is to assign the attack tasks of 119873119879geographically

dispersed ground targets to 119873119880UAVs Each UAV is equipped

with an air-to-ground missile and can attack one targetduring one flight Each UAV is assigned only one target andeach target should only be allocated to a UAV (ie 119873

119880=

119873119879) The task assignment problem needs to be solved before

the mission starts The objective of task assignment is tomaximize the total reward one example of an assignment isshown in Figure 1 In this example 3 targets were assigned to3 UAVs one target for each UAV

The major symbols used for the problem are listed inSymbols

To make an assignment we consider 3 criteria 119877119860 119862119865

and 119862119871 Their values are computed through the following 6

subcriteria (1) the value of the target (2) the value of theUAV (3) the distance between the UAV and the target (4)

the flying cost per unit distance of UAV (5) the probability

UAV

UAV

UAV

Target

Target

Target

Figure 1 Example of a task assignment 3 UAVs and 3 targets

of UAV loss and (6) the probability of killing target The 3criteria are given by the following expressions

119877119860

=

119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119901119905

119894119895V119905119895

119862119865

=

119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119889119894119895

119888119906

119894

119862119871

=

119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119901119906

119894119895V119906119894

(1)

The common model for task assignment is given as follows

max (1205961119877119860

minus 1205962119862119865

minus 1205963119862119871)

= max119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

(1205961119901119905

119894119895V119905119895

minus 1205962119889119894119895

119888119906

119894minus 1205963119901

119906

119894119895V119906119894)

(2)

st119873119880

sum

119894=1119909119894119895

= 1 (3)

119873119879

sum

119895=1119909119894119895

= 1 (4)

3sum

119896=1120596119896

= 1 (5)

119909119894119895

= 0 1 (119894 = 1 2 119873119880

119895 = 1 2 119873119879

) (6)

120596119896

isin [0 1] (119896 = 1 2 3) (7)

Objective function (2) aims to maximize the total taskreward which is the difference between the expected attackreward and the cost120596

119896(119896 = 1 2 3) are weights given bymil-

itary commanders or experts Constraints (3) and (4) ensurethat there is a one-to-one relationship between the UAV andthe target Constraint (5) ensures that theweights are normal-ized Constraints (6) and (7) give the range of the variables

Mathematical Problems in Engineering 3

The task assignment problem can be settled by solvingthe above model when all values of the parameters are givenHowever in complex battlefield scenarios much informationis severely uncertain Uncertainty increases many difficultiesfor task assignment In these conditions we need newmethods to handle uncertainty

3 Criteria Analysis

In this section we will provide an analysis of the criteriaand give methods to quantitatively describe the involveduncertainties

31 The Value of the Target The value of a target evaluatedby military commanders or experts determines whetherthe target is worth attacking For ease of calculation in thispaper the lower and upper bounds of the value of targethave been set to be 0 and 100 A higher number indicates ahigher value However it is difficult for the commanders orexperts to provide a precise value They typically prefer touse linguistic variables such as ldquovery highrdquo and ldquofairly lowrdquoThese linguistic variables are fuzzy descriptors that cannotbe used in task assignment models To solve this problemwe have defined a fuzzy linguistic set 119878 based on the work in[19 20]

119878 = 1199040 1199041 1199042 1199043 1199044 1199045 1199046 1199047 1199048

= extremely high very high high fairly high medium fairly low low very low extremely low

(8)

Each linguistic variable in 119878 matches a value intervalextremely high = [90 100] very high = [80 90] high =

[70 80] fairly high = [55 70] medium = [45 55]fairly low = [30 45] low = [20 30] very low = [10 20] andextremely low = [0 10] Using the fuzzy set the commandersor experts can use a linguistic variable to describe the value ofa target and then the variable is transformed to a numericalinterval

32 The Value of the UAV When a UAV attacks a target theUAV may also be at risk of being attacked Therefore theUAVrsquos value is an important factor that we need to considerBy contrasting it with the value of target the value ofUAV canbe accurately obtained and can be represented by a precisenumber

33 The Distance between the UAV and the Target Thedistance between the UAV and the target is an importantcriterion for task assignment To minimize the flying costsa UAV tends to select the nearest target when other variablesare equal

The distance between the UAV 119906119894and target 119905

119895is calcu-

lated by using the Euclidean distance as shown in

119889119894119895

= radic(119909119906

119894minus 119909119905

119895)2

+ (119910119906

119894minus 119910119905

119895)2 (9)

In battlefield circumstances the coordinate (119909119905

119895 119910119905

119895) of

target 119905119895is not a certain value The information we have is

the detected position (119909119905

119895

1015840

119910119905

119895

1015840

) of 119905119895 However there is usually

a deviation between a targetrsquos detected position and its realposition

Normally the real position (119909119905

119895 119910119905

119895) follows a two-

dimensional Gaussian distribution whose expectation is(119909119905

119895

1015840

119910119905

119895

1015840

) as shown in

119891 (119909119905

119895 119910119905

119895) =

1

212058712059011205902radic1 minus 1205882

sdot 119890minus(12(1minus1205882))[(119909119905

119895minus119909119905

119895

1015840

)2120590

21minus2120588((119909

119905

119895minus119909119905

119895

1015840

)1205901)((119910119905

119895minus119910119905

119895

1015840

)21205902)+(119910

119905

119895minus119910119905

119895

1015840

)2120590

22 ]

(10)

Because 119909119905

119895and 119910

119905

119895are independent we let 120588 = 0 and we

assume 1205901

= 1205902

= 1 Then (10) can be simplified as

119891 (119909119905

119895 119910119905

119895) =

12120587

119890minus(12)[(119909119905

119895minus119909119905

119895

1015840

)2+(119910119905

119895minus119910119905

119895

1015840

)2] (11)

In (11) the probability distribution of a targetrsquos realposition within the task area is shown in Figure 2 Theprobability is 0159 that a targetrsquos real position is in accordancewith the detected position

The position can be accurately detected only when theUAV approaches the target However at the time the taskassignment ismade theUAVonly has uncertain information

34 The Flying Cost per Unit Distance of UAV This criterionas well as the distance between the UAV and the targetdetermines the flying cost of a mission For a UAV the valueof this criterion is fixed

35The Probability of UAVLoss Ahigh value target is usuallyprotected by a defense system that is equipped with surface-to-air missiles When performing the attack task the UAValso faces the possibility of being destroyed Washburn andKress [21] provided the probability of 119906

119894loss when attacking

target 119905119895as

119901119906

119894119895=

exp(minus(119877 minus 1198770)

2

21198872) 119903

min119895

le 119877 le 119903max119895

0 119877 gt 119903max119895

or 119877 lt 119903min119895

(12)

where 119877 denotes the real-time distance between 119905119895and 119906119894and

1198770and 119887 are parameters calculated using

1198770 =

(119903min119895

+ 119903max119895

)

2

119887 = radic2 (119903max119895

minus 119903min119895

)

(13)

Equation (12) indicates that the probability of 119906119894loss is

0 when being out of the range of a surface-to-air missile

4 Mathematical Problems in Engineering

02

015

01

005

0

(xt998400

j yt998400

j )

Figure 2 Probability distribution of target position

within the range of the missile the probability approximatesa Gaussian distribution

To calculate 119901119906

119894119895 we reasonably assume the following (1)

the defense system will attack a UAV at the same time theUAV attacks the target (2) a UAV always attacks a targetat the UAVrsquos best striking distance which depends on theUAVrsquos onboard weapon Different weapons have their ownbest strike distances

Based on the above assumptions we substitute 119877 in (12)with 119889

str119894 119889

str119894

typically is an interval such as [20 km 25 km]Consequently 119901

119906

119894119895is also an interval like [050 065] and so

forth

36 The Probability of Killing Target As mentioned abovethe UAV always attacks the target at the UAVrsquos best strikingdistance Washburn and Kress [21] presented the probability119901119905

119894119895of UAV 119906

119894killing target 119905

119895as

119901119905

119894119895= 1minus (05)

(119903dam2119894cepdam

2119894)

(14)

where cep represents the circular error probability thedefinition of which is the following If the probability thatthe weaponrsquos two-dimensional shooting error falls within thecircle is 05 the radius of circle will be called the circular errorprobability

4 Assignment Model

The SMAA-2 method was employed to solve the taskassignment problem The SMAA-2 method can handle mul-tiattribute decision making problems in which uncertaincriteria values and uncertain weights exist simultaneouslyThe SMAA-2 method has been applied in many aspects ofthe real world Kangas et al [22] used the SMAA-2 methodfor strategic forest planning Hokkanen et al [23] used theSMAA-2 method in a technology competition for cleaningpolluted soil in Helsinki Menou et al [24] gave a decisionsupport for centralizing cargo at a Moroccan airport hubusing the SMAA-2 method Rahman et al [25] used theSMAA-2 method to evaluate the choices for sustainablerural electrification in developing countries Pesola et al [26]

applied the method on alternatives for remote monitoringsystems of municipal buildings

41 The SMAA-2 Method Considering 119898 alternatives 119860 =

1199091 1199092 119909119898

and 119899 evaluation criteria 120596 is a weightvector for the decision maker (DM) to express his subjectivepreference 120596

119896represents the weight of criterion 119896 satisfying

sum119899

119896=1 120596119896

= 1 A weight distribution with density function119891(120596) is used to represent the DMrsquos partially known orunknown preference The weight distribution is in the setof feasible weights 119882 defined as 119882 = 120596 isin 119877

119899 120596 ge

0 and sum119899

119896=1 120596119896

= 1 120585119894119896represents the uncertain or imprecise

value of criterion 119896 for alternative 119909119894 Similarly a joint

probability distribution with density function 119891(120585) in thespace 119883 is used to represent the uncertain or imprecisecriteria values The utility of alternative 119909

119894is defined as

119880119894

= 119880 (120585119894 120596) =

119899

sum

119896=1120596119896119880119896

(120585119894119896

) (15)

where 119880119896(120585119894119896

) is the utility function of criterion 119896The SMAA-2 method defines a ranking function

rank(120585119894 120596) which is given by the following to represent the

rank of alternative 119909119894

rank (120585119894 120596) = 1+

119898

sum

ℎ=1120588 (119880 (120585

ℎ 120596) gt 119880 (120585

119894 120596)) (16)

where 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) is a judgment function If

