Classical approach and fuzzy set based model

A possible evolution of the available tools in case of uncertainty and vagueness can be represented by the definition and implementation of a particular mathematical function also based on the principles of fuzzy sets: the belief functions.

This interesting topic is the object of the next section.


Belief function framework

The knowledge of DM history, and also some statistic investigation on the DM techniques used by successful management, demonstrated that decisions are, in general, eventually taken based on an intuitive process; it deeply involves the result of previous experience, elaborated by specific ability of the decision maker, which often translates into what the decision maker believes in. Such process – must be clearly said- being basically the same that leads management to unsuccessful decisions [9].

The theory of the belief functions, among other implications and aims, provides a method to mathematically define and quantify such intuitive judgment process, dealing with its intrinsic vagueness either lack of specific knowledge (i.e. ignorance).

Belief functions (BF) - Shafer , 1976 [10]-, provide a better framework than probabilistic approach, to represent uncertainties in real decision making.

Use of BF was also introduced to overcome some paradox due to the limits of probability (see Ellsberg paradox, [6]). These limits basically stem from the difficulties arising when we want to represent or model the ignorance of a certain situation (state) using probabilities.

It is worth to recall the main difference between the probability and belief theories approaches.

In probability theory, uncertainty is assigned to each individual element of a set of states under consideration (which is often called frame Θ ); uncertainty assigned to each element of the frame is the probability of occurrence of that element.

In the BF framework, uncertainty is not only assigned to single elements but also to all subsets of the frame and to entire frame itself. Such uncertainties are called m-values ( basic probability assignment functions).

In this article, only the main fundamentals of BF theory have been recalled. For a comprehensive overview see also [11].

We can express what above mentioned as follow:


A  → state A

˜A → state ‘ not A’

frame Θ = { A,  ˜A}

we have:

P(A) ≥ 0 ; P(˜A) ≥ 0 (probability framework)

m(A)  ≥ 0; m(˜A) ≥ 0; m ({A, ˜A}) ≥ 0 (BF framework)

The following simple example will further clarify the concepts.

Let’ assume that a planner has carried out a deep analysis relevant to one task duration and finds no major difference between the calculated timeframe and the forecast given by the responsible for that task. Based on his findings, he feels that the recorded value appears reasonable; however the planner does not want to put so much confidence on this, since the task is relevant to an experimental activity. The planner quantifies this level of belief as 0,3 ( in a 0-1 scale). Let’s express this situation in BF formalism:

A : assignment of correct duration of the task

m(A)= 0,3 ; m(˜A) =0 ; m({A, ˜A)} = 0,7

Note that 0,7 represents the ignorance of the planner, while he does not think the estimate can be wrong (in fact we have m(˜A) =0 ).

It must be appreciated that, had we used the probability framework, situation would be:

P(A)= 0,3

P(˜A) = 0,7

Which implies there is 70% chance the duration is not correct; of course this is not what the planner means.

Even after this trivial example, it comes out as BF differs from probabilities in representing ignorance.


Definition of belief functions

A belief function for a set A is defined as


Bel(A) is thus equal to m(A) plus the sum of all the m-values for the set of elements contained in A. It is now possible to mathematically represent the concept difference between probability and belief function frames:

Bel({a}) + Bel ({˜a}) ≤ 1

P({a}) +P({˜a}) = 1


Dempster’s rule

Dempster’s rule-here reported in its essential feature for  two statements of evidences only-, is used mainly to combine belief functions; it is applied in particular when we have to deal with multiple decision makers, or, similarly, multiple sources of information either forms of judgment.

In case of two statements, the rule is expressed as:


Being K a constant, which express the “conflict” between the two statements of evidence:

K =1 , no conflict : K = 0 total conflict, no combination possible.

Before closing the section, it is worth to draw reader’s attention to an interesting analogy.

We can in fact consider that as the Baye’s formula (2) is an evolution- aggregation of probability- of the ‘simple’ probability, the Dempster’s role is similarly an evolution – aggregation of belief functions- of BF approach.


Example of Dempster’s rule

In order to minimize problems during start up activities in a project, a Faultless Right First Time Start Up (FRFTSU) program is launched. A considerable number of risks –hundred of them- have been identified during the program. Since the available resource to follow up the possible mitigations plan would be necessary, a restricted team of expert has been appointed to select for the Project Management team the areas/risks where to focus utmost attention and eventually where to concentrate the available limited resources (commissioning people).

We set three m-values:

  • ma  belief the risk deserves focused control
  • mb  belief the risk does not deserve focused control
  • mc  uncertainty as to whether the risk deserves focused control

The problem is summarized in the following table (in a3 risks and 3 expert frame):

A two step approach is adopted [7] combining assessments of expert 1 and 2 and the result of this eventually combined with the assessment of expert 3.

