subjective effects-based assessment

987

Subjective Effects-Based Assessment

Johan SchubertDepartment of Decision Support Systems

Division of Command and Control SystemsSwedish Defence Research Agency

SE-164 90 Stockholm, [email protected]

http://www.foi.se/fusion/

Abstract - In this paper we develop a subjective Effects-BasedAssessment method. This method takes subjective assessmentsregarding the activities of a plan as inputs. From theseassessments and a cross impact matrix that represents the impactbetween all elements of the plan we calculate assessments for allother plan elements. The method is based on belief functions andtheir combination under a new generalization of the discountingoperation. The method is implemented in a CollaborationSynchronization Management Tool (CSMT).

Keywords: Effects-Based Approach to Operations, Effects-Based Assessment, Subjective assessment, Dempster-Shafertheory, belief function, pseudo belief function, decomposition,inverse support function.

1 Introduction

In this paper we develop a subjective method for Effects-Based Assessment (EBA) based on belief functions [1−6]and a cross impact matrix (CIM) [7, 8]. This work extendsour previous work [9] on analyzing plans developed withinthe Effects-Based Planning (EBP) [10] process.

A CIM can be used on the operational command levelby the staff of a joint task force headquarter in an Effects-Based Approach to Operations [11] during planning,execution and assessment of an operation. The CIMconsists of all activities (A), supporting effects (SE),decisive conditions (DC) and military end state (MES) ofthe plan. It is created by a broad working group whichmust assess how each activity impacts every other activityand supporting effect, how each supporting effect impactsevery decisive condition (and possibly other supportingeffects), and how every decisive condition impacts themilitary end state (and possibly other decisive conditions).In this paper we use British concepts [12].

The CIM can be used during assessment of theoperation as it should contain the most current view ofwhat impact all supporting effects have on the decisiveconditions and what impact all decisive conditions have onthe military end state.

Accepting human subjective assessments regarding thesuccessful outcome of activities of the plan, we can use theimpacts between plan elements as described by the CIM tocalculate similar subjective assessments of all desiredsupporting effects, decisive conditions and the military end

state. Using this methodology we get an early assessmentof all plan elements during Effects-Based Execution (EBE)and may early on observe if activities and desired effectsare developing according to plan. By observing the changeover time of these subjective assessments of effects andconditions as assessments of activities are updated, wenotice if trends are moving in the right direction as moreactivities are further executed.

In Sec. 2 we describe the construction of a CIM. In Sec.3 we develop an algorithm for assessment of plan elementsusing a CIM, and show how this may be used forsubjective assessment of all desired effects. In Sec. 4 wedecompose [13−15] assessments into separate statementsfor and against the realization of the desired effects.Finally, in Sec. 5 conclusions are drawn.

2 The plan an the CIM

The cross impact matrix will initially be created during theplanning process. It should be created by a working groupcontaining key subject matter experts as required by thetype of operation planned. Before the CIM is constructed,a plan must be formed according to EBP. The plan consistsof a military end state, decisive conditions, supportingeffects and activities, Fig. 1.

Fig. 1. Effects-based planning: MES = military end state, DC =decisive condition, SE = supporting effect, A = activity.

The working group will first need to enter all plannedactivities into the CIM, and it is important that all activitiesare well defined. They will then have to decide whichpositive or negative impact each activity will have on everyother activity. It is important to note that even if activity A1has a positive impact on activity A2, A2 could have anegative impact on A1. In the next step the working group

MES

DC1

DC2

SE1

SE2

SE3

SE4

SE5

A1

A2A3A4

A5

A6A7

in Proc. 11th Int. Conf. Information Fusion (FUSION 2008),pp. 987−994, Cologne, Germany, 30 June - 3 July 2008.

988

must decide what impact all activities have on thesupporting effects, what impact all supporting effects haveon the decisive conditions and what impact the decisiveconditions have on the military end state.

In Fig. 2 we will list these elements, except the militaryend state, to the left of the CIM and list the elements,including the military end state, above the CIM. The CIMconsists of values ranging from -9 to 9, where -9 denoteslarge negative influence, 0 means no influence and 9denotes high positive influence. For example, an impactvalue of 8, i.e., “high positive influence”, might beassigned between the activity of “securing an area” and theactivity of “transporting through that area”. How muchelement i influences element j is stored in the cell (i, j) inthe CIM (for example activity A2 influences activity A4 in apositive way by a factor of 2, but A4 influences A2 in anegative way by a factor of -2). Its value is accessiblethrough the function impact(i, j).