119880(120585ℎ 120596) gt 119880(120585

119894 120596) then 120588(119880(120585

ℎ 120596) gt 119880(120585

119894 120596)) = 1

otherwise 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) = 0

The SMAA-2 method is based on analyzing the sets offavorable rank weights 119882

119903

119894(120585) Alternative 119909

119894always obtains

rank 119903 for arbitrary weight 120596 isin 119882119903

119894(120585) 119882

119903

119894(120585) is defined as

119882119903

119894(120585) = 120596 isin 119882 rank (120585

119894 120596) = 119903 (17)

The SMAA-2 method presents 3 important analysisindices All the indices are based on properties of thesestochastic sets The first index is the rank acceptability index119887119903

119894 defined as the expected volume of 119882

119903

119894(120585)

119887119903

119894= int119883

119891 (120585) int119882119903

119894(120585)

119891 (120596) 119889120596 119889120585 (18)

It is computed as a multidimensional integral over thecriteria distributions and the favorable rank weights repre-senting the acceptability or the probability of alternative 119909

119894

rank 119903The rank acceptability index is a real number in [0 1]

and is generally expressed as a percent Normally the moreacceptable alternatives should have higher acceptabilities forthe best ranks

The second index is the central weight vector 120596119888

119894 the

best single weight vector representation of a hypotheticalDM supporting alternative 119909

119894to rank first It is computed

as a double integral over the criteria distributions and thefavorable first rank weights 119882

1119894

(120585)

120596119888

119894=

int119883

119891 (120585) int119882

1119894(120585)

119891 (120596) 120596 119889120596 119889120585

1198871119894

(19)

Mathematical Problems in Engineering 5

The central weight vector 120596119888

119894is the expected center of

gravity of the favorable first rank weights 1198821119894

(120585) It can helpthe DM understand what preferences support the differentalternatives Moreover the central weight vectors are used tocompute the confidence factor

The third index is the confidence factor 119901119888

119894 representing

the probability for the alternative 119909119894ranking first if the central

weight vector is chosen It is computed as an integral over thecriteria distributions

119901119888

119894= int120585rank(120585119894120596119888119894 )=1

119891 (120585) 119889120585 (20)

The confidence factor can also be used to judge whetherthe criteria value is accurate enough to distinguish thealternatives when the central weight vector is used Providingany weight vector can also calculate the correspondingconfidence factor in a similar way

Comparing the alternatives according to their rankacceptabilities can be seen as a ldquosecond-orderrdquo multicriteriadecision problem [13] The SMAA-2 method additionallydefines the holistic acceptability index to provide ameasure ofthe overall acceptability of each alternative It is representedas a weighted sum of the rank acceptabilities

119886ℎ

119894=

119898

sum

119903=1120572119903119887119903

119894 (21)

where 120572119903are referred to as metaweights (or rank weights) A

complete priority order between the metaweights should bewell defined Lahdelma and Salminen [13] gave three possiblechoices linear weights 120572

119903= (119898 minus 119903)(119898 minus 1) inverse weights

120572119903

= 1119903 and centroid weights 120572119903

= sum119898

119894=1199031119894 sum

119898

119894=1 1119894 Aftercomparison they preferred using centroid weights

In practice accurately calculating these indices requirescomplex computation processes In order to reduce the com-putational complexity Tervonen and Lahdelma [27] gave anapproximate computation method by using the Monte Carlotechnique Even when dealing with large-scale problemsthis method can quickly solve them Furthermore Tervonen[28] presented JSMAA open source software for SMAAcomputations

42 The Decision Model To solve the problem of assigningthe attack tasks on119873

119879targets to119873

119880UAVs we decompose the

problem into 119873119879parts based on the number of targets For

target 119905119895

(119895 = 1 2 119873119879

) let the 119873119880UAVs be alternatives

and use 119887119903

119894119895 119886ℎ

119894119895 120596119888

119894119895 and 119901

119888

119894119895to respectively represent the

rank acceptability index holistic acceptability index centralweight vector and confidence factor of alternative 119906

119894 These

indices are calculated by using the SMAA-2 method On thebasis of the indices the DMs make the assignment decisions

In this study the DMs have weight intervals of thecriteria and the widths of the intervals are small The mainconsideration of the DMs is the overall acceptability Sothe holistic acceptability index is the most suitable index

Step 1 get all the criteria information and the a priori weightinformation and define a utility function for each all criteria

Step 4 use the Monte Carlo simulation method to calculateYes

No

Step 6 input the indices into the task assignment model to getthe optimal assignment scheme

the indices brij wcij

let j = 1Step 2

if j le NTStep 3

do j = j + 1Step 5

pcij and ahij

Figure 3 The steps of task assignment

for making decisions Then the task assignment model inSection 2 is modified as

max119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119886ℎ

119894119895(22)

st119873119880

sum

119894=1119909119894119895

= 1

119873119879

sum

119895=1119909119894119895

= 1

119909119894119895

= 0 1

(119894 = 1 2 119873119880

119895 = 1 2 119873119879

)

(23)

In this model objective function (22) aims to maximizethe holistic acceptability of the assignment schemes Tocalculate the holistic acceptability the DMs select centroidweights 120572

119903= sum119898

119894=1199031119894 sum

119898

119894=1 1119894The steps of task assignment for multiple UAVs under

severe uncertainty are given in Figure 3In step 1 linear utility functions are used Let 120585

119894119896(119896 =

1 2 3) respectively represent the values of criteria 119877119860 119862119865

and 119862119871for alternative 119906

119894 Their values can be calculated using

(1) 119877119860is an income-type criterion 119862

119865and 119862

119871are cost-type

criteriaTheir utility functions can respectively be defined as

119880119896

(120585119894119896

) =120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 1

119880119896

(120585119894119896

) = minus120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 2 3

(24)

where 120585min119894119896

= min1le119894le119898120585119894119896and 120585

max119894119896

= max1le119894le119898120585119894119896The effects

of the utility functions are to normalize the criteria valuesBased on the work of [27] the detailed procedure of Step

4 is given in Algorithm 1

6 Mathematical Problems in Engineering

(1) Initialize the data(2) For 119897 = 1 to 119868

119861do 119868

119861is the number of iterations

(3) Randomly generate a weight vector 120596 = (1205961 1205962 1205963) based on the prior weight information(4) Randomly generate a set of sub-criteria values (119901119905

119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(5) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(6) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(7) Calculate 119880119894for each 119906

119894using (15)

(8) Sort 119906119894according to the size of 119880

119894 getting ℎ

119894119895119897 ℎ119894119895119897

represents the sorting of 119906119894in iteration 119897

(9) If 119906119894rank the first then 120596

119888

119894119895= 120596119888

119894119895+ 120596

(10) End for(11) For 119894 = 1 to 119873

119880do

(12) For 119903 = 1 to 119873119880do

(13) ℎ119903

119894119895= sum119868119861

119897=1(ℎ119894119895119897

= 1199031 0) ℎ119903

119894119895is the total number of times that 119906

119894obtains rank 119903

(14) 119887119903

119894119895= ℎ119903

119894119895119868119861

(15) End for(16) Calculate 119886

ℎ

119894119895according to (21)

(17) End for(18) For 119894 = 1 to 119873

119880do

(19) 120596119888

119894119895= 120596119888

119894119895ℎ

1119894119895

(20) End for(21) For 119897 = 1 to 119868

119875do 119868

119875is the number of iterations

(22) Randomly generate a set of sub-criteria values (119901119905119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(23) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(24) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(25) For 119894 = 1 to 119873119880do

(26) 120596 = 120596119888

119894119895

(27) Calculate 119880119894for each 119906

119894using (15)

(28) Sort 119906119894according to the size of 119880

119894

(29) If 119906119894rank the first then 119901

119888

119894119895= 119901119888

119894119895+ 1

(30) End for(31) End for(32) For 119894 = 1 to 119873

119880do

(33) 119901119888

119894119895= 119901119888

119894119895119868119875

(34) End for

Algorithm 1 Calculating the indices for target 119905119895

Table 1 Attribute values of the UAVs

UAV (119909119906

119894 119910119906

119894) 119903

dam119894

cepdam119894

V119906119894

119889str119894

119888119906

119894

1199061

(755 760) 20 12 80 [500 600] 141199062

(750 720) 15 8 75 [505 530] 151199063

(785 775) 15 10 90 [500 550] 131199064

(770 740) 13 8 85 [520 540] 14

5 Simulation Experiments

The proposed task assignment method was tested by sim-ulation experiments In the experiments the task area wasrepresented by 1000 lowast 1000 grid which was populated by 4UAVs and 4 targets as seen in Figure 4The experiments wererun on a computer with an Intel Core 2 Duo E7500 293GHzprocessor and 2GB RAM

The parameters used for the simulations are summarizedin Tables 1 and 2

In these simulations the DMrsquos preference has beendivided into 2 categories as follows

Target t1

Target t2

Target t3

Target t4

UAVu1

UAVu2

UAVu3

UAVu4

Figure 4 Task area

(1) The primary purpose of the mission is to kill allthe targets The DM considers criterion 119877

119860to be the most

Mathematical Problems in Engineering 7

12

34

1

23

4

0

20

40

60

80

100

Rank

Target 1

Alternative (UAV)

Acce

ptab

ility

Figure 5 Rank acceptability indices 1198871ndash1198874 for target 119905

1

Table 2 Attribute values of the targets

Target (119909119905

119895

1015840

119910119905

119895

1015840

) V119905119895

119903max119895

119903min119895

1199051

(80 95) (90 100) 600 3001199052

(70 64) (80 90) 800 3501199053

(52 80) (90 100) 600 4001199054

(58 68) (90 100) 700 400

Table 3 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 04 01 025120596max 055 02 04

important 119862119871to be the second most important and 119862

119865to

be the third most important The weight intervals are givenin Table 3

Using the Monte Carlo technique the results are shownin Table 4 The rank acceptability indices are presented inFigures 5ndash8

Using (22)-(23) the final results of task assignment arelisted in Table 5 where 1 indicates that the target was assignedto the UAV and 0 otherwise

(2) The primary purpose of the mission was to kill all thetargets in the case of ensuring that the UAVs underwent zerodamage as far as possible The DM considers criterion 119862

119871to

be the most important 119877119860to be the second most important

and 119862119865to be the third most important The weight intervals

are given in Table 6 The results are shown in Table 7The final results of task assignment are listed in Table 8

6 Conclusion

This paper presented a task assignment method for multipleUAVs under severe uncertainty conditions in which the

