Classical approach and fuzzy set based model

This paper briefly recalls the basic steps in risk and uncertain conditions and describes a simple application of fuzzy set in a decision making (DM) development process, highlighting the main difference of the approaches. Taking into account that many management decisions have to be taken based on imprecise or even subjective determinations of preference constrains and goals, the condition of vagueness and indetermination may become of great interest and a relevant systematic model may be useful.


The subject of the decision making is basically the quantitative study of how decisions are made and how they can be optimized in order to achieve a certain target.

Decision making is modeled according to the character of the problem or system (mathematically called space) we are dealing with. A basic simplified classification may be roughly the following:

  • Deterministic conditions (certain conditions of the system)
  • Risk and uncertainty of system conditions
  • Vagueness or indetermination

In particular uncertain states of nature are represented by probability distributions and each possible state is assigned a value or utility. The best decision is the one that yelds the greatest expected utility. By enumerating all available choices an assessing probabilities and utilities of the states of nature that may result, one can mechanically determine the optimal sequence of actions to be taken. In practice these simple requirements are hard to satisfy.

Sometimes reliable estimates of the probabilities involved are hard to come by. For example few statistics are available for determining a robust probability distribution of a completely new problem.

Classic approaches (in a nutshell)

Classical decision making is based on a set of possible alternatives (choices or strategies, Si )and a set of possible expected states (Ej) and a  utilities (objective) function or conditioned result for each selected couple strategy- state: C(Si|Ej).

Such results are usually ordered in the so called pay off matrix, in the form

If the decision is made in condition of certainty, the alternative C(Ek|Sh) which yields the highest utility is selected.

In case of decision is made under condition of risk, a probability distribution is assigned to the outcome states as p(Ej).

In these conditions the choice can be done taking the average value for each strategy (Expected Monetary Value):


The probability can be a “prior” one , based on previous statistical info coming from a database, either, in a more complex form, a probability calculated after a further investigation aimed to gather more accuracy of data, for example a market investigation in the actual condition of the problem to be modeled (conditional probability).

In this case, a more reliable probability distribution may be built up, and the Bayes’s formula can be applied.

Bayes’s formula is an important method for computing conditional probabilities. It is often used to compute posterior probabilities (as opposed to prior probability, like those input in the (1)) given observations. Herewith the general statement is reported:


Where, basically, P(B|A) means “probability that B happens, given A happens”.

This posterior probability can then plugged in (1).

A very simple example [3] will clarify the concept of posterior (or conditioned) probability expressed by (2).

We suppose a project manager asks the question: what’s the probability that my project will finish on time ? There are only two possibilities here: either the project finishes on time or it doesn’t.  Let’s express this formally.  Denoting the event the project finishes on time by T, the event the project does not finish on time by T1 and the probabilities of the two by P(T) and  P(T1) respectively.

Now, there are several variables that can affect project completion time. For sake of simplicity, we consider just one of them: scope change. Let’s denote the event “there is a major change of scope” by C and the complementary event (that there is no major change of scope) by C1 .

We want to know what is the probability of finishing on time, given that there has been a major scope change. This is a conditional probability because it represents the likelihood that something will happen (on-time completion) on the condition that something else has already happened (scope change).

We’re interested in the probability that the project finishes on time given that it has suffered a major change in scope. In the notation of conditional probability, this is denoted by P(T|C).

Let’s assume historical data on projects that have been carried out within  the organisation.  On analyzing the data, it is found that 60% of all projects finished on time. This implies:

P(T) = 0.6


P(T1)= 0.4

Let us assume that our organisation also tracks how many projects not on time had a scope change and it is found they are 80%

P(C|T1) = 0.8

Finally, let us that an analysis of data shows that 30% on time projects have undergone at least one major scope change. This gives:

P(C|T) = 0.3

Plugging the numbers in the following equation:

Summing up, in this organization, if a project undergoes a major change then there’s a 36% probability that it will finish on time.  Compare this to the 60% (unconditional) probability of finishing on time.  Bayes’s theorem enables us to quantify the impact of change in scope on project completion time, providing the relevant historical data are available and reliable: this last sentence is a key point.

Before treating the uncertainty of information it is worth to recall that other main classic tools are widely used such as linear programming , principal operation research method, decision making trees , Monte Carlo simulation , variants of the pay off matrix, just to quote some.

Uncertainty of available data: Shannon entropy

Some considerations can be outlined relevant to the above mentioned approaches. The probability distribution used in (1) can be considered reliable only if it comes from a consistent and numerous statistical data relevant to the problem in object.

On the other hand, the conditional probability is of course a step forward to recover a more reliable content on the probability distribution but in order to do that, further info have to be gathered in a significant amount:sometimes easier to say that to do.

In the daily decision making practice, the availability of massive relaible statistical data may be difficult as well as the manipulation of such set of data.

Further, another issue shall be taken into account: the uncertainty of the probability distribution itself. The reference to an alternative to the use of probability distribution, becomes more suitable when the distribution is characterized by an high degree of uncertainty, which means when the probability has a “flat” profile ; it is evident that when all the probailities are close one another , the content of the information given is quite poor.