Fig. 2. The CIM contains a military end state, decisiveconditions, supporting effects and activities (dark gray cellsalways contain zeros). The CIM is the impact(i, j) matrix.

At the initial stage of the construction of the CIM weinclude the basic elements of the plan meaning that allactivities should be performed and all supporting effectsand decisive conditions should be reached.

3 Assessment of plan elements

The CIM is a model of influence between elements of theplan. In assessment, our interest is on the impact betweenactivities on the lowest level and supporting effects on thenext level, and so forth. We receive subjective assessmentsregarding activities as user input. These are in the form ofbasic belief assignments (bbas) that express support forand against the success of that activity, encoded as AdPand ¬AdP, respectively.

3.1 Combining assessments

In this problem we have a simple frame of discernment

(1)

on each hierarchical level of the plan, where AdP means anAdequate Plan.

We have a set of n bbas each with three bodies ofevidence, i.e., , ,

, where, e.g.,is the first body of evidence of the ith bba

giving support to AdP. Thus, for the ith bba we have,

. (2)

The CIM contains all information regarding the impact ofeach activity on all supporting effects. When the impact ona particular supporting effect SEj is less than full wediscount the bba mi in relation to its degree of impact onSEj

. (3)

For the sake of simplicity in combination of bbas, they arefirst transformed to commonalities using

. (4)

Transforming all bbas to commonalitiesusing Eq. (4), we have

. (5)

Let us now combine all commonalities , usingDempster’s rule for commonalities. We get

(6)

where K is a normalizing constant.Commonalities can be transformed back to bbas using

. (7)

Mili

tary

End

Sta

te

DC1 DC2 SE1 SE2 A1 A2 A3 A4

DC1 5 0 6 0 0 0 0 0 0

DC2 8 6 0 0 0 0 0 0 0

SE1 0 5 2 0 0 0 0 0 0

SE2 0 5 8 0 0 0 0 0 0

A1 0 0 0 3 3 0 4 -2 -3

A2 0 0 0 3 -2 3 0 0 2

A3 0 0 0 3 6 0 8 0 0

A4 0 0 0 4 -2 -7 -2 0 0

Θ AdP AdP¬,{ }=

AdP mi AdP( ),( ){{ AdP¬ mi AdP¬( ),( )Θ 1 mi AdP( ) mi AdP¬( )––,( ) } }

i 1=n

AdP mi AdP( ),( )

mi A( )

mi AdP( ), A AdP=

mi AdP¬( ), A AdP¬=

1 mi AdP( ) mi AdP¬( )–– , A Θ=⎩⎪⎪⎨⎪⎪⎧

=

mi

αij A( )

αijmiAdP( ), A AdP=

αijmiAdP¬( ), A AdP¬=

1 αijmiAdP( ) αijmi

AdP¬( )–– , A Θ=⎩⎪⎪⎨⎪⎪⎧

=

Q A( ) m B( )B A⊇∑=

mi

αij B( ) Qi

αij A( )

Qi

αij A( )

1 α– ijmiAdP¬( ), A AdP=

1 α– ijmiAdP( ), A AdP¬=


AdP¬( )–– , A Θ=⎩⎪⎪⎨⎪⎪⎧

=

Qi

αij

Q⊕ mi{ }i 1=

n

αij A( ) =

K 1 α– ijmiAdP¬( )[ ]

i∏ , A AdP=

K 1 α– ijmiAdP( )[ ]

i∏ , A AdP¬=

K 1 αijmiAdP( ) αijmi

AdP¬( )––[ ]i

∏ , A Θ=

⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧

=

m A( ) 1–( ) B A– Q B( )B A⊇∑=

989

Transforming back from commonality to bba, we get

(8)

where

(9)

Thus, Eq. (8) becomes the subjective assessment of SEj ascalculated using the subjective input assessments of allactivities Ai that impact upon SEj.