Target 2

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 6 Rank acceptability indices 1198871ndash1198874 for target 119905

2

Target 3

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 7 Rank acceptability indices 1198871ndash1198874 for target 119905

3

values of the criteria essential to task assignment wererandom fuzzy or unknown Taking advantage of the SMAA-2 method this paper established the solving model and thesolution process

In the simulations we selected different weight infor-mation for 2 simulation experiments In the experimentsthere was little difference in the central weight vector foreach assignment so we only needed to select alternativeson the basis of the holistic acceptability index However ifthe central weight vectors vary widely the selection shouldinstead be based on the holistic acceptability index thecentral weight vector and the confidence factor

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

The task assignment problem can be settled by solvingthe above model when all values of the parameters are givenHowever in complex battlefield scenarios much informationis severely uncertain Uncertainty increases many difficultiesfor task assignment In these conditions we need newmethods to handle uncertainty

3 Criteria Analysis

In this section we will provide an analysis of the criteriaand give methods to quantitatively describe the involveduncertainties

31 The Value of the Target The value of a target evaluatedby military commanders or experts determines whetherthe target is worth attacking For ease of calculation in thispaper the lower and upper bounds of the value of targethave been set to be 0 and 100 A higher number indicates ahigher value However it is difficult for the commanders orexperts to provide a precise value They typically prefer touse linguistic variables such as ldquovery highrdquo and ldquofairly lowrdquoThese linguistic variables are fuzzy descriptors that cannotbe used in task assignment models To solve this problemwe have defined a fuzzy linguistic set 119878 based on the work in[19 20]

119878 = 1199040 1199041 1199042 1199043 1199044 1199045 1199046 1199047 1199048

= extremely high very high high fairly high medium fairly low low very low extremely low

(8)

Each linguistic variable in 119878 matches a value intervalextremely high = [90 100] very high = [80 90] high =

[70 80] fairly high = [55 70] medium = [45 55]fairly low = [30 45] low = [20 30] very low = [10 20] andextremely low = [0 10] Using the fuzzy set the commandersor experts can use a linguistic variable to describe the value ofa target and then the variable is transformed to a numericalinterval

32 The Value of the UAV When a UAV attacks a target theUAV may also be at risk of being attacked Therefore theUAVrsquos value is an important factor that we need to considerBy contrasting it with the value of target the value ofUAV canbe accurately obtained and can be represented by a precisenumber

33 The Distance between the UAV and the Target Thedistance between the UAV and the target is an importantcriterion for task assignment To minimize the flying costsa UAV tends to select the nearest target when other variablesare equal

The distance between the UAV 119906119894and target 119905

119895is calcu-

lated by using the Euclidean distance as shown in

119889119894119895

= radic(119909119906

119894minus 119909119905

119895)2

+ (119910119906

119894minus 119910119905

119895)2 (9)

In battlefield circumstances the coordinate (119909119905

119895 119910119905

119895) of

target 119905119895is not a certain value The information we have is

the detected position (119909119905

119895

1015840

119910119905

119895

1015840

) of 119905119895 However there is usually

a deviation between a targetrsquos detected position and its realposition

Normally the real position (119909119905

119895 119910119905

119895) follows a two-

dimensional Gaussian distribution whose expectation is(119909119905

119895

1015840

119910119905

119895

1015840

) as shown in

119891 (119909119905

119895 119910119905

119895) =

1

212058712059011205902radic1 minus 1205882

sdot 119890minus(12(1minus1205882))[(119909119905

119895minus119909119905

119895

1015840

)2120590

21minus2120588((119909

119905

119895minus119909119905

119895

1015840

)1205901)((119910119905

119895minus119910119905

119895

1015840

)21205902)+(119910

119905

119895minus119910119905

119895

1015840

)2120590

22 ]

(10)

Because 119909119905

119895and 119910

119905

119895are independent we let 120588 = 0 and we

assume 1205901

= 1205902

= 1 Then (10) can be simplified as

119891 (119909119905

119895 119910119905

119895) =

12120587

119890minus(12)[(119909119905

119895minus119909119905

119895

1015840

)2+(119910119905

119895minus119910119905

119895

1015840

)2] (11)

In (11) the probability distribution of a targetrsquos realposition within the task area is shown in Figure 2 Theprobability is 0159 that a targetrsquos real position is in accordancewith the detected position

The position can be accurately detected only when theUAV approaches the target However at the time the taskassignment ismade theUAVonly has uncertain information

34 The Flying Cost per Unit Distance of UAV This criterionas well as the distance between the UAV and the targetdetermines the flying cost of a mission For a UAV the valueof this criterion is fixed

35The Probability of UAVLoss Ahigh value target is usuallyprotected by a defense system that is equipped with surface-to-air missiles When performing the attack task the UAValso faces the possibility of being destroyed Washburn andKress [21] provided the probability of 119906

119894loss when attacking

target 119905119895as

119901119906

119894119895=

exp(minus(119877 minus 1198770)

2

21198872) 119903

min119895

le 119877 le 119903max119895

0 119877 gt 119903max119895

or 119877 lt 119903min119895

(12)

where 119877 denotes the real-time distance between 119905119895and 119906119894and

1198770and 119887 are parameters calculated using

1198770 =

(119903min119895

+ 119903max119895

)

2

119887 = radic2 (119903max119895

minus 119903min119895

)

(13)

Equation (12) indicates that the probability of 119906119894loss is

0 when being out of the range of a surface-to-air missile

4 Mathematical Problems in Engineering

02

015

01

005

0

(xt998400

j yt998400

j )

Figure 2 Probability distribution of target position

within the range of the missile the probability approximatesa Gaussian distribution

To calculate 119901119906

119894119895 we reasonably assume the following (1)

the defense system will attack a UAV at the same time theUAV attacks the target (2) a UAV always attacks a targetat the UAVrsquos best striking distance which depends on theUAVrsquos onboard weapon Different weapons have their ownbest strike distances

Based on the above assumptions we substitute 119877 in (12)with 119889

str119894 119889

str119894

typically is an interval such as [20 km 25 km]Consequently 119901

119906

119894119895is also an interval like [050 065] and so

forth

36 The Probability of Killing Target As mentioned abovethe UAV always attacks the target at the UAVrsquos best strikingdistance Washburn and Kress [21] presented the probability119901119905

119894119895of UAV 119906

119894killing target 119905

119895as

119901119905

119894119895= 1minus (05)

(119903dam2119894cepdam

2119894)

(14)

where cep represents the circular error probability thedefinition of which is the following If the probability thatthe weaponrsquos two-dimensional shooting error falls within thecircle is 05 the radius of circle will be called the circular errorprobability

4 Assignment Model

The SMAA-2 method was employed to solve the taskassignment problem The SMAA-2 method can handle mul-tiattribute decision making problems in which uncertaincriteria values and uncertain weights exist simultaneouslyThe SMAA-2 method has been applied in many aspects ofthe real world Kangas et al [22] used the SMAA-2 methodfor strategic forest planning Hokkanen et al [23] used theSMAA-2 method in a technology competition for cleaningpolluted soil in Helsinki Menou et al [24] gave a decisionsupport for centralizing cargo at a Moroccan airport hubusing the SMAA-2 method Rahman et al [25] used theSMAA-2 method to evaluate the choices for sustainablerural electrification in developing countries Pesola et al [26]

applied the method on alternatives for remote monitoringsystems of municipal buildings

41 The SMAA-2 Method Considering 119898 alternatives 119860 =

1199091 1199092 119909119898

and 119899 evaluation criteria 120596 is a weightvector for the decision maker (DM) to express his subjectivepreference 120596

119896represents the weight of criterion 119896 satisfying

sum119899

119896=1 120596119896

= 1 A weight distribution with density function119891(120596) is used to represent the DMrsquos partially known orunknown preference The weight distribution is in the setof feasible weights 119882 defined as 119882 = 120596 isin 119877

119899 120596 ge

0 and sum119899

119896=1 120596119896

= 1 120585119894119896represents the uncertain or imprecise

value of criterion 119896 for alternative 119909119894 Similarly a joint

probability distribution with density function 119891(120585) in thespace 119883 is used to represent the uncertain or imprecisecriteria values The utility of alternative 119909

119894is defined as

119880119894

= 119880 (120585119894 120596) =

119899

sum

119896=1120596119896119880119896

(120585119894119896

) (15)

where 119880119896(120585119894119896

) is the utility function of criterion 119896The SMAA-2 method defines a ranking function

rank(120585119894 120596) which is given by the following to represent the

rank of alternative 119909119894

rank (120585119894 120596) = 1+

119898

sum

ℎ=1120588 (119880 (120585

ℎ 120596) gt 119880 (120585

119894 120596)) (16)

where 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) is a judgment function If

119880(120585ℎ 120596) gt 119880(120585

119894 120596) then 120588(119880(120585

ℎ 120596) gt 119880(120585

119894 120596)) = 1

otherwise 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) = 0

The SMAA-2 method is based on analyzing the sets offavorable rank weights 119882

119903

119894(120585) Alternative 119909

119894always obtains

rank 119903 for arbitrary weight 120596 isin 119882119903

119894(120585) 119882

119903

119894(120585) is defined as

119882119903

119894(120585) = 120596 isin 119882 rank (120585

119894 120596) = 119903 (17)

The SMAA-2 method presents 3 important analysisindices All the indices are based on properties of thesestochastic sets The first index is the rank acceptability index119887119903

119894 defined as the expected volume of 119882

119903

119894(120585)

119887119903

119894= int119883

119891 (120585) int119882119903

119894(120585)

119891 (120596) 119889120596 119889120585 (18)

It is computed as a multidimensional integral over thecriteria distributions and the favorable rank weights repre-senting the acceptability or the probability of alternative 119909

119894

rank 119903The rank acceptability index is a real number in [0 1]

and is generally expressed as a percent Normally the moreacceptable alternatives should have higher acceptabilities forthe best ranks

The second index is the central weight vector 120596119888

119894 the

best single weight vector representation of a hypotheticalDM supporting alternative 119909

119894to rank first It is computed

as a double integral over the criteria distributions and thefavorable first rank weights 119882

1119894

(120585)

120596119888

119894=

int119883

119891 (120585) int119882

1119894(120585)

119891 (120596) 120596 119889120596 119889120585

1198871119894

(19)

Mathematical Problems in Engineering 5

The central weight vector 120596119888

119894is the expected center of

gravity of the favorable first rank weights 1198821119894

(120585) It can helpthe DM understand what preferences support the differentalternatives Moreover the central weight vectors are used tocompute the confidence factor