There are many methods available to measure such uncertainty; here we briefly recall the Shannon Entrophy method. Given the propability distribution {p} = [ p1…pn] = (p(x)|x € X) , it is expressed by

It is easy to find that the maximum of uncertainty is obtained when

H(p1…pn)= H(1/n,…,1/n)


When  H → HMAX, we basically are dealing with conditions of uncertain vagueness, a structural indefinition of the states.

In these conditions, an analysis based on probability risks to be quite weak. In this case the decision maker should switch to other approaches, among which for example to a fuzzy logic based one.

Fuzzy decision theories attempt to deal with this vagueness or fuzziness inherent to imprecise determinations of preferences, constraints or goal.

A fuzzy approach to decision making (Zadeh et al.[4])

The uncertainty  or vagueness inherent to the states of a system, can be analyzed by modeling the states themselves and the utilities assigned  to each state by fuzzy sets.

The notion of fuzzy set is here briefly reported, starting from basic concept of crisp set – i.e. classical set to which an element belongs or does not belong-. A fuzzy set can be seen as a generalization of a crisp set , by defining a membership function.

The characteristic function of a classical crisp set assigns a value of either 1 or 0 to each element of the set itself, thus discriminating between elements belonging or not belonging to the set.

The function can be generalized (called membership function, µ) so that the values assigned to the element of one set indicate the membership grade of these elements to the set in question.

The set defined by the membership function is called fuzzy set.

If X is a set then the membership µA by which a fuzzy set in A is generally defined is:

µA : X→[0,1]

larger values of µA , denote higher degree of membership.

Generally , a decision making project involves certain constraints C and goals or targets G. If we treat them as fuzzy sets we have

µC : X→[0,1]

µG : X→[0,1]

where X is the set of alternative choices.

This fuzziness allows the decision maker to set the goals and constraints in vague, linguistic terms, which can more realistically reflect the actual state of the knowledge of preference concerning these, rather than, for example, a partial approximate or erroneous probability distribution.

This model can be extended in order to have the goals and constraints to be defined in different sets, for example

X → set of possible actions

Y → set of possible effects/outcomes

The model is then described as

µC : X→[0,1]

µG : Y→[0,1]

a function f can be then defined as a mapping one from set of actions  X to the set of outcomes Y.

f: X→Y

so that a goal G defined on set Y has a corresponding goal G’ on set X. Thus

µG’(x)= µG(f(x))

A fuzzy decision D may be found that satisfy the goals G and the constraints C. Translating this concept into Boolean logic, we have an “AND” operation, while in set logic it can be surely seen as an intersection

D = G ∩ C

which is valid for more goals and constraints.

We can thus similarly obtain µD as an intersection

µD = min[µG(x), µC(x)]

This fuzzy model can be extended to take into account many goals and constraints by use of weighing coefficients .

In this case the membership functions relevant to the decision , can be written as:

Where ui and vj are weights attached to the goal Gi and to the constraint Cj.

A simple application- Selection of vendor

The fuzzy model above described can be illustrated by the following very simple example.

Suppose that for a certain equipment we must chose one of five possible vendors a, b, c, d, e, f, all of them guaranteeing the same performances, the economical quotations given by:

Q(a)= 30,000 $

Q(b)= 35,000 $

Q(c)= 32,000 $

Q(d)= 40,000 $

Q(e)= 38,000 $

Our goal is to select the vendor who will give to our company the lowest cost given the constraints that the vendor has good reputation and its workload is pretty far from saturation limit (i.e. 100% workload).

The first constraint is represented by the fuzzy set C1, where values of µ are assigned to each vendor to indicate it fits the statement ‘ good reputation’.

This value is obtained by an expert judgment, previous experience either a result of a dedicated investigation.

The second constraint, represented by C2 is recovered by specific and documented information by the vendor, either a market search or similar.

C1=0.3 (a) + 0.9 (b) +0.6(c) + 0.7(d)  + 0.6(e)

C2=0.4 (a) + 0.4(b) +0.8(c) + 0.9(d)  + 0.5(e)

The fuzzy goal of a low economical figure, is defined on the set X of quotations by the membership function:

It shall be noted that the proposed µG function, is a classic shape of membership function in fuzzy logic analysis (trapezoid shape), and is a basic one, being linear/proportional.


Other functions can be selected, among which the parabolic one either exponential depending on the attitude or preference of the decision maker with respect to, in this case, the cost variation.

For a function like that depicted in fig 2, an increment of cost result in a quite important decrease of membership function, signifying a great focus of the DM on the cost itself.

(fig. 2)

We can now calculate G’.

G’ = 0.75 / a + 0.5 /b +0.65/c + 0.25/d + 0.35/e


D = 0.3/a+0.4/b+ 0.6/c + 0.25/d +0.35/e

Finally we take the maximum of this set to obtain alternative c as the choice that seems to satisfy both goals and constraints.

It is worth just to mention that 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 will be possibly the object of another investigation.

Conclusions and remarks

Even if a not trivial mathematical theory is behind, the method reported is rather easy and quick to implement.

