a) An investment decision

An investor has to choose the most cautious strategy regarding three possible market scenarios s1, s2, s3 with the LPI-Fuzzy distribution, considered as a forecast of the experts Table 1:

LPI(p): 1/2 > p1 > p2 > p3

For the output situations "a" the investor would estimate preference order:

a₆Pr   a₅Pr   a₄Pr   a₃Pr   a₂Pr   a₁Pr

where   Pr = preference   and   a₆ = the highest.

    The PDP-procedure:

The extreme distribution matrix:

 

The investor estimates amongst the possible forecasts (a₄, 2/3) as the highest. Therefore the investment x1 is PDP-optimal with the risk at the most:

1 - 2/3 = 33 1/3 %

 

  Some other interpretations of Table (1)

    b) The relation therapy-diagnosis

Dependent on the tree possible diagnosis s1, s2, s3 three therapies are available for the physician: x1, x2, x3. The fuzzy data are the same. Then the most cautious therapy is x1, with the guaranteed patient situation at least a₄ and a risk at the most 33 1/3 % (according to the LPI-restrictions).

    c) The antagonistic conflict situation

The strategy of a player I is: ( x1, x2, x3 ), of player II (an antagonistic one) (s1, s2, s3 ). The data in the Table1 are the same: from the viewpoint of the player I by the PDP-procedure, tha most cautious strategy is x1 with with the guaranteed situation at least a₄ and a risk at the most 33 1/3 % (according to the LPI-restrictions).

    d) Linear programming models

In /12/ Linear Programming models under fuzziness using Fuzzy Sets methods, are considered. The application of the Bellman-Zadeh principle and other extended methods, based on corresponding membership functions, is difficult. On the contrary, the LPI-methods are simple and consistent: in the fuzzy model:

Ax < b, x > 0, f(x) = c' x = max

any fuzziness can be expressed by corresponding LPI-fuzzy numbers. These numbers can then be transformed in (Emin, Emax) intervals /6/. The application of the Convolution Theorem /7/, regarding the interval endpoints and then the PDP-procedure, leads to an approximate solution of the considered fuzzy program.

    e) Application in LPI-forecasting /13/

Let us analyze a possible forecast for the GNP growth of a national economy, under fuzziness. In the formula:

GNP = C + I + G + ( Ex - Im )

where: C = consumption, I = investments, G = governement spendings, Ex = exports, Im = imports

the quantities on the right side are fuzzy. Opposite to the usual econometric models, here we consider possible scenarios with fuzzy distributions which we receive from experts. In that example we shall regard three scenarios (z1, z2, z3) with the corresponding fuzzy realization probabilities (p1, p2, p3). By convolution of the corresponding fuzzy distributions, the approximate forecasts for the minimal value of the economic growth and the corresponding risk can be determined. This procedure can be extended by sequential partitions of the corresponding intervals. By some additional assumptions the convergence of the extended procedure to the resulting convolution of the continous fuzzy distributions, can be analyzed.

    Example:  LPI(p): a<p1<p2<p3

A good illustration here would be the LPI(p) for realization probability p of scenarios z1, z2, z3 with different values of a, corresponding to
C, I, G, (Ex - Im).