The third index is the confidence factor 119901119888

119894 representing

the probability for the alternative 119909119894ranking first if the central

weight vector is chosen It is computed as an integral over thecriteria distributions

119901119888

119894= int120585rank(120585119894120596119888119894 )=1

119891 (120585) 119889120585 (20)

The confidence factor can also be used to judge whetherthe criteria value is accurate enough to distinguish thealternatives when the central weight vector is used Providingany weight vector can also calculate the correspondingconfidence factor in a similar way

Comparing the alternatives according to their rankacceptabilities can be seen as a ldquosecond-orderrdquo multicriteriadecision problem [13] The SMAA-2 method additionallydefines the holistic acceptability index to provide ameasure ofthe overall acceptability of each alternative It is representedas a weighted sum of the rank acceptabilities

119886ℎ

119894=

119898

sum

119903=1120572119903119887119903

119894 (21)

where 120572119903are referred to as metaweights (or rank weights) A

complete priority order between the metaweights should bewell defined Lahdelma and Salminen [13] gave three possiblechoices linear weights 120572

119903= (119898 minus 119903)(119898 minus 1) inverse weights

120572119903

= 1119903 and centroid weights 120572119903

= sum119898

119894=1199031119894 sum

119898

119894=1 1119894 Aftercomparison they preferred using centroid weights

In practice accurately calculating these indices requirescomplex computation processes In order to reduce the com-putational complexity Tervonen and Lahdelma [27] gave anapproximate computation method by using the Monte Carlotechnique Even when dealing with large-scale problemsthis method can quickly solve them Furthermore Tervonen[28] presented JSMAA open source software for SMAAcomputations

42 The Decision Model To solve the problem of assigningthe attack tasks on119873

119879targets to119873

119880UAVs we decompose the

problem into 119873119879parts based on the number of targets For

target 119905119895

(119895 = 1 2 119873119879

) let the 119873119880UAVs be alternatives

and use 119887119903

119894119895 119886ℎ

119894119895 120596119888

119894119895 and 119901

119888

119894119895to respectively represent the

rank acceptability index holistic acceptability index centralweight vector and confidence factor of alternative 119906

119894 These

indices are calculated by using the SMAA-2 method On thebasis of the indices the DMs make the assignment decisions

In this study the DMs have weight intervals of thecriteria and the widths of the intervals are small The mainconsideration of the DMs is the overall acceptability Sothe holistic acceptability index is the most suitable index

Step 1 get all the criteria information and the a priori weightinformation and define a utility function for each all criteria

Step 4 use the Monte Carlo simulation method to calculateYes

No

Step 6 input the indices into the task assignment model to getthe optimal assignment scheme

the indices brij wcij

let j = 1Step 2

if j le NTStep 3

do j = j + 1Step 5

pcij and ahij

Figure 3 The steps of task assignment

for making decisions Then the task assignment model inSection 2 is modified as

max119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119886ℎ

119894119895(22)

st119873119880

sum

119894=1119909119894119895

= 1

119873119879

sum

119895=1119909119894119895

= 1

119909119894119895

= 0 1

(119894 = 1 2 119873119880

119895 = 1 2 119873119879

)

(23)

In this model objective function (22) aims to maximizethe holistic acceptability of the assignment schemes Tocalculate the holistic acceptability the DMs select centroidweights 120572

119903= sum119898

119894=1199031119894 sum

119898

119894=1 1119894The steps of task assignment for multiple UAVs under

severe uncertainty are given in Figure 3In step 1 linear utility functions are used Let 120585

119894119896(119896 =

1 2 3) respectively represent the values of criteria 119877119860 119862119865

and 119862119871for alternative 119906

119894 Their values can be calculated using

(1) 119877119860is an income-type criterion 119862

119865and 119862

119871are cost-type

criteriaTheir utility functions can respectively be defined as

119880119896

(120585119894119896

) =120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 1

119880119896

(120585119894119896

) = minus120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 2 3

(24)

where 120585min119894119896

= min1le119894le119898120585119894119896and 120585

max119894119896

= max1le119894le119898120585119894119896The effects

of the utility functions are to normalize the criteria valuesBased on the work of [27] the detailed procedure of Step

4 is given in Algorithm 1

6 Mathematical Problems in Engineering

(1) Initialize the data(2) For 119897 = 1 to 119868

119861do 119868

119861is the number of iterations

(3) Randomly generate a weight vector 120596 = (1205961 1205962 1205963) based on the prior weight information(4) Randomly generate a set of sub-criteria values (119901119905

119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(5) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(6) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(7) Calculate 119880119894for each 119906

119894using (15)

(8) Sort 119906119894according to the size of 119880

119894 getting ℎ

119894119895119897 ℎ119894119895119897

represents the sorting of 119906119894in iteration 119897

(9) If 119906119894rank the first then 120596

119888

119894119895= 120596119888

119894119895+ 120596

(10) End for(11) For 119894 = 1 to 119873

119880do

(12) For 119903 = 1 to 119873119880do

(13) ℎ119903

119894119895= sum119868119861

119897=1(ℎ119894119895119897

= 1199031 0) ℎ119903

119894119895is the total number of times that 119906

119894obtains rank 119903

(14) 119887119903

119894119895= ℎ119903

119894119895119868119861

(15) End for(16) Calculate 119886

ℎ

119894119895according to (21)

(17) End for(18) For 119894 = 1 to 119873

119880do

(19) 120596119888

119894119895= 120596119888

119894119895ℎ

1119894119895

(20) End for(21) For 119897 = 1 to 119868

119875do 119868

119875is the number of iterations

(22) Randomly generate a set of sub-criteria values (119901119905119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(23) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(24) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(25) For 119894 = 1 to 119873119880do

(26) 120596 = 120596119888

119894119895

(27) Calculate 119880119894for each 119906

119894using (15)

(28) Sort 119906119894according to the size of 119880

119894

(29) If 119906119894rank the first then 119901

119888

119894119895= 119901119888

119894119895+ 1

(30) End for(31) End for(32) For 119894 = 1 to 119873

119880do

(33) 119901119888

119894119895= 119901119888

119894119895119868119875

(34) End for

Algorithm 1 Calculating the indices for target 119905119895

Table 1 Attribute values of the UAVs

UAV (119909119906

119894 119910119906

119894) 119903

dam119894

cepdam119894

V119906119894

119889str119894

119888119906

119894

1199061

(755 760) 20 12 80 [500 600] 141199062

(750 720) 15 8 75 [505 530] 151199063

(785 775) 15 10 90 [500 550] 131199064

(770 740) 13 8 85 [520 540] 14

5 Simulation Experiments

The proposed task assignment method was tested by sim-ulation experiments In the experiments the task area wasrepresented by 1000 lowast 1000 grid which was populated by 4UAVs and 4 targets as seen in Figure 4The experiments wererun on a computer with an Intel Core 2 Duo E7500 293GHzprocessor and 2GB RAM

The parameters used for the simulations are summarizedin Tables 1 and 2

In these simulations the DMrsquos preference has beendivided into 2 categories as follows

Target t1

Target t2

Target t3

Target t4

UAVu1

UAVu2

UAVu3

UAVu4

Figure 4 Task area

(1) The primary purpose of the mission is to kill allthe targets The DM considers criterion 119877

119860to be the most

Mathematical Problems in Engineering 7

12

34

1

23

4

0

20

40

60

80

100

Rank

Target 1

Alternative (UAV)

Acce

ptab

ility

Figure 5 Rank acceptability indices 1198871ndash1198874 for target 119905

1

Table 2 Attribute values of the targets

Target (119909119905

119895

1015840

119910119905

119895

1015840

) V119905119895

119903max119895

119903min119895

1199051

(80 95) (90 100) 600 3001199052

(70 64) (80 90) 800 3501199053

(52 80) (90 100) 600 4001199054

(58 68) (90 100) 700 400

Table 3 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 04 01 025120596max 055 02 04

important 119862119871to be the second most important and 119862

119865to

be the third most important The weight intervals are givenin Table 3

Using the Monte Carlo technique the results are shownin Table 4 The rank acceptability indices are presented inFigures 5ndash8

Using (22)-(23) the final results of task assignment arelisted in Table 5 where 1 indicates that the target was assignedto the UAV and 0 otherwise

(2) The primary purpose of the mission was to kill all thetargets in the case of ensuring that the UAVs underwent zerodamage as far as possible The DM considers criterion 119862

119871to

be the most important 119877119860to be the second most important

and 119862119865to be the third most important The weight intervals

are given in Table 6 The results are shown in Table 7The final results of task assignment are listed in Table 8

6 Conclusion

This paper presented a task assignment method for multipleUAVs under severe uncertainty conditions in which the

Target 2

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 6 Rank acceptability indices 1198871ndash1198874 for target 119905

2

Target 3

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 7 Rank acceptability indices 1198871ndash1198874 for target 119905

3

values of the criteria essential to task assignment wererandom fuzzy or unknown Taking advantage of the SMAA-2 method this paper established the solving model and thesolution process

In the simulations we selected different weight infor-mation for 2 simulation experiments In the experimentsthere was little difference in the central weight vector foreach assignment so we only needed to select alternativeson the basis of the holistic acceptability index However ifthe central weight vectors vary widely the selection shouldinstead be based on the holistic acceptability index thecentral weight vector and the confidence factor

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

02

015

01

005

0

(xt998400

j yt998400

j )

Figure 2 Probability distribution of target position

within the range of the missile the probability approximatesa Gaussian distribution

To calculate 119901119906

119894119895 we reasonably assume the following (1)

the defense system will attack a UAV at the same time theUAV attacks the target (2) a UAV always attacks a targetat the UAVrsquos best striking distance which depends on theUAVrsquos onboard weapon Different weapons have their ownbest strike distances

Based on the above assumptions we substitute 119877 in (12)with 119889

str119894 119889

str119894

typically is an interval such as [20 km 25 km]Consequently 119901

119906

119894119895is also an interval like [050 065] and so

forth

36 The Probability of Killing Target As mentioned abovethe UAV always attacks the target at the UAVrsquos best strikingdistance Washburn and Kress [21] presented the probability119901119905

119894119895of UAV 119906

119894killing target 119905

119895as

119901119905

119894119895= 1minus (05)

(119903dam2119894cepdam

2119894)