It 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. In this case a preliminary analysis can be run 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.

It avoids time wasting in demanding data elaborations based on poor starting database.

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

  • Some main simplified methods of DM
  • Considerations and limits of probability distributions
  • Fuzzy logic analysis as a tool in case of vagueness, uncertainty lack of probability data
  • Easiness of implementation of fuzzy approach by a simple practical example

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



[1] L.Schiavina- Metodi e strumenti per la modellizzazione aziendale- Franco Angeli Milano -2006
[2] T.Zheng – Bayes’s Theorem – Cornell University Math ( )- 2008
[3] K.Awati – Bayes theorem for project managers- Eight to Late- 2010
[4]  G.Kler/ T.Folger -Fuzzy sets, uncertainty and Information- Prentice Hall – 2008
[5] K.Rao/S.Mishra- Operations Research – Narosa 2008


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.



(commento 1)
Massimiliano Arena (07/08/2010)

Ricevo dal Professor Vito Ozzola alcune interessanti osservazioni, che volentieri pubblico:

  • Nel paragrafo 2, nell’espressione di C scambiare S ed E, si ottiene C(S/E). IMPORTANTE OSSERVAZIONE: nel caso di decisioni in condizioni di certezza si ha un solo stato e la matrice è a una colonna.
  • E' piu' corretto parlare di scelta tra vendors piuttosto che di selection of a supplier

Ringrazio davvero il Prof .Ozzola per i Suoi consigli e l'attenzione che mi ha voluto dedicare. Lo ringrazio anche per avermi appassionato con il suo libro ' Divertimenti sui temi di ricerca operativa' -Ed Alinea-, rigoroso scientificamente e amabile letterariamente.

Massimiliano Arena

(commento 2)

Gianluca di Castri (09/09/2010)

Mi congratulo innanzi tutto per il lavoro di ricerca svolto dall'autore e per le sue capacità di sintesi.

A mio parere il modello basato su una logica fuzzy, che prevede in ultima analisi delle categorie intermedie fra vero e falso, è applicabile in fase di pianificazione o, se si preferisce, in regime stocastico; dal punto di vista applicativo esiste una certa analogia fra porre un coefficiente di probabilità su una durata o su un vincolo oppure applicare una loguca fuzzy; ribadisco, da un punto di vista applicativo, perchè l'impostazione teorica è differente. Si deve evitare la tentazione di usare queste tecniche in regime deterministico perchè potrebbero creare confusione, specialmente in sede di determinazione delle cause di un ritardo o di contenzioso.

D'altra parte, l'articolo di Arena evidenzia questo fatto al secondo paragrafo, in cui si riferisce a situazione di tale incertezza da rendere inapplicabile persino il concetto di probabilità e per i quali si possono introdurre metodi basati sull'approccio fuzzy. Sempre dal punto di vista applicativo, tuttavia, viene spontaneo chiedersi se, in questo caso, non sia consigliabile di tentare un approfondimento della conoscenza delle variabili e delle condizioni al contorno prima di assumere la decisione.

La conoscenza della storia, ed anche qualche sondaggio fatto in passato sui metodi di assunzione delle decisioni da parte di imprenditori di successo, hanno mostrato che in genere le decisioni vengono alla fine assunte in base ad un processo intuitivo, che è peraltro lo stesso processo che porta gli imprenditori senza successo ad assumere decisioni sbagliate. La logica fuzzy, come ogni altro metodo della scienza delle decisioni, è e deve restare uno strumento di supporto che non deve diminuire l'importanza dell'intuizione ed, in ultima analisi, della scelta discrezionale dell'imprenditore o comunque di chi è delegato ad assumere la decisione, assumendosene anche le relative responsabilità. Ove non vi sia più bisogno di discrezionalità, l'uomo può essere sostituito da una macchina sia pur sofisticata.

Mi permetto di suggerire all'autore, nelle future edizioni del suo testo, di spendere, nel paragrafo dedicato alle conclusioni, qualche parola in più sui limiti del metodo.

Gianluca di Castri


(commento 3)

Vito Ozzola (23/09/2010)

Ribadisco i complimenti, espressi via e-mail direttamente all’Ing. Arena, per il taglio rigoroso e sintetico dell’articolo e per la panoramica sullo stato dell’arte che avvolge i temi più brillanti della teoria.

Concordo con l’Ing. Di Castri. La decisione finale su un problema aziendale è di stretta competenza manageriale.  La soluzione proposta dall’applicazione delle tecniche decisionali costituisce unicamente un tassello nel mosaico delle conoscenze ed esperienze del decisore.

Le tecniche decisionali presentano un’enorme gamma di applicazioni, pescano nel torbido di ogni attività umana. In proposito segnalo una mia esperienza maturata nella linea di confine. In anni lontani mi sono dedicato a ricerche di Computer music e ho applicato i metodi decisionali alla composizione musicale per mezzo del calcolatore e all’analisi di un criterio per la valutazione del “tasso estetico” di un brano musicale.

Con viva cordialità.

Vito Ozzola

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