Applying the Dempster’s rule twice we obtain:

Between Expert 1 and Expert 2:

ma’= ma1* ma2+ma1*mc2+ma2*mc1



and between previous result and Expert 3

ma’’= ma’*ma3+m’a*mc3+ma3*m’c

m”b= m’b*mb3*m’b*mc3+mb3*m’c


By applying the above reported expressions, we have the following results for Risk 1:

ma’ = 0.98; mb’ =0; mc’ =0.2; and ma’’ =0.896; m”b =0.002; m”c=0.004

Note that for Risk 1 the beliefs of the three experts are quite aligned; in this case we can assume K=1.

For Risk 2 (mainly) and for Risk 3, this is no longer true and K< 1 shall be selected to model this feature.

Repeating above reported procedure, m” values can be calculated for Risks 2 and 3 and eventually select the risk which deserves more resources picking the higher ma”; having to do with risks, a careful investigation on mc” (uncertainty) shall be mandatory.


Application of BF to the evaluation of Expected Value Interval (EVI)

A very interesting and widely used application of the BF, is the determination of the Expected Value Interval (EVI).

In this case, rather than a single expected value –typically the value of an utility- an interval of it is calculated , taking into account the ‘ignorance content’ of the analysis under examination.

The interval is defined as:



A simple example will briefly make understand the variety of possible application of this tool.


Example of EVI calculation- ROI preliminary forecast –concept phase

In order to support a concept decision, the management wants a rough preliminary figure of the ROI (Return On Investment) for a certain investment, based on previous similar projects.

After searching company database, the analysts have drawn the following table:

Further, according to analysts, it is pretty improbable the ROI can be lower or higher than the figures reported.

The data reported in the table are quite consistent and reliable, while the total number of events (projects) does not sum to 1.

If we look at the available data as a mass function distribution, using the properties of the BF we can write:

m(0.23)=0.4; m(0.28)=0.3;m(0.35)=0.1;m(0.4)=0.1 and m{(0.23,0.28,0.35,0.40)}=0.1

Based on the (5) we can calculate the expected ROI interval:

EVI(ROI)=[0.4*0.23 +0.3*0.28 +0.1*0.35+0.1*0.4* +0.1*0.23,

0.4*0.23 +0.3*0.28 +0.1*0.35+0.1*0.4* +0.1*0.4]= [0.274;0.291]

In order to conclude the brief review of EVI, it is worth to recall an elaboration of the (5); introducing the parameter  , which is defined as the probability that ambiguity will be resolved in the most favorable way; the EVI is calculated as:


Conclusions and remarks

Even if a not trivial mathematical theory is behind, the non probabilistic methods reported are rather easy and quick to implement.

They can then be used as a first step in a decision making analysis to focus on the choice(s) that better fit a certain constraints/goals group, enabling a preliminary analysis based on a repeatable and rational basis.

This approach seems to be more consistent in case of lack of completeness either reliability of statistical data.

Summing up, the main points have been briefly recalled and discussed:

  • Some main simplified methods of DM based on imprecise information
  • Considerations on limits of probability distributions
  • Fuzzy logic analysis and BF as a tool in case of vagueness, uncertainty lack of probability data
  • Easiness of implementation by simple practical examples

One of the main concepts coming out from this discussion is that the Decision Maker should seriously take into account an alternative approach to decision when dealing with uncertainty, especially if time and resources do not allow a detailed modeling; in this case the described approaches can be a very useful tool.



[4] G.Kler/ T.Folger -Fuzzy sets, uncertainty and Information- Prentice Hall – 2008
[5] K.Rao/S.Mishra- Operations Research – Narosa 2008
[6] R.P.Srivastava- Decision under ambiguity-Archives of Control Science- 1997
[7] T.W.Lin/H.Lu – A cost variance investigation using belief functions- 2005
[8] T.Strat- Decision analysis using belief functions- International journal of approximate reasoning-1990
[9] G.Di Castri- Discussion on decision making techniques- AICE website- -2010
[10] G.Shafer - A Mathematical Theory of Evidence, Princeton University Press-1976
[11] K.Sentz/S.Ferson - Combination of Evidence in Dempster-Shafer Theory -2002




Massimiliano Arena, Aerospace Engineer, Project Manager (IPMA Certified), Certified Cost Engineer (Certified Cost Engineer CCE/ICECA), has been working for years in the project management of industrial plants leading Engineering and Main Contracting companies operating worldwide.


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