What is calculated for supporting effects fromsubjective assessment of activities can in a second phase becalculated for decisive conditions using the newlycalculated assessments of supporting effects. In the sameway we can calculate the subjective assessment of themilitary end state from the assessment of decisiveconditions.

3.2 Combining assessments regarding planelements using the CIM.

At the activities level we have a frame of discernment

. (10)

In order to map this onto the problem of combiningassessments, Sec. 3.1, we must first generalize thediscounting operation.

The discounting operation was introduced to handle thecase when the source of some piece of evidence is lackingin credibility [3]. The credibility of the source, 0 < α < 1,also became the credibility of the piece of evidence. Thesituation was handled by discounting each supported

proposition other than Θ with the credibility α and byadding the discounted mass to Θ;

. (11)

We generalize the discounting operation by allowing thecredibility to take values in the interval .

Definition 1. Let be a bba where. Then

. (12)

is a generalized discounting of m where is aninverse simple support function (ISSF) whenever α < 0.

Definition 2. An inverse simple support function on aframe of discernment is a functioncharacterized by a weight and a focal element

, such that , andwhen .

Let us recall the meaning of simple supportfunctions (SSFs) and ISSFs, [13]: An SSF

represents a state of belief that “Youhave some reason to believe that the actual world is inA (and nothing more)”. An ISSF , onthe other hand, represents a state of belief that “Youhave some reason not to believe that the actual worldis in A”.

Before combining the mass functions we discount themusing the impact values of the CIM. This ensures that eachactivity influences the supporting effect to its properdegree.

For SEj and Ai we have

(13)

where the discounting factor is defined as

. (14)

This is a generalization of the discounting operator wherediscounting factors may assume values less than 0, i.e.,

.We combine all bbas on the activities level and bring the

result to the supporting effects level. At the supportingeffects level we have a similar frame of discernment

. (15)

Using Eq. (8), Eq. (13) and Eq. (14), we define

m⊕ mi{ }i 1=

n

αij A( ) =

K 1 α– ijmiAdP¬( )[ ]

i∏

⎩⎨⎧


AdP¬( )––[ ]⎭⎬⎫

,i

∏– A AdP=

K 1 α– ijmiAdP( )[ ]

i∏

⎩⎨⎧


AdP¬( )––[ ]⎭⎬⎫

,i

∏– A AdP¬=

K 1 αijmiAdP( ) αijmi

AdP¬( )––[ ]i

∏ , A Θ=

⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧

=

K 1– 1 α– ijmiAdP¬( )[ ]

i∏=


AdP¬( )––[ ]i

∏–

1 α– ijmiAdP( )[ ]

i∏+


AdP¬( )––[ ]i

∏–


AdP¬( )––[ ].i

∏+

ΘA AdP AdP¬,{ }=

m%

A( ) αm A( ), A Θ≠1 α– αm Θ( ),+ A Θ=

⎩⎪⎨⎪⎧

=

1– α 1<≤m:2Θ 0 1,[ ]→

1– α 1<≤

m%

A( ) αm A( ), A Θ≠1 α– αm Θ( ),+ A Θ=

⎩⎪⎨⎪⎧

=

m%

A( )

Θ m:2Θ ∞– ∞,( )→w 1 ∞,( )∈

A Θ⊆ m Θ( ) w= m A( ) 1 w–= m X( ) 0=X A Θ,{ }∉

m1 A( ) 0 1,[ ]∈

m2 A( ) ∞– 0,( )∈

mAi

αkj A( )

αAiSE jm

AiAdP( ), A AdP=

αAiSE jm

AiAdP¬( ), A AdP¬=

1 αAiSE jm

AiAdP( ) αAiSE j

mAi

AdP¬( )–– , A Θ=

⎩⎪⎪⎪⎨⎪⎪⎪⎧

=

αAiSE j

Δimpact Ai SE j,( )

10--------------------------------------=

αkj 0.9– 0.8– 0.7– …,0.9, , ,{ }=

ΘSE AdP AdP¬,{ }=

990

(16)

and

(17)

with

(18)

For simplicity, we may manage the normalization by firstcalculating

(19)

using Eq. (16), Eq. (17), and Eq. (18), followed bycalculating

(20)

where any may be ≤ 0.When this is the case is called a pseudo belieffunction [13].