(14)

where cep represents the circular error probability thedefinition of which is the following If the probability thatthe weaponrsquos two-dimensional shooting error falls within thecircle is 05 the radius of circle will be called the circular errorprobability

4 Assignment Model

The SMAA-2 method was employed to solve the taskassignment problem The SMAA-2 method can handle mul-tiattribute decision making problems in which uncertaincriteria values and uncertain weights exist simultaneouslyThe SMAA-2 method has been applied in many aspects ofthe real world Kangas et al [22] used the SMAA-2 methodfor strategic forest planning Hokkanen et al [23] used theSMAA-2 method in a technology competition for cleaningpolluted soil in Helsinki Menou et al [24] gave a decisionsupport for centralizing cargo at a Moroccan airport hubusing the SMAA-2 method Rahman et al [25] used theSMAA-2 method to evaluate the choices for sustainablerural electrification in developing countries Pesola et al [26]

applied the method on alternatives for remote monitoringsystems of municipal buildings

41 The SMAA-2 Method Considering 119898 alternatives 119860 =

1199091 1199092 119909119898

and 119899 evaluation criteria 120596 is a weightvector for the decision maker (DM) to express his subjectivepreference 120596

119896represents the weight of criterion 119896 satisfying

sum119899

119896=1 120596119896

= 1 A weight distribution with density function119891(120596) is used to represent the DMrsquos partially known orunknown preference The weight distribution is in the setof feasible weights 119882 defined as 119882 = 120596 isin 119877

119899 120596 ge

0 and sum119899

119896=1 120596119896

= 1 120585119894119896represents the uncertain or imprecise

value of criterion 119896 for alternative 119909119894 Similarly a joint

probability distribution with density function 119891(120585) in thespace 119883 is used to represent the uncertain or imprecisecriteria values The utility of alternative 119909

119894is defined as

119880119894

= 119880 (120585119894 120596) =

119899

sum

119896=1120596119896119880119896

(120585119894119896

) (15)

where 119880119896(120585119894119896

) is the utility function of criterion 119896The SMAA-2 method defines a ranking function

rank(120585119894 120596) which is given by the following to represent the

rank of alternative 119909119894

rank (120585119894 120596) = 1+

119898

sum

ℎ=1120588 (119880 (120585

ℎ 120596) gt 119880 (120585

119894 120596)) (16)

where 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) is a judgment function If

119880(120585ℎ 120596) gt 119880(120585

119894 120596) then 120588(119880(120585

ℎ 120596) gt 119880(120585

119894 120596)) = 1

otherwise 120588(119880(120585ℎ 120596) gt 119880(120585

119894 120596)) = 0

The SMAA-2 method is based on analyzing the sets offavorable rank weights 119882

119903

119894(120585) Alternative 119909

119894always obtains

rank 119903 for arbitrary weight 120596 isin 119882119903

119894(120585) 119882

119903

119894(120585) is defined as

119882119903

119894(120585) = 120596 isin 119882 rank (120585

119894 120596) = 119903 (17)

The SMAA-2 method presents 3 important analysisindices All the indices are based on properties of thesestochastic sets The first index is the rank acceptability index119887119903

119894 defined as the expected volume of 119882

119903

119894(120585)

119887119903

119894= int119883

119891 (120585) int119882119903

119894(120585)

119891 (120596) 119889120596 119889120585 (18)

It is computed as a multidimensional integral over thecriteria distributions and the favorable rank weights repre-senting the acceptability or the probability of alternative 119909

119894

rank 119903The rank acceptability index is a real number in [0 1]

and is generally expressed as a percent Normally the moreacceptable alternatives should have higher acceptabilities forthe best ranks

The second index is the central weight vector 120596119888

119894 the

best single weight vector representation of a hypotheticalDM supporting alternative 119909

119894to rank first It is computed

as a double integral over the criteria distributions and thefavorable first rank weights 119882

1119894

(120585)

120596119888

119894=

int119883

119891 (120585) int119882

1119894(120585)

119891 (120596) 120596 119889120596 119889120585

1198871119894

(19)

Mathematical Problems in Engineering 5

The central weight vector 120596119888

119894is the expected center of

gravity of the favorable first rank weights 1198821119894

(120585) It can helpthe DM understand what preferences support the differentalternatives Moreover the central weight vectors are used tocompute the confidence factor

The third index is the confidence factor 119901119888

119894 representing

the probability for the alternative 119909119894ranking first if the central

weight vector is chosen It is computed as an integral over thecriteria distributions

119901119888

119894= int120585rank(120585119894120596119888119894 )=1

119891 (120585) 119889120585 (20)

The confidence factor can also be used to judge whetherthe criteria value is accurate enough to distinguish thealternatives when the central weight vector is used Providingany weight vector can also calculate the correspondingconfidence factor in a similar way

Comparing the alternatives according to their rankacceptabilities can be seen as a ldquosecond-orderrdquo multicriteriadecision problem [13] The SMAA-2 method additionallydefines the holistic acceptability index to provide ameasure ofthe overall acceptability of each alternative It is representedas a weighted sum of the rank acceptabilities

119886ℎ

119894=

119898

sum

119903=1120572119903119887119903

119894 (21)

where 120572119903are referred to as metaweights (or rank weights) A

complete priority order between the metaweights should bewell defined Lahdelma and Salminen [13] gave three possiblechoices linear weights 120572

119903= (119898 minus 119903)(119898 minus 1) inverse weights

120572119903

= 1119903 and centroid weights 120572119903

= sum119898

119894=1199031119894 sum

119898

119894=1 1119894 Aftercomparison they preferred using centroid weights

In practice accurately calculating these indices requirescomplex computation processes In order to reduce the com-putational complexity Tervonen and Lahdelma [27] gave anapproximate computation method by using the Monte Carlotechnique Even when dealing with large-scale problemsthis method can quickly solve them Furthermore Tervonen[28] presented JSMAA open source software for SMAAcomputations

42 The Decision Model To solve the problem of assigningthe attack tasks on119873

119879targets to119873

119880UAVs we decompose the

problem into 119873119879parts based on the number of targets For

target 119905119895

(119895 = 1 2 119873119879

) let the 119873119880UAVs be alternatives

and use 119887119903

119894119895 119886ℎ

119894119895 120596119888

119894119895 and 119901

119888

119894119895to respectively represent the

rank acceptability index holistic acceptability index centralweight vector and confidence factor of alternative 119906

119894 These

indices are calculated by using the SMAA-2 method On thebasis of the indices the DMs make the assignment decisions

In this study the DMs have weight intervals of thecriteria and the widths of the intervals are small The mainconsideration of the DMs is the overall acceptability Sothe holistic acceptability index is the most suitable index

Step 1 get all the criteria information and the a priori weightinformation and define a utility function for each all criteria

Step 4 use the Monte Carlo simulation method to calculateYes

No

Step 6 input the indices into the task assignment model to getthe optimal assignment scheme

the indices brij wcij

let j = 1Step 2

if j le NTStep 3

do j = j + 1Step 5

pcij and ahij

Figure 3 The steps of task assignment

for making decisions Then the task assignment model inSection 2 is modified as

max119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119886ℎ

119894119895(22)

st119873119880

sum

119894=1119909119894119895

= 1

119873119879

sum

119895=1119909119894119895

= 1

119909119894119895

= 0 1

(119894 = 1 2 119873119880

119895 = 1 2 119873119879

)

(23)

In this model objective function (22) aims to maximizethe holistic acceptability of the assignment schemes Tocalculate the holistic acceptability the DMs select centroidweights 120572

119903= sum119898

119894=1199031119894 sum

119898

119894=1 1119894The steps of task assignment for multiple UAVs under

severe uncertainty are given in Figure 3In step 1 linear utility functions are used Let 120585

119894119896(119896 =

1 2 3) respectively represent the values of criteria 119877119860 119862119865

and 119862119871for alternative 119906

119894 Their values can be calculated using

(1) 119877119860is an income-type criterion 119862

119865and 119862

119871are cost-type

criteriaTheir utility functions can respectively be defined as

119880119896

(120585119894119896

) =120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 1

119880119896

(120585119894119896

) = minus120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 2 3

(24)

where 120585min119894119896

= min1le119894le119898120585119894119896and 120585

max119894119896

= max1le119894le119898120585119894119896The effects

of the utility functions are to normalize the criteria valuesBased on the work of [27] the detailed procedure of Step

4 is given in Algorithm 1

6 Mathematical Problems in Engineering

(1) Initialize the data(2) For 119897 = 1 to 119868

119861do 119868

119861is the number of iterations

(3) Randomly generate a weight vector 120596 = (1205961 1205962 1205963) based on the prior weight information(4) Randomly generate a set of sub-criteria values (119901119905

119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(5) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(6) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(7) Calculate 119880119894for each 119906

119894using (15)

(8) Sort 119906119894according to the size of 119880

119894 getting ℎ

119894119895119897 ℎ119894119895119897

represents the sorting of 119906119894in iteration 119897

(9) If 119906119894rank the first then 120596

119888

119894119895= 120596119888

119894119895+ 120596

(10) End for(11) For 119894 = 1 to 119873

119880do

(12) For 119903 = 1 to 119873119880do

(13) ℎ119903

119894119895= sum119868119861

119897=1(ℎ119894119895119897

= 1199031 0) ℎ119903

119894119895is the total number of times that 119906

119894obtains rank 119903

(14) 119887119903

119894119895= ℎ119903

119894119895119868119861

(15) End for(16) Calculate 119886

ℎ

119894119895according to (21)

(17) End for(18) For 119894 = 1 to 119873

119880do

(19) 120596119888

119894119895= 120596119888

119894119895ℎ

1119894119895

(20) End for(21) For 119897 = 1 to 119868

119875do 119868

119875is the number of iterations

(22) Randomly generate a set of sub-criteria values (119901119905119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(23) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(24) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(25) For 119894 = 1 to 119873119880do

(26) 120596 = 120596119888

119894119895

(27) Calculate 119880119894for each 119906

119894using (15)