In the same way we may calculate the support fordecisive conditions, and the military end state.

For decisive conditions, we define

(21)

and

(22)

with

(23)

As previously, we may manage the normalization by firstcalculating

(24)

mSE jAdP( ) Δ m

⊕ mAi{ }

i 1=

n

α ji AdP( )=

K 1 α– AiSE jm

AiAdP¬( )[ ]

i∏

⎩⎨⎧

=

1 αAiSE jm

AiAdP( ) αAiSE j

mAi

AdP¬( )––[ ]⎭⎬⎫

i∏–

K 1impact Ai SE j,( )

10--------------------------------------– m

Ai

AdP¬( )i

∏⎩⎨⎧

=

1impact Ai SE j,( )

10--------------------------------------m

Ai

AdP( )–i

∏–

impact Ai SE j,( )

10--------------------------------------m

Ai

AdP¬( )–⎭⎬⎫

mSE jAdP¬( ) Δ m

⊕ mAi{ }

i 1=

n

α ji AdP¬( )=

K 1 α– AiSE jm

AiAdP( )[ ]

i∏

⎩⎨⎧

=

1 αAiSE jm

AiAdP( ) αAiSE j

mAi

AdP¬( )––[ ]⎭⎬⎫

i∏–


10--------------------------------------– m

Ai

AdP( )i

∏⎩⎨⎧

=

1impact Ai SE j,( )

10--------------------------------------m

Ai

AdP( )–i

∏–

impact Ai SE j,( )

10--------------------------------------m

Ai

AdP¬( )–⎭⎬⎫

mS E jΘ( ) Δ m

⊕ mAi{ }

i 1=

n

α ji Θ( )=

K 1 αAiSE jm

AiAdP( ) αAiSE j

mAi

AdP¬( )––[ ]i

∏=


10--------------------------------------m

Ai

AdP( )–i

∏=

impact Ai SE j,( )

10--------------------------------------m

Ai

AdP¬( )–

mS E j

*AdP( )

mS E jAdP( )

K--------------------------=

mS E j

*AdP¬( )

mS E jAdP¬( )

K-------------------------------=

mS E j

* Θ( )mS E j

Θ( )

K--------------------=

mS E jAdP( )

mS E j

*AdP( )

mS E j

*AdP( ) mS E j

*AdP¬( ) mS E j

* Θ( )+ +-------------------------------------------------------------------------------------------=

mS E jAdP¬( )

mS E j

*AdP¬( )

mS E j

*AdP( ) mS E j

*AdP¬( ) mS E j

* Θ( )+ +-------------------------------------------------------------------------------------------=

mS E jΘ( )

mS E j

* Θ( )

mS E j

*AdP( ) mS E j

*AdP¬( ) mS E j

* Θ( )+ +-------------------------------------------------------------------------------------------=

mS E jAdP( ) mS E j

AdP¬( ) mS E jΘ( ), ,

mS E j

mDC jAdP( ) Δ m

⊕ mSEi{ }

i 1=

n

α ji AdP( )=

K 1impact SEi DC j,( )

10-------------------------------------------– m

SEi

AdP¬( )i

∏⎩⎨⎧

=

1impact SEi DC j,( )

10-------------------------------------------m

SEi

AdP( )–i

∏–

impact SEi DC j,( )

10-------------------------------------------m

SEi

AdP¬( )–⎭⎬⎫

mDC jAdP¬( ) Δ m

⊕ mSEi{ }

i 1=

n

α ji AdP¬( )=


10-------------------------------------------– m

SEi

AdP( )i

∏⎩⎨⎧

=

1impact SEi DC j,( )

10-------------------------------------------m

SEi

AdP( )–i

∏–

impact SEi DC j,( )

10-------------------------------------------m

SEi

AdP¬( )–⎭⎬⎫

mDC jΘ( ) Δ m

⊕ mSEi{ }

i 1=

n

α ji Θ( )=


10-------------------------------------------m

SEi

AdP( )–i

∏=

impact SEi DC j,( )

10-------------------------------------------m

SEi

AdP¬( )–

mDC j

*AdP( )

mDC jAdP( )

K----------------------------=

mDC j

*AdP¬( )

mDC jAdP¬( )

K--------------------------------=

mDC j

* Θ( )mDC j

Θ( )

K---------------------=

991


(25)

Similarly, for the military end state, we have

(26)

and

(27)

with

(28)

As before, for simplicity, we may manage thenormalization by first calculating

(29)


(30)

With these calculations we have all pieces of a subjectiveEBA algorithm (Algorithm 1).