(28) Sort 119906119894according to the size of 119880

119894

(29) If 119906119894rank the first then 119901

119888

119894119895= 119901119888

119894119895+ 1

(30) End for(31) End for(32) For 119894 = 1 to 119873

119880do

(33) 119901119888

119894119895= 119901119888

119894119895119868119875

(34) End for

Algorithm 1 Calculating the indices for target 119905119895

Table 1 Attribute values of the UAVs

UAV (119909119906

119894 119910119906

119894) 119903

dam119894

cepdam119894

V119906119894

119889str119894

119888119906

119894

1199061

(755 760) 20 12 80 [500 600] 141199062

(750 720) 15 8 75 [505 530] 151199063

(785 775) 15 10 90 [500 550] 131199064

(770 740) 13 8 85 [520 540] 14

5 Simulation Experiments

The proposed task assignment method was tested by sim-ulation experiments In the experiments the task area wasrepresented by 1000 lowast 1000 grid which was populated by 4UAVs and 4 targets as seen in Figure 4The experiments wererun on a computer with an Intel Core 2 Duo E7500 293GHzprocessor and 2GB RAM

The parameters used for the simulations are summarizedin Tables 1 and 2

In these simulations the DMrsquos preference has beendivided into 2 categories as follows

Target t1

Target t2

Target t3

Target t4

UAVu1

UAVu2

UAVu3

UAVu4

Figure 4 Task area

(1) The primary purpose of the mission is to kill allthe targets The DM considers criterion 119877

119860to be the most

Mathematical Problems in Engineering 7

12

34

1

23

4

0

20

40

60

80

100

Rank

Target 1

Alternative (UAV)

Acce

ptab

ility

Figure 5 Rank acceptability indices 1198871ndash1198874 for target 119905

1

Table 2 Attribute values of the targets

Target (119909119905

119895

1015840

119910119905

119895

1015840

) V119905119895

119903max119895

119903min119895

1199051

(80 95) (90 100) 600 3001199052

(70 64) (80 90) 800 3501199053

(52 80) (90 100) 600 4001199054

(58 68) (90 100) 700 400

Table 3 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 04 01 025120596max 055 02 04

important 119862119871to be the second most important and 119862

119865to

be the third most important The weight intervals are givenin Table 3

Using the Monte Carlo technique the results are shownin Table 4 The rank acceptability indices are presented inFigures 5ndash8

Using (22)-(23) the final results of task assignment arelisted in Table 5 where 1 indicates that the target was assignedto the UAV and 0 otherwise

(2) The primary purpose of the mission was to kill all thetargets in the case of ensuring that the UAVs underwent zerodamage as far as possible The DM considers criterion 119862

119871to

be the most important 119877119860to be the second most important

and 119862119865to be the third most important The weight intervals

are given in Table 6 The results are shown in Table 7The final results of task assignment are listed in Table 8

6 Conclusion

This paper presented a task assignment method for multipleUAVs under severe uncertainty conditions in which the

Target 2

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 6 Rank acceptability indices 1198871ndash1198874 for target 119905

2

Target 3

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 7 Rank acceptability indices 1198871ndash1198874 for target 119905

3

values of the criteria essential to task assignment wererandom fuzzy or unknown Taking advantage of the SMAA-2 method this paper established the solving model and thesolution process

In the simulations we selected different weight infor-mation for 2 simulation experiments In the experimentsthere was little difference in the central weight vector foreach assignment so we only needed to select alternativeson the basis of the holistic acceptability index However ifthe central weight vectors vary widely the selection shouldinstead be based on the holistic acceptability index thecentral weight vector and the confidence factor

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

The central weight vector 120596119888

119894is the expected center of

gravity of the favorable first rank weights 1198821119894

(120585) It can helpthe DM understand what preferences support the differentalternatives Moreover the central weight vectors are used tocompute the confidence factor

The third index is the confidence factor 119901119888

119894 representing

the probability for the alternative 119909119894ranking first if the central

weight vector is chosen It is computed as an integral over thecriteria distributions

119901119888

119894= int120585rank(120585119894120596119888119894 )=1

119891 (120585) 119889120585 (20)

The confidence factor can also be used to judge whetherthe criteria value is accurate enough to distinguish thealternatives when the central weight vector is used Providingany weight vector can also calculate the correspondingconfidence factor in a similar way

Comparing the alternatives according to their rankacceptabilities can be seen as a ldquosecond-orderrdquo multicriteriadecision problem [13] The SMAA-2 method additionallydefines the holistic acceptability index to provide ameasure ofthe overall acceptability of each alternative It is representedas a weighted sum of the rank acceptabilities

119886ℎ

119894=

119898

sum

119903=1120572119903119887119903

119894 (21)

where 120572119903are referred to as metaweights (or rank weights) A

complete priority order between the metaweights should bewell defined Lahdelma and Salminen [13] gave three possiblechoices linear weights 120572

119903= (119898 minus 119903)(119898 minus 1) inverse weights

120572119903

= 1119903 and centroid weights 120572119903

= sum119898

119894=1199031119894 sum

119898

119894=1 1119894 Aftercomparison they preferred using centroid weights

In practice accurately calculating these indices requirescomplex computation processes In order to reduce the com-putational complexity Tervonen and Lahdelma [27] gave anapproximate computation method by using the Monte Carlotechnique Even when dealing with large-scale problemsthis method can quickly solve them Furthermore Tervonen[28] presented JSMAA open source software for SMAAcomputations

42 The Decision Model To solve the problem of assigningthe attack tasks on119873

119879targets to119873

119880UAVs we decompose the

problem into 119873119879parts based on the number of targets For

target 119905119895

(119895 = 1 2 119873119879

) let the 119873119880UAVs be alternatives

and use 119887119903

119894119895 119886ℎ

119894119895 120596119888

119894119895 and 119901

119888

119894119895to respectively represent the

rank acceptability index holistic acceptability index centralweight vector and confidence factor of alternative 119906

119894 These

indices are calculated by using the SMAA-2 method On thebasis of the indices the DMs make the assignment decisions

In this study the DMs have weight intervals of thecriteria and the widths of the intervals are small The mainconsideration of the DMs is the overall acceptability Sothe holistic acceptability index is the most suitable index

Step 1 get all the criteria information and the a priori weightinformation and define a utility function for each all criteria

Step 4 use the Monte Carlo simulation method to calculateYes

No

Step 6 input the indices into the task assignment model to getthe optimal assignment scheme

the indices brij wcij

let j = 1Step 2

if j le NTStep 3

do j = j + 1Step 5

pcij and ahij

Figure 3 The steps of task assignment

for making decisions Then the task assignment model inSection 2 is modified as

max119873119880

sum

119894=1

119873119879

sum

119895=1119909119894119895

119886ℎ

119894119895(22)

st119873119880

sum

119894=1119909119894119895

= 1

119873119879

sum

119895=1119909119894119895

= 1

119909119894119895

= 0 1

(119894 = 1 2 119873119880

119895 = 1 2 119873119879

)

(23)

In this model objective function (22) aims to maximizethe holistic acceptability of the assignment schemes Tocalculate the holistic acceptability the DMs select centroidweights 120572

119903= sum119898

119894=1199031119894 sum

119898

119894=1 1119894The steps of task assignment for multiple UAVs under

severe uncertainty are given in Figure 3In step 1 linear utility functions are used Let 120585

119894119896(119896 =

1 2 3) respectively represent the values of criteria 119877119860 119862119865

and 119862119871for alternative 119906

119894 Their values can be calculated using

(1) 119877119860is an income-type criterion 119862

119865and 119862

119871are cost-type

criteriaTheir utility functions can respectively be defined as

119880119896

(120585119894119896

) =120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 1

119880119896

(120585119894119896

) = minus120585119894119896

minus 120585min119894119896

120585max119894119896

minus 120585min119894119896

119896 = 2 3

(24)

where 120585min119894119896

= min1le119894le119898120585119894119896and 120585

max119894119896

= max1le119894le119898120585119894119896The effects

of the utility functions are to normalize the criteria valuesBased on the work of [27] the detailed procedure of Step

4 is given in Algorithm 1

6 Mathematical Problems in Engineering

(1) Initialize the data(2) For 119897 = 1 to 119868

119861do 119868

119861is the number of iterations

(3) Randomly generate a weight vector 120596 = (1205961 1205962 1205963) based on the prior weight information(4) Randomly generate a set of sub-criteria values (119901119905

119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(5) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(6) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(7) Calculate 119880119894for each 119906

119894using (15)

(8) Sort 119906119894according to the size of 119880

119894 getting ℎ

119894119895119897 ℎ119894119895119897

represents the sorting of 119906119894in iteration 119897

(9) If 119906119894rank the first then 120596

119888

119894119895= 120596119888

119894119895+ 120596

(10) End for(11) For 119894 = 1 to 119873

119880do

(12) For 119903 = 1 to 119873119880do

(13) ℎ119903

119894119895= sum119868119861

119897=1(ℎ119894119895119897

= 1199031 0) ℎ119903

119894119895is the total number of times that 119906

119894obtains rank 119903

(14) 119887119903

119894119895= ℎ119903

119894119895119868119861

(15) End for(16) Calculate 119886

ℎ

119894119895according to (21)

(17) End for(18) For 119894 = 1 to 119873

119880do

(19) 120596119888

119894119895= 120596119888

119894119895ℎ

1119894119895

(20) End for(21) For 119897 = 1 to 119868

119875do 119868

119875is the number of iterations

(22) Randomly generate a set of sub-criteria values (119901119905119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(23) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(24) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(25) For 119894 = 1 to 119873119880do

(26) 120596 = 120596119888

119894119895

(27) Calculate 119880119894for each 119906

119894using (15)

(28) Sort 119906119894according to the size of 119880

119894

(29) If 119906119894rank the first then 119901

119888

119894119895= 119901119888

119894119895+ 1

(30) End for(31) End for(32) For 119894 = 1 to 119873

119880do

(33) 119901119888

119894119895= 119901119888

119894119895119868119875

(34) End for

Algorithm 1 Calculating the indices for target 119905119895

Table 1 Attribute values of the UAVs

UAV (119909119906

119894 119910119906

119894) 119903

dam119894

cepdam119894

V119906119894

119889str119894

119888119906

119894

1199061

(755 760) 20 12 80 [500 600] 141199062

(750 720) 15 8 75 [505 530] 151199063

(785 775) 15 10 90 [500 550] 131199064

(770 740) 13 8 85 [520 540] 14

5 Simulation Experiments

The proposed task assignment method was tested by sim-ulation experiments In the experiments the task area wasrepresented by 1000 lowast 1000 grid which was populated by 4UAVs and 4 targets as seen in Figure 4The experiments wererun on a computer with an Intel Core 2 Duo E7500 293GHzprocessor and 2GB RAM

The parameters used for the simulations are summarizedin Tables 1 and 2

In these simulations the DMrsquos preference has beendivided into 2 categories as follows

Target t1

Target t2

Target t3

Target t4

UAVu1

UAVu2

UAVu3

UAVu4

Figure 4 Task area

(1) The primary purpose of the mission is to kill allthe targets The DM considers criterion 119877

119860to be the most

Mathematical Problems in Engineering 7

12

34

1

23

4

0

20

40

60

80

100

Rank

Target 1

Alternative (UAV)

Acce

ptab

ility

Figure 5 Rank acceptability indices 1198871ndash1198874 for target 119905

1

Table 2 Attribute values of the targets

Target (119909119905

119895

1015840

119910119905

119895

1015840

) V119905119895

119903max119895

119903min119895

1199051

(80 95) (90 100) 600 3001199052

(70 64) (80 90) 800 3501199053

(52 80) (90 100) 600 4001199054

(58 68) (90 100) 700 400

Table 3 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 04 01 025120596max 055 02 04

important 119862119871to be the second most important and 119862

119865to

be the third most important The weight intervals are givenin Table 3

Using the Monte Carlo technique the results are shownin Table 4 The rank acceptability indices are presented inFigures 5ndash8

Using (22)-(23) the final results of task assignment arelisted in Table 5 where 1 indicates that the target was assignedto the UAV and 0 otherwise

(2) The primary purpose of the mission was to kill all thetargets in the case of ensuring that the UAVs underwent zerodamage as far as possible The DM considers criterion 119862

119871to

be the most important 119877119860to be the second most important

and 119862119865to be the third most important The weight intervals

are given in Table 6 The results are shown in Table 7The final results of task assignment are listed in Table 8

6 Conclusion

This paper presented a task assignment method for multipleUAVs under severe uncertainty conditions in which the

Target 2

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 6 Rank acceptability indices 1198871ndash1198874 for target 119905

2

Target 3

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 7 Rank acceptability indices 1198871ndash1198874 for target 119905

3

values of the criteria essential to task assignment wererandom fuzzy or unknown Taking advantage of the SMAA-2 method this paper established the solving model and thesolution process

In the simulations we selected different weight infor-mation for 2 simulation experiments In the experimentsthere was little difference in the central weight vector foreach assignment so we only needed to select alternativeson the basis of the holistic acceptability index However ifthe central weight vectors vary widely the selection shouldinstead be based on the holistic acceptability index thecentral weight vector and the confidence factor

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

(1) Initialize the data(2) For 119897 = 1 to 119868

119861do 119868

119861is the number of iterations

(3) Randomly generate a weight vector 120596 = (1205961 1205962 1205963) based on the prior weight information(4) Randomly generate a set of sub-criteria values (119901119905

119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(5) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(6) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(7) Calculate 119880119894for each 119906

119894using (15)

(8) Sort 119906119894according to the size of 119880

119894 getting ℎ

119894119895119897 ℎ119894119895119897

represents the sorting of 119906119894in iteration 119897

(9) If 119906119894rank the first then 120596

119888

119894119895= 120596119888

119894119895+ 120596

(10) End for(11) For 119894 = 1 to 119873

119880do

(12) For 119903 = 1 to 119873119880do

(13) ℎ119903

119894119895= sum119868119861

119897=1(ℎ119894119895119897

= 1199031 0) ℎ119903

119894119895is the total number of times that 119906

119894obtains rank 119903

(14) 119887119903

119894119895= ℎ119903

119894119895119868119861

(15) End for(16) Calculate 119886

ℎ

119894119895according to (21)

(17) End for(18) For 119894 = 1 to 119873

119880do

(19) 120596119888

119894119895= 120596119888

119894119895ℎ

1119894119895

(20) End for(21) For 119897 = 1 to 119868

119875do 119868

119875is the number of iterations

(22) Randomly generate a set of sub-criteria values (119901119905119894119895 119901119906

119894119895 V119906119894 V119905119895 119889119894119895 119888119906

119894)

(23) Calculate 120585119894119896(119896 = 1 2 3) for each 119906

119894 using (1)

(24) Calculate 119880119896(120585119894119896

) (119896 = 1 2 3) for each 119906119894 using (24)

(25) For 119894 = 1 to 119873119880do

(26) 120596 = 120596119888

119894119895

(27) Calculate 119880119894for each 119906

119894using (15)

(28) Sort 119906119894according to the size of 119880

119894

(29) If 119906119894rank the first then 119901

119888

119894119895= 119901119888

119894119895+ 1

(30) End for(31) End for(32) For 119894 = 1 to 119873

119880do

(33) 119901119888

119894119895= 119901119888

119894119895119868119875

(34) End for

Algorithm 1 Calculating the indices for target 119905119895

Table 1 Attribute values of the UAVs

UAV (119909119906

119894 119910119906

119894) 119903

dam119894

cepdam119894

V119906119894

119889str119894

119888119906

119894

1199061

(755 760) 20 12 80 [500 600] 141199062

(750 720) 15 8 75 [505 530] 151199063

(785 775) 15 10 90 [500 550] 131199064

(770 740) 13 8 85 [520 540] 14

5 Simulation Experiments

The proposed task assignment method was tested by sim-ulation experiments In the experiments the task area wasrepresented by 1000 lowast 1000 grid which was populated by 4UAVs and 4 targets as seen in Figure 4The experiments wererun on a computer with an Intel Core 2 Duo E7500 293GHzprocessor and 2GB RAM

The parameters used for the simulations are summarizedin Tables 1 and 2

In these simulations the DMrsquos preference has beendivided into 2 categories as follows

Target t1

Target t2

Target t3

Target t4

UAVu1

UAVu2

UAVu3

UAVu4

Figure 4 Task area

(1) The primary purpose of the mission is to kill allthe targets The DM considers criterion 119877

119860to be the most

Mathematical Problems in Engineering 7

12

34

1

23

4

0

20

40

60

80

100

Rank

Target 1

Alternative (UAV)

Acce

ptab

ility

Figure 5 Rank acceptability indices 1198871ndash1198874 for target 119905

1

Table 2 Attribute values of the targets

Target (119909119905

119895

1015840

119910119905

119895

1015840

) V119905119895

119903max119895

119903min119895

1199051

(80 95) (90 100) 600 3001199052

(70 64) (80 90) 800 3501199053

(52 80) (90 100) 600 4001199054

(58 68) (90 100) 700 400

Table 3 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 04 01 025120596max 055 02 04

important 119862119871to be the second most important and 119862

119865to

be the third most important The weight intervals are givenin Table 3

Using the Monte Carlo technique the results are shownin Table 4 The rank acceptability indices are presented inFigures 5ndash8

Using (22)-(23) the final results of task assignment arelisted in Table 5 where 1 indicates that the target was assignedto the UAV and 0 otherwise

(2) The primary purpose of the mission was to kill all thetargets in the case of ensuring that the UAVs underwent zerodamage as far as possible The DM considers criterion 119862

119871to

be the most important 119877119860to be the second most important

and 119862119865to be the third most important The weight intervals

are given in Table 6 The results are shown in Table 7The final results of task assignment are listed in Table 8

6 Conclusion

This paper presented a task assignment method for multipleUAVs under severe uncertainty conditions in which the

Target 2

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 6 Rank acceptability indices 1198871ndash1198874 for target 119905

2

Target 3

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 7 Rank acceptability indices 1198871ndash1198874 for target 119905

3

values of the criteria essential to task assignment wererandom fuzzy or unknown Taking advantage of the SMAA-2 method this paper established the solving model and thesolution process

In the simulations we selected different weight infor-mation for 2 simulation experiments In the experimentsthere was little difference in the central weight vector foreach assignment so we only needed to select alternativeson the basis of the holistic acceptability index However ifthe central weight vectors vary widely the selection shouldinstead be based on the holistic acceptability index thecentral weight vector and the confidence factor

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

12

34

1

23

4

0

20

40

60

80

100

Rank

Target 1

Alternative (UAV)

Acce

ptab

ility

Figure 5 Rank acceptability indices 1198871ndash1198874 for target 119905

1

Table 2 Attribute values of the targets

Target (119909119905

119895

1015840

119910119905

119895

1015840

) V119905119895

119903max119895

119903min119895

1199051

(80 95) (90 100) 600 3001199052

(70 64) (80 90) 800 3501199053

(52 80) (90 100) 600 4001199054

(58 68) (90 100) 700 400

Table 3 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 04 01 025120596max 055 02 04

important 119862119871to be the second most important and 119862

119865to

be the third most important The weight intervals are givenin Table 3

Using the Monte Carlo technique the results are shownin Table 4 The rank acceptability indices are presented inFigures 5ndash8

Using (22)-(23) the final results of task assignment arelisted in Table 5 where 1 indicates that the target was assignedto the UAV and 0 otherwise

(2) The primary purpose of the mission was to kill all thetargets in the case of ensuring that the UAVs underwent zerodamage as far as possible The DM considers criterion 119862

119871to

be the most important 119877119860to be the second most important

and 119862119865to be the third most important The weight intervals

are given in Table 6 The results are shown in Table 7The final results of task assignment are listed in Table 8

6 Conclusion

This paper presented a task assignment method for multipleUAVs under severe uncertainty conditions in which the

Target 2

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 6 Rank acceptability indices 1198871ndash1198874 for target 119905

2

Target 3

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 7 Rank acceptability indices 1198871ndash1198874 for target 119905

3

values of the criteria essential to task assignment wererandom fuzzy or unknown Taking advantage of the SMAA-2 method this paper established the solving model and thesolution process

In the simulations we selected different weight infor-mation for 2 simulation experiments In the experimentsthere was little difference in the central weight vector foreach assignment so we only needed to select alternativeson the basis of the holistic acceptability index However ifthe central weight vectors vary widely the selection shouldinstead be based on the holistic acceptability index thecentral weight vector and the confidence factor

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 4 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