Algorithm 1: Subjective EBA

• For all calculate:

, , using Eq. (20);


, , using Eq. (25);

• Calculate:

, , using Eq. (30);

• Return all calculated values.

In Fig. 3 the calculated values of Algorithm 1 are presentedin the upper part labelled “Effects”, together with theinitial subjective assessments , , and

in the lower part labelled “Activities” within theCSMT. Obviously, m(AdP) is indicated by the green part,m(¬AdP) by the red part and the uncommitted m(Θ) by thegray part.

In order to further enhance the usability it may be ofvalue to include a diagram of the change over time forthese assessments. In Fig. 4 this is exemplified for theMilitary End State as calculated by Eq. (30) at differenttimes.

4 Decomposing assessments

In this section we will decompose the belief functions (orpseudo belief functions), calculated in Algorithm 1 by Eq.(20), Eq. (25) and Eq. (30), into its separate componentsfor and against the plan using the decompositionintroduced by Smets [13], i.e.,

. (31)

This is done in order to observe the strength for and againstthe plan as if they are two separate pieces of evidence.Their combination is the result already calculated inAlgorithm 1. Thus, decomposition is the inverse ofcombination.

mDC jAdP( )

mDC j

*AdP( )

mDC j

*AdP( ) mDC j

*AdP¬( ) mDC j

* Θ( )+ +----------------------------------------------------------------------------------------------=

mDC jAdP¬( )

mDC j

*AdP¬( )

mDC j

*AdP( ) mDC j

*AdP¬( ) mDC j

* Θ( )+ +----------------------------------------------------------------------------------------------=

mDC jΘ( )

mDC j

* Θ( )

mDC j

*AdP( ) mDC j

*AdP¬( ) mDC j

* Θ( )+ +----------------------------------------------------------------------------------------------=

mMES AdP( ) Δ m⊕ mDCi

{ }i 1=

n

α ji AdP( )=

K 1impact DCi MES,( )

10----------------------------------------------– m

DCi

AdP¬( )i

∏⎩⎨⎧

=

1impact DCi MES,( )

10----------------------------------------------m

DCi

AdP( )–i

∏–

impact DCi MES,( )

10----------------------------------------------m

DCi

AdP¬( )–⎭⎬⎫

mMES AdP¬( ) Δ m⊕ mDCi

{ }i 1=

n

α ji AdP¬( )=


10----------------------------------------------– m

DCi

AdP( )i

∏⎩⎨⎧

=

1impact DCi MES,( )

10----------------------------------------------m

DCi

AdP( )–i

∏–

impact DCi MES,( )

10----------------------------------------------m

DCi

AdP¬( )–⎭⎬⎫

mMES Θ( ) Δ m⊕ mDCi

{ }i 1=

n

α ji Θ( )=


10----------------------------------------------m

DCi

AdP( )–i

∏=

impact DCi MES,( )

10----------------------------------------------m

DCi

AdP¬( )–

mMES*

AdP( )mMES AdP( )

K-----------------------------=

mMES*

AdP¬( )mMES AdP¬( )

K---------------------------------=

mMES* Θ( )

mMES Θ( )

K-----------------------=

mMES AdP( )mMES

*AdP( )

mMES*

AdP( ) mMES*

AdP¬( ) mMES* Θ( )+ +

--------------------------------------------------------------------------------------------------=

mMES AdP¬( )mMES

*AdP¬( )

mMES*

AdP( ) mMES*

AdP¬( ) mMES* Θ( )+ +

--------------------------------------------------------------------------------------------------=

mMES Θ( )mMES

* Θ( )

mMES*

AdP( ) mMES*

AdP¬( ) mMES* Θ( )+ +

--------------------------------------------------------------------------------------------------=

SE j

mS E jAdP( ) mS E j

AdP¬( ) mS E jΘ( )

DC j

mDC jAdP( ) mDC j

AdP¬( ) mDC jΘ( )

mMES AdP( ) mMES AdP¬( ) mMES Θ( )

mA jAdP( ) mA j

AdP¬( )mA j

Θ( )

mi A( ) 1 Q⊕ mi{ }i 1=

nα

B( ) 1–( ) B A– 1+

B A⊇∏–=

Fig. 3. Subjective Effect-Based Assessment (EBA) in the Collaborative Synchronization Management Tool (CSMT).

992

Fig. 4. Subjective assessments over time of Military End State.

First, we begin by calculating commonalities for allsupporting effects using Eq. (4). We have

(32)

where , and are calculatedusing Eq. (20).

Secondly, using Eq. (31) and Eq. (32), we get

(33)

and

. (34)

where and are two different ISSFs, i.e.,, with

. (35)

Here Eq. (33) and Eq. (34) are the two sought aftercomponents bringing evidence for ( ) and against( ) the plan, respectively.

If it turns out that or thenwe may instead calculate the degree to which we do notbelieve in that proposition. When we have,

(36)

where is a SSF.The support for “not believe in AdP” should be

interpreted as the additional support needed before you areuncommitted towards AdP, i.e., . Only whenyou receive more than additionalsupport for AdP should you start believing in it.

We should note that, usually

MES

Time

100%

0%

QSE jAdP( ) mSE j

AdP( ) mSE jΘ( )+=

QSE jAdP¬( ) mSE j

AdP¬( ) mSE jΘ( )+=

QSE jΘ( ) mSE j

Θ( )=

mSE jAdP( ) mSE j

AdP¬( ) mSE jΘ( )

mSE j

+AdP( ) 1

QSE jΘ( )

QSE jAdP( )

--------------------------–=

mSE j

+ Θ( ) 1 QSE jΘ( )–=

mSE j

-AdP¬( ) 1

QSE jΘ( )

QSE jAdP¬( )

------------------------------–=

mSE j

- Θ( ) 1 QSE jΘ( )–=

mSE j

+mSE j

-

mSE j

+AdP( ) mSE j

-AdP¬( ), ∞– ∞,( )∈

mSE jmSE j

+mSE j

-⊕=

mSE j

+

mSE j

-

mSE j

+AdP( ) 0< mSE j

-AdP¬( ) 0<

mSE j

+AdP( ) 0<

mSE j

+notnot believe in AdP( ) 1

1

1 mSE j

+AdP( )–

-----------------------------------–=

mSE j

+not Θ( ) 1 mSE j

+notnot believe in AdP( )–=

mSE j

+not0 1,[ ]∈

mSE j

+ Θ( ) 1=mSE j

+notnot believe in AdP( )

993

(37)

as the left hand side of Eq. (37) can be rewritten as

(38)

which is in general different from the right hand side ofEq. (37),

. (39)

Similarly, when we have,

(40)where is also a SSF.

These two decomposed SSFs are then two independentsubjective assessments for and against the successfuloutcome of SEj, respectively.

We may perform the same type of decomposition for theDecisive Conditions. Using Eq. (4), we get

(41)

where , and can becalculated using Eq. (25).

Furthermore, using Eq. (31) and Eq. (41) we have

(42)

and

. (43)

When we have,

(44)

and when we have,

(45)where and are SSFs.

Finally, we perform a single decomposition for theMilitary End State, again using Eq. (4). We get,

(46)

where , and can becalculated using Eq. (30).

In the same manner as before we get, using Eq. (31) andEq. (46),

(47)

and

. (48)

If then we calculate support in

(49)

and if we calculate

(50)

where and are SSFs.As before these propositions should be interpreted as

the additional belief needed before we are completelyuncommitted towards these propositions, i.e., before

and , respectively.These equations, Eq. (47) and Eq. (48), are the two parts

expressing support for and against the overall plan.Together with their combination

, (51)

already calculated directly by Eq. (30), they express thebest subjective assessment on the outcome of the plan.

mSE j

+notnot believe in AdP( ) mSE j

-AdP¬( )≠

mSE j


1

1 mSE j

+AdP( )–

-----------------------------------–=

11

1 1QSE j

Θ( )