01320 08678 00001 00001 04959 01556 03485 00455 052651199062

04395 00685 03161 01759 05237 01631 03132 1 052811199063

04284 00636 03432 01648 04684 01544 03772 0 051971199064

00001 00001 03406 06592 05215 01137 03648 0 00621

1199052

1199061

00504 09493 00002 00001 05050 01587 03362 0 048191199062

03616 00263 03145 02976 05258 01663 03079 1 043071199063

05879 00242 02904 00975 04767 01532 03701 0 065171199064

00001 00002 03949 06048 05172 01592 03236 0 00720

1199053

1199061

01631 08366 00002 00001 05011 01539 03450 00894 054341199062

03790 00849 03293 02068 05259 01651 03090 1 047751199063

04578 00784 03271 01367 04690 01547 03763 0 055291199064

00001 00001 03434 06564 05205 01969 02826 0 00626

1199054

1199061

00841 09156 00001 00002 04959 01568 03473 0 050031199062

04516 00439 03288 01757 05226 01627 03147 1 053131199063

04642 00404 03503 01451 04696 01547 03757 0 054631199064

00001 00001 03208 06790 05491 01521 02988 0 00585

Target 4

12

34

1

23

4Rank

Alternative (UAV)

0

20

40

60

80

100

Acce

ptab

ility

Figure 8 Rank acceptability indices 1198871ndash1198874 for target 119905

4

Table 5 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 0 1 01199062

0 0 0 11199063

0 1 0 01199064

1 0 0 0

Another advantage to the SMAA-2 method is that itallows using any preference models commonly acceptedby DMs in practical problems This paper uses the linearutility function because it is reasonably easy to handle both

Table 6 The weight intervals

Attribute 119877119860

119862119865

119862119871

120596min 025 01 045120596max 035 02 055

theoretically and computationally also DMs understood iteasily

The proposed method provides a newmethod to apply totask assignment It can provide an acceptable task allocationscheme before a mission when a great deal of information isuncertain

Symbols

119906119894

(119894 = 1 2 119873119880

) UAVs119905119895

(119895 = 1 2 119873119879

) Targets119877119860 Expected attack rewards

119862119865 Flying costs of UAVs

119862119871 Expected losses of UAVs

(119909119906

119894 119910119906

119894) Two-dimensional coordinate of 119906

119894

(119909119905

119895 119910119905

119895) Two-dimensional coordinate of 119905

119895

(119909119905

119895

1015840

119910119905

119895

1015840

) Predetected two-dimensionalcoordinate of 119905

119895

V119906119894 Value of 119906

119894

V119905119895 Value of 119905

119895

119888119906

119894 Flying cost per unit distance of 119906

119894

119889119894119895 Distance between UAV 119906

119894and target

119905119895

119901119905

119894119895 Probability of 119906

119894killing 119905

119895

119901119906

119894119895 Probability of 119906

119894loss

119889str119894 Best strike distance of 119906

119894rsquos onboard

weapon

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 7 Rank acceptability indices and holistic acceptability indices

Target UAV 1198871119894119895

1198872119894119895

1198873119894119895

1198874119894119895

120596119888

119894119895119901119888

119894119895119886ℎ

119894119895

1199051

1199061

0 09997 00002 00001 0 0 0 0 045441199062

0 0 00001 09999 0 0 0 0 000001199063

09999 00001 0 0 03163 01668 05169 09976 099991199064

00001 00002 09997 0 03149 01847 05004 0 01820

1199052

1199061

09997 00002 00001 0 03165 01666 05168 1 099981199062

0 00001 09999 0 0 0 0 0 018181199063

0 0 0 10000 0 0 0 0 000001199064

00003 09997 0 0 03093 01687 05220 0 04547

1199053

1199061

0 09997 00003 0 0 0 0 0 045451199062

10000 0 0 0 03164 01667 05169 1 100001199063

0 0 0 10000 0 0 0 0 000001199064

0 00003 09997 0 0 0 0 1 01819

1199054

1199061

0 09997 00003 0 0 0 0 0 045451199062

0 0 0 10000 0 0 0 0 01199063

10000 0 0 0 03163 01672 05165 1 100001199064

0 00003 09997 0 0 0 0 1 01819

Table 8 The final result of assignment

UAV Target1199051

1199052

1199053

1199054

1199061

0 1 0 01199062

0 0 1 01199063

0 0 0 11199064

1 0 0 0

119903dam119894

Damage radius of 119906119894rsquos onboard weapon

cepdam119894

Circular error probability of 119906119894rsquos

onboard weapon119903max119895

Maximum range of 119905119895rsquos defence missile

119903min119895

Minimum range of 119905119895rsquos defence missile

119909119894119895 Decision variables 119909

119894119895= 1 if 119905

119895is

assigned to 119906119894and is 0 otherwise

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partly supported by the National NaturalScience Foundation of China (nos 71401048 and 71131002)and the Humanities and Social Science Projects of Ministryof Education of China (no 13YJC630051)

References

[1] C Schumacher P R Chandler M Pachter and L S PachterldquoOptimization of air vehicles operations using mixed-integer

linear programmingrdquo Journal of the Operational Research Soci-ety vol 58 no 4 pp 516ndash527 2007

[2] C C Murray and M H Karwan ldquoAn extensible modelingframework for dynamic reassignment and rerouting in cooper-ative airborne operationsrdquo Naval Research Logistics vol 57 no7 pp 634ndash652 2010

[3] M Alighanbari and J P How ldquoCooperative task assignmentof unmanned aerial vehicles in adversarial environmentsrdquo inProceedings of the American Control Conference (ACC rsquo05) pp4661ndash4666 June 2005

[4] Z Lian and A Deshmukh ldquoPerformance prediction of anunmanned airborne vehicle multi-agent systemrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 680ndash6952006

[5] T Shima S J Rasmussen A G Sparks and K M PassinoldquoMultiple task assignments for cooperating uninhabited aerialvehicles using genetic algorithmsrdquo Computers and OperationsResearch vol 33 no 11 pp 3252ndash3269 2006

[6] E Edison and T Shima ldquoIntegrated task assignment and pathoptimization for cooperating uninhabited aerial vehicles usinggenetic algorithmsrdquo Computers amp Operations Research vol 38no 1 pp 340ndash356 2011

[7] V K Shetty M Sudit and R Nagi ldquoPriority-based assignmentand routing of a fleet of unmanned combat aerial vehiclesrdquoComputers amp Operations Research vol 35 no 6 pp 1813ndash18282008

[8] H-L Choi L Brunet and J P How ldquoConsensus-based decen-tralized auctions for robust task allocationrdquo IEEE Transactionson Robotics vol 25 no 4 pp 912ndash926 2009

[9] L F Bertuccelli H L Choi P Cho and J P How ldquoReal-timemulti-UAV task assignment in dynamic and uncertain environ-mentsrdquo in Proceedings of the AIAA Guidance Navigation andControl Conference August 2009

[10] M Alighanbari and J P How ldquoA robust approach to the UAVtask assignment problemrdquo International Journal of Robust andNonlinear Control vol 18 no 2 pp 118ndash134 2008

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[11] M Alighanbari L F Bertuccelli and J P How ldquoFilter-embedded UAV task assignment algorithms for dynamic envi-ronmentsrdquo in AIAA Guidance Navigation and Control Confer-ence and Exhibit pp 1ndash15 2004

[12] R Lahdelma J Hokkanen and P Salminen ldquoSMAAmdashStochastic multiobjective acceptability analysisrdquo European Jour-nal of Operational Research vol 106 no 1 pp 137ndash143 1998

[13] R Lahdelma and P Salminen ldquoSMAA-2 stochastic multi-criteria acceptability analysis for group decisionmakingrdquoOper-ations Research vol 49 no 3 pp 444ndash454 2001

[14] R Lahdelma and P Salminen ldquoPseudo-criteria versus linearutility function in stochastic multi-criteria acceptability analy-sisrdquo European Journal of Operational Research vol 141 no 2 pp454ndash469 2002

[15] R Lahdelma P Salminen and J Hokkanen ldquoLocating a wastetreatment facility by using stochastic multicriteria acceptabilityanalysis with ordinal criteriardquo European Journal of OperationalResearch vol 142 no 2 pp 345ndash356 2002

[16] R Lahdelma K Miettinen and P Salminen ldquoReference pointapproach for multiple decision makersrdquo European Journal ofOperational Research vol 164 no 3 pp 785ndash791 2005

[17] T Tervonen R Lahdelma J A Dias J Figueira and P Salmi-nen ldquoSMAA-TRIrdquo in Environmental Security in Harbors andCoastal Areas NATO Security through Science Series pp 217ndash231 Springer Amsterdam The Netherlands 2007

[18] R Lahdelma and P Salminen ldquoProspect theory and stochasticmulti-criteria acceptability analysis (SMAA)rdquo Omega vol 37no 5 pp 961ndash971 2009

[19] H Liao Z Xu and X-J Zeng ldquoDistance and similarity mea-sures for hesitant fuzzy linguistic term sets and their applicationin multi-criteria decision makingrdquo Information Sciences vol271 pp 125ndash142 2014

[20] J Q Wang J T Wu J Wang H Y Zhang and X H ChenldquoInterval-valued hesitant fuzzy linguistic sets and their applica-tions in multi-criteria decision-making problemsrdquo InformationSciences vol 288 pp 55ndash72 2014

[21] A Washburn and M Kress Combat Modeling Springer NewYork NY USA 2009

[22] A S Kangas J Kangas R Lahdelma and P Salminen ldquoUsingSMAA-2 method with dependent uncertainties for strategicforest planningrdquo Forest Policy and Economics vol 9 no 2 pp113ndash125 2006

[23] J Hokkanen R Lahdelma and P Salminen ldquoMulti-criteriadecision support in a technology competition for cleaning pol-luted soil in Helsinkirdquo Journal of Environmental Managementvol 60 no 4 pp 339ndash348 2000

[24] A Menou A Benallou R Lahdelma and P Salminen ldquoDeci-sion support for centralizing cargo at a Moroccan airport hubusing stochastic multicriteria acceptability analysisrdquo EuropeanJournal of Operational Research vol 204 no 3 pp 621ndash6292010

[25] M M Rahman J V Paatero and R Lahdelma ldquoEvaluation ofchoices for sustainable rural electrification in developing coun-tries a multi-criteria approachrdquo Energy Policy vol 59 pp 589ndash599 2013

[26] A Pesola A Serkkola R Lahdelma and P Salminen ldquoMulticri-teria evaluation of alternatives for remotemonitoring systems ofmunicipal buildingsrdquo Energy and Buildings vol 72 pp 229ndash2372014

[27] T Tervonen and R Lahdelma ldquoImplementing stochastic multi-criteria acceptability analysisrdquo European Journal of OperationalResearch vol 178 no 2 pp 500ndash513 2007

[28] T Tervonen ldquoJSMAAOpen source software for SMAAcompu-tationsrdquo International Journal of Systems Science vol 45 no 1pp 69ndash81 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of