QSE jAdP( )

--------------------------–⎝ ⎠⎜ ⎟⎛ ⎞

–

--------------------------------------------------–=

1QSE j

AdP( )

QSE jΘ( )

--------------------------–=

mSE j

-AdP¬( ) 1

QSE jΘ( )

QSE jAdP¬( )

------------------------------–=

mSE j

-AdP¬( ) 0<

mSE j

-notnot believe in AdP¬( ) 1

1

1 mSE j

-AdP( )–

-----------------------------------–=

mSE j

-not Θ( ) 1 mSE j

-notnot believe in AdP¬( )–=

mSE j

-not0 1,[ ]∈

QDC jAdP( ) mDC j

AdP( ) mDC jΘ( )+=

QDC jAdP¬( ) mDC j

AdP¬( ) mDC jΘ( )+=

QDC jΘ( ) mDC j

Θ( )=

mDC jAdP( ) mDC j

AdP¬( ) mDC jΘ( )

mDC j

+AdP( ) 1

QDC jΘ( )

QDC jAdP( )

----------------------------–=

mDC j

+ Θ( ) 1 QDC jΘ( )–=

mDC j

-AdP¬( ) 1

QDC jΘ( )

QDC jAdP¬( )

--------------------------------–=

mDC j

- Θ( ) 1 QDC jΘ( )–=

mDC j

+AdP( ) 0<

mDC j


1

1 mDC j

+AdP( )–

-------------------------------------–=

mDC j

+not Θ( ) 1 mDC j


mDC j

-AdP¬( ) 0<

mDC j

-notnot believe in AdP¬( ) 1

1

1 mDC j

-AdP( )–

-------------------------------------–=

mDC j

-not Θ( ) 1 mDC j


mDC j

+notmDC j

-not

QMES AdP( ) mMES AdP( ) mMES Θ( )+=

QMES AdP¬( ) mMES AdP¬( ) mMES Θ( )+=

QMES Θ( ) mMES Θ( )=


mMES+

AdP( ) 1QMES Θ( )

QMES AdP( )-----------------------------–=

mMES+ Θ( ) 1 QMES Θ( )–=

mMES-

AdP¬( ) 1QMES Θ( )

QMES AdP¬( )---------------------------------–=

mMES- Θ( ) 1 QMES Θ( )–=

mMES+

AdP( ) 0<

mMES+not

not believe in AdP( ) 11

1 mMES+

AdP( )–--------------------------------------–=

mMES+not Θ( ) 1 mMES


mMES-

AdP¬( ) 0<

mMES-not

not believe in AdP¬( ) 11

1 mMES-

AdP( )–--------------------------------------–=

mMES-not Θ( ) 1 mMES


mMES+not

mMES-not

mMES+not Θ( ) 1= mMES

-not Θ( ) 1=

mMES mMES+not

mMES-not⊕=

994

All of this can be put together into Algorithm 2 forcalculating subjective EBA with decomposed support forand against each elements of the overall plan.

Performing and updating Algorithm 2 whenever newsubjective assessments regarding activities are received,should give planners the earliest indication on futureoutcomes of all plan elements and any opportunity toreplan during execution when necessary.

Algorithm 2: Subjective EBA with decomposition


, , using Eq. (20);

, , using Eq. (33);

, using Eq. (34);

If calculate:

, using Eq. (36);

If calculate:

, using Eq. (40);


, , using Eq. (25);

, , using Eq. (42);

, using Eq. (43);

If calculate:

, using Eq. (44);

If calculate:

, using Eq. (45);

• Calculate , , and

using Eq. (30).

, , using Eq. (47);

, using Eq. (48);

If calculate:

, using Eq. (49);

If calculate:

, using Eq. (50);

• Return all calculated values.

5 Conclusions

We have developed a subjective Effects-Based Assessmentmethod for making subjective assessment of plans andplan elements within the Effects-Based Approach toOperations.

We have shown that such subjective assessments can beperformed of all supporting effects, decisive conditions

and the military end state by taking human subjectiveassessments about activities as input and extending thoseassessments to all other plan elements using a cross impactmatrix.

References

[1] Arthur P. Dempster, Upper and lower probabilities inducedby a multivalued mapping, The Annals of MathematicalStatistics, Vol. 38, No. 2, 325−339, April 1967.

[2] Arthur P. Dempster, A generalization of Bayesianinference, Journal of the Royal Statistical Society Series B, Vol.30, No. 2, 205−247, 1968.

[3] Glenn Shafer, A Mathematical Theory of Evidence,Princeton University Press, Princeton, NJ, 1976.

[4] Glenn Shafer, Perspectives on the theory and practice ofbelief functions, International Journal of ApproximateReasoning, Vol. 4, No. 5/6, pp. 323−362, September−November1990.

[5] Philippe Smets and Robert Kennes, The transferable beliefmodel, Artificial Intelligence, Vol. 66, No. 2. pp. 191−234,March 1994.

[6] Philippe Smets, Practical uses of belief functions, inProceedings of the Fifteenth Conference on Uncertainty inArtificial Intelligence, Stockholm, Sweden, 30 July−1 August1999, pp. 612−621.

[7] Jerome C. Glenn and Theodore J. Gordon, FuturesResearch Methodology − Version 2.0. American Council for theUnited Nations University, Washington, DC, USA, 2003.

[8] Theodore J. Gordon and H. Hayward, Initial experimentswith the cross-impact matrix method of forecasting, Futures, Vol.1, No. 2, pp. 100−116, 1968.

[9] Johan Schubert, Mattias Wallén, and Johan Walter,Morphological Refinement of Effect Based Planning, inStockholm Contributions in Military Technology 2007, SwedishNational Defence College, Stockholm, Sweden, 2008, to appear.

[10] Edward A. Smith, Complexity, networking, and effects-based approaches to operations, U.S. Department of DefenseCCRP, Washington, DC, 2006.

[11] Effects-based approach to multinational operations,Concept of operations (CONOPS) with implementationprocedures. Version 1.0, United States Joint Forces Command,Suffolk, VA, USA, 31 July 2006.

[12] Incorporating and extending the UK military effects-basedapproach. Joint Doctrine Note (JDN 7/06), Development,Concepts and Doctrine Centre (DCDC), Ministry of Defence,Shrivenham, Swindon, UK, 2006.

[13] Philippe Smets, The canonical decomposition of a weightedbelief, in Proceedings of the 14th International Joint Conferenceon Artificial Intelligence, Montréal, Canada, 20−25 August 1995,Vol. 2, pp. 1896−1901.

[14] Johan Schubert, Managing decomposed belief functions, inUncertainty and Intelligent Information Systems, B. Bouchon-Meunier, C. Marsala, M. Rifqi, R.R. Yager, Eds., WorldScientific Publishing, Singapore, 2008, to appear.

[15] Johan Schubert, Clustering decomposed belief functionsusing generalized weights of conflict, International Journal ofApproximate Reasoning, Vol. 48, No. 2, pp. 466−480, June 2008.

SE j

mS E jAdP( ) mS E j

AdP¬( ) mS E jΘ( )

mSE j

+AdP( ) mSE j

+ Θ( )

mSE j

-AdP¬( ) mSE j

- Θ( )

mSE j

+AdP( ) 0<

mSE j

+notnot believe in AdP( ) mSE j

+not Θ( )

mSE j

-AdP¬( ) 0<

mSE j

-notnot believe in AdP¬( ) mSE j

+not Θ( )

DC j

mDC jAdP( ) mDC j

AdP¬( ) mDC jΘ( )

mDC j

+AdP( ) mDC j

+ Θ( )

mDC j

-AdP¬( ) mDC j

- Θ( )

mDC j

+AdP( ) 0<

mDC j

+notnot believe in AdP( ) mDC j

+not Θ( )

mDC j

-AdP¬( ) 0<

mDC j

-notnot believe in AdP¬( ) mDC j

+not Θ( )


mMES+

AdP( ) mMES+ Θ( )

mMES-

AdP¬( ) mMES- Θ( )

mMES+

AdP( ) 0<

mMES+not

not believe in AdP( ) mMES+not Θ( )

mMES-

AdP¬( ) 0<

mMES-not

not believe in AdP¬( ) mMES+not Θ( )

subjective effects-based assessment

Documents